![April 4, 2005 10:3 3086 DK2961˙C001
Chapter 1
Asymptotic Rates of Blowup for the Minimal Energy
Function for the Null Controllability of
Thermoelastic Plates: The Free Case
George Avalos1
University of Nebraska-Lincoln
Irena Lasiecka2
University of Virginia
1.1 Introduction ............................................................................ 2
1.1.1 Motivation ...................................................................... 2
1.1.2 Description of the PDE Model and Statement of the Problem ...................... 4
1.1.3 Main Result ..................................................................... 8
1.2 The Necessary Observability Inequality .................................................. 9
1.3 Some Preliminary Machinery ........................................................... 10
1.4 A Singular Trace Estimate ............................................................... 12
1.5 Proof of Theorem 1.1(1) ................................................................ 14
1.5.1 Estimating the Mechanical Velocity .............................................. 14
1.5.2 Estimating the Mechanical Displacement ......................................... 17
1.5.3 Conclusion of the Proof of Theorem 1.1(1) ....................................... 20
1.6 Proof of Theorem 1.1(2) ................................................................ 20
1.6.1 A First Supporting Estimate ...................................................... 20
1.6.2 Conclusion of the Proof of Theorem 1.1(2) ....................................... 23
1.7 Proof of Theorem 1.1(3) ................................................................ 24
References .............................................................................. 24
Abstract Continuing the analysis undertaken in References 8 and 9, we consider the null-
controllability problem for thermoelastic plate partial differential equations (PDEs) models in the
absence of rotational inertia, defined on a two-dimensional domain , and subject to the free mechan-
ical boundary conditions of second and third order. It is now known that such uncontrolled systems
generate analytic semigroups on finite energy spaces. Consequently, the concept of null controllabil-
ity is indeed an appropriate question for consideration. It is shown that all finite energy states can be
driven to zero by means of L2
[(0, T ) × ] controls in either the mechanical or thermal component.
However, the main intent of the paper is to quantify the singularity, as T ↓ 0, of the minimal energy
function relative to null controllability. In particular we shall show that in the case of one control
function acting upon the system, the singularity of minimal energy is optimal; it is in fact of order
O(T − 5
2 ), which is the same rate of blowup as that of any finite dimensional approximation of the
problem. The PDE estimates, which are obtained in the process of deriving this sharp numerology,
will have a strong bearing on regularity properties of related stochastic differential equations.
1The research of George Avalos was partially supported by NSF DMS-0208121.
2The research of Irena Lasiecka was partially supported by NSF DMS-01043 and ARO DAAD19-02-1-0179.
1](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-21-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
2 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
1.1 Introduction
In this chapter we address specific questions related to the null controllability of thermoelastic
plates subject to free mechanical boundary conditions, these being represented by shear forces and
moments. These particular boundary conditions are of particular interest in the control theory of
plates [22, 24, 23]. As we shall see below, the model under consideration is one which corresponds
to an infinite speed of propagation; accordingly, null controllability—in arbitrarily short time—is
an appropriate topic for study in regard to these plates. We will give at length a full and precise
description of our thermoelastic control problem; but for the benefit of the reader and in order to
motivate the specific problem under study, we will first provide a few opening remarks.
1.1.1 Motivation
There are several ways of controlling a given plate dynamic. This control can be accomplished by
using: 1. internal controls, 2. boundary controls, or 3. controls localized on an open subset of . In
addition, one may use either one control action (be it thermal or mechanical) or simultaneous me-
chanical and thermal controls (i.e., controls located on both the mechanical and thermal components
of the system). Depending on the objective to be achieved, one framework of control might be more
advantageous than another. For instance, if the particular issue at hand is to guarantee the minimal
support of control functions, then boundary control would be the most appropriate control situation.
However, if one is concerned with the cost of control—or equivalently, with quantifying the associ-
ated “minimal energy”—then internal controls should be considered. In this connection, a question
of both practical and mathematical relevance is the question of finding the optimal asymptotics that
describe the singularity of the associated minimal energy, as T ↓ 0. Since the work of T. Seidman in
Reference 34, the optimal asymptotics are well defined and well known for finite dimensional control
systems. In fact, these asymptotics are given by the sharp formula T −k− 1
2 where index k corresponds
to the Kalman rank condition and measures the defect of controllability (see below). The above for-
mula actually gives a lower bound for the singularity of the minimal energy associated with any PDE
system.
Given then the existence of formula in Reference 34 for controlled finite dimensional systems,
we are in a position to loosely define the “optimal” singularity for any controlled PDE. In fact,
for a given infinite dimensional system, the “optimal” rate of singularity of its associated minimal
energy will be the rate of singularity enjoyed by approximating (or truncated) finite dimensional
systems (assuming of course that each finite dimensional truncation has the same Kalman rank).
For example, scalar first order (in time) models will have its optimal rate of blowup of minimal
energy as being O(T − 3
2 ); in general, the optimal singularity for vectorial coupled structures will
depend on the number of controls used with respect to number of interactions. Thus, in the case
of thermal plates with one control only, the optimal singularity of any finite dimensional truncation
is T − 5
2 (this is seen below). In the case of two controls used (both thermal and mechanical) the
optimal singularity is T − 3
2 . Whether, however, the minimal energy asymptotics actually obeys the
optimal rate of singularity (predicted from finite dimensions) is an altogether different matter. Indeed,
in References 34 and 36 (highly nontrivial) finite-dimensional estimates are derived and can be
subsequently applied to finite-dimensional truncations of infinite-dimensional systems; however, the
delicate estimates are controlled by a constant Cn, say, where n stands for the dimensionality of the
respective approximation. These constants may well tend to infinity as n goes to infinity. In such an
event (as seen in References 14, 6, and 40) the optimal asymptotics for the original PDE are lost.
This brings us to the key question asked in this chapter: Is it possible to achieve the optimal rate of
singularity for a (fully infinite dimensional) controlled PDE model?](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-22-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.1 Introduction 3
The answer to the above question—in the negative—has been known for many years in the case of
the heat equation with either boundary or localized controls. Indeed, the rate for boundary control of
the heat equation is the exponential blowup rate eO( 1
T )
; see References 35 and 37. This rate is known
to be sharp [20]. A similar negative answer has been provided in the case of thermoelastic systems
under the influence of boundary controls—in fact, such boundary controls likewise lead to eO( 1
T )
exponential blowup [25]. Therefore, in light of the rational rates of minimal energy blowup exhibited
by finite-dimensional controlled systems (as shown in Reference 34) and of the definition given above
for optimal rates of minimal energy blowup for controlled PDEs, it is manifest that thermoelastic
plates under the influence of boundary or localized controls will not give rise to minimal energies
that exhibit an optimal (finite dimensional) singularity. Thus, in searching for PDE control situations,
which will yield up the optimal algebraic singularity enjoyed by finite dimensional truncations, the
only reasonable choice left is the implementation of internal controls. In the specific context of
our thermoelastic PDE, the relevant question then becomes: Do the minimal energies of internally
controlled (fully infinite dimensional) thermoelastic plates exhibit the optimal rate of blowup O(T − 5
2 )
by either mechanical or thermal control?
The relevance of this question should not be underestimated from both a practical and mathematical
point of view. Indeed, from a practical point of view one would like to know whether a given
finite-dimensional approximation of the system contains critical information and moreover reflects
controllability properties of the original PDE model. From a mathematical point of view, the solution
to the null controllability problem is not only of interest in its own right as an issue in control theory,
but this solution can also give rise to deep and significant connections between the algebraic optimal
singularity of minimal energy and other fields of analysis, including stochastic analysis. In point
of fact, within the field of stochastic differential equations, there is an acute need to know of those
PDE control environments that will yield up optimal (and algebraic) rates of singularity of minimal
energy. These particular rates are critical in finding the regularity and solvability of certain stochastic
differential equations [14, 15, 19], as well as in setting conditions for the hypoellipticity of certain
degenerate infinite dimensional elliptic problems [32]. It is shown in Reference 32 that Hormander’s
hypoellipticity condition is strongly linked to the singularity of the minimal energy function. Null
controllabilityisalsorelatedtotheanalysisofregularityoftheBellman’sfunction,whichisassociated
with the minimal time control problem. Indeed, as eloquently described in References 14 and 15,
this property bears a close relation to the regularity of some Markov semigroups, including Orstein–
Uhlenbeck processes and related Kolmogorov equations. For some of these semigroups (see, e.g.,
Reference 15—Theorem 8.3.3) the minimal energy singularity associated with null controllability
describes differentiability properties and regularizing effects of the Orstein–Uhlenbeck process.
Moreover, the regularity of solutions to the Kolmogorov equation depends on the singularity of the
minimal energy as T ↓ 0. Also, as shown in Reference 14, optimal estimates for the norms of
controls are critical in being able to prove Liouville’s property for harmonic functions of Markov
processes (see p. 108 in Reference 15). In sum, there is an abundance of examples from the literature
that clearly illustrate that, in the context of computing optimal minimal energy asymptotics as T ↓ 0,
the tools of controllability can potentially enable a mathematical control theorist to transcend his or
her deterministic realm so as to solve fundamental problems in other areas of analysis, including
stochastic PDEs.
In addition, the procurement of optimal algebraic estimates for the minimal energy allows one to
clearly explain the role of the hyperbolic-parabolic coupling within the PDE structure (in Eq. (1.1)
below). In particular, it has been shown recently in Reference 25 that, owing to optimal algebraic
singularitiesofminimalenergy,itispossibletooffsetthesingularityofminimalenergybyintroducing
a very strong coupling within the system. Thus, in some sense, the lack of a second control in the
system may be quantitatively compensated for by taking large values of the coupling parameter “α.”
From our remarks above, it is clear that this compensatory phenomenon will not be observed with
boundary or partially supported controls, which, as we have said, lead to blowups of exponential type.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-23-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
4 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
Having decribed the goal and motivation for the problem considered, we shall describe the main
contribution of this chapter within the context of recent work in that area. The problem of controlla-
bility/reachability for thermal plates has attracted considerable attention in recent years with many
contributions available in the literature [22, 23, 24, 1, 2, 3, 10, 18, 16, 17, 11], but we shall focus
particularly on works related directly to singular behavior, as T ↓ 0, of the minimal energy relative
to null controllability.
The study of optimal singularity for thermoelastic plates with internal controls started in Refer-
ences 8, 9, and 40, where for the first time the optimal rates T − 5
2 were established for the “commu-
tative” case (i.e., plates with hinged mechanical boundary conditions). The proof given in Reference
40 is based on a spectral method that exploits the commutativity in an essential way, whereas the
proof given in Reference 8 is based on weighted energy estimates, thereby giving one the chance
to extend this method to other noncommutative models (e.g., clamped or free mechanical bound-
ary conditions). The “commutative” case (hinged boundary conditions) has been also treated in
Reference 11, where null controllability with thermal controls of partially localized support was
proved. For this commutative model under boundary control (either thermal or boundary), the ex-
ponentially blowing up and sharp asymptotics eO( 1
T )
have been shown [25]. The techniques used in
these papers rely critically on spectral analysis and commutativity.
It turns out that a proper and necessarily more technical extension of the method introduced in
Reference 8 will allow the consideration of noncommutative models. (By “noncommutative models”
we mean those models wherein the domains of the respective spatial differential operators of the
plate and heat dynamics do not necessarily enjoy any sort of compatibility.) In particular, the optimal
singularity of the (null control) minimal energy is proved in Reference 9 for clamped plates with one
control only. It should be stressed that the proof in the noncommutative case depends in an essential
manner upon estimates provided by the analyticity of the underlying thermoelastic semigroup; this
property of analyticity was discovered for the clamped case in References 31 and 28 and for free
case in Reference 27. The most challenging case is, of course, that of the free mechanical bound-
ary conditions (introduced in the context of control theory in Reference 22), in which a coupling
between thermal and mechanical variables also occurs on the boundary. This additional coupling
compels us to develop below a delicate string of trace estimates that measure the singularity at the
boundary.
The main aim of this chapter is to provide a complete analysis of the free case. We shall show
that in the case of mechanical control one still obtains the optimal singularity. Instead, in the case
of thermal control the estimate is “off” by 3/4. A question whether this estimate can be improved,
thereby leading to the optimal singularity T − 5
2 , still remains an open question.
1.1.2 Description of the PDE Model and Statement of the Problem
Having given our general remarks above, we now proceed to precisely describe the present prob-
lem under consideration; this work will continue and extend the analysis that has been previously
undertaken in References 6, 7, 8, 9 through and 40. We will consider throughout the two-dimensional
PDE system of thermoelasticity in the absence of rotational inertia. As we have already stated, it
is now known that for all possible mechanical boundary conditions, the thermoelastic PDE model
is associated with the generator of an analytic C0-semigroup (see References 31, 28, 39, and 27).
Given then that the underlying PDE dynamics are “parabolic-like,” it is natural to consider the
null controllability problem for the thermoelastic system, namely, can one find L2
(Q) controls
(mechanical or thermal) that steer the solution of the PDE from the initial data to the zero state?
(We shall make this control theoretic notion more precise below. As usual, Q here denotes the
cylinder × (0, T ).) Having established L2
(Q)-null controllability for the PDE, and moreover
assuming that the controllability time is arbitrary, we can subsequently proceed to measure the rate
of blowup, as T ↓ 0, for the minimal energy function that is associated to null controllability. As
is well known, and as we shall see below, this work is very much tied up with obtaining the sharp](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-24-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.1 Introduction 5
observability inequality associated with null controllability; moreover, this analysis is rather sensi-
tive to the mechanical boundary conditions imposed. In Reference 8—as well as in Reference 40
via a very different methodology—the problem of blowup for the minimal energy function was
undertaken in the canonical case of hinged mechanical boundary conditions; in Reference 9, we
revisit this problem for the more difficult clamped case. In this paper, we complete the picture
by analyzing the singularity of minimal energy for the case of the thermoelastic PDE under the
so-called free boundary conditions. In general, the analyses involved in the attainment of (null and
exact control) observability inequalities for thermoelastic systems are profoundly sensitive to the
particular set of boundary conditions are being imposed. But the free case, presently under consid-
eration, will give rise to the most problematic scenario of all. This situation is due to the high degree
of coupling between the mechanical and the thermal variables, with the coupling taking place in the
PDE itself and in the free mechanical boundary conditions.
We describe the problem in detail. Let be a bounded open set of R2
, with smooth boundary . For
the free case, following [22, 23] the corresponding model PDE is as follows: the (mechanical) vari-
ables [ω(t, x), ωt (t, x)] and the (thermal) variable θ(t, x) solve, for given data {[ω0, ω1, θ0], u1, u2},
the PDE system
ωtt + 2
ω + α θ = a1u1
θt − θ − α ωt = a2u2
on (0, T ) ×
ω + (1 − µ)B1ω + αθ = 0
∂ ω
∂ν
+ (1 − µ)
∂ B2ω
∂τ
− ω + α
∂θ
∂ν
= 0
on (0, T ) ×
∂θ
∂ν
+ λθ = 0 on (0, T ) × , where λ 0
ω(t = 0) = ω0; ωt (t = 0) = ω1; θ(t = 0) = θ0 on .
(1.1)
Here, α 0 is the parameter that couples the disparate dynamics (i.e., the heat equation vs. the
Euler plate equation); the constant µ ∈ (0, 1) is Poisson’s ratio. Also, the (control) parameters a1
and a2 satisfy a1 ≥ 0, a2 ≥ 0 and a1 + a2 0 (in other words, at least one of the controls, be it
thermal or mechanical, is always present.) The (free) boundary operators Bi are given by
B1w ≡ 2ν1ν2
∂2
w
∂x∂y
− ν2
1
∂2
w
∂y2
− ν2
2
∂2
w
∂x2
;
(1.2)
B2w ≡
ν2
1 − ν2
2
∂2
w
∂x∂y
+ ν1ν2
∂2
w
∂y2
−
∂2
w
∂x2
.
The PDE Eq. (1.1) is the model explicitly derived and analyzed in References 24 and 22 in the
“limit case.” That is to say, we are considering the two-dimensional thermoelastic system in the
absence of rotational forces; the small and nonnegative, classical parameter γ is taken here to be
zero. As we stated at the outset, it is now well known that the lack of rotational inertia in the model
Eq. (1.1) will result in the corresponding dynamics having their evolution described by the generator
of an analytic semigroup on the associated basic space of finite energy. In short, the present case
γ = 0 corresponds to parabolic-like dynamics; this is in stark contrast to the case γ 0—as analyzed
in the control papers [22], [23], [3] and myriad others—for which the corresponding PDE manifests
hyperbolic-like dynamics.
In fact, if we define
H ≡ H2
() × L2
() × L2
(), (1.3)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-25-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
6 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
then one can proceed to show by the Lumer Phillips theorem that the thermoelastic plate model can
be associated with the generator of a C0-semigroup of contractions on H. That is to say, there exists
A : D(A) ⊂ H → H, and {eAt
}t≥0 ⊂ L(H) such that [ω, ωt , θ] satisfies the PDE (1.1) if and only
if [ω, ωt , θ] satisfies the abstract ODE
d
dt
ω(t)
ωt (t)
θ(t)
= A
ω(t)
ωt (t)
θ(t)
+
0
a1u1(t)
a2u2(t)
;
ω(0)
ωt (0)
θ(0)
=
ω0
ω1
θ0
.
In consequence of this relation, we have immediately from classical semigroup theory that
{[ω0, ω1, θ0], [u1, u2]} ∈ H × [L2
(Q)]2
⇒ [ω, ωt , θ] ∈ C([0, T ]; H). (1.4)
Because of the underlying analyticity, which will ultimately mean that there are smoothing effects
associated with the application of the semigroup {eAt
}t≥0, the null controllability problem for the
controlled PDE Eq. (1.1)—with respect to internal L2
-controls—is an appropriate one to study.
Moreover, one might speculate that, as in the case of the canonical heat equation [12], should the
PDE Eq. (1.1) in fact be null controllable, it will be so in arbitrary small time (because of the
underlying infinite speed of propagation). It is this speculation that motivates our working definition
of null controllability for the present paper.
DEFINITION 1.1 The PDE (1.1) is said to be null controllable if, for any T 0 and arbitrary
initial data x ≡ [ω0, ω1, θ0] ∈ H, there exists a control function [u1, u2] ∈ [L2
(Q)]2
such that the
corresponding solution [ω, ωt , θ] ∈ C([0, T ]; H) satisfies [ω(T ), ωt (T ), θ(T )] = [0, 0, 0].
However, the issue of null controllability, although certainly an important part of this paper, is
subordinate to our main objective, which is to measure the rate of singularity of the associated
minimal energy function.
We develop this notion of “minimal energy.” Assume for the time being that the Eq. (1.1) is
null controllable within the class of [L2
(Q)]2
-controls, in the sense of the Definition 1.1. Subse-
quently, one can then speak of the associated minimal norm control, relative to given initial data
x ≡ [ω0, ω1, θ0] ∈ H and given terminal time T . That is to say, we can consider the problem of finding
a control u0
T (x) that steers the solution [ω, ωt , θ] of Eq. (1.1) (with [u1, u2] = u0
T (x) therein) from
initial data x to zero in arbitrary time T and minimizes the L2
norm. In fact, by standard convex
optimization arguments (see, e.g., Reference 13), given any x ∈ H and fixed T , one can find a control
u0
T (x) which solves the problem
u0
T (x)
[L2(Q)]2 = min u[L2(Q)]2 ,
where, above, the minimum is taken with respect to all possible null controllers u = [u1, u2] ∈
[L2
(Q)]2
of the PDE (1.1) (which steer initial data x to rest at time t = T ). Subsequently, we can
define the minimal energy function Emin(T ) as
Emin(T ) ≡ sup
xH=1
u0
T (x)
[L2(Q)]2 . (1.5)
Under the assumption of null controllability, as defined in Definition 1.1, we have that Emin(T ) is
bounded away from zero. A natural follow-up question is “how does Emin(T ) behave as terminal time
T ↓ 0, or equivalently (by Eq. (1.5)), for given time T , how exactly does the quantity u0
T (x)[L2(Q)]2
grow as T ↓ 0?”](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-26-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.1 Introduction 7
The problem of studying the rate of blowup for minimal norm controls is a classical one and has
its origins from the finite dimensional setting. In fact, a very complete and satisfactory solution has
been given in Reference 34 for the following controlled ODE in Rn
:
d
dt
y(t) = A
y(t) + B
u(t),
y0 ∈ Rn
(1.6)
where
u ∈ L2
(0, T ; Rm
) and A (resp., B) is an n × n (resp. n × m) matrix, with m ≤ n (so
consequently the solution
y ∈ C([0, T ]; Rn
). The problem in this finite dimensional milieu, like that
for our controlled PDE (1.1), is to ascertain the rate of singularity for the associated minimal energy
function, which is defined in the same way as in Eq. (1.5). The solution to this problem is tied up
with the classical Kalman’s rank condition. Namely, a beautifully simple (though highly nontrivial)
formula in Reference 34—an alternative constructive proof of this formula is given in Reference 40;
see also Reference 36—yields that the minimal energy function associated to the null controllability
of Eq. (1.6) is O(T −k− 1
2 ), where k is the Kalman’s rank of the system Eq. (1.6) (that is, k is the
smallest integer such that rank ([B, AB, . . . , Ak
B]) = n; see Reference 41).
By a formal application of Seidman’s finite dimensional result, one can get an inkling of the
numerology involved in the computation of the minimal energy Emin(T ) for the PDE system Eq. (1.1).
Forexample,letusconsiderthethermoelasticEq.(1.1)butwithnow ω satisfyingthecanonicalhinged
mechanical/Dirichlet thermal boundary conditions
ω| = ω| = θ| = 0 on . (1.7)
In this case, it is shown in Reference 27 that when, say, thermal control only is implemented (i.e.,
a1 = 0 in Eq. (1.1)), the thermoelastic PDE under the hinged boundary conditions Eq. (1.7) may be
associated with the ordinary differential equation (ODE) (1.6), with
A =
0 1 0
−1 0 1
0 −1 −1
, and B =
0
0
a2
. (1.8)
This ODE in three space dimensions is a direct consequence of the analysis undertaken for the
canonical hinged case in Reference 26. By way of obtaining the ODE (1.6), we have formally
“factored out” the Laplacian from the (rearranged) infinitesimal generator of the thermoelastic
semigroup, which is given in (Section 1.2.2) of Reference 27 (see also Reference 28, p. 311).
Considering now finite dimensional truncations of (by making use of the spectral resolution of
the Laplacian under Dirichlet boundary conditions) and applying the algorithm of Seidman to the
given controllability pair [A, B] in Eq. (1.8), we compute readily that the minimal energy func-
tion associated with the null controllability of the finite dimensional Eq. (1.6)—an approximation
in some sense of the thermoelastic system under the hinged boundary conditions—blows up at a
rate on the order of T − 5
2 . These numerics lead to the following question: Does the minimal en-
ergy Eq. (1.5) (i.e., the minimal energy for the full-fledged infinite dimensional system) obey the
law Emin(T ) = O(T − 5
2 )? Of course, Seidman’s formula for matrices gives no conclusive proof as
to what is actually happening for the fully infinite dimensional model. In fact, it is well known
that the minimal energy of a given infinite dimensional system may bear no relation to the limit
of minimal energies of any given sequence of finite dimensional approximations. For example, it
was shown in Reference 14 that the growth of the minimal energy function for a given infinite
dimensional system may be arbitrarily large, even when Kalman’s rank k = 1 and spectral diag-
onal systems are being considered. Moreover, in Reference 35 it is shown that for the case of the
boundary controlled heat equation, the sharp observability inequality corresponding to the (null)
minimal energy of a given heat operator’s finite dimensional truncation obeys rational rates of](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-27-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
8 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
singularity. On the other hand, the asymptotics of the minimal energy, which are obtained for the
(infinite dimensional) heat equation, are of exponential type. A similar phenomenon is observed in
References 40 and 6, wherein strongly damped wave equations under internal control are considered.
In this situation, with the damping operator given by Aβ
, the asymptotics of minimal energy be-
have as T − β
2(1−β) for any β 3
4
. Thus, when the damping operator approaches Kelvin’s Voight
damping, the singularity loses its algebraic character with β
2(1−β)
↑ ∞. Instead, for β ≤ 3
4
, the
singularity is optimal (i.e., the same as that for finite dimensional truncations) and is equal
to T − 3
2 .
But as formal as the application of Seidman’s finite dimensional algorithm may seem in the present
context, there is in fact a relevance here to the thermoelastic PDE, which is approximately described
by controllability pair [A, B]. The minimal energy function with respect to null controllability of the
thermoelastic PDE, under the hinged boundary conditions Eq. (1.7), does indeed obey the singular
rate Emin(T ) = O(T − 5
2 ). This minimal energy analysis for the hinged case was shown independently
in References 8 and 40 (and most recently in Reference 25 where the asymptotics with respect to the
coupling α are also provided). In Reference 40, it is of prime importance that the hinged boundary
conditions Eq. (1.7) be in play, for these mechanical boundary conditions allow a fortuitous spectral
resolution of the underlying thermoelastic generator. With the eigenfunctions of the thermoelastic
dynamics in hand, it is shown in Reference 40 via a constructive class of suboptimal steering controls
that the delicate observability estimates for solutions for the spectrally truncated adjoint problem—
adjoint with respect to null controllability—are preserved; as a consequence, a rational rate of
singularity for the infinite dimensional null minimal energy is obtained in the limit. However, for
other sets of mechanical boundary conditions, including the physically relevant clamped and (above
all) free boundary conditions under consideration at present, there will be no such available spectral
decomposition.
On the other hand, the methodology employed in References 8 and 9, and the present work, is
“eigenfunction independent”; in particular, we blend a weighted multiplier method of Carleman’s
type with boundary trace estimates exhibiting singular behavior of the boundary traces. This rather
special behavior is a consequence of the underlying analyticity. In principle, our work in Reference 8
to estimate the blowup of the “minimal norm control” as T ↓ 0 is applicable to a variety of
dynamics. (In fact, our method of proof in Reference 8 and in the present work is used in
Reference 7 to estimate the minimal norm control of the abstract wave equation under Kelvin–
Voight damping.) Moreover, the robustness of our method allows us in Reference 9 to analyze
the rate of singularity of the minimal energy function for the null controllability of thermoelastic
plates in the case of clamped boundary conditions. As we said above, there is no spectral de-
composition or factorization of the thermoelastic generator in the case of mechanical boundary
conditions other than the canonical hinged case and thus no rigorous association with the abstract
ODE (1.8). Still, we show in Reference 9 that for the clamped case, the minimal energy obeys the
singular rate “predicted” in Reference 34, namely, Emin(T ) = O(T − 5
2 ). Our intent in this paper
is to bring the story to a close by investigating the minimal energy function for the null control-
lability of thermoelastic systems under the high-order free boundary conditions that are present in
Eq. (1.1).
1.1.3 Main Result
In regards to our stated problem, the main result is as follows:
THEOREM 1.1
Let terminal time T 0 be arbitrary and a1, a2 ≥ 0 with a1 + a2 0. Then, given initial data
[ω0, ω1, θ0] ∈ H, there exist control(s) [u1, u2] ∈ [L2
(Q)]2
such that the corresponding solution
[ω, ωt , θ] of (1.1) satisfies [ω(T ), ωt (T ), θ(T )] = [0, 0, 0]. (That is to say, the PDE model Eq. (1.1)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-28-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.2 The Necessary Observability Inequality 9
is null controllable within the class of [L2
(Q)]2
—controls in arbitrary short time.) Moreover, We
have the following rates of blowup for the minimal energy function:
1. (thermal control) If a1 = 0, then Emin(T ) = O(T − 13
4 −
) for all 0;
2. (mechanical control) If a2 = 0, then Emin(T ) = O(T − 5
2 );
3. If a1 0 and a2 0, then Emin(T ) = O(T − 3
2 ).
REMARK 1.1 The null controllability of thermoelastic plates with free boundary conditions and
under one internal control (be it mechanical or thermal) appears to be, as far as we know, a new
result in the literature. The Theorem 1.1 above, in addition to asserting the said null controllability
property, provides the asymptotics for the singularity of the associated minimal energy function.
These asymptotics are optimal in the case of a single mechanical control and in the case of two
controls acting upon the system. In the case of a single thermal control the estimate is “off” by 3/4
with respect to the desired “finite dimensional prediction” in Reference 34. Whether this estimate
can be improved upon is an open question.
Our method of proof of Theorem 1.1 is based on weighted energy estimates that are flexible
enough to accomodate analytic estimates and the resulting singularity. The proof has the following
main technical ingredients:
1. special weighted nonlocal multipliers introduced in Reference 4 and subsequently invoked in
References 3, 5, 29, and 6, and elsewhere;
2. the analyticity of semigroups associated with thermoelastic PDE models in the absence of
rotational forces, as demonstrated in References 31, 26, 27, and 28;
3. new singular estimates for boundary traces of solutions of Eq. (1.9), which are of their own
intrinsic interest and which are needed to handle the boundary terms resulting from the
weighted estimates employed.
1.2 The Necessary Observability Inequality
The proof of Theorem 1.1 is based on the derivation of the observability inequality associated
with the null controllability of the PDE (1.1) with respect to thermal or mechanical control or both.
This inequality is formulated in terms of the solution of the homogeneous PDE, which is “dual”
or “adjoint” to that in Eq. (1.1). Namely, we shall consider solutions [φ, φt , ϑ] to the following
system:
φtt + 2
φ + α ϑ = 0 on (0, T ) ×
ϑt − ϑ − α φt = 0 on (0, T ) ×
φ + (1 − µ)B1φ + αϑ = 0
∂ φ
∂ν
+ (1 − µ)
∂ B2φ
∂τ
− φ + α
∂ϑ
∂ν
= 0
on
∂ϑ
∂ν
+ λϑ = 0 on , λ 0
[φ(0), −φt (0), ϑ(0)] = [φ0, φ1, ϑ0] ∈ H.
(1.9)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-29-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
10 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
If we define the bilinear form a(·, ·) : H2
() × H2
() → R by
a(w, w̃) ≡
[wxx w̃xx + wyyw̃yy + µ(wxx w̃yy + wyyw̃xx ) + 2(1 − µ)wxyw̃xy] d,
then we can state the “Green’s formula,” which involves this bilinear form (see Reference 22) and
which is valid for functions w, w̃ (smooth enough):
( 2
w)w̃d = a(w, w̃) +
∂ w
∂ν
+ (1 − µ)
∂ B2w
∂τ
w̃d
−
[ w + (1 − µ)B1w]
∂w̃
∂ν
d . (1.10)
Let E(t) denote the energy of the adjoint system Eq. (1.9), where
E(t) ≡
1
2
a(φ(t), φ(t))d +
1
2
|φ(t)|2
d +
1
2
|ϑ(t)|2
d. (1.11)
In terms of this energy, then one can show by classical functional analytical arguments (see, e.g.,
References 41 and 3) that the PDE (1.1) is null controllable, in the sense of 1, if and only if the
adjoint variables [φ, φt , ϑ] of Eq. (1.9) satisfy the following continuous observability inequality, for
some constant CT :
[φ(T ), φt (T ), ϑ(T )]H ≤ CT (a1φt L2(Q) + a2ϑL2(Q)). (1.12)
Having worked to establish the sharp constant CT in the observability inequality Eq. (1.12), one
can proceed through an algorithmic procedure—using an explicit representation of the minimal norm
control, by convex optimization—so as to have that for all terminal time T 0,
Emin(T ) = O(CT ).
Because the details of this argument are known and have been previously spelled out (see, e.g.,
References 9 and 8), we defer from repeating them here.
Because of this characterization of the behavior of Emin(T ) with the constant CT in Eq. (1.12), our
work will accordingly be geared toward establishing this inequality (where, again, control parameters
ai satisfy a1, a2 ≥ 0, and a1 + a2 0).
1.3 Some Preliminary Machinery
Inthissection,weexplicitlydefinetheunderlyinggeneratorA : D(A) ⊂ H → H,whichdescribes
the thermoelastic flow. Subsequently, a proposition is derived with which to associate powers of this
generator with specific Sobolev spaces. This characterization of the powers will be critical in work.
r To start, we define the linear operator AD : D(AD) ⊂ L2
() → L2
() by
AD ≡ − ;
(1.13)
D(AD) = H2
() ∩ H1
0 ().
r We will also need the following (Dirichlet) map D : L2
( ) → L2
():
Df = g ⇔ g = 0 on and g| = f on . (1.14)
By the classical elliptic regularity, we have that D ∈ L(Hs
( ), Hs+ 1
2 ()) for all s (see
Reference 30).](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-30-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.3 Some Preliminary Machinery 11
r We also define the linear operator Å : D(AD) ⊂ L2
() → L2
() by setting Å = 2
,
for ∈ D(Å), where
D(Å) =
∈ H4
() : [ + (1 − µ)B1] = 0
and
∂
∂ν
+ (1 − µ)
∂ B2
∂τ
−
= 0
,
where the boundary operators Bi are as defined in Eq. 1.3.
This operator is densely defined, positive definite, and self-adjoint. Consequently by
Reference 21, one has the characterization
D(Å
1
2 ) ≈ H2
(); with moreover
Å
1
2 φ
2
L2()
= a(φ, φ) +
φ2
d .
r Moreover, we define the elliptic operators Gi by
G1h = v ⇔
2
v = 0 on
v + (1 − µ)B1v = h
∂ v
∂ν
+ (1 − µ)
∂ B2v
∂τ
− v = 0
on
;
G2h = v ⇔
2
v = 0 on
v + (1 − µ)B1v = 0
∂ v
∂ν
+ (1 − µ)
∂ B2v
∂τ
− v = h
on
. (1.15)
By elliptic regularity (see, e.g., Reference 30) one has that for all real s,
G1 ∈ L(Hs
( ), Hs+ 5
2 ()); G2 ∈ L(Hs
( ), Hs+ 7
2 ()). (1.16)
With these operators defined above, we have that the generator A : D(A) ⊂ H → H of the
thermoelastic semigroup may be given the explicit representation
A =
0 I 0
−Å 0 α(AD(I − Dγ0) − ÅG1γ0 + λÅG2γ0)
0 −αAD(I − Dγ0) −αAD(I − Dγ0)
;
D(A) =
[ω0, ω1, θ0] ∈ H2
() × H2
() × H2
() : Å [ω0 + α (G1γ0 − λG2γ0) θ0] ∈ L2
()
and
∂θ0
∂ν
+ λθ0
= 0
(1.17)
(here, γ0 ∈ L(H1
(), H
1
2 ( )) is the classical Sobolev trace map; i.e., γ0 f = f | for f ∈ C∞
()).
As we have said, it is now known that the generator A : D(A) ⊂ H → H for the thermoelas-
tic plate, with free mechanical boundary conditions, is associated with an analytic C0-semigroup
{eAt
}t≥0 of contractions on H (see Reference 27 and references therein), with moreover A−1
being
bounded on H.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-31-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
12 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
For this realization of the generator, we now proceed to show the following:
PROPOSITION 1.1
Let integer k = 1, 2, . . . . Then D(Ak
) ⊂ H2k+2
() × H2k
() × H2k
().
PROOF OF PROPOSITION 1.1 Let first [ω0, ω1, θ0] ∈ D(A). Then by definition, ω1, θ0 ∈
H2
(). Moreover, from the abstract representation in Eq. (1.17), we have
Åω0 + αÅG1 θ0| − αλÅG2 θ0| + α θ0 = f ∈ L2
().
Because θ0| ∈ H
3
2 ( ), then consequently from the elliptic regularity results posted in Eq. (1.16),
ω0 = Å−1
f − αG1 θ0| + αλG2 θ0| − αÅ−1
θ0 ∈ H4
().
So the assertion is true for k = 1.
Proceeding now by induction, suppose that the result holds true for integer k − 1, k ≥ 2, and let
[ω0, ω1, θ0] ∈ D(Ak
). Then, because
A
ω0
ω1
θ0
∈ D(Ak−1
),
we have
ω1 ∈ H2k
();
Åω0 + αÅG1 θ0| − αλÅG2 θ0| + α θ0 = f ∈ H2k−2
();
ARθ0 − α ω1 = g ∈ H2k−2
(). (1.18)
Here, AR : D(AR) ⊂ L2
() → L2
() is the elliptic operator defined by
AR f = − f ; D(AR) =
f ∈ H2
() :
∂ f
∂ν
+ λf = 0, λ 0
. (1.19)
Reading off the third equation in Eq. (1.18), we obtain, after using elliptic regularity,
θ0 = A−1
R (g + Dγ0θ0 − α ω1) ∈ H2k
().
In turn, we can use again the result in Eq. (1.16) to have that
ω0 = Å−1
f − αG1γ0θ0 + αλG2γ0θ0 − αÅ−1
θ0 ∈ H2k+2
().
This concludes the proof of Proposition 1.1.
1.4 A Singular Trace Estimate
In this section, we exploit the underlying analyticity of the thermoelastic semigroup so as to
generate pointwise (in time) estimate of boundary traces of the adjoint variables φt (t) and ϑ(t)
of Eq. (1.9). These estimates will be of use to us in the proof of Theorem 1.1, inasmuch as they](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-32-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.4 A Singular Trace Estimate 13
each reflect a proper “distribution” between the measurement E(t) of the energy and the observation
term—be it φt or ϑ. The price to pay for these benefical estimates is the appearance therein of singular
weights of the form 1
ts , where parameter s will depend on the order of derivatives present.
LEMMA 1.1
Let
x(t) ≡ [φ(t), φt (t), ϑ(t)] denote the solution of the adjoint system Eq. (1.9), subject to the initial
condition
x(0) = [φ0, −φ1, ϑ0] ∈ H. Let,moreover, Dm be a differential operator of order m ≥ 0
with respect to the interior variables. Then for integers k = 1, 2, . . . , and all t 0 we have
1. Dmϑ(t)L( ) ≤
Ck
t
m
2 + 1
4
eA t
2
x0
1
2k
H ϑ(t)
1− 1
2k
L2() ;
2. Dmφt (t)L( ) ≤
Ck
t
m
2 + 1
4
eA t
2
x0
1
2k
H φt (t)
1− 1
2k
L2() ;
3. D1φtt (t)L( ) ≤
Ck
t
7
4
eA t
2
x0
H
.
PROOF OF LEMMA 1.1 By a trace interpolation result (see, e.g., Reference 38) and the iterative
use of a classical PDE moment inequality, we have the following string of estimates, which is valid
for any g ∈ H2k+1
(m+1)
():
Dm gL( ) ≤ C Dm g
1
2
L() Dm g
1
2
H1() ≤ C g
1
2
Hm () g
1
2
Hm+1()
≤ C g
1
2
L() g
1
4
H2m () g
1
4
H2(m+1)() ≤ C g
3
4
L() g
1
8
H4m () g
1
8
H4(m+1)()
≤ . . . ≤ C g
1− 1
2k
L() g
1
2k+1
H2k m ()
g
1
2k+1
H2k (m+1)()
. (1.20)
Now by virtue of the analyticity of the thermoelastic semigroup {eAt
}t≥0 and Proposition 1.1, we
have for all t 0,
[φ(t), φt (t), ϑ(t)] ∈ D(A2k−1
m
) ⇒ [φt (t), ϑ(t)] ∈ [H2km
()]2
. (1.21)
Setting now g ≡ ϑ(t) (resp., φt (t)) in Eq. (1.20), we obtain
Dmϑ(t)L( ) ≤ C ϑ(t)
1− 1
2k
L() ϑ(t)
1
2k+1
H2k m ()
ϑ(t)
1
2k+1
H2k (m+1)()
≤ C
A2k−1
m
x(t)
1
2k+1
H
A2k−1
(m+1)
x(t)
1
2k+1
H
ϑ(t)
1− 1
2k
L()
= C
A2k−1
m
eA t
2 eA t
2
x0
1
2k+1
H
A2k−1
(m+1)
eA t
2 eA t
2
x0
1
2k+1
H
ϑ(t)
1− 1
2k
L() . (1.22)
At this point, we can invoke the well known pointwise estimate that is valid for any generator of
an analytic semigroup: for all time t 0 and integer m = 1, 2, . . . ,
Am
eAt
L(H) ≤
Cm
tm
, (1.23)
where constant C is independent of m (see, e.g., Reference 33, p. 70). Applying this estimate to the
chain Eq. (1.22), we have
Dmϑ(t)L( ) ≤
C
t
m
2 + 1
4
eA t
2
x0
1
2k
H ϑ(t)
1− 1
2k
L() .](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-33-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
14 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
This gives (Lemma 1.1, Step 1) (Step 2) is obtained in the very same way, by setting g = φt in
Eq. (1.22) and then invoking the containment Eq. (1.21). For (Step 3), we have along the same lines,
by means of the trace interpolation inequality in Reference 38 and the containment Eq. (1.21),
D1φtt (t)L2( ) ≤ C φtt (t)
1
2
H1() φtt (t)
1
2
H2() ≤ C
A
1
2
xt (t)
1
2
H
A
xt (t)
1
2
H
≤ CA2
x(t)
1
2
H ≤
C
t
7
4
eA t
2
x0
1
2
H
,
which completes the proof.
1.5 Proof of Theorem 1.1(1)
1.5.1 Estimating the Mechanical Velocity
In what follows, we will have need of the polynomial weight h(t), defined by
h(t) ≡ ts
(T − t)s
. (1.24)
For the proof of Theorem 1.1(1), we will take s ≡ 6.
In terms of the the solution [φ, φt , ϑ] of Eq. (1.9) and its corresponding energy E(t), the necessary
inequality for the case of thermal control is
E(T ) ≤ CT ϑL2(Q) . (1.25)
It is the derivation of this inequality that will drive the proof of Theorem 1.1.
We will start by applying the Laplacian to both sides of the heat equation in Eq. (1.9). This gives
ϑt − 2
ϑ − α 2
φt = 0 in .
From this expression and the free boundary conditions in Eq. (1.9), we have that the velocity term
φt satisfies the following elliptic problem for all t 0:
2
φt (t) =
1
α
ϑt (t) −
1
α
2
ϑ(t) in
φt (t) + (1 − µ)B1φt (t) = −αϑt
∂ φt (t)
∂ν
+ (1 − µ)
∂ B2φt (t)
∂τ
− φt (t) = αλϑt
on .
(1.26)
Using this Green’s map defined in Eq. (1.15), we have from Eq. (1.26) that the velocity φt may
be written explicitly as
φt (t) =
1
α
Å−1
ϑt (t) − 2
ϑ(t)
− αG1γ0(ϑt (t)) + αλG2γ0(ϑt (t)). (1.27)
From this, we have
T
0
h φt 2
L2() dt
=
T
0
h
1
α
Å−1
[ ϑt (t) − 2
ϑ(t)] − αG1γ0(ϑt (t)) + αλG2γ0(ϑt (t)), φt
L2()
dt, (1.28)
where h(t) is the polynomial weight described in Eq. (1.24).](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-34-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.5 Proof of Theorem 1.1(1) 15
Analysis of the right-hand side of Eq. (1.28).
1.
T
0
h ([G1 − λG2] γ0ϑt , φt )L2() dt = −
T
0
h
([G1 − λG2] γ0ϑ, φt )L2() dt
−
T
0
h([G1 − λG2] γ0ϑ, φtt )L2() dt. (1.29)
a. By the regularity posted in Eq. (1.16) and an application of Lemma 1.1 (with m = 0 and
k = 2, say) we have
T
0
h
([G1 − λG2] γ0ϑ, φt )L2() dt
≤ C
T
0
|h
| ϑL2( ) φt L2() dt
≤ C
T
0
|h
|
t
1
4
h(t)
h(t)
5
8
ϑ(t)
3
4
L2()
eA t
2
x0
5
4
H
dt.
Invoking Hölder’s inequality to this right hand side, with Hölder conjugates (8
3
, 8
5
), we
obtain now the estimate
T
0
h
([G1 − λG2] γ0ϑ, φt )L2() dt
≤ C T
26
3
T
0
h(t) ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.30)
b. Proceeding as above, with m = 0 and k = 2 in Lemma 1.1, we have
T
0
h([G1 − λG2] γ0ϑ, φtt )L2() dt
≤ C
T
0
h(t)
t
1
4
ϑ(t)
3
4
L2()
eA t
2
x0
1
4
H
A
xt (t)L2() dt
≤ C
T
0
h(t)
t
5
4
ϑ(t)
3
4
L2()
eA t
2
x0
5
4
H
dt
≤ C T
26
3
T
0
h(t) ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.31)
Combining Eq. (1.30) and Eq. (1.31) now gives
T
0
h([G1 − λG2] γ0ϑt , φt )L2() dt
≤ C T
20
3
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.32)
2. Next,
T
0
h
Å−1
( ϑt ), φt
L2()
dt = −
T
0
h
Å−1
( ϑ), φt
L2()
dt
−
T
0
h
Å−1
( ϑ), φtt
L2()
dt. (1.33)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-35-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
16 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
a. By Green’s theorem and the Lemma 1.1, with m = 0 and k = 2, we have
T
0
h
Å−1
( ϑ), φt
L2()
dt
=
T
0
h
ϑ,
∂
∂ν
+ I Å−1
φt
L2( )
dt −
T
0
h
ϑ, Å−1
φt
L2()
dt
≤ C T
26
3
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.34)
b. Likewise, by Green’s Theorem, the analyticity of the semigroup and Lemma 1.1, with
m = 0 and k = 2, we have
T
0
h(t)
ϑ, Å−1
φtt
L2()
dt
≤ C T
26
3
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.35)
Applying the estimates Eq. (1.34) and Eq. (1.35) to Eq. (1.33) now yields
T
0
h( ϑt , φt )L2() dt
≤ C T
26
3
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.36)
3. By the Green’s identity posted in Eq. (1.10), we have
T
0
h
2
ϑ, Å−1
φt
L2()
dt =
T
0
h(t)(ϑ, φt )L2() dt
=
T
0
h(t)
∂
∂ν
+ (1 − µ)
∂ B2
∂τ
ϑ, Å−1
φt
L2( )
−
[ + (1 − µ)B1] ϑ,
∂
∂ν
Å−1
φt
L2( )
dt.
Applying once more the Lemma 1.1 (e.g., with m = 3, k = 3) we have
T
0
h
2
ϑ, Å−1
φt
L2()
dt
≤ C T 8
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.37)
Combining the expression Eq. (1.28) with the estimates of Eqs. (1.32 ), (1.36), and (1.37) gives
us the following estimate for the mechanical velocity:
LEMMA 1.2
With s = 6 in Eq. (1.24), the solution [φ, φt , ϑ] of (1.9) satisfies the following estimate for all 0:
T
0
h(t) φt 2
L2() dt ≤ C T 8
T
0
ϑ2
L2() dt +
T
0
E
t
2
dt.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-36-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.5 Proof of Theorem 1.1(1) 17
1.5.2 Estimating the Mechanical Displacement
Here, we shall show the following:
LEMMA 1.3
The solution [φ, φt , ϑ] of Eq. (1.9) satisfies the following estimate for all , δ 0:
T
0
h(t)
Å
1
2 φ
2
L2()
dt ≤ CT
13
2 −δ
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt +
T
0
h(t)E(t)dt.
PROOF OF LEMMA 1.3 We start by applying the multiplier h(t)φ(t) to the mechanical
component in Eq. (1.9). We arrive at the relation
T
0
h(t)
Å
1
2 φ
2
L2()
dt
=
T
0
h
(t)(φt , φ)L2() dt +
T
0
h(t) φt 2
L2() dt
+ αλ
T
0
h(t)
Å
1
2 G2γ0ϑ, Å
1
2 φ
L2()
dt − αλ
T
0
h(t)
Å
1
2 G2γ0ϑ, Å
1
2 φ
L2()
dt
− α
T
0
h(t)(ϑ, φ)L2()dt + α
T
0
h(t)
(ϑ, λφ +
∂φ
∂ν L2( )
dt. (1.38)
Now, using the elliptic regularity posted in Eq. (1.16) and the usage of Lemma 1.1, with m = 0
and k = 3, we obtain
T
0
h(t)
Å
1
2 G2γ0ϑ, Å
1
2 φ
L2()
dt − αλ
T
0
h(t)
Å
1
2 G2γ0ϑ, Å
1
2 φ
L2()
dt
− α
T
0
h(t)(ϑ, φ)L2() dt + α
T
0
h(t)
(ϑ, λφ +
∂φ
∂ν L2( )
dt
≤ C T
26
3
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.39)
Combining this estimate with that in Lemma 1.2 then gives the preliminary estimate
T
0
h(t)
Å
1
2 φ
2
L2()
dt ≤
T
0
h
(t)(φt , φ)L2() dt
+ CT 8
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.40)
Apparently, we must estimate the first term on the right-hand side of Eq. (1.40). To this end, we
use the pointwise expression for φt in Eq. (1.27):
T
0
h
(φt , φ)L2() dt =
T
0
h
1
α
Å−1
( ϑt − 2
ϑ) − αG1γ0ϑt + αλG2γ0ϑt , φ
L2()
dt
(1.41)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-37-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
18 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
1. The abstract Green’s Theorem gives
T
0
h
( 2
ϑ, Å−1
φ)L2()dt =
T
0
h
(ϑ, φ)L2() +
∂
∂ν
+ (1 − µ)
∂ B2
∂τ
ϑ, Å−1
φ
L2( )
−
[ + (1 − µ)B1] ϑ,
∂
∂ν
Å−1
φ
L2( )
dt.
Applying now the Lemma 1.1 with m = 3, 2 yields
T
0
h
( 2
ϑ, Å−1
φ)L2()dt
≤ C
T
0
|h
|
t
7
4
ϑ
1− 1
2k
L2()
eA t
2
x0
1+ 1
2k
H dt.
Let k ≥ 4. Then, because h
(t) = 6t5
(T − 2t)(T − t)5
, we can apply now Hölder’s inequality
with Hölder conjugates (2 2k
2k −1
, 1
1
2 +2−k−1 ) so as to have
T
0
h
( 2
ϑ, Å−1
φ)L2()dt
≤ C
T
0
t
2k−1−6
2k −1 |T − 2t|
2k+1
2k −1 (T − t)
2k+2−6
2k −1 ϑ2
L2() dt
+
T
0
h(t)E
t
2
dt ≤ CT
13×2k−1−12
2k −1
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt
(again this inequality being valid for k ≥ 4). Now for any δ 0, we can rechoose integer k
large enough so as to have 13×2k−1
−12
2k −1
≥ 13
2
− δ. This gives, then, for T 1,
T
0
h
( 2
ϑ, Å−1
φ)L2()dt
≤ CT
13
2 −δ
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt. (1.42)
(This is the term which ultimately dictates the singularity.)
2. Next,
T
0
h
[(G1 − λG2)γ0ϑt , φ]L2()dt = −
T
0
h
[(G1 − λG2)γ0ϑ, φ]L2() dt
−
T
0
h
[(G1 − λG2)γ0ϑ, φt ]L2() dt. (1.43)
a. By the regularity posted in Eq. (1.16) and Lemma 1.1,
T
0
h
[(G1 − λG2)γ0ϑ, φ]L2() dt
≤ C
T
0
|h
|
t
1
4
ϑ
1− 1
2k
L2()
eA t
2
x0
1+ 1
2k
H dt.
Applying Hölder’s inequality to the right-hand side, with Hölder conjugates (2 2k
2k −1
,
1
1
2 +2−k−1 ) now yields
T
0
h
[(G1 − λG2)γ0ϑ, φ]L2()dt
≤ C
T
0
|h
|
t
1
4
h(t)
h(t)
1
2 +2−k−1
× ϑ
1− 1
2k
L2()
eA t
2
x0
1+ 1
2k
H dt ≤ C T
3 5×2k−1−4
2k −1
T
0
ϑ2
L2() +
T
0
h(t)E
t
2
dt.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-38-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.5 Proof of Theorem 1.1(1) 19
Because for any δ 0, we can choose integer k large enough so that 35×2k−1
−4
2k −1
≥ 15
2
−δ,
we then get
T
0
h
[(G1 − λG2)γ0ϑ, φ]L2() dt
≤ C T
15
2 −
T
0
ϑ2
L2() +
T
0
h(t)E
t
2
dt.
(1.44)
b. In the same way as above, we have for integer k large enough in Lemma 1.1,
T
0
h
[(G1 − λG2)γ0ϑ, φt ]L2() dt
≤ C T
15
2 −δ
T
0
ϑ2
L2() +
T
0
h(t)E
t
2
dt.
(1.45)
The estimates Eqs. (1.44) and (1.45), applied to the relation Eq. (1.43) now give
T
0
h
[(G1 − λG2)γ0ϑt , φ]L2() dt
≤ C T
15
2 −
T
0
ϑ2
L2() +
T
0
h(t)E
t
2
dt.
(1.46)
for integer k large enough.
3.
T
0
h
Å−1
ϑt , φ
L2()
dt = −
T
0
h
( ϑ, Å−1
φ)L2() dt −
T
0
h
ϑ, Å−1
φt
L2()
dt
(1.47)
a. By Green’s Theorem and Lemma 1.5, we have in a fashon similar to that in (1.a.),
T
0
h
( ϑ, Å−1
φ)L2() dt
=
−
T
0
h
(θ, Å−1
φ)L2()
+
T
0
h
θ,
∂
∂ν
+ λ Å−1
φ
L2( )
≤ C T
3 5×2k−1−4
2k −1
T
0
ϑ2
L2()
+
T
0
h(t)E
t
2
dt ≤ CT
15
2 −δ
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt,
(1.48)
for integer k large enough.
b. In the same way,
T
0
h
ϑ, Å−1
φt
L2()
dt
≤ CT
15
2 −δ
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.49)
Eqs. (1.47), (1.48), and (1.49) together give the estimate
T
0
h
ϑt , Å−1
φ
L2()
dt
≤ CT
15
2 −δ
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.50)
Combining Eqs. (1.40), (1.41), (1.46), (1.50), and (1.42) will complete the proof of Lemma 1.3.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-39-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
20 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
1.5.3 Conclusion of the Proof of Theorem 1.1(1)
Combining Lemmas 1.2 and 1.3 gives the following estimate for the energy:
T
0
E(t) dt ≤ C T
13
2 −δ
T
0
ϑ2
L2() dt +
T
0
h(t)E
t
2
dt;
or after changing limits of integration,
T
2
0
[(1 − )h(t) − 2h(2t)] E(t) dt + (1 − )
T
T
2
h(t)E(t)dt ≤ C T
13
2 −δ
T
0
ϑ2
L2() dt.
For 0 small enough, this yields then
T
T
2
h(t)E(t) dt ≤ C T
13
2 −δ
T
0
ϑ2
L2() dt.
Using the dissipation inherent in the thermoelastic system (i.e., E(t) ≤ E(s) for s ≤ t), we finally
obtain
E(T ) ≤ CT
13
2 −δ−13
T
0
ϑ2
L2() dt.
This establishes the inequality Eq. (1.25), with CT = CT −q
, where q = 13
4
− δ
2
, for any δ 0. This
concludes the proof of Theorem 1.1(1).
1.6 Proof of Theorem 1.1(2)
1.6.1 A First Supporting Estimate
In what follows, we will again make use of the polynomial weight h(t) in Eq. (1.24), with s = 4
therein.
In the present case of mechanical control, the necessary inequality (Eq. (1.12)) becomes
E(T ) ≤ CT φt L2(Q) . (1.51)
to be valid for all finite energy solutions to Eq. (1.9).
We start by establishing the following estimate:
PROPOSITION 1.2
The solution [φ, φt , ϑ] of Eq. (1.9) satisfies the relation
T
0
h(t)
A−1
R ϑt , ϑ
L2()
dt
≤ C T 4
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt.
PROOF OF PROPOSITION 1.2 From the mechanical component of Eq. (1.9) we have, after
an extra differentiation in time, the expression −α ϑt = ∂3
∂t3 φ + 2
φt ; whence we obtain
A−1
R ϑt =
1
α
A−2
R
∂3
∂t3
φ +
1
α
A−2
R
2
φt ,](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-40-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
1.6 Proof of Theorem 1.1(2) 21
where the positive definite, self-adjoint operator AR : D(AR) : L2
() → L2
() is as defined in
Eq. (1.19). Subsequently, we will have the following relation:
T
0
h(t)
A−1
R ϑt , ϑ
L2()
dt =
1
α
T
0
h(t)
∂3
∂t3
φ, A−2
R ϑ
L2()
dt
+
1
α
T
0
h(t)
2
φt , A−2
R ϑ
L2()
dt. (1.52)
We need to estimate the right-hand side of this expression.
1. For the first term on the right-hand side of Eq. (1.52), integration by parts gives
T
0
h(t)
∂3
∂t3
φ, A−2
R ϑ
L2()
dt =
T
0
h(t)
φt , A−2
R ϑtt
L2()
dt
+ 2
T
0
h
(t)
φt , A−2
R ϑt
L2()
dt +
T
0
h
(t)
φt , A−2
R ϑ
L2()
dt. (1.53)
We proceed to scrutinize each term on the right-hand side. To this end, we introduce the
(Robin) map R ∈ L[L2
( ), L2
()], defined by
R f = g ⇔ g = 0 on and
∂g
∂ν
+ λg = f on (1.54)
(by elliptic regularity, we have in fact that R ∈ L[Hs
( ), Hs+ 3
2 ()] for all real s). Using this
quantity with the heat equation in Eq. (1.9), we will then have the relations
A−2
R ϑt = −A−1
R ϑ − αA−1
R
I − R
λγ0 +
∂
∂ν
φt ;
(1.55)
A−2
R ϑtt = ϑ + α
I − R
λγ0 +
∂
∂ν
φt − αA−1
R
I − R
λγ0 +
∂
∂ν
φtt .
a. From Eq. (1.56), we have
T
0
h(t)
φt , A−2
R ϑtt
L2()
dt
≤
T
0
h(t)
×
φt , ϑ + α
I − R
λγ0 +
∂
∂ν
φt
L2()
dt
+
T
0
h(t)
φt , αA−1
R
I − R
λγ0 +
∂
∂ν
φtt
L2()
dt. (1.56)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-41-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
22 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
To handle the most problematic term on the right-hand side of this expression (with again
x(t) = [φ(t), φt (t), ϑ(t)]), we use the singular trace estimate in Lemma 1.1(3):
T
0
h(t)
φt , A−1
R R
∂
∂ν
φtt
L2()
dt ≤ C
T
0
h(t) φt L2()
∂
∂ν
φtt
L2( )
dt
≤ C
T
0
h(t)
t
3
4 t
φt L2()
eA t
2
x(0)
H
dt
≤ C
T
0
h(t)
t
7
2
φt 2
L2() dt
+
T
0
h(t)E
t
2
dt
≤ C T
9
2
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt.
Applying this estimate to Eq. (1.56) and treating in like fashion the other terms on the
right-hand side thereof, we have
T
0
h(t)
φt , A−2
R ϑtt
L2()
dt
≤ C T
9
2
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.57)
b. Using the first relation in Eq. (1.56), we have, analogously to what was obtained in (1.a),
T
0
h
(t)
φt , A−2
R ϑ
L2()
dt + 2
T
0
h
(t)
φt , A−2
R ϑt
L2()
dt
≤ C
T
0
|h
(t)| +
|h
(t)|
t
3
4
φt L2()
eA t
2
x(0)
H
dt
≤ C T 4
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt. (1.58)
Combining Eqs. (1.56) and (1.58) now gives
T
0
h(t)
∂3
∂t3
φ, A−2
R ϑ
L2()
dt
≤ C T 4
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt.
(1.59)
2. By the “Green’s” formula in Eq. (1.10), we have
T
0
h(t)
2
φt , A−2
R ϑ
L2()
dt =
T
0
h(t)a
φt , A−2
R ϑ
L2()
dt
+
T
0
h(t)
αλϑt + φt , A−2
R ϑ
L2( )
dt −
T
0
h(t)
φt + (1−µ)B1φt ,
∂
∂ν
A−2
R ϑ
L2( )
dt
= −
T
0
h(t)
φt + (1 − µ)B1φt ,
λI +
∂
∂ν
A−2
R ϑ
L2( )
dt
+
T
0
h(t)
φt , 2
A−2
R ϑ
L2()
dt +
T
0
h(t)
∂
∂ν
φt , [ + (1 − µ)B1] A−2
R ϑ
L2( )
dt
−
T
0
h(t)
φt ,
∂
∂ν
+ (1 − µ)
∂ B2
∂τ
− I
A−2
R ϑ
L2( )
dt. (1.60)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-42-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
24 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
2. Estimating the Mechanical Component. Here, we apply the multiplier intrinsic to uncoupled
plates and beams. To wit, from the mechanical component of Eq. (1.9), we have via h(t)φ(t)
and an invocation of the Green’s Theorem Eq. (1.10) the expression
T
0
Å
1
2 φ
2
L2()
dt = −α
T
0
h(t)(ϑ, φ) dt +
T
0
h
(t)(φt , φ) dt +
T
0
h(t) φt 2
L2() dt.
(1.65)
Applying the estimate of Eq. (1.64) (available for the thermal component) now gives
T
0
Å
1
2 φ
2
L2()
dt ≤ C T 4
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt. (1.66)
Combining the estimates of Eqs. (1.64) and (1.66) now give the estimate for the energy
T
0
h(t)E(t) dt ≤ C T 4
T
0
φt 2
L2() dt +
T
0
h(t)E
t
2
dt.
With this in hand, we can proceed as in the previous case so as to have the observability
inequality Eq. (1.51), with CT = T − 5
2 . Subsequently, we will determine that in the present case
ofmechanicalcontrol,onehasEmin(T ) = O(T − 5
2 ).ThisconcludestheproofofTheorem1.1(2)
with free boundary conditions and one control.
1.7 Proof of Theorem 1.1(3)
Here we set the index s = 2 in Eq. (1.24). In this present case of dual—mechanical and thermal—
control, the necessary inequality is
E(T ) ≤ CT (φt L2(Q) + ϑL2(Q)), (1.67)
where again [φ, φt , ϑ] solve the homogeneous system Eq. (1.9). Using the relation Eq. (1.65),
we have
(1 − )
T
0
h(t)
Å
1
2 φ
2
L2()
dt ≤ C
T
0
h(t) +
[h
(t)]2
h(t)
φt 2
L2() + ϑ2
L2()
dt
≤ CT 2
T
0
φt 2
L2() + ϑ2
L2()
dt.
This then gives
T
0
h(t)E(t) dt ≤ CT 2
T
0
φt 2
L2() + ϑ2
L2()
dt,
whence we obtain the inequality Eq. (1.67). From here, we can use the usual algorithmic argument
so as to have Emin(T ) = O(T − 3
2 ). This concludes the proof of Theorem 1.1(3).
References
[1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled
parabolic-hyperbolic system, Electron. J. Differential Equations, 2000, No. 22 (2000),
1–15.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-44-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
References 25
[2] G. Avalos, Exact controllability of a thermoelastic system with control in the thermal compo-
nent only, Differential Integral Equations, 13, (2000), 613–630.
[3] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free bound-
ary conditions, SIAM J. Control Optim., 38, No. 2 (2000), 337–383.
[4] G. Avalos and I. Lasiecka, Exponential Stability of a Thermoelastic System without Mechanical
Dissipation, Rendiconti dell’Istituto di Matematica dll’Università di Trieste, Vol. XXVIII,
Supplemento (1996), 1–28.
[5] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boun-
dary conditions without mechanical dissipation, SIAM J. Math. Anal., 29, No. 1 (1998), 155–
182.
[6] G. Avalos and I. Lasiecka, A Note on the Null Controllability of a Thermoelastic Plates and
Singularity of the Associated Minimal Energy Function, Scuola Normale Superiore (Pisa),
Preprints di Matematica, n. 10 (Giugno 2002).
[7] G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the
strongly damped abstract wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), II (2003),
601–616.
[8] G. Avalos and I. Lasiecka, Mechanical and thermal null controllability of thermoelastic plates
and singularity of the associated minimal energy function, Control Cybern. special volume
dedicated to K. Malanowski. 32, No. 3 (2003), 473–491.
[9] G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of
the associated minimal energy function, J. Math. Anal. Appl., 294, (2004), 34–61.
[10] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free bound-
ary conditions, SIAM J. Control Optim., 38, No. 2 (2000), 337–383.
[11] A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl.
Anal., in press.
[12] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinte
Dimensional Systems, Vol. II, Birkhäuser, Boston, (1993).
[13] J. Cea, Lectures on Optimization—Theory and Algorithms Published for the Tata Institute of
Fundamental Research, Springer-Verlag, New York, (1978).
[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University
Press, New York, New York, (1992).
[15] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore,
(2001).
[16] M. Eller, I. Lasiecka, and R. Triggiani. Simultaneous exact-approximate boundary controlla-
bility of thermoelastic plates with variabale thermal coefficients and moment control. J. Math.
Anal. Appl., 251, (2000), 452–478.
[17] M. Eller, I. Lasiecka and R. Triggiabi. Simultaneous exact controllability of thermoelastic
plates with variable thermal coefficients and clamped/Dirichlet boundary controls, Discrete
Continuous Dynamical Syst., 7, No. 2 (2001), 283–302.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-45-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
26 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability
[18] S. Hansen and B. Zhang, Boundary control of a thermoelastic beam, J. Math. Anal. Appl., 210,
(1997), 182–205.
[19] F.GozziandP.Loreti,Regularityoftheminimumtimefunctionandminimumenergyproblems,
SIAM J. Control Optim., in press.
[20] B. Guichal, A lower bound of the norm of the control operator for the heat equation. J. Math.
Anal. Appl. 110, No. 2 (1985), 519–527.
[21] P. Grisvard, Caracterization de quelques espaces d’interpolation, Arch. Rational Mech. Anal.,
25, (1967), 40–63.
[22] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math., 10, SIAM,
Philadelphia, (1989).
[23] J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal.,
112, (1990), 223–267.
[24] J. Lagnese and J.L. Lions, Modelling, Analysis and Control of Thin Plates, Masson Paris,
(1988).
[25] I. Lasiecka and T. Seidman, Thermal boundary control of a thermoelastic system,
preprint (2004).
[26] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with clamped/hinged
B.C., Abst. Appl. Anal., 3, No. 2 (1998), 153–169.
[27] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled BC. Part II:
The case of free BC, Annali Scuola Normale di Pisa, Classes Scienze (Serie IV, Fasicolo 3–4),
Vol. XXVII (1998[c]), 457–497.
[28] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous
and Approximation Theories. I: Abstract Parabolic Systems, Cambridge University Press,
New York, (2000).
[29] I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity
with free boundary conditions and partial boundary dissipation, Commun. Partial Differential
Equations, 24, 910 (1999), 1801–1847.
[30] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications,
Vol. 1, Springer-Verlag, New York, (1972).
[31] Z. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8,
No. 3 (1995), 1–6.
[32] A. Lunardi, Schauder’s estimates for a class of degenerate elliptic and parabolic operators with
unbounded coefficients in Rn
, Ann. Scuola Sup. Pisa (IV), XXIV (1997), 133–164.
[33] T.I. Seidman, How fast are violent controls? Math. Controls Signals Syst., (1988), 89–95.
[34] T.I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math.
Optim., 11, (1984), 145–152.
[35] T.I. Seidman and J. Yong, How fast are violent controls?, II, Math. Controls Signals Syst., 9,
(1997), 327–340.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-46-320.jpg)
![April 4, 2005 10:3 3086 DK2961˙C001
References 27
[36] T. Seidman, S. Avdonian, and S. Ivanov, The window problem for series of complex exponen-
tials, J. Fourier’s Anal., 6, (2000), 235–254.
[37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag,
New York, (1984).
[38] R. Triggiani, Analyticity, and lack thereof, of semigroups arising from thermo-elastic plates, in
the special volume entitled Computational Science for the 21st Century, Chapter on Control:
Theory and Numerics, Wiley 1997; Proceedings in honor of R. Glowinski, (May, 1997).
[39] R. Triggiani, Optimal estimates of norms of fast controls in exact null controllability of two
non-classical abstract parabolic systems, Adv. Differential Equations, 8, No. 2 (2003), 189–229.
[40] J. Zabczyk, Mathematical Control Theory, Birkhäuser, Boston, (1992).](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-47-320.jpg)
![April 5, 2005 13:52 3086 DK2961˙C002
Chapter 2
Interior and Boundary Stabilization
of Navier-Stokes Equations
Viorel Barbu
Alexandru Ioan Cuza University
2.1 Introduction ............................................................................ 29
2.2 Part I: Interior Control [4] ............................................................... 30
2.2.1 Introduction ..................................................................... 30
2.2.2 Main Results .................................................................... 32
2.3 Part II: Boundary Control [3] ............................................................ 35
2.3.1 Introduction ..................................................................... 35
2.3.2 Main Results (Case d = 3) ....................................................... 39
References .............................................................................. 41
Abstract We report on very recent work on the stabilization of the steady-state solutions to
Navier-Stokes equations on an open bounded domain ⊂ Rd
, d = 2, 3, by either interior or else
boundary control.
More precisely, as to the interior case, we obtain that the steady-state solutions to Navier-Stokes
equations on ⊂ Rd
, d = 2, 3, with no-slip boundary conditions, are locally exponentially stabi-
lizable by a finite-dimensional feedback controller with support in an arbitrary open subset ω ⊂
of positive measure. The (finite) dimension of the feedback controller is minimal and is related to
the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation.
Second, as to the boundary case, we obtain that the steady-state solutions to Navier-Stokes
equations on a bounded domain ⊂ Rd
, d = 2, 3, are locally exponentially stabilizable by a
boundary closed-loop feedback controller, acting tangentially on the boundary ∂, in the Dirichlet
boundaryconditions. If d = 3,thenonlinearityimposesanddictatestherequirementthatstabilization
must occur in the space [H
3
2 +
()]3
, 0, a high topological level. A first implication thereof is
that,ford = 3,theboundaryfeedbackstabilizingcontrollermustbeinfinitedimensional.Moreover,it
generally acts on the entire boundary ∂. Instead, for d = 2, where the topological level for stabilizat-
ion is [H
3
2 −
()]2
, the boundary feedback stabilizing controller can be chosen to act on an arbitrarily
small portion of the boundary. Moreover, still for d = 2, it may even be finite dimensional, and this
occurs if the linearized operator is diagonalizable over its finite-dimensional unstable subspace.
2.1 Introduction
We hereby report on recent joint work on the stabilization of steady-state solutions to Navier-
Stokes equations on an open bounded domain ⊂ Rd
, d = 2, 3, by either interior feedback control
or else boundary feedback control. The case of interior control is taken from the joint work with
Triggiani in Reference 4. The case of boundary control is taken from the joint work with Lasiecka
and Triggiani in Reference 3. To enhance readability, we provide independent accounts of each case.
29](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-49-320.jpg)
![April 5, 2005 13:52 3086 DK2961˙C002
30 Interior and Boundary Stabilization of Navier-Stokes Equations
2.2 Part I: Interior Control [4]
2.2.1 Introduction
The controlled Navier-Stokes equations
Consider the controlled Navier-Stokes equations (see Reference 6, p. 45, and Reference 13, p. 253
for the uncontrolled case u ≡ 0) with the non-slip Dirichlet B.C.:
yt (x, t) − ν y(x, t) + (y · ∇)y(x, t) = m(x)u(x, t) + fe(x)
+ ∇ p(x, t) in Q = × (0, ∞), (2.1)
∇ · y=0 in Q;
y=0 on = ∂ × (0, ∞);
y(x, 0)=y0(x) in .
Here, is an open, smooth, bounded domain of Rd
, d = 2, 3; m is the characteristic function of an
open smooth subset ω ⊂ of positive measure; u is the control input; and y = (y1, y2, . . . , yd) is
the state (velocity) of the system. The function v = mu can be viewed itself as an internal controller
with support in Qω = ω×(0, ∞). The functions y0, fe ∈ [L2
()]d
are given, the latter being a body
force, whereas p is the unknown pressure.
Let (ye, pe) ∈ [(H2
()]d
∩ V ) × H1
() be a steady-state solution to Eq. (2.1), that is,
−ν ye + (ye · ∇)ye = fe + ∇ pe in ; (2.2)
∇ · ye = 0 in ;
ye = 0 on ∂.
The steady-state solution is known to exist for d = 2, 3, (see Reference 6, Theorem 7.3, p. 59). Here
[6, p. 9], [13, p. 18]
V =
y ∈
H1
0 ()
d
; ∇ · y = 0
, with norm yV
≡ y
=
|∇y(x)|2
d
1
2
. (2.3)
Literature
According to some recent results of Imanuvilov [9] (see also Reference 1) any such solution ye
is locally exactly controllable on every interval [0, T ] with controller u with support in Qω. More
precisely, if the distance ye − y0H2() is sufficiently small, then there is a solution (y, p, u) to
Eq. (2.1) of appropriate regularity such that y(T ) ≡ ye. The steering control is open-loop and
depends on the initial condition. Subsequently, Reference 2 proved that any steady-state solution
ye is locally exponentially stabilizable by means of an infinite-dimensional feedback controller by
using the controllability of the linear Stokes equation. In contrast, here we shall prove, via the state
decomposition technique of References 14 and 15 and the first-order stabilization Riccati equation
method developed in our previous work (Reference 2; see also Reference 5 still in the parabolic
case, as well as Reference 11 in the hyperbolic case), that any steady-state solution ye is locally
exponentially stabilizable by a finite-dimensional closed-loop feedback controller of the form
u = −
2K
i=1
(RN (y − ye), ψi )ωψi , (2.4)](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-50-320.jpg)
![priests and friars coming in from abroad, should be liable to
castration”; and he adds in a note:
“Not, certainly, as implying a charge of immorality. Amidst the
multitude of accusations which I have seen brought against the Irish
priests of the last century, I have never, save in a single instance,
encountered a charge of unchastity. Rather the exceptional and signal
purity of Irish Catholic women of the lower class, unparalleled probably
in the civilized world, and not characteristic of the race, which in the
sixteenth century was no less distinguished for licentiousness, must be
attributed wholly and entirely to the influence of the Catholic clergy.”
Mr. Froude cannot be wholly generous and honest in a matter of
this kind, but what is true in this is sufficient for our purpose
without inquiring into what is false. It is plain from his own words
that the one thing that saved the Irish people from perdition, body
and soul, was their Catholic faith. Yet this is the man who, having
thus testified to the rival effects of Catholicity and Protestantism on
a people, has the effrontery to tell us in the “Revival of Romanism”
that
“If by this [conversions] or any other cause the Catholic Church
anywhere recovers her ascendency, she will again exhibit the detestable
features which have invariably attended her supremacy. Her rule will
once more be found incompatible either with justice or intellectual
growth, and our children will be forced to recover by some fresh
struggle the ground which our forefathers conquered for us, and which
we by our pusillanimity surrendered” (p. 103).
With his own testimony before us we may well ask in
amazement, Of which church is he writing? It would seem as
though Heaven, which through all ages has looked down upon and
permitted martyrdom for the faith, had in this instance called upon,
not a tender virgin or a strong youth, not an old man tottering into
the grave or an innocent child, to step into the arena and offer up
their life and blood for the cause of Christ, but a whole people. And
the martyrdom of this people was not for a day or an hour; it was
the slow torture of centuries. A legacy of martyrdom was
“bequeathed from bleeding sire to son.” Life was hopeless to the](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-53-320.jpg)
![One ray of light in the universal darkness now enshrouding
Protestantism shines before the eyes of Mr. Froude. It falls on the
present German Empire. Here at least the weary watchman crying
out the hours of heaven may call “All is well” to the sleepers. Here
Protestantism had its true birth; here it finds its true home. In this
blessed land lies hope and salvation for a lost world. But the picture
is so graphic that we give it in Mr. Froude’s own words:
“As the present state of France,” he says, “is the measure of the value
of the Catholic revival, so Northern Germany, spiritually, socially, and
politically, is the measure of the power of consistent Protestantism.
Germany was the cradle of the Reformation. In Germany it moves
forward to its manhood; and there, and not elsewhere, will be found the
intellectual solution of the speculative perplexities which are now
dividing and bewildering us” (pp. 130–131).
“Luther was the root in which the intellect of the modern Germans
took its rise. In the spirit of Luther this mental development has gone
forward ever since. The seed changes its form when it develops leaves
and flowers. But the leaves and flowers are in the seed, and the
thoughts of the Germany of to-day lay in germs in the great reformer.
Thus Luther has remained through later history the idol of the nation
whom he saved. The disputes between religion and science, so baneful
in their effects elsewhere, have risen into differences there, but never
into quarrels” (p. 132).
“Protestant Germany stands almost alone, with hands and head alike
clear. Her theology is undergoing change. Her piety remains unshaken.
Protestant she is, Protestant she means to be.... By the mere weight of
superior worth the Protestant states have established their ascendency
over Catholic Austria and Bavaria, and compel them, whether they will
or not, to turn their faces from darkness to light.[106] ... German religion
may be summed up in the word which is at once the foundation and the
superstructure of all religion—Duty! No people anywhere or at any time
have understood better the meaning of duty; and to say that is to say
all” (pp. 134–135).
These glowing periods are very tempting to the critic; but it is a
mark of cruelty and savagery to gloat over an easy prey. We forbear
all verbal criticism, then, and simply deny in toto the truth of Mr.
Froude’s statement. It is so very wrong that we can only think he
wrote from his imagination—a weakness from which he suffers](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-57-320.jpg)
![being moved thereto “by a terrible remonstrance against Christmas
day grounded upon divine Scripture, wherein Christmas is called
Antichrists masse, and those masse-mongers and Papists who
observe it,” and after much time “spent in consultation about the
abolition of Christmas day, passed order to that effect, and resolved
to sit upon the following day, which was commonly called Christmas
day.” Whether this latter resolution was carried into effect we do
not know. If so, let us hope that their Christmas dinners disagreed
with them horribly, and that the foul fiend Nightmare kept hideous
vigil by every Parliamentary pillow.
But think of such an atrocious sentiment being heard at all in
Westminster! How must the very echoes of the hall have shrunk
from repeating that monstrous proposition—how shuddered and
fled away into remotest corners and crevices as that
“Hideous hum
Ran through the arch’d roof in words deceiving”!
How must they have disbelieved their ears, and tossed the impious
utterance back and forth from one to another in agonized
questioning, growing feebler and fainter at each repulse, until their
voices, faltering through doubt into dismay, grew dumb with horror!
How must “Rufus’ Roaring Hall”[107]
have roared again outright with
rage and grief over that strange, that unhallowed profanation! What
wan phantoms of old-time mummeries and maskings, what dusty
and crumbling memories of royal feast and junketing, must have
hovered about the heads of those audacious innovators, shrieking
at them what unsyllabled reproaches from voiceless lips, shaking at
them what shadowy fingers of entreaty or menace! And if the
proverb about ill words and burning ears be true, how those crop-
ears must have tingled!
Within those very walls England’s kings for generations had kept
their Christmas-tide most royally with revelry and dance and](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-66-320.jpg)
![the nobility and gentry, to one hundred and twenty dishes served
by as many knights, while the mayor, who sat at a side-table, no
doubt, had to his own share no fewer than twenty-four dishes,
followed, it is to be feared, if he ate them all, by as many
nightmares; how that meek and exemplary Christian monarch,
Henry VIII., “welcomed the coming, sped the parting” wife at
successive Christmas banquets of as much splendor as the spoils of
something over a thousand monasteries could furnish forth;[108]
how
good Queen Bess, who had her own private reading of the doctrine
“it is more blessed to give than to receive,” sat in state there at this
festival season to accept the offerings of her loyal lieges, high and
low, gentle and simple, from prime minister to kitchen scullion, until
she was able to add to the terrors of death by having to leave
behind her something like three thousand dresses and some
trunkfuls of jewels in Christmas gifts; or what gorgeous revels and
masques—Inigo Jones (Inigo Marquis Would-be), Ben Jonson, and
Master Henry Lawes (he of “the tuneful and well-measured song”)
thereto conspiring—made the holidays joyous under James and
Charles. Some ghostly savor of those bygone banquets might, one
would think, have made even Praise-God Barebone’s mouth water,
and melted his surly virtue into tolerance of other folks’ cakes and
ale—what virtue, however ascetic, could resist the onslaught of two
thousand French cooks? Some faint, far echo of all these vanished
jollities should have won the ear, if not the heart, of the grimmest
“saint” among them. Or if they were proof against the
blandishments of the world’s people, if they fled from the
abominations of Baal, could not their own George Wither move
them to spare the cheery, harmless frivolities, the merry pranks of
Yule? Jovially as any Cavalier, shamelessly as any Malignant of them
all, he sings their praises in his
“CHRISTMAS CAROL.](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-69-320.jpg)
![d t e e o e et s be e y
“Hark! now the wags abroad do call
Each other forth to rambling;
Anon you’ll see them in the hall,
For nuts and apples scrambling.
Hark! how the roofs with laughter sound;
Anon they’ll think the house goes round,
For they the cellar’s depths have found.
And there they will be merry.
“The wenches with the wassail-bowls
About the streets are singing;
The boys are come to catch the owls,
The wild mare[109] in is bringing.
Our kitchen-boy hath broke his box,
And to the kneeling of the ox
Our honest neighbors come by flocks,
And here they will be merry.
“Now kings and queens poor sheep-cotes have,
And mate with everybody;
The honest now may play the knave,
And wise men play at noddy.
Some youths will now a-mumming go,
Some others play at Rowland-boe,
And twenty other gambols moe,
Because they will be merry.
“Then wherefore, in these merry days,
Should we, I pray, be duller?
No, let us sing some roundelays,
To make our mirth the fuller;
And, while we thus inspired sing,
Let all the streets with echoes ring—
Woods and hills and everything
Bear witness we are merry.”
Or Master Milton, again, Latin secretary to the council, author of
the famous Iconoclastes, shield (or, as some would have put it,
official scold) of the Commonwealth, the scourge of prelacy and
conqueror of Salmasius—he was orthodox surely; yet what of
Arcades and Cornus? Master Milton, too, had written holiday](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-71-320.jpg)
![“Christmas is come—the beautiful festival, the one I love most, and
which gives me the same joy as it gave the shepherds of Bethlehem. In
real truth, one’s whole soul sings with joy at this beautiful coming of
God upon earth—a coming which here is announced on all sides of us
by music and by our charming nadalet[110] Nothing at Paris can give you
a notion of what Christmas is with us. You have not even the midnight
Mass. We all of us went to it, papa at our head, on the most perfect
night possible. Never was there a finer sky than ours was that midnight
—so fine that papa kept perpetually throwing back the hood of his
cloak, that he might look up at the sky. The ground was white with
hoar-frost, but we were not cold; besides, the air, as we met it, was
warmed by the bundles of blazing torchwood which our servants carried
in front of us to light us on our way. It was delightful, I do assure you;
and I should like you to have seen us there on our road to church, in
those lanes with the bushes along their banks as white as if they were
in flower. The hoar-frost makes the most lovely flowers. We saw a long
spray so beautiful that we wanted to take it with us as a garland for the
communion-table, but it melted in our hands; all flowers fade so soon! I
was very sorry about my garland; it was mournful to see it drop away
and get smaller and smaller every minute.”
It is Eugénie de Guérin who writes thus—that pure and delicate
spirit so well fitted to feel and value all that is beautiful and
touching in this most beautiful and touching service of the church.
To come from the one reading to the other is like being lifted
suddenly out of a narrow valley to the free air and boundless views
of a mountain-top; like coming from the gaslight into the starlight;
it is like hearing the song of the skylark after the twitter of the robin
—a sound pleasant and cheery enough in itself, but not elevating,
not inspiring, not in any way satisfying to that hunger after ideal
excellence which is the true life of the spirit, and which strikes the
true key-note of this festal time.
But Eugénie de Guérin is perhaps too habitual a dweller on those
serene heights to furnish a fair comparison; let us take a homelier
picture from a lower level. It is still in France; this time in Burgundy,
as the other was in Languedoc:
“Every year, at the approach of Advent, people refresh their
memories, clear their throats, and begin preluding, in the long evenings](https://image.slidesharecdn.com/85676-250224195711-da46204f/85/Control-theory-of-partial-differential-equations-1st-Edition-Guenter-Leugering-75-320.jpg)
Control theory of partial differential equations 1st Edition Guenter Leugering
- 1. Visit https://ebookultra.com to download the full version and explore more ebooks Control theory of partial differential equations 1st Edition Guenter Leugering _____ Click the link below to download _____ https://ebookultra.com/download/control-theory-of- partial-differential-equations-1st-edition-guenter- leugering/ Explore and download more ebooks at ebookultra.com
- 2. Here are some suggested products you might be interested in. Click the link to download Handbook of Differential Equations Stationary Partial Differential Equations Volume 6 1st Edition Michel Chipot https://ebookultra.com/download/handbook-of-differential-equations- stationary-partial-differential-equations-volume-6-1st-edition-michel- chipot/ Applied Partial Differential Equations John Ockendon https://ebookultra.com/download/applied-partial-differential- equations-john-ockendon/ Stochastic Partial Differential Equations 2nd Edition Chow https://ebookultra.com/download/stochastic-partial-differential- equations-2nd-edition-chow/ Introduction to Partial Differential Equations Rao K.S https://ebookultra.com/download/introduction-to-partial-differential- equations-rao-k-s/
- 3. Partial Differential Equations 2nd Edition Lawrence C. Evans https://ebookultra.com/download/partial-differential-equations-2nd- edition-lawrence-c-evans/ Elliptic Partial Differential Equations 2nd Edition Qing Han https://ebookultra.com/download/elliptic-partial-differential- equations-2nd-edition-qing-han/ Stable Solutions of Elliptic Partial Differential Equations 1st Edition Louis Dupaigne https://ebookultra.com/download/stable-solutions-of-elliptic-partial- differential-equations-1st-edition-louis-dupaigne/ Numerical solution of hyperbolic partial differential equations John A. Trangenstein https://ebookultra.com/download/numerical-solution-of-hyperbolic- partial-differential-equations-john-a-trangenstein/ Partial differential equations An introduction 2nd Edition Strauss W.A. https://ebookultra.com/download/partial-differential-equations-an- introduction-2nd-edition-strauss-w-a/
- 5. Control theory of partial differential equations 1st Edition Guenter Leugering Digital Instant Download Author(s): Guenter Leugering, Oleg Imanuvilov, Bing-Yu Zhang, Roberto Triggiani ISBN(s): 9781420028317, 1420028316 Edition: 1 File Details: PDF, 2.54 MB Year: 2005 Language: english
- 7. Control Theory of Partial Differential Equations
- 8. M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Masse Yale University Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EDITORIAL BOARD EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida
- 9. LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, HopfAlgebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations
- 10. April 18, 2005 9:24 3086 FM
- 11. Boca Raton London New York Singapore Control Theory of Partial Differential Equations Oleg Imanuvilov Iowa State University Ames, Iowa, USA Guenter Leugering University of Erlangen Nuremberg, Germany Roberto Triggiani University of Virginia Charlottesville, Virginia, USA Bing-Yu Zhang University of Cincinnati Cincinnati, Ohio, USA
- 12. Published in 2005 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2546-8 (Softcover) International Standard Book Number-13: 978-0-8247-2546-4 (Softcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Taylor & Francis Group is the Academic Division of T&F Informa plc. DK2961_Discl.fm Page 1 Thursday, April 21, 2005 10:16 AM
- 13. April 25, 2005 11:56 3086 FM Preface The present volume contains contributions by participants in the “Conference on Control Theory for Partial Differential Equations,” which was held over a two-and-a-half day period, May 30 to June 1, 2003, at Georgetown University, Washington, D.C. The conference was dedicated to the occasion of the retirement of Professor Jack Lagnese from the Mathematics Department of Georgetown University. It seemed most appropriate to honor the productive and successful scientific career of Jack Lagnese by convening a conference that would bring together a select group of international specialists in the theory of partial differential equations and their control. Over the years, many of the invitees have enjoyed a personal and professional association with Jack. The lasting impact of Jack’s contributions to control theory of partial differential equations and applied mathematics is well documented by over 80 research articles and three books. In addition, Jack served the scientific community for many years in his capacity, at various times, as a program director in the Applied Mathematics Program within the National Science Foundation, as an editor on the boards of several journals, as editor-in-chief of the SIAM Journal on Control and Optimization, and as president of the SIAM Activity Group on Control and Systems Theory. He was also a consultant to The National Institute for Standards and Technology for a number of years. Control theory for distributed parameter systems, and specifically for systems governed by partial differential equations, has been a research field of its own for more than three decades. Although having a distinctive identity and philosophy within the theory of dynamical systems, this field has also contributed to the general theory of partial differential equations. Optimal interior and boundary regularity of mixed problems, global uniqueness issues for over-determined problems and related Carleman estimates, various types of a priori inequalities, and stability and long-time behavior are just some examples of important developments in the theory of partial differential equations arising from control theoretic considerations. In recent years, the field has broadened considerably as more realistic models have been introduced and investigated in areas such as elasticity, thermoelasticity, and aeroelasticity; in problems involving interactions between fluids and elastic structures; and in other problems of fluid dynamics, to name but a few. These new models present fresh mathematical challenges. For example, the mathematical foundations of fundamental theoretical issues have to be developed, and conceptual insights that are useful to the designer and the practitioner need to be provided. This process leads to novel numerical challenges that must also be addressed. The papers contained in this volume provide a broad range of significant recent developments, new discoveries, and mathematical tools in the field and further point to challenging open problems. The conference was made possible through generous financial support by the National Science Foundation and Georgetown University, whose sponsorship is greatly appreciated. We wish to thank Marcel Dekker for agreeing to include this volume in its well-known and highly regarded series “Lecture Notes in Pure and Applied Mathematics” and for its high professional standards in handling this volume. The Scientific Committee: Oleg Imanuvilov Guenter Leugering Roberto Triggiani (Chair) Bing-Yu Zhang vii
- 14. April 25, 2005 11:56 3086 FM
- 15. April 25, 2005 11:56 3086 FM Contributors George Avalos Department of Mathematics University of Nebraska-Lincoln Lincoln, Nebraska Viorel Barbu Department of Mathematics Alexandru Ioan Cuza University Iasi, Romania Mikhail I. Belishev Saint Petersburg Department Steklov Institute of Mathematics Saint Petersburg, Russia Igor Chueshov Department of Mathematics and Mechanics Kharkov University Kharkov, Ukraine Michel C. Delfour Department de Mathematiques et de Statistique Universite de Montreal Montreal, Quebec Canada Nicolas Doyon Department de Mathematiques et de Statistique Universite Laval Quebec City, Quebec Canada Matthias M. Eller Department of Mathematics Georgetown University Washington, D.C. Oleg Imanuvilov Department of Mathematics Iowa State University Ames, Iowa Victor Isakov Department of Mathematics and Statistics Wichita State University Wichita, Kansas Jack E. Lagnese Department of Mathematics Georgetown University Washington, D.C. Irena Lasiecka Department of Mathematics University of Virginia Charlottesville, Virginia Guenter Leugering Institute of Applied Mathematics University of Erlangen-Nuremberg Erlangen, Germany Wei Li School of Mathematics Sichuan University Chengdu, China Walter Littman School of Mathematics University of Minnesota Minneapolis, Minnesota Zhuangyi Liu Department of Mathematics and Statistics University of Minnesota, Duluth Duluth, Minnesota Michael Renardy Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia David L. Russell Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia E.J.P. Georg Schmidt Department of Mathematics and Statistics McGill University Montreal, Quebec Canada ix
- 16. April 25, 2005 11:56 3086 FM x Contributors Thomas I. Seidman Department of Mathematics and Statistics University of Maryland Baltimore, Maryland Ralph E. Showalter Department of Mathematics Oregon State University Corvallis, Oregon Marianna A. Shubov Department of Mathematics and Statistics Texas Tech University Lubbock, Texas Jürgen Sprekels Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany Dan Tiba Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany Roberto Triggiani Department of Mathematics University of Virginia Charlottesville, Virginia Masahiro Yamamoto Department of Mathematical Sciences University of Tokyo Tokyo, Japan Yiming Yang Department of Mathematics and Physics Beijing Technology and Business University Beijing, China Jiongmin Yong Department of Mathematics Fudan University Shanghai, China Bing-Yu Zhang Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio Xu Zhang School of Mathematics Sichuan University Chengdu, China Jean-Paul Zolésio Centre National de Recherche Scientifique (CNRS) and Institut National de Recherche en Informatique et en Automatique (INRIA) Sophia Antipolis, France
- 17. April 25, 2005 11:56 3086 FM Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability of Thermoelastic Plates: The Free Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 George Avalos and Irena Lasiecka 2 Interior and Boundary Stabilization of Navier-Stokes Equations. . . . . . . . . . . . . . . . . . . . . . . 29 Viorel Barbu 3 On Approximating Properties of Solutions of the Heat Equation. . . . . . . . . . . . . . . . . . . . . . . 43 Mikhail I. Belishev 4 Kolmogorov’s ε-Entropy for a Class of Invariant Sets and Dimension of Global Attractors for Second-Order Evolution Equations with Nonlinear Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Igor Chueshov and Irena Lasiecka 5 Extension of the Uniform Cusp Property in Shape Optimization . . . . . . . . . . . . . . . . . . . . . . . 71 Michel C. Delfour, Nicolas Doyon, and Jean-Paul Zolésio 6 Gårding’s Inequality on Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Matthias M. Eller 7 An Inverse Problem for the Dynamical Lame System with Two Sets of Local Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Victor Isakov 8 On Singular Perturbations in Problems of Exact Controllability of Second-Order Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Jack E. Lagnese xi
- 18. April 25, 2005 11:56 3086 FM xii Contents 9 Domain Decomposition in Optimal Control Problems for Partial Differential Equations Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Guenter Leugering 10 Controllability of Parabolic and Hyperbolic Equations: Toward a Unified Theory . . . . 157 Wei Li and Xu Zhang 11 A Remark on Boundary Control on Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Walter Littman 12 Model Structure and Boundary Stabilization of an Axially Moving Elastic Tape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Zhuangyi Liu and David L. Russell 13 Nonlinear Perturbations of Partially Controllable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Michael Renardy 14 On Junctions in a Network of Canals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 E.J.P. Georg Schmidt 15 On Uniform Null Controllability and Blowup Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Thomas I. Seidman 16 Poroelastic Filtration Coupled to Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Ralph E. Showalter 17 Operator-Valued Analytic Functions Generated by Aircraft Wing Model (Subsonic Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Marianna A. Shubov 18 Optimal Design of Mechanical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Jürgen Sprekels and Dan Tiba 19 Global Exact Controllability on H1 Γ0 (Ω) × L2(Ω) of Semilinear Wave Equations with Neumann L2(0,T;L2(Γ1))-Boundary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Roberto Triggiani 20 Carleman Estimates for the Three-Dimensional Nonstationary Lamé System and Application to an Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Oleg Imanuvilov and Masahiro Yamamoto
- 19. April 25, 2005 11:56 3086 FM Contents xiii 21 Forced Oscillations of a Damped Benjamin-Bona-Mahony Equation in a Quarter Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Yiming Yang and Bing-Yu Zhang 22 Exact Controllability of the Heat Equation with Hyperbolic Memory Kernel . . . . . . . . . 387 Jiongmin Yong and Xu Zhang
- 20. April 25, 2005 11:56 3086 FM
- 21. April 4, 2005 10:3 3086 DK2961˙C001 Chapter 1 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability of Thermoelastic Plates: The Free Case George Avalos1 University of Nebraska-Lincoln Irena Lasiecka2 University of Virginia 1.1 Introduction ............................................................................ 2 1.1.1 Motivation ...................................................................... 2 1.1.2 Description of the PDE Model and Statement of the Problem ...................... 4 1.1.3 Main Result ..................................................................... 8 1.2 The Necessary Observability Inequality .................................................. 9 1.3 Some Preliminary Machinery ........................................................... 10 1.4 A Singular Trace Estimate ............................................................... 12 1.5 Proof of Theorem 1.1(1) ................................................................ 14 1.5.1 Estimating the Mechanical Velocity .............................................. 14 1.5.2 Estimating the Mechanical Displacement ......................................... 17 1.5.3 Conclusion of the Proof of Theorem 1.1(1) ....................................... 20 1.6 Proof of Theorem 1.1(2) ................................................................ 20 1.6.1 A First Supporting Estimate ...................................................... 20 1.6.2 Conclusion of the Proof of Theorem 1.1(2) ....................................... 23 1.7 Proof of Theorem 1.1(3) ................................................................ 24 References .............................................................................. 24 Abstract Continuing the analysis undertaken in References 8 and 9, we consider the null- controllability problem for thermoelastic plate partial differential equations (PDEs) models in the absence of rotational inertia, defined on a two-dimensional domain , and subject to the free mechan- ical boundary conditions of second and third order. It is now known that such uncontrolled systems generate analytic semigroups on finite energy spaces. Consequently, the concept of null controllabil- ity is indeed an appropriate question for consideration. It is shown that all finite energy states can be driven to zero by means of L2 [(0, T ) × ] controls in either the mechanical or thermal component. However, the main intent of the paper is to quantify the singularity, as T ↓ 0, of the minimal energy function relative to null controllability. In particular we shall show that in the case of one control function acting upon the system, the singularity of minimal energy is optimal; it is in fact of order O(T − 5 2 ), which is the same rate of blowup as that of any finite dimensional approximation of the problem. The PDE estimates, which are obtained in the process of deriving this sharp numerology, will have a strong bearing on regularity properties of related stochastic differential equations. 1The research of George Avalos was partially supported by NSF DMS-0208121. 2The research of Irena Lasiecka was partially supported by NSF DMS-01043 and ARO DAAD19-02-1-0179. 1
- 22. April 4, 2005 10:3 3086 DK2961˙C001 2 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability 1.1 Introduction In this chapter we address specific questions related to the null controllability of thermoelastic plates subject to free mechanical boundary conditions, these being represented by shear forces and moments. These particular boundary conditions are of particular interest in the control theory of plates [22, 24, 23]. As we shall see below, the model under consideration is one which corresponds to an infinite speed of propagation; accordingly, null controllability—in arbitrarily short time—is an appropriate topic for study in regard to these plates. We will give at length a full and precise description of our thermoelastic control problem; but for the benefit of the reader and in order to motivate the specific problem under study, we will first provide a few opening remarks. 1.1.1 Motivation There are several ways of controlling a given plate dynamic. This control can be accomplished by using: 1. internal controls, 2. boundary controls, or 3. controls localized on an open subset of . In addition, one may use either one control action (be it thermal or mechanical) or simultaneous me- chanical and thermal controls (i.e., controls located on both the mechanical and thermal components of the system). Depending on the objective to be achieved, one framework of control might be more advantageous than another. For instance, if the particular issue at hand is to guarantee the minimal support of control functions, then boundary control would be the most appropriate control situation. However, if one is concerned with the cost of control—or equivalently, with quantifying the associ- ated “minimal energy”—then internal controls should be considered. In this connection, a question of both practical and mathematical relevance is the question of finding the optimal asymptotics that describe the singularity of the associated minimal energy, as T ↓ 0. Since the work of T. Seidman in Reference 34, the optimal asymptotics are well defined and well known for finite dimensional control systems. In fact, these asymptotics are given by the sharp formula T −k− 1 2 where index k corresponds to the Kalman rank condition and measures the defect of controllability (see below). The above for- mula actually gives a lower bound for the singularity of the minimal energy associated with any PDE system. Given then the existence of formula in Reference 34 for controlled finite dimensional systems, we are in a position to loosely define the “optimal” singularity for any controlled PDE. In fact, for a given infinite dimensional system, the “optimal” rate of singularity of its associated minimal energy will be the rate of singularity enjoyed by approximating (or truncated) finite dimensional systems (assuming of course that each finite dimensional truncation has the same Kalman rank). For example, scalar first order (in time) models will have its optimal rate of blowup of minimal energy as being O(T − 3 2 ); in general, the optimal singularity for vectorial coupled structures will depend on the number of controls used with respect to number of interactions. Thus, in the case of thermal plates with one control only, the optimal singularity of any finite dimensional truncation is T − 5 2 (this is seen below). In the case of two controls used (both thermal and mechanical) the optimal singularity is T − 3 2 . Whether, however, the minimal energy asymptotics actually obeys the optimal rate of singularity (predicted from finite dimensions) is an altogether different matter. Indeed, in References 34 and 36 (highly nontrivial) finite-dimensional estimates are derived and can be subsequently applied to finite-dimensional truncations of infinite-dimensional systems; however, the delicate estimates are controlled by a constant Cn, say, where n stands for the dimensionality of the respective approximation. These constants may well tend to infinity as n goes to infinity. In such an event (as seen in References 14, 6, and 40) the optimal asymptotics for the original PDE are lost. This brings us to the key question asked in this chapter: Is it possible to achieve the optimal rate of singularity for a (fully infinite dimensional) controlled PDE model?
- 23. April 4, 2005 10:3 3086 DK2961˙C001 1.1 Introduction 3 The answer to the above question—in the negative—has been known for many years in the case of the heat equation with either boundary or localized controls. Indeed, the rate for boundary control of the heat equation is the exponential blowup rate eO( 1 T ) ; see References 35 and 37. This rate is known to be sharp [20]. A similar negative answer has been provided in the case of thermoelastic systems under the influence of boundary controls—in fact, such boundary controls likewise lead to eO( 1 T ) exponential blowup [25]. Therefore, in light of the rational rates of minimal energy blowup exhibited by finite-dimensional controlled systems (as shown in Reference 34) and of the definition given above for optimal rates of minimal energy blowup for controlled PDEs, it is manifest that thermoelastic plates under the influence of boundary or localized controls will not give rise to minimal energies that exhibit an optimal (finite dimensional) singularity. Thus, in searching for PDE control situations, which will yield up the optimal algebraic singularity enjoyed by finite dimensional truncations, the only reasonable choice left is the implementation of internal controls. In the specific context of our thermoelastic PDE, the relevant question then becomes: Do the minimal energies of internally controlled (fully infinite dimensional) thermoelastic plates exhibit the optimal rate of blowup O(T − 5 2 ) by either mechanical or thermal control? The relevance of this question should not be underestimated from both a practical and mathematical point of view. Indeed, from a practical point of view one would like to know whether a given finite-dimensional approximation of the system contains critical information and moreover reflects controllability properties of the original PDE model. From a mathematical point of view, the solution to the null controllability problem is not only of interest in its own right as an issue in control theory, but this solution can also give rise to deep and significant connections between the algebraic optimal singularity of minimal energy and other fields of analysis, including stochastic analysis. In point of fact, within the field of stochastic differential equations, there is an acute need to know of those PDE control environments that will yield up optimal (and algebraic) rates of singularity of minimal energy. These particular rates are critical in finding the regularity and solvability of certain stochastic differential equations [14, 15, 19], as well as in setting conditions for the hypoellipticity of certain degenerate infinite dimensional elliptic problems [32]. It is shown in Reference 32 that Hormander’s hypoellipticity condition is strongly linked to the singularity of the minimal energy function. Null controllabilityisalsorelatedtotheanalysisofregularityoftheBellman’sfunction,whichisassociated with the minimal time control problem. Indeed, as eloquently described in References 14 and 15, this property bears a close relation to the regularity of some Markov semigroups, including Orstein– Uhlenbeck processes and related Kolmogorov equations. For some of these semigroups (see, e.g., Reference 15—Theorem 8.3.3) the minimal energy singularity associated with null controllability describes differentiability properties and regularizing effects of the Orstein–Uhlenbeck process. Moreover, the regularity of solutions to the Kolmogorov equation depends on the singularity of the minimal energy as T ↓ 0. Also, as shown in Reference 14, optimal estimates for the norms of controls are critical in being able to prove Liouville’s property for harmonic functions of Markov processes (see p. 108 in Reference 15). In sum, there is an abundance of examples from the literature that clearly illustrate that, in the context of computing optimal minimal energy asymptotics as T ↓ 0, the tools of controllability can potentially enable a mathematical control theorist to transcend his or her deterministic realm so as to solve fundamental problems in other areas of analysis, including stochastic PDEs. In addition, the procurement of optimal algebraic estimates for the minimal energy allows one to clearly explain the role of the hyperbolic-parabolic coupling within the PDE structure (in Eq. (1.1) below). In particular, it has been shown recently in Reference 25 that, owing to optimal algebraic singularitiesofminimalenergy,itispossibletooffsetthesingularityofminimalenergybyintroducing a very strong coupling within the system. Thus, in some sense, the lack of a second control in the system may be quantitatively compensated for by taking large values of the coupling parameter “α.” From our remarks above, it is clear that this compensatory phenomenon will not be observed with boundary or partially supported controls, which, as we have said, lead to blowups of exponential type.
- 24. April 4, 2005 10:3 3086 DK2961˙C001 4 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability Having decribed the goal and motivation for the problem considered, we shall describe the main contribution of this chapter within the context of recent work in that area. The problem of controlla- bility/reachability for thermal plates has attracted considerable attention in recent years with many contributions available in the literature [22, 23, 24, 1, 2, 3, 10, 18, 16, 17, 11], but we shall focus particularly on works related directly to singular behavior, as T ↓ 0, of the minimal energy relative to null controllability. The study of optimal singularity for thermoelastic plates with internal controls started in Refer- ences 8, 9, and 40, where for the first time the optimal rates T − 5 2 were established for the “commu- tative” case (i.e., plates with hinged mechanical boundary conditions). The proof given in Reference 40 is based on a spectral method that exploits the commutativity in an essential way, whereas the proof given in Reference 8 is based on weighted energy estimates, thereby giving one the chance to extend this method to other noncommutative models (e.g., clamped or free mechanical bound- ary conditions). The “commutative” case (hinged boundary conditions) has been also treated in Reference 11, where null controllability with thermal controls of partially localized support was proved. For this commutative model under boundary control (either thermal or boundary), the ex- ponentially blowing up and sharp asymptotics eO( 1 T ) have been shown [25]. The techniques used in these papers rely critically on spectral analysis and commutativity. It turns out that a proper and necessarily more technical extension of the method introduced in Reference 8 will allow the consideration of noncommutative models. (By “noncommutative models” we mean those models wherein the domains of the respective spatial differential operators of the plate and heat dynamics do not necessarily enjoy any sort of compatibility.) In particular, the optimal singularity of the (null control) minimal energy is proved in Reference 9 for clamped plates with one control only. It should be stressed that the proof in the noncommutative case depends in an essential manner upon estimates provided by the analyticity of the underlying thermoelastic semigroup; this property of analyticity was discovered for the clamped case in References 31 and 28 and for free case in Reference 27. The most challenging case is, of course, that of the free mechanical bound- ary conditions (introduced in the context of control theory in Reference 22), in which a coupling between thermal and mechanical variables also occurs on the boundary. This additional coupling compels us to develop below a delicate string of trace estimates that measure the singularity at the boundary. The main aim of this chapter is to provide a complete analysis of the free case. We shall show that in the case of mechanical control one still obtains the optimal singularity. Instead, in the case of thermal control the estimate is “off” by 3/4. A question whether this estimate can be improved, thereby leading to the optimal singularity T − 5 2 , still remains an open question. 1.1.2 Description of the PDE Model and Statement of the Problem Having given our general remarks above, we now proceed to precisely describe the present prob- lem under consideration; this work will continue and extend the analysis that has been previously undertaken in References 6, 7, 8, 9 through and 40. We will consider throughout the two-dimensional PDE system of thermoelasticity in the absence of rotational inertia. As we have already stated, it is now known that for all possible mechanical boundary conditions, the thermoelastic PDE model is associated with the generator of an analytic C0-semigroup (see References 31, 28, 39, and 27). Given then that the underlying PDE dynamics are “parabolic-like,” it is natural to consider the null controllability problem for the thermoelastic system, namely, can one find L2 (Q) controls (mechanical or thermal) that steer the solution of the PDE from the initial data to the zero state? (We shall make this control theoretic notion more precise below. As usual, Q here denotes the cylinder × (0, T ).) Having established L2 (Q)-null controllability for the PDE, and moreover assuming that the controllability time is arbitrary, we can subsequently proceed to measure the rate of blowup, as T ↓ 0, for the minimal energy function that is associated to null controllability. As is well known, and as we shall see below, this work is very much tied up with obtaining the sharp
- 25. April 4, 2005 10:3 3086 DK2961˙C001 1.1 Introduction 5 observability inequality associated with null controllability; moreover, this analysis is rather sensi- tive to the mechanical boundary conditions imposed. In Reference 8—as well as in Reference 40 via a very different methodology—the problem of blowup for the minimal energy function was undertaken in the canonical case of hinged mechanical boundary conditions; in Reference 9, we revisit this problem for the more difficult clamped case. In this paper, we complete the picture by analyzing the singularity of minimal energy for the case of the thermoelastic PDE under the so-called free boundary conditions. In general, the analyses involved in the attainment of (null and exact control) observability inequalities for thermoelastic systems are profoundly sensitive to the particular set of boundary conditions are being imposed. But the free case, presently under consid- eration, will give rise to the most problematic scenario of all. This situation is due to the high degree of coupling between the mechanical and the thermal variables, with the coupling taking place in the PDE itself and in the free mechanical boundary conditions. We describe the problem in detail. Let be a bounded open set of R2 , with smooth boundary . For the free case, following [22, 23] the corresponding model PDE is as follows: the (mechanical) vari- ables [ω(t, x), ωt (t, x)] and the (thermal) variable θ(t, x) solve, for given data {[ω0, ω1, θ0], u1, u2}, the PDE system ωtt + 2 ω + α θ = a1u1 θt − θ − α ωt = a2u2 on (0, T ) × ω + (1 − µ)B1ω + αθ = 0 ∂ ω ∂ν + (1 − µ) ∂ B2ω ∂τ − ω + α ∂θ ∂ν = 0 on (0, T ) × ∂θ ∂ν + λθ = 0 on (0, T ) × , where λ 0 ω(t = 0) = ω0; ωt (t = 0) = ω1; θ(t = 0) = θ0 on . (1.1) Here, α 0 is the parameter that couples the disparate dynamics (i.e., the heat equation vs. the Euler plate equation); the constant µ ∈ (0, 1) is Poisson’s ratio. Also, the (control) parameters a1 and a2 satisfy a1 ≥ 0, a2 ≥ 0 and a1 + a2 0 (in other words, at least one of the controls, be it thermal or mechanical, is always present.) The (free) boundary operators Bi are given by B1w ≡ 2ν1ν2 ∂2 w ∂x∂y − ν2 1 ∂2 w ∂y2 − ν2 2 ∂2 w ∂x2 ; (1.2) B2w ≡ ν2 1 − ν2 2 ∂2 w ∂x∂y + ν1ν2 ∂2 w ∂y2 − ∂2 w ∂x2 . The PDE Eq. (1.1) is the model explicitly derived and analyzed in References 24 and 22 in the “limit case.” That is to say, we are considering the two-dimensional thermoelastic system in the absence of rotational forces; the small and nonnegative, classical parameter γ is taken here to be zero. As we stated at the outset, it is now well known that the lack of rotational inertia in the model Eq. (1.1) will result in the corresponding dynamics having their evolution described by the generator of an analytic semigroup on the associated basic space of finite energy. In short, the present case γ = 0 corresponds to parabolic-like dynamics; this is in stark contrast to the case γ 0—as analyzed in the control papers [22], [23], [3] and myriad others—for which the corresponding PDE manifests hyperbolic-like dynamics. In fact, if we define H ≡ H2 () × L2 () × L2 (), (1.3)
- 26. April 4, 2005 10:3 3086 DK2961˙C001 6 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability then one can proceed to show by the Lumer Phillips theorem that the thermoelastic plate model can be associated with the generator of a C0-semigroup of contractions on H. That is to say, there exists A : D(A) ⊂ H → H, and {eAt }t≥0 ⊂ L(H) such that [ω, ωt , θ] satisfies the PDE (1.1) if and only if [ω, ωt , θ] satisfies the abstract ODE d dt ω(t) ωt (t) θ(t) = A ω(t) ωt (t) θ(t) + 0 a1u1(t) a2u2(t) ; ω(0) ωt (0) θ(0) = ω0 ω1 θ0 . In consequence of this relation, we have immediately from classical semigroup theory that {[ω0, ω1, θ0], [u1, u2]} ∈ H × [L2 (Q)]2 ⇒ [ω, ωt , θ] ∈ C([0, T ]; H). (1.4) Because of the underlying analyticity, which will ultimately mean that there are smoothing effects associated with the application of the semigroup {eAt }t≥0, the null controllability problem for the controlled PDE Eq. (1.1)—with respect to internal L2 -controls—is an appropriate one to study. Moreover, one might speculate that, as in the case of the canonical heat equation [12], should the PDE Eq. (1.1) in fact be null controllable, it will be so in arbitrary small time (because of the underlying infinite speed of propagation). It is this speculation that motivates our working definition of null controllability for the present paper. DEFINITION 1.1 The PDE (1.1) is said to be null controllable if, for any T 0 and arbitrary initial data x ≡ [ω0, ω1, θ0] ∈ H, there exists a control function [u1, u2] ∈ [L2 (Q)]2 such that the corresponding solution [ω, ωt , θ] ∈ C([0, T ]; H) satisfies [ω(T ), ωt (T ), θ(T )] = [0, 0, 0]. However, the issue of null controllability, although certainly an important part of this paper, is subordinate to our main objective, which is to measure the rate of singularity of the associated minimal energy function. We develop this notion of “minimal energy.” Assume for the time being that the Eq. (1.1) is null controllable within the class of [L2 (Q)]2 -controls, in the sense of the Definition 1.1. Subse- quently, one can then speak of the associated minimal norm control, relative to given initial data x ≡ [ω0, ω1, θ0] ∈ H and given terminal time T . That is to say, we can consider the problem of finding a control u0 T (x) that steers the solution [ω, ωt , θ] of Eq. (1.1) (with [u1, u2] = u0 T (x) therein) from initial data x to zero in arbitrary time T and minimizes the L2 norm. In fact, by standard convex optimization arguments (see, e.g., Reference 13), given any x ∈ H and fixed T , one can find a control u0 T (x) which solves the problem u0 T (x) [L2(Q)]2 = min u[L2(Q)]2 , where, above, the minimum is taken with respect to all possible null controllers u = [u1, u2] ∈ [L2 (Q)]2 of the PDE (1.1) (which steer initial data x to rest at time t = T ). Subsequently, we can define the minimal energy function Emin(T ) as Emin(T ) ≡ sup xH=1 u0 T (x) [L2(Q)]2 . (1.5) Under the assumption of null controllability, as defined in Definition 1.1, we have that Emin(T ) is bounded away from zero. A natural follow-up question is “how does Emin(T ) behave as terminal time T ↓ 0, or equivalently (by Eq. (1.5)), for given time T , how exactly does the quantity u0 T (x)[L2(Q)]2 grow as T ↓ 0?”
- 27. April 4, 2005 10:3 3086 DK2961˙C001 1.1 Introduction 7 The problem of studying the rate of blowup for minimal norm controls is a classical one and has its origins from the finite dimensional setting. In fact, a very complete and satisfactory solution has been given in Reference 34 for the following controlled ODE in Rn : d dt y(t) = A y(t) + B u(t), y0 ∈ Rn (1.6) where u ∈ L2 (0, T ; Rm ) and A (resp., B) is an n × n (resp. n × m) matrix, with m ≤ n (so consequently the solution y ∈ C([0, T ]; Rn ). The problem in this finite dimensional milieu, like that for our controlled PDE (1.1), is to ascertain the rate of singularity for the associated minimal energy function, which is defined in the same way as in Eq. (1.5). The solution to this problem is tied up with the classical Kalman’s rank condition. Namely, a beautifully simple (though highly nontrivial) formula in Reference 34—an alternative constructive proof of this formula is given in Reference 40; see also Reference 36—yields that the minimal energy function associated to the null controllability of Eq. (1.6) is O(T −k− 1 2 ), where k is the Kalman’s rank of the system Eq. (1.6) (that is, k is the smallest integer such that rank ([B, AB, . . . , Ak B]) = n; see Reference 41). By a formal application of Seidman’s finite dimensional result, one can get an inkling of the numerology involved in the computation of the minimal energy Emin(T ) for the PDE system Eq. (1.1). Forexample,letusconsiderthethermoelasticEq.(1.1)butwithnow ω satisfyingthecanonicalhinged mechanical/Dirichlet thermal boundary conditions ω| = ω| = θ| = 0 on . (1.7) In this case, it is shown in Reference 27 that when, say, thermal control only is implemented (i.e., a1 = 0 in Eq. (1.1)), the thermoelastic PDE under the hinged boundary conditions Eq. (1.7) may be associated with the ordinary differential equation (ODE) (1.6), with A = 0 1 0 −1 0 1 0 −1 −1 , and B = 0 0 a2 . (1.8) This ODE in three space dimensions is a direct consequence of the analysis undertaken for the canonical hinged case in Reference 26. By way of obtaining the ODE (1.6), we have formally “factored out” the Laplacian from the (rearranged) infinitesimal generator of the thermoelastic semigroup, which is given in (Section 1.2.2) of Reference 27 (see also Reference 28, p. 311). Considering now finite dimensional truncations of (by making use of the spectral resolution of the Laplacian under Dirichlet boundary conditions) and applying the algorithm of Seidman to the given controllability pair [A, B] in Eq. (1.8), we compute readily that the minimal energy func- tion associated with the null controllability of the finite dimensional Eq. (1.6)—an approximation in some sense of the thermoelastic system under the hinged boundary conditions—blows up at a rate on the order of T − 5 2 . These numerics lead to the following question: Does the minimal en- ergy Eq. (1.5) (i.e., the minimal energy for the full-fledged infinite dimensional system) obey the law Emin(T ) = O(T − 5 2 )? Of course, Seidman’s formula for matrices gives no conclusive proof as to what is actually happening for the fully infinite dimensional model. In fact, it is well known that the minimal energy of a given infinite dimensional system may bear no relation to the limit of minimal energies of any given sequence of finite dimensional approximations. For example, it was shown in Reference 14 that the growth of the minimal energy function for a given infinite dimensional system may be arbitrarily large, even when Kalman’s rank k = 1 and spectral diag- onal systems are being considered. Moreover, in Reference 35 it is shown that for the case of the boundary controlled heat equation, the sharp observability inequality corresponding to the (null) minimal energy of a given heat operator’s finite dimensional truncation obeys rational rates of
- 28. April 4, 2005 10:3 3086 DK2961˙C001 8 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability singularity. On the other hand, the asymptotics of the minimal energy, which are obtained for the (infinite dimensional) heat equation, are of exponential type. A similar phenomenon is observed in References 40 and 6, wherein strongly damped wave equations under internal control are considered. In this situation, with the damping operator given by Aβ , the asymptotics of minimal energy be- have as T − β 2(1−β) for any β 3 4 . Thus, when the damping operator approaches Kelvin’s Voight damping, the singularity loses its algebraic character with β 2(1−β) ↑ ∞. Instead, for β ≤ 3 4 , the singularity is optimal (i.e., the same as that for finite dimensional truncations) and is equal to T − 3 2 . But as formal as the application of Seidman’s finite dimensional algorithm may seem in the present context, there is in fact a relevance here to the thermoelastic PDE, which is approximately described by controllability pair [A, B]. The minimal energy function with respect to null controllability of the thermoelastic PDE, under the hinged boundary conditions Eq. (1.7), does indeed obey the singular rate Emin(T ) = O(T − 5 2 ). This minimal energy analysis for the hinged case was shown independently in References 8 and 40 (and most recently in Reference 25 where the asymptotics with respect to the coupling α are also provided). In Reference 40, it is of prime importance that the hinged boundary conditions Eq. (1.7) be in play, for these mechanical boundary conditions allow a fortuitous spectral resolution of the underlying thermoelastic generator. With the eigenfunctions of the thermoelastic dynamics in hand, it is shown in Reference 40 via a constructive class of suboptimal steering controls that the delicate observability estimates for solutions for the spectrally truncated adjoint problem— adjoint with respect to null controllability—are preserved; as a consequence, a rational rate of singularity for the infinite dimensional null minimal energy is obtained in the limit. However, for other sets of mechanical boundary conditions, including the physically relevant clamped and (above all) free boundary conditions under consideration at present, there will be no such available spectral decomposition. On the other hand, the methodology employed in References 8 and 9, and the present work, is “eigenfunction independent”; in particular, we blend a weighted multiplier method of Carleman’s type with boundary trace estimates exhibiting singular behavior of the boundary traces. This rather special behavior is a consequence of the underlying analyticity. In principle, our work in Reference 8 to estimate the blowup of the “minimal norm control” as T ↓ 0 is applicable to a variety of dynamics. (In fact, our method of proof in Reference 8 and in the present work is used in Reference 7 to estimate the minimal norm control of the abstract wave equation under Kelvin– Voight damping.) Moreover, the robustness of our method allows us in Reference 9 to analyze the rate of singularity of the minimal energy function for the null controllability of thermoelastic plates in the case of clamped boundary conditions. As we said above, there is no spectral de- composition or factorization of the thermoelastic generator in the case of mechanical boundary conditions other than the canonical hinged case and thus no rigorous association with the abstract ODE (1.8). Still, we show in Reference 9 that for the clamped case, the minimal energy obeys the singular rate “predicted” in Reference 34, namely, Emin(T ) = O(T − 5 2 ). Our intent in this paper is to bring the story to a close by investigating the minimal energy function for the null control- lability of thermoelastic systems under the high-order free boundary conditions that are present in Eq. (1.1). 1.1.3 Main Result In regards to our stated problem, the main result is as follows: THEOREM 1.1 Let terminal time T 0 be arbitrary and a1, a2 ≥ 0 with a1 + a2 0. Then, given initial data [ω0, ω1, θ0] ∈ H, there exist control(s) [u1, u2] ∈ [L2 (Q)]2 such that the corresponding solution [ω, ωt , θ] of (1.1) satisfies [ω(T ), ωt (T ), θ(T )] = [0, 0, 0]. (That is to say, the PDE model Eq. (1.1)
- 29. April 4, 2005 10:3 3086 DK2961˙C001 1.2 The Necessary Observability Inequality 9 is null controllable within the class of [L2 (Q)]2 —controls in arbitrary short time.) Moreover, We have the following rates of blowup for the minimal energy function: 1. (thermal control) If a1 = 0, then Emin(T ) = O(T − 13 4 − ) for all 0; 2. (mechanical control) If a2 = 0, then Emin(T ) = O(T − 5 2 ); 3. If a1 0 and a2 0, then Emin(T ) = O(T − 3 2 ). REMARK 1.1 The null controllability of thermoelastic plates with free boundary conditions and under one internal control (be it mechanical or thermal) appears to be, as far as we know, a new result in the literature. The Theorem 1.1 above, in addition to asserting the said null controllability property, provides the asymptotics for the singularity of the associated minimal energy function. These asymptotics are optimal in the case of a single mechanical control and in the case of two controls acting upon the system. In the case of a single thermal control the estimate is “off” by 3/4 with respect to the desired “finite dimensional prediction” in Reference 34. Whether this estimate can be improved upon is an open question. Our method of proof of Theorem 1.1 is based on weighted energy estimates that are flexible enough to accomodate analytic estimates and the resulting singularity. The proof has the following main technical ingredients: 1. special weighted nonlocal multipliers introduced in Reference 4 and subsequently invoked in References 3, 5, 29, and 6, and elsewhere; 2. the analyticity of semigroups associated with thermoelastic PDE models in the absence of rotational forces, as demonstrated in References 31, 26, 27, and 28; 3. new singular estimates for boundary traces of solutions of Eq. (1.9), which are of their own intrinsic interest and which are needed to handle the boundary terms resulting from the weighted estimates employed. 1.2 The Necessary Observability Inequality The proof of Theorem 1.1 is based on the derivation of the observability inequality associated with the null controllability of the PDE (1.1) with respect to thermal or mechanical control or both. This inequality is formulated in terms of the solution of the homogeneous PDE, which is “dual” or “adjoint” to that in Eq. (1.1). Namely, we shall consider solutions [φ, φt , ϑ] to the following system: φtt + 2 φ + α ϑ = 0 on (0, T ) × ϑt − ϑ − α φt = 0 on (0, T ) × φ + (1 − µ)B1φ + αϑ = 0 ∂ φ ∂ν + (1 − µ) ∂ B2φ ∂τ − φ + α ∂ϑ ∂ν = 0 on ∂ϑ ∂ν + λϑ = 0 on , λ 0 [φ(0), −φt (0), ϑ(0)] = [φ0, φ1, ϑ0] ∈ H. (1.9)
- 30. April 4, 2005 10:3 3086 DK2961˙C001 10 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability If we define the bilinear form a(·, ·) : H2 () × H2 () → R by a(w, w̃) ≡ [wxx w̃xx + wyyw̃yy + µ(wxx w̃yy + wyyw̃xx ) + 2(1 − µ)wxyw̃xy] d, then we can state the “Green’s formula,” which involves this bilinear form (see Reference 22) and which is valid for functions w, w̃ (smooth enough): ( 2 w)w̃d = a(w, w̃) + ∂ w ∂ν + (1 − µ) ∂ B2w ∂τ w̃d − [ w + (1 − µ)B1w] ∂w̃ ∂ν d . (1.10) Let E(t) denote the energy of the adjoint system Eq. (1.9), where E(t) ≡ 1 2 a(φ(t), φ(t))d + 1 2 |φ(t)|2 d + 1 2 |ϑ(t)|2 d. (1.11) In terms of this energy, then one can show by classical functional analytical arguments (see, e.g., References 41 and 3) that the PDE (1.1) is null controllable, in the sense of 1, if and only if the adjoint variables [φ, φt , ϑ] of Eq. (1.9) satisfy the following continuous observability inequality, for some constant CT : [φ(T ), φt (T ), ϑ(T )]H ≤ CT (a1φt L2(Q) + a2ϑL2(Q)). (1.12) Having worked to establish the sharp constant CT in the observability inequality Eq. (1.12), one can proceed through an algorithmic procedure—using an explicit representation of the minimal norm control, by convex optimization—so as to have that for all terminal time T 0, Emin(T ) = O(CT ). Because the details of this argument are known and have been previously spelled out (see, e.g., References 9 and 8), we defer from repeating them here. Because of this characterization of the behavior of Emin(T ) with the constant CT in Eq. (1.12), our work will accordingly be geared toward establishing this inequality (where, again, control parameters ai satisfy a1, a2 ≥ 0, and a1 + a2 0). 1.3 Some Preliminary Machinery Inthissection,weexplicitlydefinetheunderlyinggeneratorA : D(A) ⊂ H → H,whichdescribes the thermoelastic flow. Subsequently, a proposition is derived with which to associate powers of this generator with specific Sobolev spaces. This characterization of the powers will be critical in work. r To start, we define the linear operator AD : D(AD) ⊂ L2 () → L2 () by AD ≡ − ; (1.13) D(AD) = H2 () ∩ H1 0 (). r We will also need the following (Dirichlet) map D : L2 ( ) → L2 (): Df = g ⇔ g = 0 on and g| = f on . (1.14) By the classical elliptic regularity, we have that D ∈ L(Hs ( ), Hs+ 1 2 ()) for all s (see Reference 30).
- 31. April 4, 2005 10:3 3086 DK2961˙C001 1.3 Some Preliminary Machinery 11 r We also define the linear operator Å : D(AD) ⊂ L2 () → L2 () by setting Å = 2 , for ∈ D(Å), where D(Å) = ∈ H4 () : [ + (1 − µ)B1] = 0 and ∂ ∂ν + (1 − µ) ∂ B2 ∂τ − = 0 , where the boundary operators Bi are as defined in Eq. 1.3. This operator is densely defined, positive definite, and self-adjoint. Consequently by Reference 21, one has the characterization D(Å 1 2 ) ≈ H2 (); with moreover Å 1 2 φ 2 L2() = a(φ, φ) + φ2 d . r Moreover, we define the elliptic operators Gi by G1h = v ⇔ 2 v = 0 on v + (1 − µ)B1v = h ∂ v ∂ν + (1 − µ) ∂ B2v ∂τ − v = 0 on ; G2h = v ⇔ 2 v = 0 on v + (1 − µ)B1v = 0 ∂ v ∂ν + (1 − µ) ∂ B2v ∂τ − v = h on . (1.15) By elliptic regularity (see, e.g., Reference 30) one has that for all real s, G1 ∈ L(Hs ( ), Hs+ 5 2 ()); G2 ∈ L(Hs ( ), Hs+ 7 2 ()). (1.16) With these operators defined above, we have that the generator A : D(A) ⊂ H → H of the thermoelastic semigroup may be given the explicit representation A = 0 I 0 −Å 0 α(AD(I − Dγ0) − ÅG1γ0 + λÅG2γ0) 0 −αAD(I − Dγ0) −αAD(I − Dγ0) ; D(A) = [ω0, ω1, θ0] ∈ H2 () × H2 () × H2 () : Å [ω0 + α (G1γ0 − λG2γ0) θ0] ∈ L2 () and ∂θ0 ∂ν + λθ0 = 0 (1.17) (here, γ0 ∈ L(H1 (), H 1 2 ( )) is the classical Sobolev trace map; i.e., γ0 f = f | for f ∈ C∞ ()). As we have said, it is now known that the generator A : D(A) ⊂ H → H for the thermoelas- tic plate, with free mechanical boundary conditions, is associated with an analytic C0-semigroup {eAt }t≥0 of contractions on H (see Reference 27 and references therein), with moreover A−1 being bounded on H.
- 32. April 4, 2005 10:3 3086 DK2961˙C001 12 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability For this realization of the generator, we now proceed to show the following: PROPOSITION 1.1 Let integer k = 1, 2, . . . . Then D(Ak ) ⊂ H2k+2 () × H2k () × H2k (). PROOF OF PROPOSITION 1.1 Let first [ω0, ω1, θ0] ∈ D(A). Then by definition, ω1, θ0 ∈ H2 (). Moreover, from the abstract representation in Eq. (1.17), we have Åω0 + αÅG1 θ0| − αλÅG2 θ0| + α θ0 = f ∈ L2 (). Because θ0| ∈ H 3 2 ( ), then consequently from the elliptic regularity results posted in Eq. (1.16), ω0 = Å−1 f − αG1 θ0| + αλG2 θ0| − αÅ−1 θ0 ∈ H4 (). So the assertion is true for k = 1. Proceeding now by induction, suppose that the result holds true for integer k − 1, k ≥ 2, and let [ω0, ω1, θ0] ∈ D(Ak ). Then, because A ω0 ω1 θ0 ∈ D(Ak−1 ), we have ω1 ∈ H2k (); Åω0 + αÅG1 θ0| − αλÅG2 θ0| + α θ0 = f ∈ H2k−2 (); ARθ0 − α ω1 = g ∈ H2k−2 (). (1.18) Here, AR : D(AR) ⊂ L2 () → L2 () is the elliptic operator defined by AR f = − f ; D(AR) = f ∈ H2 () : ∂ f ∂ν + λf = 0, λ 0 . (1.19) Reading off the third equation in Eq. (1.18), we obtain, after using elliptic regularity, θ0 = A−1 R (g + Dγ0θ0 − α ω1) ∈ H2k (). In turn, we can use again the result in Eq. (1.16) to have that ω0 = Å−1 f − αG1γ0θ0 + αλG2γ0θ0 − αÅ−1 θ0 ∈ H2k+2 (). This concludes the proof of Proposition 1.1. 1.4 A Singular Trace Estimate In this section, we exploit the underlying analyticity of the thermoelastic semigroup so as to generate pointwise (in time) estimate of boundary traces of the adjoint variables φt (t) and ϑ(t) of Eq. (1.9). These estimates will be of use to us in the proof of Theorem 1.1, inasmuch as they
- 33. April 4, 2005 10:3 3086 DK2961˙C001 1.4 A Singular Trace Estimate 13 each reflect a proper “distribution” between the measurement E(t) of the energy and the observation term—be it φt or ϑ. The price to pay for these benefical estimates is the appearance therein of singular weights of the form 1 ts , where parameter s will depend on the order of derivatives present. LEMMA 1.1 Let x(t) ≡ [φ(t), φt (t), ϑ(t)] denote the solution of the adjoint system Eq. (1.9), subject to the initial condition x(0) = [φ0, −φ1, ϑ0] ∈ H. Let,moreover, Dm be a differential operator of order m ≥ 0 with respect to the interior variables. Then for integers k = 1, 2, . . . , and all t 0 we have 1. Dmϑ(t)L( ) ≤ Ck t m 2 + 1 4 eA t 2 x0 1 2k H ϑ(t) 1− 1 2k L2() ; 2. Dmφt (t)L( ) ≤ Ck t m 2 + 1 4 eA t 2 x0 1 2k H φt (t) 1− 1 2k L2() ; 3. D1φtt (t)L( ) ≤ Ck t 7 4 eA t 2 x0 H . PROOF OF LEMMA 1.1 By a trace interpolation result (see, e.g., Reference 38) and the iterative use of a classical PDE moment inequality, we have the following string of estimates, which is valid for any g ∈ H2k+1 (m+1) (): Dm gL( ) ≤ C Dm g 1 2 L() Dm g 1 2 H1() ≤ C g 1 2 Hm () g 1 2 Hm+1() ≤ C g 1 2 L() g 1 4 H2m () g 1 4 H2(m+1)() ≤ C g 3 4 L() g 1 8 H4m () g 1 8 H4(m+1)() ≤ . . . ≤ C g 1− 1 2k L() g 1 2k+1 H2k m () g 1 2k+1 H2k (m+1)() . (1.20) Now by virtue of the analyticity of the thermoelastic semigroup {eAt }t≥0 and Proposition 1.1, we have for all t 0, [φ(t), φt (t), ϑ(t)] ∈ D(A2k−1 m ) ⇒ [φt (t), ϑ(t)] ∈ [H2km ()]2 . (1.21) Setting now g ≡ ϑ(t) (resp., φt (t)) in Eq. (1.20), we obtain Dmϑ(t)L( ) ≤ C ϑ(t) 1− 1 2k L() ϑ(t) 1 2k+1 H2k m () ϑ(t) 1 2k+1 H2k (m+1)() ≤ C A2k−1 m x(t) 1 2k+1 H A2k−1 (m+1) x(t) 1 2k+1 H ϑ(t) 1− 1 2k L() = C A2k−1 m eA t 2 eA t 2 x0 1 2k+1 H A2k−1 (m+1) eA t 2 eA t 2 x0 1 2k+1 H ϑ(t) 1− 1 2k L() . (1.22) At this point, we can invoke the well known pointwise estimate that is valid for any generator of an analytic semigroup: for all time t 0 and integer m = 1, 2, . . . , Am eAt L(H) ≤ Cm tm , (1.23) where constant C is independent of m (see, e.g., Reference 33, p. 70). Applying this estimate to the chain Eq. (1.22), we have Dmϑ(t)L( ) ≤ C t m 2 + 1 4 eA t 2 x0 1 2k H ϑ(t) 1− 1 2k L() .
- 34. April 4, 2005 10:3 3086 DK2961˙C001 14 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability This gives (Lemma 1.1, Step 1) (Step 2) is obtained in the very same way, by setting g = φt in Eq. (1.22) and then invoking the containment Eq. (1.21). For (Step 3), we have along the same lines, by means of the trace interpolation inequality in Reference 38 and the containment Eq. (1.21), D1φtt (t)L2( ) ≤ C φtt (t) 1 2 H1() φtt (t) 1 2 H2() ≤ C A 1 2 xt (t) 1 2 H A xt (t) 1 2 H ≤ CA2 x(t) 1 2 H ≤ C t 7 4 eA t 2 x0 1 2 H , which completes the proof. 1.5 Proof of Theorem 1.1(1) 1.5.1 Estimating the Mechanical Velocity In what follows, we will have need of the polynomial weight h(t), defined by h(t) ≡ ts (T − t)s . (1.24) For the proof of Theorem 1.1(1), we will take s ≡ 6. In terms of the the solution [φ, φt , ϑ] of Eq. (1.9) and its corresponding energy E(t), the necessary inequality for the case of thermal control is E(T ) ≤ CT ϑL2(Q) . (1.25) It is the derivation of this inequality that will drive the proof of Theorem 1.1. We will start by applying the Laplacian to both sides of the heat equation in Eq. (1.9). This gives ϑt − 2 ϑ − α 2 φt = 0 in . From this expression and the free boundary conditions in Eq. (1.9), we have that the velocity term φt satisfies the following elliptic problem for all t 0: 2 φt (t) = 1 α ϑt (t) − 1 α 2 ϑ(t) in φt (t) + (1 − µ)B1φt (t) = −αϑt ∂ φt (t) ∂ν + (1 − µ) ∂ B2φt (t) ∂τ − φt (t) = αλϑt on . (1.26) Using this Green’s map defined in Eq. (1.15), we have from Eq. (1.26) that the velocity φt may be written explicitly as φt (t) = 1 α Å−1 ϑt (t) − 2 ϑ(t) − αG1γ0(ϑt (t)) + αλG2γ0(ϑt (t)). (1.27) From this, we have T 0 h φt 2 L2() dt = T 0 h 1 α Å−1 [ ϑt (t) − 2 ϑ(t)] − αG1γ0(ϑt (t)) + αλG2γ0(ϑt (t)), φt L2() dt, (1.28) where h(t) is the polynomial weight described in Eq. (1.24).
- 35. April 4, 2005 10:3 3086 DK2961˙C001 1.5 Proof of Theorem 1.1(1) 15 Analysis of the right-hand side of Eq. (1.28). 1. T 0 h ([G1 − λG2] γ0ϑt , φt )L2() dt = − T 0 h ([G1 − λG2] γ0ϑ, φt )L2() dt − T 0 h([G1 − λG2] γ0ϑ, φtt )L2() dt. (1.29) a. By the regularity posted in Eq. (1.16) and an application of Lemma 1.1 (with m = 0 and k = 2, say) we have T 0 h ([G1 − λG2] γ0ϑ, φt )L2() dt ≤ C T 0 |h | ϑL2( ) φt L2() dt ≤ C T 0 |h | t 1 4 h(t) h(t) 5 8 ϑ(t) 3 4 L2() eA t 2 x0 5 4 H dt. Invoking Hölder’s inequality to this right hand side, with Hölder conjugates (8 3 , 8 5 ), we obtain now the estimate T 0 h ([G1 − λG2] γ0ϑ, φt )L2() dt ≤ C T 26 3 T 0 h(t) ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.30) b. Proceeding as above, with m = 0 and k = 2 in Lemma 1.1, we have T 0 h([G1 − λG2] γ0ϑ, φtt )L2() dt ≤ C T 0 h(t) t 1 4 ϑ(t) 3 4 L2() eA t 2 x0 1 4 H A xt (t)L2() dt ≤ C T 0 h(t) t 5 4 ϑ(t) 3 4 L2() eA t 2 x0 5 4 H dt ≤ C T 26 3 T 0 h(t) ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.31) Combining Eq. (1.30) and Eq. (1.31) now gives T 0 h([G1 − λG2] γ0ϑt , φt )L2() dt ≤ C T 20 3 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.32) 2. Next, T 0 h Å−1 ( ϑt ), φt L2() dt = − T 0 h Å−1 ( ϑ), φt L2() dt − T 0 h Å−1 ( ϑ), φtt L2() dt. (1.33)
- 36. April 4, 2005 10:3 3086 DK2961˙C001 16 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability a. By Green’s theorem and the Lemma 1.1, with m = 0 and k = 2, we have T 0 h Å−1 ( ϑ), φt L2() dt = T 0 h ϑ, ∂ ∂ν + I Å−1 φt L2( ) dt − T 0 h ϑ, Å−1 φt L2() dt ≤ C T 26 3 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.34) b. Likewise, by Green’s Theorem, the analyticity of the semigroup and Lemma 1.1, with m = 0 and k = 2, we have T 0 h(t) ϑ, Å−1 φtt L2() dt ≤ C T 26 3 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.35) Applying the estimates Eq. (1.34) and Eq. (1.35) to Eq. (1.33) now yields T 0 h( ϑt , φt )L2() dt ≤ C T 26 3 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.36) 3. By the Green’s identity posted in Eq. (1.10), we have T 0 h 2 ϑ, Å−1 φt L2() dt = T 0 h(t)(ϑ, φt )L2() dt = T 0 h(t) ∂ ∂ν + (1 − µ) ∂ B2 ∂τ ϑ, Å−1 φt L2( ) − [ + (1 − µ)B1] ϑ, ∂ ∂ν Å−1 φt L2( ) dt. Applying once more the Lemma 1.1 (e.g., with m = 3, k = 3) we have T 0 h 2 ϑ, Å−1 φt L2() dt ≤ C T 8 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.37) Combining the expression Eq. (1.28) with the estimates of Eqs. (1.32 ), (1.36), and (1.37) gives us the following estimate for the mechanical velocity: LEMMA 1.2 With s = 6 in Eq. (1.24), the solution [φ, φt , ϑ] of (1.9) satisfies the following estimate for all 0: T 0 h(t) φt 2 L2() dt ≤ C T 8 T 0 ϑ2 L2() dt + T 0 E t 2 dt.
- 37. April 4, 2005 10:3 3086 DK2961˙C001 1.5 Proof of Theorem 1.1(1) 17 1.5.2 Estimating the Mechanical Displacement Here, we shall show the following: LEMMA 1.3 The solution [φ, φt , ϑ] of Eq. (1.9) satisfies the following estimate for all , δ 0: T 0 h(t) Å 1 2 φ 2 L2() dt ≤ CT 13 2 −δ ϑ2 L2() dt + T 0 h(t)E t 2 dt + T 0 h(t)E(t)dt. PROOF OF LEMMA 1.3 We start by applying the multiplier h(t)φ(t) to the mechanical component in Eq. (1.9). We arrive at the relation T 0 h(t) Å 1 2 φ 2 L2() dt = T 0 h (t)(φt , φ)L2() dt + T 0 h(t) φt 2 L2() dt + αλ T 0 h(t) Å 1 2 G2γ0ϑ, Å 1 2 φ L2() dt − αλ T 0 h(t) Å 1 2 G2γ0ϑ, Å 1 2 φ L2() dt − α T 0 h(t)(ϑ, φ)L2()dt + α T 0 h(t) (ϑ, λφ + ∂φ ∂ν L2( ) dt. (1.38) Now, using the elliptic regularity posted in Eq. (1.16) and the usage of Lemma 1.1, with m = 0 and k = 3, we obtain T 0 h(t) Å 1 2 G2γ0ϑ, Å 1 2 φ L2() dt − αλ T 0 h(t) Å 1 2 G2γ0ϑ, Å 1 2 φ L2() dt − α T 0 h(t)(ϑ, φ)L2() dt + α T 0 h(t) (ϑ, λφ + ∂φ ∂ν L2( ) dt ≤ C T 26 3 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.39) Combining this estimate with that in Lemma 1.2 then gives the preliminary estimate T 0 h(t) Å 1 2 φ 2 L2() dt ≤ T 0 h (t)(φt , φ)L2() dt + CT 8 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.40) Apparently, we must estimate the first term on the right-hand side of Eq. (1.40). To this end, we use the pointwise expression for φt in Eq. (1.27): T 0 h (φt , φ)L2() dt = T 0 h 1 α Å−1 ( ϑt − 2 ϑ) − αG1γ0ϑt + αλG2γ0ϑt , φ L2() dt (1.41)
- 38. April 4, 2005 10:3 3086 DK2961˙C001 18 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability 1. The abstract Green’s Theorem gives T 0 h ( 2 ϑ, Å−1 φ)L2()dt = T 0 h (ϑ, φ)L2() + ∂ ∂ν + (1 − µ) ∂ B2 ∂τ ϑ, Å−1 φ L2( ) − [ + (1 − µ)B1] ϑ, ∂ ∂ν Å−1 φ L2( ) dt. Applying now the Lemma 1.1 with m = 3, 2 yields T 0 h ( 2 ϑ, Å−1 φ)L2()dt ≤ C T 0 |h | t 7 4 ϑ 1− 1 2k L2() eA t 2 x0 1+ 1 2k H dt. Let k ≥ 4. Then, because h (t) = 6t5 (T − 2t)(T − t)5 , we can apply now Hölder’s inequality with Hölder conjugates (2 2k 2k −1 , 1 1 2 +2−k−1 ) so as to have T 0 h ( 2 ϑ, Å−1 φ)L2()dt ≤ C T 0 t 2k−1−6 2k −1 |T − 2t| 2k+1 2k −1 (T − t) 2k+2−6 2k −1 ϑ2 L2() dt + T 0 h(t)E t 2 dt ≤ CT 13×2k−1−12 2k −1 T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt (again this inequality being valid for k ≥ 4). Now for any δ 0, we can rechoose integer k large enough so as to have 13×2k−1 −12 2k −1 ≥ 13 2 − δ. This gives, then, for T 1, T 0 h ( 2 ϑ, Å−1 φ)L2()dt ≤ CT 13 2 −δ T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.42) (This is the term which ultimately dictates the singularity.) 2. Next, T 0 h [(G1 − λG2)γ0ϑt , φ]L2()dt = − T 0 h [(G1 − λG2)γ0ϑ, φ]L2() dt − T 0 h [(G1 − λG2)γ0ϑ, φt ]L2() dt. (1.43) a. By the regularity posted in Eq. (1.16) and Lemma 1.1, T 0 h [(G1 − λG2)γ0ϑ, φ]L2() dt ≤ C T 0 |h | t 1 4 ϑ 1− 1 2k L2() eA t 2 x0 1+ 1 2k H dt. Applying Hölder’s inequality to the right-hand side, with Hölder conjugates (2 2k 2k −1 , 1 1 2 +2−k−1 ) now yields T 0 h [(G1 − λG2)γ0ϑ, φ]L2()dt ≤ C T 0 |h | t 1 4 h(t) h(t) 1 2 +2−k−1 × ϑ 1− 1 2k L2() eA t 2 x0 1+ 1 2k H dt ≤ C T 3 5×2k−1−4 2k −1 T 0 ϑ2 L2() + T 0 h(t)E t 2 dt.
- 39. April 4, 2005 10:3 3086 DK2961˙C001 1.5 Proof of Theorem 1.1(1) 19 Because for any δ 0, we can choose integer k large enough so that 35×2k−1 −4 2k −1 ≥ 15 2 −δ, we then get T 0 h [(G1 − λG2)γ0ϑ, φ]L2() dt ≤ C T 15 2 − T 0 ϑ2 L2() + T 0 h(t)E t 2 dt. (1.44) b. In the same way as above, we have for integer k large enough in Lemma 1.1, T 0 h [(G1 − λG2)γ0ϑ, φt ]L2() dt ≤ C T 15 2 −δ T 0 ϑ2 L2() + T 0 h(t)E t 2 dt. (1.45) The estimates Eqs. (1.44) and (1.45), applied to the relation Eq. (1.43) now give T 0 h [(G1 − λG2)γ0ϑt , φ]L2() dt ≤ C T 15 2 − T 0 ϑ2 L2() + T 0 h(t)E t 2 dt. (1.46) for integer k large enough. 3. T 0 h Å−1 ϑt , φ L2() dt = − T 0 h ( ϑ, Å−1 φ)L2() dt − T 0 h ϑ, Å−1 φt L2() dt (1.47) a. By Green’s Theorem and Lemma 1.5, we have in a fashon similar to that in (1.a.), T 0 h ( ϑ, Å−1 φ)L2() dt = − T 0 h (θ, Å−1 φ)L2() + T 0 h θ, ∂ ∂ν + λ Å−1 φ L2( ) ≤ C T 3 5×2k−1−4 2k −1 T 0 ϑ2 L2() + T 0 h(t)E t 2 dt ≤ CT 15 2 −δ T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt, (1.48) for integer k large enough. b. In the same way, T 0 h ϑ, Å−1 φt L2() dt ≤ CT 15 2 −δ T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.49) Eqs. (1.47), (1.48), and (1.49) together give the estimate T 0 h ϑt , Å−1 φ L2() dt ≤ CT 15 2 −δ T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt. (1.50) Combining Eqs. (1.40), (1.41), (1.46), (1.50), and (1.42) will complete the proof of Lemma 1.3.
- 40. April 4, 2005 10:3 3086 DK2961˙C001 20 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability 1.5.3 Conclusion of the Proof of Theorem 1.1(1) Combining Lemmas 1.2 and 1.3 gives the following estimate for the energy: T 0 E(t) dt ≤ C T 13 2 −δ T 0 ϑ2 L2() dt + T 0 h(t)E t 2 dt; or after changing limits of integration, T 2 0 [(1 − )h(t) − 2h(2t)] E(t) dt + (1 − ) T T 2 h(t)E(t)dt ≤ C T 13 2 −δ T 0 ϑ2 L2() dt. For 0 small enough, this yields then T T 2 h(t)E(t) dt ≤ C T 13 2 −δ T 0 ϑ2 L2() dt. Using the dissipation inherent in the thermoelastic system (i.e., E(t) ≤ E(s) for s ≤ t), we finally obtain E(T ) ≤ CT 13 2 −δ−13 T 0 ϑ2 L2() dt. This establishes the inequality Eq. (1.25), with CT = CT −q , where q = 13 4 − δ 2 , for any δ 0. This concludes the proof of Theorem 1.1(1). 1.6 Proof of Theorem 1.1(2) 1.6.1 A First Supporting Estimate In what follows, we will again make use of the polynomial weight h(t) in Eq. (1.24), with s = 4 therein. In the present case of mechanical control, the necessary inequality (Eq. (1.12)) becomes E(T ) ≤ CT φt L2(Q) . (1.51) to be valid for all finite energy solutions to Eq. (1.9). We start by establishing the following estimate: PROPOSITION 1.2 The solution [φ, φt , ϑ] of Eq. (1.9) satisfies the relation T 0 h(t) A−1 R ϑt , ϑ L2() dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. PROOF OF PROPOSITION 1.2 From the mechanical component of Eq. (1.9) we have, after an extra differentiation in time, the expression −α ϑt = ∂3 ∂t3 φ + 2 φt ; whence we obtain A−1 R ϑt = 1 α A−2 R ∂3 ∂t3 φ + 1 α A−2 R 2 φt ,
- 41. April 4, 2005 10:3 3086 DK2961˙C001 1.6 Proof of Theorem 1.1(2) 21 where the positive definite, self-adjoint operator AR : D(AR) : L2 () → L2 () is as defined in Eq. (1.19). Subsequently, we will have the following relation: T 0 h(t) A−1 R ϑt , ϑ L2() dt = 1 α T 0 h(t) ∂3 ∂t3 φ, A−2 R ϑ L2() dt + 1 α T 0 h(t) 2 φt , A−2 R ϑ L2() dt. (1.52) We need to estimate the right-hand side of this expression. 1. For the first term on the right-hand side of Eq. (1.52), integration by parts gives T 0 h(t) ∂3 ∂t3 φ, A−2 R ϑ L2() dt = T 0 h(t) φt , A−2 R ϑtt L2() dt + 2 T 0 h (t) φt , A−2 R ϑt L2() dt + T 0 h (t) φt , A−2 R ϑ L2() dt. (1.53) We proceed to scrutinize each term on the right-hand side. To this end, we introduce the (Robin) map R ∈ L[L2 ( ), L2 ()], defined by R f = g ⇔ g = 0 on and ∂g ∂ν + λg = f on (1.54) (by elliptic regularity, we have in fact that R ∈ L[Hs ( ), Hs+ 3 2 ()] for all real s). Using this quantity with the heat equation in Eq. (1.9), we will then have the relations A−2 R ϑt = −A−1 R ϑ − αA−1 R I − R λγ0 + ∂ ∂ν φt ; (1.55) A−2 R ϑtt = ϑ + α I − R λγ0 + ∂ ∂ν φt − αA−1 R I − R λγ0 + ∂ ∂ν φtt . a. From Eq. (1.56), we have T 0 h(t) φt , A−2 R ϑtt L2() dt ≤ T 0 h(t) × φt , ϑ + α I − R λγ0 + ∂ ∂ν φt L2() dt + T 0 h(t) φt , αA−1 R I − R λγ0 + ∂ ∂ν φtt L2() dt. (1.56)
- 42. April 4, 2005 10:3 3086 DK2961˙C001 22 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability To handle the most problematic term on the right-hand side of this expression (with again x(t) = [φ(t), φt (t), ϑ(t)]), we use the singular trace estimate in Lemma 1.1(3): T 0 h(t) φt , A−1 R R ∂ ∂ν φtt L2() dt ≤ C T 0 h(t) φt L2() ∂ ∂ν φtt L2( ) dt ≤ C T 0 h(t) t 3 4 t φt L2() eA t 2 x(0) H dt ≤ C T 0 h(t) t 7 2 φt 2 L2() dt + T 0 h(t)E t 2 dt ≤ C T 9 2 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. Applying this estimate to Eq. (1.56) and treating in like fashion the other terms on the right-hand side thereof, we have T 0 h(t) φt , A−2 R ϑtt L2() dt ≤ C T 9 2 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. (1.57) b. Using the first relation in Eq. (1.56), we have, analogously to what was obtained in (1.a), T 0 h (t) φt , A−2 R ϑ L2() dt + 2 T 0 h (t) φt , A−2 R ϑt L2() dt ≤ C T 0 |h (t)| + |h (t)| t 3 4 φt L2() eA t 2 x(0) H dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. (1.58) Combining Eqs. (1.56) and (1.58) now gives T 0 h(t) ∂3 ∂t3 φ, A−2 R ϑ L2() dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. (1.59) 2. By the “Green’s” formula in Eq. (1.10), we have T 0 h(t) 2 φt , A−2 R ϑ L2() dt = T 0 h(t)a φt , A−2 R ϑ L2() dt + T 0 h(t) αλϑt + φt , A−2 R ϑ L2( ) dt − T 0 h(t) φt + (1−µ)B1φt , ∂ ∂ν A−2 R ϑ L2( ) dt = − T 0 h(t) φt + (1 − µ)B1φt , λI + ∂ ∂ν A−2 R ϑ L2( ) dt + T 0 h(t) φt , 2 A−2 R ϑ L2() dt + T 0 h(t) ∂ ∂ν φt , [ + (1 − µ)B1] A−2 R ϑ L2( ) dt − T 0 h(t) φt , ∂ ∂ν + (1 − µ) ∂ B2 ∂τ − I A−2 R ϑ L2( ) dt. (1.60)
- 43. April 4, 2005 10:3 3086 DK2961˙C001 1.6 Proof of Theorem 1.1(2) 23 For the first term on the right-hand side of Eq. (1.60), we apply the Lemma 1.1(1) (with m = 2 and D2 ≡ + (1 − µ)B1 therein) so as to have T 0 h(t) φt + (1 − µ)B1φt , λI + ∂ ∂ν A−2 R ϑ L2( ) dt ≤ C T 0 h(t) t 5 4 φt 1− 1 2k H2() eA t 2 x0 1 2k H ϑL2() dt ≤ C T 0 h(t) t 5 4 φt 1− 1 2k H2() eA t 2 x0 1+ 1 2k H . Now letting k = 2, say, we can invoke Hölder’s inequality, with Hölder conjugates 8 3 , 5 3 , to obtain the estimate T 0 h(t) φt + (1 − µ)B1φt , λI + ∂ ∂ν A−2 R ϑ L2( ) dt ≤ C T 14 3 T 0 h(t) φt 2 L2() dt + T 0 h(t)E(t/2) dt. (1.61) Applying this estimate to the right-hand side of Eq. (1.60) and subsequently handling the other terms thereof in a similar way—via the use of Lemma 1.1—we will have T 0 h(t) 2 φt , A−1 R ϑ L2() dt ≤ C T 14 3 T 0 h(t) φt 2 L2() dt + T 0 h(t)E(t) dt. (1.62) Combining Eqs. (1.52), (1.59), and (1.62) concludes the proof of Proposition 1.2. 1.6.2 Conclusion of the Proof of Theorem 1.1(2) 1. Estimating the Thermal Component. Applying the multiplier h(t)A−1 R ϑ(t) to the heat compo- nent of the system Eq. (1.9) and subsequently invoking Proposition 1.2, we have T 0 h(t) ϑ2 L2() = − T 0 h(t) A−1 R ϑt , ϑ dt − α T 0 h(t) I − R λγ0 + ∂ ∂ν φt , ϑ dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt + T 0 h(t) λφt + ∂ ∂ν φt L2( ) ϑL2() dt. (1.63) Via the Lemma 1.1 (with m = 1, D1 = λI + ∂ ∂ν , and k = 1, say), we can estimate the third term on the right-hand side of Eq. (1.63) as T 0 h(t) λφt + ∂ ∂ν φt L2( ) ϑL2() dt ≤ CT 5 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. Combining this estimate with Eq. (1.63), we now obtain T 0 h(t) ϑ2 L2() ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. (1.64)
- 44. April 4, 2005 10:3 3086 DK2961˙C001 24 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability 2. Estimating the Mechanical Component. Here, we apply the multiplier intrinsic to uncoupled plates and beams. To wit, from the mechanical component of Eq. (1.9), we have via h(t)φ(t) and an invocation of the Green’s Theorem Eq. (1.10) the expression T 0 Å 1 2 φ 2 L2() dt = −α T 0 h(t)(ϑ, φ) dt + T 0 h (t)(φt , φ) dt + T 0 h(t) φt 2 L2() dt. (1.65) Applying the estimate of Eq. (1.64) (available for the thermal component) now gives T 0 Å 1 2 φ 2 L2() dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. (1.66) Combining the estimates of Eqs. (1.64) and (1.66) now give the estimate for the energy T 0 h(t)E(t) dt ≤ C T 4 T 0 φt 2 L2() dt + T 0 h(t)E t 2 dt. With this in hand, we can proceed as in the previous case so as to have the observability inequality Eq. (1.51), with CT = T − 5 2 . Subsequently, we will determine that in the present case ofmechanicalcontrol,onehasEmin(T ) = O(T − 5 2 ).ThisconcludestheproofofTheorem1.1(2) with free boundary conditions and one control. 1.7 Proof of Theorem 1.1(3) Here we set the index s = 2 in Eq. (1.24). In this present case of dual—mechanical and thermal— control, the necessary inequality is E(T ) ≤ CT (φt L2(Q) + ϑL2(Q)), (1.67) where again [φ, φt , ϑ] solve the homogeneous system Eq. (1.9). Using the relation Eq. (1.65), we have (1 − ) T 0 h(t) Å 1 2 φ 2 L2() dt ≤ C T 0 h(t) + [h (t)]2 h(t) φt 2 L2() + ϑ2 L2() dt ≤ CT 2 T 0 φt 2 L2() + ϑ2 L2() dt. This then gives T 0 h(t)E(t) dt ≤ CT 2 T 0 φt 2 L2() + ϑ2 L2() dt, whence we obtain the inequality Eq. (1.67). From here, we can use the usual algorithmic argument so as to have Emin(T ) = O(T − 3 2 ). This concludes the proof of Theorem 1.1(3). References [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 2000, No. 22 (2000), 1–15.
- 45. April 4, 2005 10:3 3086 DK2961˙C001 References 25 [2] G. Avalos, Exact controllability of a thermoelastic system with control in the thermal compo- nent only, Differential Integral Equations, 13, (2000), 613–630. [3] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free bound- ary conditions, SIAM J. Control Optim., 38, No. 2 (2000), 337–383. [4] G. Avalos and I. Lasiecka, Exponential Stability of a Thermoelastic System without Mechanical Dissipation, Rendiconti dell’Istituto di Matematica dll’Università di Trieste, Vol. XXVIII, Supplemento (1996), 1–28. [5] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boun- dary conditions without mechanical dissipation, SIAM J. Math. Anal., 29, No. 1 (1998), 155– 182. [6] G. Avalos and I. Lasiecka, A Note on the Null Controllability of a Thermoelastic Plates and Singularity of the Associated Minimal Energy Function, Scuola Normale Superiore (Pisa), Preprints di Matematica, n. 10 (Giugno 2002). [7] G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), II (2003), 601–616. [8] G. Avalos and I. Lasiecka, Mechanical and thermal null controllability of thermoelastic plates and singularity of the associated minimal energy function, Control Cybern. special volume dedicated to K. Malanowski. 32, No. 3 (2003), 473–491. [9] G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294, (2004), 34–61. [10] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free bound- ary conditions, SIAM J. Control Optim., 38, No. 2 (2000), 337–383. [11] A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., in press. [12] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinte Dimensional Systems, Vol. II, Birkhäuser, Boston, (1993). [13] J. Cea, Lectures on Optimization—Theory and Algorithms Published for the Tata Institute of Fundamental Research, Springer-Verlag, New York, (1978). [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, New York, (1992). [15] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, (2001). [16] M. Eller, I. Lasiecka, and R. Triggiani. Simultaneous exact-approximate boundary controlla- bility of thermoelastic plates with variabale thermal coefficients and moment control. J. Math. Anal. Appl., 251, (2000), 452–478. [17] M. Eller, I. Lasiecka and R. Triggiabi. Simultaneous exact controllability of thermoelastic plates with variable thermal coefficients and clamped/Dirichlet boundary controls, Discrete Continuous Dynamical Syst., 7, No. 2 (2001), 283–302.
- 46. April 4, 2005 10:3 3086 DK2961˙C001 26 Asymptotic Rates of Blowup for the Minimal Energy Function for the Null Controllability [18] S. Hansen and B. Zhang, Boundary control of a thermoelastic beam, J. Math. Anal. Appl., 210, (1997), 182–205. [19] F.GozziandP.Loreti,Regularityoftheminimumtimefunctionandminimumenergyproblems, SIAM J. Control Optim., in press. [20] B. Guichal, A lower bound of the norm of the control operator for the heat equation. J. Math. Anal. Appl. 110, No. 2 (1985), 519–527. [21] P. Grisvard, Caracterization de quelques espaces d’interpolation, Arch. Rational Mech. Anal., 25, (1967), 40–63. [22] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math., 10, SIAM, Philadelphia, (1989). [23] J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal., 112, (1990), 223–267. [24] J. Lagnese and J.L. Lions, Modelling, Analysis and Control of Thin Plates, Masson Paris, (1988). [25] I. Lasiecka and T. Seidman, Thermal boundary control of a thermoelastic system, preprint (2004). [26] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with clamped/hinged B.C., Abst. Appl. Anal., 3, No. 2 (1998), 153–169. [27] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled BC. Part II: The case of free BC, Annali Scuola Normale di Pisa, Classes Scienze (Serie IV, Fasicolo 3–4), Vol. XXVII (1998[c]), 457–497. [28] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I: Abstract Parabolic Systems, Cambridge University Press, New York, (2000). [29] I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Commun. Partial Differential Equations, 24, 910 (1999), 1801–1847. [30] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, (1972). [31] Z. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8, No. 3 (1995), 1–6. [32] A. Lunardi, Schauder’s estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in Rn , Ann. Scuola Sup. Pisa (IV), XXIV (1997), 133–164. [33] T.I. Seidman, How fast are violent controls? Math. Controls Signals Syst., (1988), 89–95. [34] T.I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11, (1984), 145–152. [35] T.I. Seidman and J. Yong, How fast are violent controls?, II, Math. Controls Signals Syst., 9, (1997), 327–340.
- 47. April 4, 2005 10:3 3086 DK2961˙C001 References 27 [36] T. Seidman, S. Avdonian, and S. Ivanov, The window problem for series of complex exponen- tials, J. Fourier’s Anal., 6, (2000), 235–254. [37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, New York, (1984). [38] R. Triggiani, Analyticity, and lack thereof, of semigroups arising from thermo-elastic plates, in the special volume entitled Computational Science for the 21st Century, Chapter on Control: Theory and Numerics, Wiley 1997; Proceedings in honor of R. Glowinski, (May, 1997). [39] R. Triggiani, Optimal estimates of norms of fast controls in exact null controllability of two non-classical abstract parabolic systems, Adv. Differential Equations, 8, No. 2 (2003), 189–229. [40] J. Zabczyk, Mathematical Control Theory, Birkhäuser, Boston, (1992).
- 48. April 4, 2005 10:3 3086 DK2961˙C001
- 49. April 5, 2005 13:52 3086 DK2961˙C002 Chapter 2 Interior and Boundary Stabilization of Navier-Stokes Equations Viorel Barbu Alexandru Ioan Cuza University 2.1 Introduction ............................................................................ 29 2.2 Part I: Interior Control [4] ............................................................... 30 2.2.1 Introduction ..................................................................... 30 2.2.2 Main Results .................................................................... 32 2.3 Part II: Boundary Control [3] ............................................................ 35 2.3.1 Introduction ..................................................................... 35 2.3.2 Main Results (Case d = 3) ....................................................... 39 References .............................................................................. 41 Abstract We report on very recent work on the stabilization of the steady-state solutions to Navier-Stokes equations on an open bounded domain ⊂ Rd , d = 2, 3, by either interior or else boundary control. More precisely, as to the interior case, we obtain that the steady-state solutions to Navier-Stokes equations on ⊂ Rd , d = 2, 3, with no-slip boundary conditions, are locally exponentially stabi- lizable by a finite-dimensional feedback controller with support in an arbitrary open subset ω ⊂ of positive measure. The (finite) dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. Second, as to the boundary case, we obtain that the steady-state solutions to Navier-Stokes equations on a bounded domain ⊂ Rd , d = 2, 3, are locally exponentially stabilizable by a boundary closed-loop feedback controller, acting tangentially on the boundary ∂, in the Dirichlet boundaryconditions. If d = 3,thenonlinearityimposesanddictatestherequirementthatstabilization must occur in the space [H 3 2 + ()]3 , 0, a high topological level. A first implication thereof is that,ford = 3,theboundaryfeedbackstabilizingcontrollermustbeinfinitedimensional.Moreover,it generally acts on the entire boundary ∂. Instead, for d = 2, where the topological level for stabilizat- ion is [H 3 2 − ()]2 , the boundary feedback stabilizing controller can be chosen to act on an arbitrarily small portion of the boundary. Moreover, still for d = 2, it may even be finite dimensional, and this occurs if the linearized operator is diagonalizable over its finite-dimensional unstable subspace. 2.1 Introduction We hereby report on recent joint work on the stabilization of steady-state solutions to Navier- Stokes equations on an open bounded domain ⊂ Rd , d = 2, 3, by either interior feedback control or else boundary feedback control. The case of interior control is taken from the joint work with Triggiani in Reference 4. The case of boundary control is taken from the joint work with Lasiecka and Triggiani in Reference 3. To enhance readability, we provide independent accounts of each case. 29
- 50. April 5, 2005 13:52 3086 DK2961˙C002 30 Interior and Boundary Stabilization of Navier-Stokes Equations 2.2 Part I: Interior Control [4] 2.2.1 Introduction The controlled Navier-Stokes equations Consider the controlled Navier-Stokes equations (see Reference 6, p. 45, and Reference 13, p. 253 for the uncontrolled case u ≡ 0) with the non-slip Dirichlet B.C.: yt (x, t) − ν y(x, t) + (y · ∇)y(x, t) = m(x)u(x, t) + fe(x) + ∇ p(x, t) in Q = × (0, ∞), (2.1) ∇ · y=0 in Q; y=0 on = ∂ × (0, ∞); y(x, 0)=y0(x) in . Here, is an open, smooth, bounded domain of Rd , d = 2, 3; m is the characteristic function of an open smooth subset ω ⊂ of positive measure; u is the control input; and y = (y1, y2, . . . , yd) is the state (velocity) of the system. The function v = mu can be viewed itself as an internal controller with support in Qω = ω×(0, ∞). The functions y0, fe ∈ [L2 ()]d are given, the latter being a body force, whereas p is the unknown pressure. Let (ye, pe) ∈ [(H2 ()]d ∩ V ) × H1 () be a steady-state solution to Eq. (2.1), that is, −ν ye + (ye · ∇)ye = fe + ∇ pe in ; (2.2) ∇ · ye = 0 in ; ye = 0 on ∂. The steady-state solution is known to exist for d = 2, 3, (see Reference 6, Theorem 7.3, p. 59). Here [6, p. 9], [13, p. 18] V = y ∈ H1 0 () d ; ∇ · y = 0 , with norm yV ≡ y = |∇y(x)|2 d 1 2 . (2.3) Literature According to some recent results of Imanuvilov [9] (see also Reference 1) any such solution ye is locally exactly controllable on every interval [0, T ] with controller u with support in Qω. More precisely, if the distance ye − y0H2() is sufficiently small, then there is a solution (y, p, u) to Eq. (2.1) of appropriate regularity such that y(T ) ≡ ye. The steering control is open-loop and depends on the initial condition. Subsequently, Reference 2 proved that any steady-state solution ye is locally exponentially stabilizable by means of an infinite-dimensional feedback controller by using the controllability of the linear Stokes equation. In contrast, here we shall prove, via the state decomposition technique of References 14 and 15 and the first-order stabilization Riccati equation method developed in our previous work (Reference 2; see also Reference 5 still in the parabolic case, as well as Reference 11 in the hyperbolic case), that any steady-state solution ye is locally exponentially stabilizable by a finite-dimensional closed-loop feedback controller of the form u = − 2K i=1 (RN (y − ye), ψi )ωψi , (2.4)
- 51. Another Random Scribd Document with Unrelated Content
- 52. dislike of labor, and enured to wretchedness and hunger; and, on every failure of the potato crop, hundreds of thousands were starving.” Horrible as such a picture is, it is but a faint sketch of the reality. All readers of Irish history know it, and no student of English legislation should forget or pass over that dark chapter in England’s history. Our own readers have seen the whole system vividly sketched in these pages recently in the series of papers on “English Rule in Ireland.” What, in human nature and human possibilities, was to become of a people thus submitted to so long and unbending and systematic a course of degradation? They had nothing left but their faith, and the eternal truth of the promise that this is the victory which overcometh the world; and that our faith shall make us free was never more gloriously and wondrously made manifest than in the case of the Irish people. Ignorance was made compulsory by this Protestant government. The statute law of Ireland forbade Catholics to open schools or to teach in them. The Irish people, of all peoples, have ever had a craving for knowledge. What was left to them to do? “The Catholics,” says Mr. Froude, “with the same steady courage and unremitting zeal with which they had maintained and multiplied the number of their priests, had established open schools in places like Killarney, where the law was a dead-letter. In the more accessible counties, where open defiance was dangerous, they extemporized class teachers under ruined walls or in the dry ditches by the roadside, where ragged urchins, in the midst of their poverty, learnt English and the elements of arithmetic, and even to read and construe Ovid and Virgil. With institutions which showed a vitality so singular and so spontaneous repressive acts of Parliament contended in vain.” Ignorance is esteemed to be the prolific mother of vice. The social condition of the Irish people was made as bad as legislation could make it. Where was the room for morality in such a case? In vainly trying to explain away that most brutal project of law for the mutilation of the Irish priests, Mr. Froude says (vol. i. p. 557): “They (the Lord Lieutenant and Privy Council) did propose, not that all the Catholic clergy in Ireland, as Plowden says, but that unregistered
- 53. priests and friars coming in from abroad, should be liable to castration”; and he adds in a note: “Not, certainly, as implying a charge of immorality. Amidst the multitude of accusations which I have seen brought against the Irish priests of the last century, I have never, save in a single instance, encountered a charge of unchastity. Rather the exceptional and signal purity of Irish Catholic women of the lower class, unparalleled probably in the civilized world, and not characteristic of the race, which in the sixteenth century was no less distinguished for licentiousness, must be attributed wholly and entirely to the influence of the Catholic clergy.” Mr. Froude cannot be wholly generous and honest in a matter of this kind, but what is true in this is sufficient for our purpose without inquiring into what is false. It is plain from his own words that the one thing that saved the Irish people from perdition, body and soul, was their Catholic faith. Yet this is the man who, having thus testified to the rival effects of Catholicity and Protestantism on a people, has the effrontery to tell us in the “Revival of Romanism” that “If by this [conversions] or any other cause the Catholic Church anywhere recovers her ascendency, she will again exhibit the detestable features which have invariably attended her supremacy. Her rule will once more be found incompatible either with justice or intellectual growth, and our children will be forced to recover by some fresh struggle the ground which our forefathers conquered for us, and which we by our pusillanimity surrendered” (p. 103). With his own testimony before us we may well ask in amazement, Of which church is he writing? It would seem as though Heaven, which through all ages has looked down upon and permitted martyrdom for the faith, had in this instance called upon, not a tender virgin or a strong youth, not an old man tottering into the grave or an innocent child, to step into the arena and offer up their life and blood for the cause of Christ, but a whole people. And the martyrdom of this people was not for a day or an hour; it was the slow torture of centuries. A legacy of martyrdom was “bequeathed from bleeding sire to son.” Life was hopeless to the
- 54. Irish people under the Penal Laws; the world a wide prison; the earth a grave. They could only lift their eyes and hearts to heaven and wait patiently for merciful death to come. This was the supreme test of faith to a noble and passionate race, as it was faith’s supremest testimony. No work of the saints, no writings of the fathers, no Heaven-illumined mind ever brought to the aid of faith stronger reason for conviction than this. As words pale before deeds, as the blood of a martyr speaks more loudly to men, and cries more clamorously to heaven, than all that divine philosophy can utter or inspired poet sing, so the attitude of the Irish people, so opposed to all the instincts of their quick and passionate nature, bore the very noblest testimony to the reality of the Christian religion. A world looked down into that dark arena and waited for some sign of faltering in the victim, for some sign of pity in the persecutor. Neither came. The victim refused to die or sacrifice to the gods; the persecutor to relent. The struggle ended at length through the sheer weariness of the latter, and brighter times came because darker could not be devised. Faith conquered. The Irish people arose from its grave, and at once spread abroad over the world to preach the Gospel and to plant the church which for two centuries it had watered with its blood. The Act of Catholic Emancipation was the first real sign of resurrection, and that was only passed in 1829. So much for Protestantism having “ceased to be aggressive after the middle of the seventeenth century.” How aggressive are certain Protestant powers to-day all men know. Another thing happened to Protestantism after the middle of the seventeenth century: “It no longer produced men conspicuously nobler and better than Romanism,” says Mr. Froude, “and therefore it no longer made converts. As it became established, it adapted itself to the world, laid aside its harshness, confined itself more and more to the enforcement of particular doctrines” (of no doctrines in particular, we should be inclined to say), “and abandoned, at first tacitly and afterward deliberately, the pretence to interfere with private life or practical business.”
- 55. In plainer words, Protestantism, having secured its place in this world, left the next world to take care of itself, and left men free to go to the devil or not just as they pleased. Mr. Froude faithfully pictures the result: “Thus Protestant countries are no longer able to boast of any special or remarkable moral standard; and the effect of the creed on the imagination is analogously impaired. Protestant nations show more energy than Catholic nations because the mind is left more free, and the intellect is undisturbed by the authoritative instilment of false principles” (p. 111). This strikes us as a very easy manner of begging a very important question. However, we are less concerned now with Mr. Froude’s Catholics than with his Protestants. “But,” he goes on, “Protestant nations have been guilty, as nations, of enormous crimes. Protestant individuals, who profess the soundest of creeds, seem, in their conduct, to have no creed at all, beyond a conviction that pleasure is pleasant, and that money will purchase it. Political corruption grows up; sharp practice in trade grows up— dishonest speculations, short weights and measures, and adulteration of food. The commercial and political Protestant world, on both sides of the Atlantic, has accepted a code of action from which morality has been banished; and the clergy have for the most part sat silent, and occupy themselves in carving and polishing into completeness their schemes of doctrinal salvation. They shrink from offending the wealthy members of their congregation.” (We believe we heard concordant testimony to this from distinguished members of the late Protestant Episcopalian Convention and Congress.) “They withdraw into the affairs of the other world, and leave the present world to the men of business and the devil.” Mr. Froude having thus placidly handed Protestantism over to the devil, we might as well leave it there, as the devil is proverbially reported to know and take care of his own. And certainly, if Protestantism be only half what Mr. Froude depicts it, it is the devil’s, and a more active and fruitful agent of evil he could not well desire. One thing is beyond dispute: if Protestantism be what so ardent an advocate as Mr. Froude says it is, it is high time for a
- 56. change. It is time for some one or something to step in and dispute the devil’s absolute sovereignty. If this is the result of the Protestant mind being “left more free” than the Catholic, the sooner such freedom is curtailed the better. It is the freedom of lethargy and license which has yielded up even the little that it had of real freedom and truth to its own child, Materialism, the modern name for paganism. “They” (the Protestant clergy), says Mr. Froude, “have allowed the Gospel to be superseded by the new formulas of political economy. This so-called science is the most barefaced attempt that has ever yet been openly made on this earth to regulate human society without God or recognition of the moral law. The clergy have allowed it to grow up, to take possession of the air, to penetrate schools and colleges, to control the actions of legislatures, without even so much as opening their lips in remonstrance.” Yes, because they had nothing better to offer in its place. And this Mr. Froude advances with much truth as one of the causes of the “Revival of Romanism”: “I once ventured,” he tells us, “to say to a leading Evangelical preacher in London that I thought the clergy were much to blame in these matters. If the diseases of society were unapproachable by human law, the clergy might at least keep their congregations from forgetting that there was a law of another kind which in some shape or other would enforce itself. He told me very plainly that he did not look on it as part of his duty. He could not save the world, nor would he try. The world lay in wickedness, and would lie in wickedness to the end. His business was to save out of it individual souls by working on their spiritual emotions, and bringing them to what he called the truth. As to what men should do or not do, how they should occupy themselves, how and how far they might enjoy themselves, on what principles they should carry on their daily work—on these and similar subjects he had nothing to say. “I needed no more to explain to me why Evangelical preachers were losing their hold on the more robust intellects, or why Catholics, who at least offered something which at intervals might remind men that they had souls, should have power to win away into their fold many a tender conscience which needed detailed support and guidance” (pp. 112– 113).
- 57. One ray of light in the universal darkness now enshrouding Protestantism shines before the eyes of Mr. Froude. It falls on the present German Empire. Here at least the weary watchman crying out the hours of heaven may call “All is well” to the sleepers. Here Protestantism had its true birth; here it finds its true home. In this blessed land lies hope and salvation for a lost world. But the picture is so graphic that we give it in Mr. Froude’s own words: “As the present state of France,” he says, “is the measure of the value of the Catholic revival, so Northern Germany, spiritually, socially, and politically, is the measure of the power of consistent Protestantism. Germany was the cradle of the Reformation. In Germany it moves forward to its manhood; and there, and not elsewhere, will be found the intellectual solution of the speculative perplexities which are now dividing and bewildering us” (pp. 130–131). “Luther was the root in which the intellect of the modern Germans took its rise. In the spirit of Luther this mental development has gone forward ever since. The seed changes its form when it develops leaves and flowers. But the leaves and flowers are in the seed, and the thoughts of the Germany of to-day lay in germs in the great reformer. Thus Luther has remained through later history the idol of the nation whom he saved. The disputes between religion and science, so baneful in their effects elsewhere, have risen into differences there, but never into quarrels” (p. 132). “Protestant Germany stands almost alone, with hands and head alike clear. Her theology is undergoing change. Her piety remains unshaken. Protestant she is, Protestant she means to be.... By the mere weight of superior worth the Protestant states have established their ascendency over Catholic Austria and Bavaria, and compel them, whether they will or not, to turn their faces from darkness to light.[106] ... German religion may be summed up in the word which is at once the foundation and the superstructure of all religion—Duty! No people anywhere or at any time have understood better the meaning of duty; and to say that is to say all” (pp. 134–135). These glowing periods are very tempting to the critic; but it is a mark of cruelty and savagery to gloat over an easy prey. We forbear all verbal criticism, then, and simply deny in toto the truth of Mr. Froude’s statement. It is so very wrong that we can only think he wrote from his imagination—a weakness from which he suffers
- 58. oftenest when he wishes most to be effective. Had he searched the world he could not have found a worse instance to prove his point than North Germany. Prussia is the leading North German and Protestant state, and in various passages Mr. Froude shows that he takes it as his beau- ideal of a Protestant power. How stands Protestantism in Prussia to- day? The indications for more than a quarter of a century past have been that Protestantism in Prussia was little more than the shadow of a once mighty name. These indications have become more marked of late years, especially since the consolidation of the new German Empire. Earnest German Protestants are continually deploring the fact; the press proclaims it; the Protestant ministers avow it, and all the world knew of it, save, apparently, Mr. Froude. “Protestantism in Prussia” formed the subject of a letter from the Berlin correspondent of the London Times as recently as Sept. 7, 1877. His testimony on such a subject could scarcely be called in question, but even if it could be the facts narrated speak for themselves. “Forty years ago,” he says, “the clergy of the Established Church of this country, including the leading divines and the members of the ecclesiastical government, almost to a man were under the influence of free-thinking theories. “It was the time when German criticism first undertook to dissect the Bible. History seemed to have surpassed theology, and divines had recourse to ‘interpreting’ what they thought they could no longer maintain according to the letter. The movement extended from the clergy to the educated classes, gradually reaching the lower orders, and ultimately pervaded the entire nation. At this juncture atheism sprang forward to reap the harvest sown by latitudinarians. Then reaction set in. The clergy reverted to orthodoxy, and their conversion to the old faith happening to coincide with the return of the government to political conservatism, subsequent to the troublous period of 1848, the stricter principles embraced by the cloth were systematically enforced by consistory and school....
- 59. “The clergy turned orthodox twenty-five years ago; the laity did not. The servants of the altar, having realized the melancholy effect of opposite tenets, resolutely fell back upon the ancient dogmas of Christianity; the congregations declined to follow suit. Hence the few ‘liberal’ clergymen remaining after the advent of the orthodox period had the consolation of knowing themselves to be in accord, if not with their clerical brethren, at least with the majority of the educated, and, perhaps, even the uneducated, classes.” He proceeds to mention various cases of prominent Lutheran clergymen who denied the divinity of Christ, or other doctrines equally necessary to be maintained by men professing to be Christians, and of the unsuccessful attempts made to silence them. As the correspondent says “irreverent liberal opinion on the case is well reflected in an article in the Berlin Volks-Zeitung,” which is so instructive that we quote it for the especial benefit of Mr. Froude: “As long as Protestant clergymen are appointed by provincial consistories officiating in behalf of the crown our congregations will have to put up with any candidates that may be forced upon them. They may, perhaps, be allowed to nominate their pastors, but they will be impotent to exact the confirmation of their choice from the ecclesiastical authorities. Nor do we experience any particular curiosity as to the result of the inquiry instituted against Herr Hossbach. In matters of this delicate nature judicious evasions have been too often resorted to by clever accused, and visibly favored by ordained judges of the faith, for us to care much for the result of the suit opened. A sort of fanciful and imaginative prevarication has always flourished in theological debate, and the old artifice, it is to be foreseen, will be employed with fresh versatility in the present instance. Should the election of Herr Hossbach be confirmed, the consistorial decree will be garnished with so many ‘ifs’ and ‘althoughs’ that the brilliant ray of truth will be dimmed by screening assumptions, like a candle placed behind a colored glass. Similarly, should the consistory decline to ratify the choice of the vestry, the refusal is sure to be rendered palatable by the employment of particularly mild and euphonious language. In either case the triumph of the victorious party will be but half a triumph.... It is not a little remarkable that the Protestant Church in this country should be kept under the control of superimposed authorities, while Roman Catholics and Jews are free to preach what they like. The power of the Catholic hierarchy has been broken by the new laws. Catholic clergymen
- 60. deviating from the approved doctrine of the Church are protected by the Government from the persecution of their bishops. Catholic congregations are positively urged and instigated to profit by the privileges accorded them, and assert their independence against bishop and priest. Jewish rabbis, too, are free to disseminate any doctrine without being responsible for their teaching to spiritual or secular judges. Only Protestant congregations enjoy the doubtful advantage of having the election of their clergy controlled, and the candor of their clergy made the theme of penal inquiry.... And yet Protestant congregations have a ready means of escape at their disposal. Let them leave the church, and they are free to elect whomsoever they may choose as their minister. As it is, the indecision of the congregations maintains the status quo by forcing liberal clergymen into the dogmatic straight-waistcoat of the consistories.” “In the above argument one important fact is overlooked,” says the Times’ correspondent. “Among the liberals opposed to the consistories there are many atheists, but few sufficiently religious to care for reform. Hence the course taken by the consistories may be resented, but the preaching of the liberal clergy is not popular enough to create a new denomination or to compel innovation within the pale of the church. The fashionable metaphysical systems of Germany are pessimist.” A week previous to the date of this letter the Lutheran pastors held their annual meeting at Berlin. The Rev. Dr. Grau, who is referred to as “a distinguished professor of theology,” speaking of the task of the clergy in modern times—certainly a most important subject for consideration—said: “These are serious times for the church. The protection of the temporal power is no longer awarded to us to anything like the extent it formerly was. The great mass of the people is either indifferent or openly hostile to doctrinal teaching. Not a few listen to those striving to combine Christ with Belial, and to reconcile redeeming truth with modern science and culture. There are those who dream of a future church erected on the ruins of the Lutheran establishment, which by these enterprising neophytes is already regarded as dead and gone.”
- 61. “The meeting,” observes the correspondent, “by passing the resolutions proposed by Dr. Grau, endorsed the opinions of the principal speaker.” And he adds: “While giving this unmitigated verdict upon the state of religion among the people, the meeting displayed open antagonism to the leading authorities of the church. To the orthodox pastors the sober and sedative policy pursued by the Ober Kirchen Rath is a dereliction even more offensive than the downright apostasy of the liberals. To render their opposition intelligible the change that has recently supervened in high quarters should be adverted to in a few words. Soon after his accession to the throne the reigning sovereign, in his capacity as summus episcopus, recommended a lenient treatment of liberal views. Though himself strictly orthodox, as he has repeatedly taken occasion to announce, the emperor is tolerant in religion, and too much of a statesman to overlook the undesirable consequences that must ensue from permanent warfare between church and people. He therefore appointed a few moderate liberals members of the supreme council, accorded an extensive degree of self-government to the synods, at the expense of his own episcopal prerogative, and finally sanctioned civil marriage and ‘civil baptism,’ as registration is sarcastically called in this country, to the intense astonishment and dismay of the orthodox. The last two measures, it is true, were aimed at the priests of the Roman Catholic Church, who were to be deprived of the power of punishing those of their flock siding with the state in the ecclesiastical war; but, as the operation of the law could not be restricted to one denomination, Protestants were made amenable to a measure which, to the orthodox among them, was quite as objectionable as to the believing adherents of the Pope. The supreme council of the Protestant Church, having to approve these several innovations adopted by the crown, gradually accustomed itself to regard compromise and bland pacification as one of the principal duties imposed upon it.” The correspondent ends his letter thus: “When all was over orthodoxy was at feud with the people as well as with the authoritative guardians of the church. Yet neither people nor guardians remonstrated. For opposite reasons both were equally convinced they could afford to ignore the charges made.” So important was the letter that the London Times made it the subject of an editorial article, wherein it speaks of “the singular
- 62. revival of theological and ecclesiastical controversy, which is observable in all directions,” having “at last reached the slumbering Protestantism of Prussia.” It confesses that “The state of things as described by our correspondent is certainly a very anomalous one. The Prussian Protestant Church has, of late years at least, had but little hold on the respect and affections of the great majority of the people; they are at best but indifferent to it when they are not actively hostile. We are not concerned to investigate the causes of this lack of popularity; we are content to take it as a fact manifest to all who know the country and acknowledged by all observers alike.” “German Protestantism was a power and an influence,” it says, “To which the modern world is deeply indebted, and with which, now that ultramontanism is triumphant in the Church of Rome and priestcraft is again striving in all quarters to exert its sway, the friends of freedom and toleration can ill afford to dispense. There is no more ominous sign in the history of an established church than a divorce between intelligence and orthodoxy. This is what, to all appearances, has happened in Prussia.” We could corroborate this by abundance of testimony from all quarters; but surely the evidence here given is sufficient to convince any man of the deplorable state of Protestantism in Prussia. Why Mr. Froude should have chosen that country of all others for his Protestant paradise we cannot conceive, unless on the ground that he is Mr. Froude. “The world on one side, and Popery on the other,” he says, “are dividing the practical control over life and conduct. North Germany, manful in word and deed, sustains the fight against both enemies and carries the old flag to victory. A few years ago another Thirty Years’ War was feared for Germany. A single campaign sufficed to bring Austria on her knees. Protestantism, as expressed in the leadership of Prussia, assumed the direction of the German Confederation” (pp. 135–136). And whither does this leadership tend? To the devil, if the London Times, if Dr. Grau, if every observant man who has written or spoken on this subject, is to be believed. The only religion in
- 63. Prussia to-day is the Catholic; Protestantism has yielded to atheism or nothingism. The persecution has only proved and tempered the Catholic Church; not even a strong and favoring government can infuse a faint breath of life into the dead carcase of Prussian Protestantism. It is much the same story all the world over. Mr. Froude sees clearly enough what is coming. Protestantism as a religious power is dead. It has lost all semblance of reality. It had no religious reality from the beginning. It will still continue to be used as an agent by political schemers and conspirators; but in the fight between religion and irreligion it is of little worth. The fight is not here, but where Mr. Froude rightly places it—between the irreligious world and Catholicity, which “are dividing the practical control over life and conduct.” And thus heresies die out; they expire of their own corruption. Their very offspring rise up against them. Their children cry for bread and they give them a stone. The fragments of truth on which they first build are sooner or later crushed out by the great mass of falsehood. The few good seeds are choked up by the harvest of the bad, and only the ill weeds thrive, until all the space around them is desolate of fruit or light or sweetness, or anything fair under heaven. Then comes the husbandman in his own good time, and curses the barren fig-tree and clears the desolate waste. It will be with Protestantism as it has been with all the heresies; Christians will wonder, and the time would seem not to be very far distant when they will wonder that Protestantism ever should have been. It will go to its grave, the same wide grave that has swallowed up heresy after heresy. Gnosticism, Arianism, Pelagianism, Nestorianism, Monophysitism, Protestantism, all the isms, are children of the same family, live the same life, die the same death. The everlasting church buries them all, and no man mourns their loss.
- 65. A RAMBLE AFTER THE WAITS. “Christmas comes but once a year, So let us all be merry,” saith the old song. And now, as the festal season draws nigh, everybody seems bent on fulfilling the behest to the uttermost. The streets are gay with lights and laughter; the shops are all a-glitter with precious things; the markets are bursting with good cheer. The air vibrates with a babble of merry voices, until the very stars seem to catch the infection and twinkle a thought more brightly. The faces of those you meet beam with joyous expectation; huge baskets on their arms, loaded with good things for the morrow, jostle and thump you at every turn, but no one dreams of being ill- natured on Christmas Eve; mysterious bundles in each hand contain unimagined treasures for the little ones at home. And hark! do you not catch a jingle of distant sleigh-bells, a faint, far-off patter and scrunching of tiny hoofs upon the snow? It is the good St. Nicholas setting out upon his merry round; it is Dasher and Slasher and Prancer and Vixen scurrying like the wind over the house-tops. And high over all—“the poor man’s music”—the merry, merry bells of Yule, the solemn, the sacred bells, peal forth the tidings of great joy. Is it not hard to conceive that the time should have been when Christmas was not? impossible to conceive that any in a Christian land should have wished to do away with it—should have been willing, having had it, ever to forego a festival so fraught with all holy and happy memories? Yet once such men were found, and but little more than two centuries ago. It was on the 24th day of December, 1652—day for ever to be marked with the blackest of black stones, nay, with a bowlder of Plutonian nigritude—that the British House of Commons,
- 66. being moved thereto “by a terrible remonstrance against Christmas day grounded upon divine Scripture, wherein Christmas is called Antichrists masse, and those masse-mongers and Papists who observe it,” and after much time “spent in consultation about the abolition of Christmas day, passed order to that effect, and resolved to sit upon the following day, which was commonly called Christmas day.” Whether this latter resolution was carried into effect we do not know. If so, let us hope that their Christmas dinners disagreed with them horribly, and that the foul fiend Nightmare kept hideous vigil by every Parliamentary pillow. But think of such an atrocious sentiment being heard at all in Westminster! How must the very echoes of the hall have shrunk from repeating that monstrous proposition—how shuddered and fled away into remotest corners and crevices as that “Hideous hum Ran through the arch’d roof in words deceiving”! How must they have disbelieved their ears, and tossed the impious utterance back and forth from one to another in agonized questioning, growing feebler and fainter at each repulse, until their voices, faltering through doubt into dismay, grew dumb with horror! How must “Rufus’ Roaring Hall”[107] have roared again outright with rage and grief over that strange, that unhallowed profanation! What wan phantoms of old-time mummeries and maskings, what dusty and crumbling memories of royal feast and junketing, must have hovered about the heads of those audacious innovators, shrieking at them what unsyllabled reproaches from voiceless lips, shaking at them what shadowy fingers of entreaty or menace! And if the proverb about ill words and burning ears be true, how those crop- ears must have tingled! Within those very walls England’s kings for generations had kept their Christmas-tide most royally with revelry and dance and
- 67. wassail. There Henry III. on New Year’s day, 1236, to celebrate the coronation of Eleanor, his queen, entertained 6,000 of his poorer subjects of all degrees; and there twelve years later, though he himself ate his plum-pudding at Winchester, he was graciously pleased to bid his treasurer “fill the king’s Great Hall from Christmas day to the Day of Circumcision with poor people and feast them.” There, too, at a later date Edward III. had for sauce to his Christmas turkey—not to mention all sorts of cates and confections, tarts and pasties of most cunning device, rare liquors and spiced wines—no less than two captive kings, to wit, David of Scotland and John of France. Poor captive kings! Their turkey—though no doubt their princely entertainer was careful to help them to the daintiest tidbits, and to see that they had plenty of stuffing and cranberry sauce—must have been but a tasteless morsel, and their sweetbreads bitter indeed. Another Scottish king, the first James, of tuneful and unhappy memory, had even worse (pot) luck soon after. Fate, and that hospitable penchant of our English cousins in the remoter centuries for quietly confiscating all stray Scotch princes who fell in their way, as though they had been contraband of war, gave him the enviable opportunity of eating no less than a score of Christmas dinners on English soil. But he seems to have been left to eat them alone or with his jailer in “bowery Windsor’s calm retreat” or the less cheerful solitude of the Tower. It does not appear that either the fourth or the fifth Henry, his enforced hosts, ever asked him to put his royal Scotch legs under their royal English mahogany. Had Richard II. been in the place of “the ingrate and cankered Bolingbroke,” we may be sure that his northern guest would not have been treated so shabbily. In his time Westminster and his two thousand French cooks (shades of Lucullus! what an appetite he must have had, and what a broiling and a baking and a basting must they have kept up among them; the proverb of “busier than an English oven at Christmas” had reason then, at least) were not long left idle; for it was their sovereign’s jovial custom to keep open house in the holidays for as many as ten thousand a day—a comfortable tableful. It was his motto plainly to
- 68. “Be merry, for our time of stay is short.” Such a device, however, the third Richard might have made his own with still greater reason. That ill-used prince, who was no doubt a much better fellow at bottom than it has pleased Master Shakspeare to represent him—if Richmond had not been Queen Bess’ grandpapa, we should like enough have had a different story and altogether less about humps and barking dogs—made the most of a limited opportunity to show what he could do in the way of holiday dinner-giving. The only two Christmases he had to spend as king at Westminster—for him but a royal stage on his way to a more permanent residence at Bosworth Field—he celebrated with extraordinary magnificence, as became a prince “reigning,” says Philip de Comines, “in greater splendor than any king of England for the last hundred years.” On the second and last Christmas of his reign and life the revelry was kept up till the Epiphany, when “the king himself, wearing his crown, held a splendid feast in the Great Hall similar to his coronation.” Wearing his crown, poor wretch! He seems to have felt that his time was short for wearing it, and that he must put it to use while he had it. Already, indeed, as he feasted, rapacious Fortune, swooping implacable, was clawing it with skinny, insatiable claws, estimating its value and the probable cost of altering it to fit another wearer, and thinking how much better it would look on the long head of her good friend Richmond, who had privately bespoken it. No doubt some cold shadow of that awful, unseen presence fell across the banquet-table and poisoned the royal porridge. What need to tell over the long roll of Christmas jollities, whose memory from those historic walls might have pleaded with or rebuked the sour iconoclasts planning gloomily to put an end to all such for ever; how even close-fisted Henry VII.—no fear of his losing a crown, if gripping tight could keep it—feasted there the lord-mayor and aldermen of London on the ninth Christmas of his reign, sitting down himself, with his queen and court and the rest of
- 69. the nobility and gentry, to one hundred and twenty dishes served by as many knights, while the mayor, who sat at a side-table, no doubt, had to his own share no fewer than twenty-four dishes, followed, it is to be feared, if he ate them all, by as many nightmares; how that meek and exemplary Christian monarch, Henry VIII., “welcomed the coming, sped the parting” wife at successive Christmas banquets of as much splendor as the spoils of something over a thousand monasteries could furnish forth;[108] how good Queen Bess, who had her own private reading of the doctrine “it is more blessed to give than to receive,” sat in state there at this festival season to accept the offerings of her loyal lieges, high and low, gentle and simple, from prime minister to kitchen scullion, until she was able to add to the terrors of death by having to leave behind her something like three thousand dresses and some trunkfuls of jewels in Christmas gifts; or what gorgeous revels and masques—Inigo Jones (Inigo Marquis Would-be), Ben Jonson, and Master Henry Lawes (he of “the tuneful and well-measured song”) thereto conspiring—made the holidays joyous under James and Charles. Some ghostly savor of those bygone banquets might, one would think, have made even Praise-God Barebone’s mouth water, and melted his surly virtue into tolerance of other folks’ cakes and ale—what virtue, however ascetic, could resist the onslaught of two thousand French cooks? Some faint, far echo of all these vanished jollities should have won the ear, if not the heart, of the grimmest “saint” among them. Or if they were proof against the blandishments of the world’s people, if they fled from the abominations of Baal, could not their own George Wither move them to spare the cheery, harmless frivolities, the merry pranks of Yule? Jovially as any Cavalier, shamelessly as any Malignant of them all, he sings their praises in his “CHRISTMAS CAROL.
- 70. “So now is come our joyful’st feast, Let every man be jolly; Each room with ivy leaves is drest, And every post with holly. Though some churls at our mirth repine, Round your foreheads garlands twine, Drown sorrow in a cup of wine, And let us all be merry. “Now all our neighbors’ chimneys smoke, And Christmas blocks are burning; Their ovens they with bak’d meats choke, And all their spits are turning. Without the door let sorrow lie; And if for cold it hap to die, We’ll bury’t in a Christmas pye. And evermore be merry. “Now every lad is wondrous trim, And no man minds his labor; Our lasses have provided them A bagpipe and a tabor. Young men and maids, and girls and boys, Give life to one another’s joys; And you anon shall by their noise Perceive that they are merry.... “Now poor men to the justices With capons make their errants; And if they hap to fail of these, They plague them with their warrants: But now they feed them with good cheer, And what they want they take in beer; For Christmas comes but once a year, And then they shall be merry.... “The client now his suit forbears, The prisoner’s heart is eased, The debtor drinks away his cares, And for the time is pleased. Though others’ purses be more fat, Why should we pine or grieve at that? Hang sorrow! care will kill a cat, And therefore let’s be merry....
- 71. d t e e o e et s be e y “Hark! now the wags abroad do call Each other forth to rambling; Anon you’ll see them in the hall, For nuts and apples scrambling. Hark! how the roofs with laughter sound; Anon they’ll think the house goes round, For they the cellar’s depths have found. And there they will be merry. “The wenches with the wassail-bowls About the streets are singing; The boys are come to catch the owls, The wild mare[109] in is bringing. Our kitchen-boy hath broke his box, And to the kneeling of the ox Our honest neighbors come by flocks, And here they will be merry. “Now kings and queens poor sheep-cotes have, And mate with everybody; The honest now may play the knave, And wise men play at noddy. Some youths will now a-mumming go, Some others play at Rowland-boe, And twenty other gambols moe, Because they will be merry. “Then wherefore, in these merry days, Should we, I pray, be duller? No, let us sing some roundelays, To make our mirth the fuller; And, while we thus inspired sing, Let all the streets with echoes ring— Woods and hills and everything Bear witness we are merry.” Or Master Milton, again, Latin secretary to the council, author of the famous Iconoclastes, shield (or, as some would have put it, official scold) of the Commonwealth, the scourge of prelacy and conqueror of Salmasius—he was orthodox surely; yet what of Arcades and Cornus? Master Milton, too, had written holiday
- 72. masques, and, what is more, they had been acted; nay, he had even been known more than once, on no less authority than his worshipful nephew, Master Philips, “to make so bold with his body as to take a gaudy-day” with the gay sparks of Gray’s Inn. Alas! such carnal-minded effusions belonged to the unregenerate days of both these worthy brethren, when they still dwelt in the tents of the ungodly, before they had girded on the sword of Gideon and gone forth to smite the Amalekite hip and thigh. Vainly might the menaced festival look for aid in that direction. So far from saying a word in its favor, they would now have been fiercest in condemnation, if only to cover their early backsliding; if only to avert any suspicion that they still hankered after the fleshpots. Poor Christmas was doomed. So, by act of Parliament, “our joyful’st feast” was solemnly stricken out of the calendar, cashiered from its high pre-eminence among the holidays of the year, and degraded to the ranks of common days. All its quaint bravery of holly-berries and ivy-leaves was stripped from it, its jolly retinue of boars’ heads and wassail- bowls, of Yule-clogs and mistletoe-boughs, of maskers and mummers, of waits and carols, Lords of Misrule and Princes of Christmas, sent packing. Then began “the fiery persecution of poor mince-pie throughout the land; plum-porridge was denounced as mere popery, and roast-beef as anti-Christian.” ’Twas a fatal, a perfidious, a short-lived triumph. The nation, shocked in its most cherished traditions, repudiated the hideous doctrine; the British stomach, deprived of its holiday beef and pudding, so to speak, revolted. The reign of the righteous was speedily at an end. History, with her usual shallowness, ascribes to General Monk the chief part in the Restoration; it was really brought about by that short-sighted edict of the 24th of December, 1652. Charles or Cromwell, king or protector—what cared honest Hodge who ruled and robbed him? But to forego his Christmas porridge—that was a different matter; and Britons never should be slaves. So, just eight years after it had been banished, Christmas was brought back again with manifold rejoicing and bigger wassail-bowls and Yule-clogs than ever; and,
- 73. as if to make honorable amends for its brief exile, the Lord of Misrule himself was crowned and seated on the throne, where, as we all know, to do justice to his office, if he never said a foolish thing he never did a wise one. And from that time to this Christmas has remained a thoroughly British institution, as firmly entrenched in the national affections, as generally respected, and perhaps as widely appreciated as Magna Charta itself. Sit on Christmas day! A British Parliament now would as soon think of sitting on the Derby day. To how many of their constituents have the two festivals any widely differing significance perhaps it would be wise not to inquire too closely. Each is a holiday—that is, a day off work, a synonym for “a good time,” a little better dinner than usual, and considerably more beer. Like the children, “they reflect nothing at all about the matter, nor understand anything in it beyond the cake and orange.” “La justice elle-même,” says Balzac, “se traduit aux yeux de la halle par le commissaire—personage avec lequel elle se familiarise.” His epigram the author of Ginx’s Baby may translate for us—English epigrams, like English plays, being for the most part matter of importation free of duty; e.g., that famous one in Lothair about the critic being a man who has failed in literature or art, another consignment from Balzac—when he makes Ginx’s theory of government epitomize itself as a policeman. So Ginx’s notion of Christmas, we suspect, is apt to be beef and beer and Boxing-night —with perhaps a little more beer. Certainly the attachment of the British public to these features of the day—we are considering it for the moment in the light in which a majority of non-Catholics look upon it, apparently, as a merely social festival, and not at all in its religious aspect (though to a Catholic, of course, the two are as indistinguishably blended as the rose and the perfume of the rose)—has never been shaken. If one may judge from a large amount of the English fiction which at this season finds its way to the American market—and the novels of to- day, among a novel-reading people, are as straight and sure a guide to its heart as were ever its ballads in the time of old Fletcher
- 74. of Saltoun—if one may judge from much of English Christmas literature, these incidents of the day are, if not the most important, certainly the most prominent and popular. What we may call the Beef and Beer aspect of the season these stories are never tired of glorifying and exalting. Dickens is the archpriest of this idolatry, which, indeed, he in a measure invented, or at least brought into vogue; and his Christmas Stories, as most of his stories, fairly reek with the odors of the kitchen and the tap-room. Material comfort, and that, too, usually of a rather coarse kind, is the universal theme, and even the charity they are supposed to inculcate can scarcely be called a moral impulse, so much as the instinct of a physical good-nature, well-fed and content with itself and the world —of a good-humored selfishness willing to make others comfortable, because thereby it puts away from itself the discomfort of seeing them otherwise. It is a kind of charity which, in another sense than that of Scripture, has to cover a multitude of sins. One may say this of Dickens, without at all detracting from his many great qualities as a writer, that he has done more, perhaps, than any other writer to demoralize and coarsen the popular notion of what Christmas is and means; to make of his readers at best but good-humored pagans with lusty appetites for all manner of victuals and an open-handed readiness to share their good things with the first comer. These are no doubt admirable traits; but one gets a little tired of having them for ever set forth as the crown and completion of Christian excellence, the sum and substance of all that is noble and exalted in the sentiment of the season. Let us enjoy our Christmas dinner by all means; let the plum-pudding be properly boiled and the turkey done to a turn, and may we all have enough to spare a slice or two for a poorer neighbor! But must we therefore sit down and gobble turkey and pudding from morning till night? Should we hang up a sirloin and fall down and worship it? Is that all that Christmas means? Turn from the best of these books to this exquisite little picture of Christmas Eve in a Catholic land:
- 75. “Christmas is come—the beautiful festival, the one I love most, and which gives me the same joy as it gave the shepherds of Bethlehem. In real truth, one’s whole soul sings with joy at this beautiful coming of God upon earth—a coming which here is announced on all sides of us by music and by our charming nadalet[110] Nothing at Paris can give you a notion of what Christmas is with us. You have not even the midnight Mass. We all of us went to it, papa at our head, on the most perfect night possible. Never was there a finer sky than ours was that midnight —so fine that papa kept perpetually throwing back the hood of his cloak, that he might look up at the sky. The ground was white with hoar-frost, but we were not cold; besides, the air, as we met it, was warmed by the bundles of blazing torchwood which our servants carried in front of us to light us on our way. It was delightful, I do assure you; and I should like you to have seen us there on our road to church, in those lanes with the bushes along their banks as white as if they were in flower. The hoar-frost makes the most lovely flowers. We saw a long spray so beautiful that we wanted to take it with us as a garland for the communion-table, but it melted in our hands; all flowers fade so soon! I was very sorry about my garland; it was mournful to see it drop away and get smaller and smaller every minute.” It is Eugénie de Guérin who writes thus—that pure and delicate spirit so well fitted to feel and value all that is beautiful and touching in this most beautiful and touching service of the church. To come from the one reading to the other is like being lifted suddenly out of a narrow valley to the free air and boundless views of a mountain-top; like coming from the gaslight into the starlight; it is like hearing the song of the skylark after the twitter of the robin —a sound pleasant and cheery enough in itself, but not elevating, not inspiring, not in any way satisfying to that hunger after ideal excellence which is the true life of the spirit, and which strikes the true key-note of this festal time. But Eugénie de Guérin is perhaps too habitual a dweller on those serene heights to furnish a fair comparison; let us take a homelier picture from a lower level. It is still in France; this time in Burgundy, as the other was in Languedoc: “Every year, at the approach of Advent, people refresh their memories, clear their throats, and begin preluding, in the long evenings
- 76. by the fireside, those carols whose invariable and eternal theme is the coming of the Messias. They take from old pamphlets little collections begrimed with dust and smoke, ... and as soon as the first Sunday of Advent sounds they gossip, they gad about, they sit together by the fireside, sometimes at one house, sometimes at another, taking turns in paying for the chestnuts and white wine, but singing with one common voice the praises of the Little Jesus. There are very few villages, even, which during all the evenings of Advent do not hear some of these curious canticles shouted in their streets to the nasal drone of bagpipes. “More or less, until Christmas Eve, all goes on in this way among our devout singers, with the difference of some gallons of wine or some hundreds of chestnuts. But this famous eve once come, the scale is pitched upon a higher key; the closing evening must be a memorable one.... The supper finished, a circle gathers around the hearth, which is arranged and set in order this evening after a particular fashion, and which at a later hour of the night is to become the object of special interest to the children. On the burning brands an enormous log has been placed; ... it is called the Suche (the Yule-log). ‘Look you,’ say they to the children, ‘if you are good this evening Noel will rain down sugar- plums in the night.’ And the children sit demurely, keeping as quiet as their turbulent little natures will permit. The groups of older persons, not always as orderly as the children, seize this good opportunity to surrender themselves with merry hearts and boisterous voices to the chanted worship of the miraculous Noel. For this final solemnity they have kept the most powerful, the most enthusiastic, the most electrifying carols. “This last evening the merry-making is prolonged. Instead of retiring at ten or eleven o’clock, as is generally done on all the preceding evenings, they wait for the stroke of midnight; this word sufficiently proclaims to what ceremony they are going to repair. For ten minutes or a quarter of an hour the bells have been calling the faithful with a triple- bob-major; and each one, furnished with a little taper streaked with various colors (the Christmas candle), goes through the crowded streets, where the lanterns are dancing like will-o’-the-wisps at the impatient summons of the multitudinous chimes. It is the midnight Mass.” There you have fun, feasting, and frolic, as, indeed, there may fitly be to all innocent degrees of merriment, on the day which brought redemption to mankind. But there is also, behind and pervading all this rejoicing and harmless household gayety, the
- 77. religious sentiment which elevates and inspires it, which chastens it from commonplace and grossness, which gives it a meaning and a soul. The English are fond of calling the French an irreligious people, because French literature, especially French fiction, from which they judge, takes its tone from Paris, which is to a great extent irreligious. But outside of the large cities, if a balance were struck on this point between the two countries, it would scarcely be in favor of England. This, however, by way of episode and as a protest against this grovelling, material treatment of the most glorious festival of the Christian year. As we were about to say when interrupted, though Christmas regained its foothold as a national holiday at the Restoration, it came back sadly denuded of its following and shorn of most of its old-time attractions. So it fared in old England. In New England it can scarcely be said ever to have won a foothold at all, or at best no more than a foothold and a sullen toleration. Almost the first act of those excellent Pilgrim Fathers who did not land at Plymouth Rock was to anticipate by thirty years or so the action of their Parliamentary brethren at home in abolishing the sacred anniversary, which must, indeed, have been a tacit rebuke to the spirit of their creed. They landed on the 16th of December, and “on ye 25th day,” writes William Bradford, “began to erect ye first house for comone use to receive them and their goods.” And lest this might seem an exception made under stress, we find it recorded next year that “on ye day caled Christmas day ye Gov’r caled them out to worke.” So it is clear New England began with a calendar from which Christmas was expunged. In New England affections Thanksgiving day replaces it—an “institution” peculiarly acceptable, we must suppose, to the thrift which can thus wipe out its debt of gratitude to Heaven by giving one day for three hundred and sixty-four—liquidating its liabilities, so to speak, at the rate of about three mills in the dollar. In the Middle States and in the South the day has more of its time-old observance, but neither here nor elsewhere may we hope to encounter many of the quaint and cheery customs with which our fathers loved to honor it, and which
- 78. made it for them the pivot of the year. Wither has told us something of these; let a later minstrel give us a fuller picture of what Merry Christmas was in days of yore:
- 79. “And well our Christian sires of old Loved, when the year its course had rolled, And brought blithe Christmas back again, With all its hospitable train. Domestic and religious rite Gave honor to the holy night: On Christmas Eve the bells were rung; On Christmas Eve the Mass was sung; That only night of all the year Saw the stoled priest the chalice rear. The damsel donned her kirtle sheen; The hall was dressed with holly green; Forth to the wood did merry men go To gather in the mistletoe. Then opened wide the baron’s hall To vassals, tenants, serf, and all. The heir, with roses in his shoes, That night might village partner choose; The lord, underogating, share The vulgar game of ‘post and pair.’ All hailed with uncontrolled delight, And general voice, the happy night That to the cottage, as the crown, Brought tidings of salvation down. The fire, with well-dried logs supplied, Went roaring up the chimney wide; The huge hall-table’s oaken face, Scrubbed till it shone, the day to grace, Bore then upon its massive board No mark to part the squire and lord. Then was brought in the lusty brawn By old blue-coated serving-man; Then the grim boar’s head frowned on high, Crested with bays and rosemary.... The wassail round in good brown bowls, Garnished with ribbons, blithely trowls. There the huge sirloin reeked; hard by Plum-porridge stood and Christmas pye. Then came the merry masquers in And carols roared with blithesome din; If unmelodious was the song, It was a hearty note and strong. Wh li t i th i i
- 80. Who lists may in their mumming see Traces of ancient mystery.... England was merry England then— Old Christmas brought his sports again; ’Twas Christmas broached the mightiest ale; ’Twas Christmas told the merriest tale; A Christmas gambol oft would cheer A poor man’s heart through half the year.” Let Herrick supplement the picture with his “CEREMONIES FOR CHRISTMASSE. “Come, bring with a noise, My merrie, merrie boyes, The Christmas log to the firing; While my good dame, she Bids ye all be free And drink to your hearts’ desiring. “With the last yeeres brand Light the new block, and For good successe in his spending On your psaltries play, That sweet luck may Come while the log is a-teending. “Drink now the strong beere, Cut the white loafe here, The while the meate is a-shredding For the rare mince-pie, And the plums stand by To fill the paste that’s a-kneading.” Does the picture please you? Would you fain be a guest at the baron’s table, or lend a hand with jovial Herrick to fetch in the mighty Yule-log? Are you longing for a cut of that boar’s head or a draught of the wassail, or curious to explore the contents of that mysterious “Christmas pye,” which seems to differ so much from all
- 81. other pies that it has to be spelled with a y? Well, well, we must not repine. Fate, which has denied us these joys, has given us compensations. No doubt the baron, for all his Yule-logs, would sometimes have given his baronial head (when he happened to have a cold in it) for such a fire—let it be of sea-coal in a low grate and the curtains drawn—as the reader and his humble servant are this very minute toasting their toes at. Those huge open fireplaces are admirably effective in poetry, but not altogether satisfactory of a cold winter’s night, when half the heat goes up the chimney and all the winds of heaven are shrieking in through the chinks in your baronial hall and playing the very mischief with your baronial rheumatism. Or do we believe that boar’s head was such a mighty fascinating dish after all, or much, if anything, superior to the soused pig’s head with which good old Squire Bracebridge replaced it? No, every age to its own customs; we may be sure that each finds out what is best for it and for its people. Yet one custom we do begrudge a little to the past, or rather to the other lands where it still lingers here and there in the present. That is the graceful and kindly custom of the waits. These were Christmas carols, as the reader no doubt knows, chanted by singers from house to house in the rural districts during the season of Advent. In France they were called noels, and in Longfellow’s translation of one of these we may see what they were like:
- 82. “I hear along our street Pass the minstrel throngs; Hark! they play so sweet. On their hautboys, Christmas songs! Let us by the fire Ever higher Sing them till the night expire!... “Shepherds at the grange Where the Babe was born Sang with many a change Christmas carols until morn. Let us, etc. “These good people sang Songs devout and sweet; While the rafters rang, There they stood with freezing feet. Let us, etc. “Who by the fireside stands Stamps his feet and sings; But he who blows his hands Not so gay a carol brings. Let us, etc.” In some parts of rural England, too, the custom is still to some extent kept up, and the reader may find a pleasant, and we dare say faithful, description of it in a charming English story called Under the Greenwood Tree, by Mr. Thomas Hardy, a writer whose closeness of observation and precision and delicacy of touch give him a leading place among the younger writers of fiction. Very pleasant, we fancy, it must be of a Christmas Eve when one is, as aforesaid, toasting one’s toes at the fire over a favorite book, or hanging up the children’s stockings, let us say, or peering through the curtains out over the moonlit snow, and wondering how cold it is out-doors with that little perfunctory shiver which is
- 83. comfort’s homage to itself—there should always be snow upon the ground at Christmas, for then Nature “With speeches fair Woos the gentle air To hide her guilty front with innocent snow”; but let us have no wind, since “Peaceful was the night Wherein the Prince of Light His reign of peace upon the world began. The winds, with wonder whist, Smoothly the waters kist, Whispering new joys to the wild ocean, Who now hath quite forgot to rave, While birds of calm sit brooding on the charméd wave”— at such a time, we say, it would be pleasant to hear the shrill voices of the Waits cleaving the cold, starlit air in some such quaint old ditty as the “Cherry-tree Carol” or “The Three Ships.” No doubt, too, would we but confess it, there would come to us a little wicked enhancement of pleasure in the reflection that the artists without were a trifle less comfortable than the hearer within. That rogue Tibullus had a shrewd notion of what constitutes true comfort when he wrote, Quam juvat immites ventos audire cubantem—which, freely translated, means, How jolly it is to sit by the fireside and listen to other fellows singing for your benefit in the cold without! But that idea we should dismiss as unworthy, and even try to feel a little uncomfortable by way of penance; and then, when their song was ended, and we heard their departing footsteps scrunching fainter and fainter in the snow, and their voices dying away until they became the merest suggestion of an echo, we should perhaps find—for these are to be ideal Waits—that their song had left
- 84. behind it in the listener’s soul a starlit silence like that of the night without, but the stars should be heavenly thoughts. These are ideal Waits; the real ones might be less agreeable or salutary. But have we far to look for such? Are there not on the shelves yonder a score of immortal minstrels only waiting our bidding to sing the sacred glories of the time? Shall we ask grave John Milton to tune his harp for us, or gentle Father Southworth, or impassioned Crashaw, or tender Faber? These are Waits we need not scruple to listen to, nor fail to hear with profit. Milton’s Ode on the Nativity is, no doubt, the finest in the language. Considering the difficulties of a subject to which, short of inspiration, it is next to impossible to do any justice at all, it is very fine indeed. It is not all equal, however; there are in it stanzas which remind one that he was but twenty-one when he wrote it. Yet other stanzas are scarcely surpassed by anything he has written.
- 85. “Yea, Truth and Justice then Will down return to men, Orb’d in a rainbow; and, like glories wearing Mercy will sit between, Thron’d in celestial sheen, With radiant feet the tissued clouds down steering, And heaven, as at some festival, Will open wide the gates of her high palace hall. “But wisest Fate says, No, It must not yet be so; The Babe yet lies in smiling infancy That on the bitter cross Must redeem our loss, So both himself and us to glorify; Yet first to those ychained in sleep The wakeful trump of doom must thunder thro’ the deep, “With such a horrid clang As on Mount Sinai rang, While the red fire and smould’ring clouds out-brake. The aged earth, aghast With terror of that blast, Shall from the surface to the centre shake; When at the world’s last session The dreadful Judge in middle air shall spread his throne. ————— “The oracles are dumb; No voice or hideous hum Runs through the arched roof in words deceiving. Apollo from his shrine Can no more divine, With hollow shriek the steep of Delphos leaving. No nightly trance or breathèd spell Inspires the pale-eyed priest from the prophetic cell. “The lonely mountains o’er, And the resounding shore, A voice of weeping heard and loud lament. From haunted spring, and dale Edg’d with poplar pale, The parting genius is with sighing sent.
- 86. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com