![viii Theory of Stabilization for Linear Boundary Control Systems
observation/control, and are connected not only with functional analysis but
also non-harmonic analysis and classical Fourier analysis, etc. The monograph
consists of eight chapters which strongly reflects the author’s works over thirty
years except for Chapter 2: Some were taught in graduate courses at Kobe
University. The organization of the monograph is stated as follows: It begins
with the linear tabilization problem of finite dimension in Chapter 1.
Finite-dimensional models constitute pseudo-internal structures of pdes.
Although the problem is entirely solved by W. M. Wonham in 1967 [70], we
develop a much easier new approach, which has never appeared even among the
community of finite-dimensional control theory: It is based on Sylvester’s
equation. Infinite-dimensional versions of the equation appear in later chapters
as an essential tool for stabilization problems throughout the monograph.
Chapter 2 is a brief introduction of basic results on standard elliptic differential
operators L and related Sobolev spaces necessary for our control problems:
These results are well known among the pdes community, but proofs of some
results are stated for the readers’ convenience. As for results requiring much
preparation we only provide some references instead of proofs. In Chapters 3
through 7, the main topics discussed are, where stabilization problems of linear
parabolic systems are successfully solved in the boundary observation/boundary
feedback scheme. The elliptic operator L is derived from a pair of standard (but
general enough) differential operators (L , τ), and forms the coefficient of our
control systems, where L denotes a uniformly elliptic differential operator and
τ a boundary operator. The operator L is sectorial, and thus −L turns out to be
an infinitesimal generator of an analytic semigroup. One of important issues is
certainly the existence or non-existence of Riesz bases associated with L: When
an associated Riesz basis exists, a sequence of finite-dimensional approximation
models of the original pde is quantitatively justified, so that the control laws
based on the approximated finite-dimensional models effectively works. There
is an attempt to draw out a class of elliptic operators with Riesz bases (see the
footnote in the beginning of Chapter 4). However, is the class of pdes admitting
associated Riesz bases general enough or much narrower than expected? We do
not have a satisfactory solution to the question yet. Based on these observations,
our feedback laws are constructed so that they are applied to a general class of
pdes, without assuming Riesz bases.
There are two kinds of feedback schemes: One is a static feedback scheme,
and the other a dynamic feedback scheme. In Chapter 3, the stabilization
problem and related problems are discussed in the static feedback scheme, in
which the outputs of the system are directly fed back into the system through
the actuators. While the scheme has difficulties in engineering implementations,
it works as an auxiliary means in the dynamic feedback schemes. In Chapter 4,
we establish stabilization in the scheme of boundary observation/boundary
feedback. The feedback scheme is the dynamic feedback scheme, in which the
outputs on the boundary are fed back into the system through another](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-12-320.jpg)
![Preface ix
differential equation described in another abstract space. This differential
equation is called a dynamic compensator, the concept of which originates from
D. G. Luenberger’s paper [33] in 1966 for linear odes. In his paper, two kinds of
compensators are proposed: One is an identity compensator, and the other a
compensator of general type. We formulate the latter compensator in the
feedback loop to cope with the stabilization problem, and finally reduce the
compensator to a finite-dimensional one. All arguments are algebraic, and do
not depend on the kind of boundary operators τ. In Chapter 5, the problem is
discussed from another viewpoint when the system admits a Riesz basis. Since a
finite-dimensional approximation to the pde is available as a strongly effective
means, an identity compensator is installed in the feedback loop. Most
stabilization results in the literature are based on identity compensators, but
have difficulty in terms of mathematical generality. In Chapters 4 and 5,
observability and controllability conditions on sensors and actuators,
respectively, are assumed on the pseudo-internal substructure of finite
dimension. We then ask in Chapter 6 the following: What can we claim when
the observability and controllability conditions are lost? Output stabilization is
one of the answers: Assuming an associated Riesz basis, we propose sufficient
conditions on output stabilization. A related problem is also discussed, which
leads to a new problem, that is, the problem of pole allocation with constraints.
To show mathematical generality of our stabilization scheme, we generalize in
Chapter 7 the class of operators L, in which −L is a generator of eventually
differentiable semigroups: A class of delay-differential equations generates such
operators L.
In our general stabilization scheme, we solve an inverse problem associated
with the infinite-dimensional Sylvester’s equation. The problem forms a so
called ill-posed problem lacking of continuity property. Finally in Chapter 8, we
propose a numerical approximation algorhism to the inverse problem, the
solution of which is mathematically ensured. The algorhism consists of a simple
idea, but needs tedious calculations. Although the algorhism has some
restrictions at present, it is expected that it would work in more general settings
of the parameters. Numerical approximation itself is a problem independent of
our stabilization problem. However, the latter certainly leads to a development
of new problems in numerical analysis. The author hopes that willing readers
could open a new area in effective numerical algorhisms.
The author in his graduate school days had an opportunity to read papers by
Y. Sakawa, by H. O. Fattorini, and by S. Agmon and L. Nirenberg ([2, 17, 18,
57]) among others, and learned about the close relationships lying in differential
equations, functional analysis, and the theory of functions. Inspired by these
results, he had a hope to contribute to deep results of such nature, since then. He](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-13-320.jpg)
![Chapter 1
Preliminary results -
Stabilization of linear
systems of finite
dimension
1.1 Introduction
We develop in this chapter the basic problem arising from stabilization
problems of finite-dimension. Since the celebrated pole assignment theory [70]
(see also [56, 68]) for linear control systems of finite dimension appeared, the
theory has been applied to various stabilization problems both of finite
dimension and infinite dimension such as the one with boundary
output/boundary input scheme (see, e.g., [12, 13, 28, 37 – 40, 42 – 45, 47 – 50,
53, 58, 59] and the references therein). The symbol Hn, n = 1, 2, ..., hereafter
will denote a finite-dimensional Hilbert space with dim Hn = n, equipped with
inner product ⟨·, ·⟩n and norm ∥·∥. The symbol ∥·∥ is also used for the
L (Hn)-norm. Let L, G, and W be operators in L (Hn), L (CN; Hn), and
L (Hn; CN), respectively. Here and hereafter, the symbol L (R; S), R and S
being linear spaces of finite or infinite dimension, means the set of all linear
bounded operators mapping R into S. The set L (R; S) forms a linear space.
When R = S, L (R; R) is abbreviated simply as L (R). Given L, W, and any set
of n complex numbers, Z = {ζi}1⩽i⩽n, the problem is to seek a suitable G such
that σ(L − GW) = Z, where σ(L − GW) means the spectrum of the operator](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-17-320.jpg)
![2 Theory of Stabilization for Linear Boundary Control Systems
L−GW. Or alternatively, given L and G, its algebraic counterpart is to seek a W
such that σ(L − GW) = Z. Stimulated by the result of [70], various approaches
and algorhisms for computation of G or W have been proposed since then (see,
e.g., [7, 10, 14]). However, each approach needs much preparation and a deep
background in linear algebra to achieve stabilization and determine the
necessary parameters. Explicit realizations of G or W sometimes seem
complicated. One for this is no doubt the complexity of the process in
determining G or W exactly satisfying the relation, σ(L−GW) = Z.
Let us describe our control system: Our system, consisting of a state
u(·) ∈ Hn, output y = Wu ∈ CN, and input f ∈ CN, is described by a linear
differential equation in Hn,
du
dt
+Lu = Gf, y = Wu, u(0) = u0 ∈ Hn. (1.1)
Here,
Gf =
N
∑
k=1
fkgk for f = (f1 ... fN)T
∈ CN
,
Wu =
(
⟨u, w1⟩n ... ⟨u, wN⟩n
)T
for u ∈ Hn,
(1.2)
(...)T denoting the transpose of vectors or matrices throughout the monogtaph.
The vectors wk ∈ Hn denote given weights of the observation (output); and gk ∈
Hn are actuators to be constructed. By setting f = y in (1.1), the control system
yields a feedback system,
du
dt
+(L−GW)u = 0, u(0) = u0 ∈ Hn. (1.3)
According to the choice of a basis for Hn, the operators L, G, and W are
identified with matrices of respective size. We hereafter employ the above
symbols somewhat different from those familiar in the control theory
community of finite dimension, in which state of the system, for example,
would be often represented as x(·); output Cx; input u; and equation
dx
dt
= Ax+Bu = (A+BC)x, u = Cx.
The reason for employing present symbols is that they are consistent with those
in systems of infinite dimension discussed in later chapters.
Let us assume that σ(L) ∩ C− ̸= ∅, so that the system (1.1) with f = 0 is
unstable. Given a µ 0, the stabilization problem for the finite dimensional
control system (1.3) is to seek a G or W such that
e−t(L−GW)
⩽ const e−µt
, t ⩾ 0. (1.4)
The pole assignment theory [70] plays a fundamental role in the above problem,
and has been applied so far to various linear systems. The theory is concretely](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-18-320.jpg)
![Preliminary results - Stabilization of linear systems of finite dimension 3
stated as follows: Let Z = {ζi}1⩽i⩽n be any set of n complex numbers, where some
ζi may coincide. Then, there exists an operator G such that σ(L − GW) = Z,
if and only if the pair (W, L) is observable. Thus, if the set Z is chosen such
that minζ∈Z Re ζ, say µ (= Re ζ1) is positive, and if there is no generalized
eigenspace of L−GW corresponding to ζ1, we obtain the decay estimate (1.4).
Now we ask: Do we need all information on σ(L − GW) for stabilization?
In fact, to obtain the decay estimate (1.4), it is not necessary to designate all
elements of the set Z: What is really necessary is the number,
µ = minζi∈Z Re ζi, say = Re ζ1, and the spectral property that ζ1 does not allow
any generalized eigenspace; the latter is the requirement that no factor of
algebraic growth in time is added to the right-hand side of (1.4). In fact, when
an algebraic growth is added, the decay property becomes a little worse, and
the gain constant (⩾ 1) in (1.4) increases. The above operator L − GW also
appears, as a pseudo-substructure, in the stabilization problems of infinite
dimensional linear systems such as parabolic systems and/or retarded systems
(see, e.g., [16]): These systems are decomposed into two, and understood as
composite systems consisting of two states; one belonging to a finite
dimensional subspace, and the other to an infinite dimensional one. It is
impossible, however, to manage the infinite dimensional substructures. Thus, no
matter how precisely the finite dimensional spectrum σ(L − GW) could be
assigned, it does not exactly dominate the whole structure of infinite
dimension. In other words, the assigned spectrum of finite dimension is not
necessarily a subset of the spectrum of the infinite-dimensional feedback
control system.
In view of the above observations, our aim in this chapter is to develop a
new approach much simpler than those in existing literature, which allows us to
construct a desired operator G or a set of actuators gk ensuring the decay (1.4)
in a simpler and more explicit manner (see (2.10) just below Lemma 2.2). The
result is, however, not as sharp as in [70] in the sense that it does not generally
provide the precise location of the assigned eigenvalues. From the above
viewpoint of infinite-dimensional control theory, however, the result would be
meaningful enough, and satisfactory for stabilization. We note that our result
exactly coincides with the standard pole assignment theory in the case where we
can choose N = 1 (see Proposition 2.3 in Section 2). The results of this chapter
are based on those discussed in [48, 51, 52].
Our approach is based on Sylvester’s equation of finite dimension.
Sylvester’s equation in infinite-dimensional spaces has also been studied
extensively (see, e.g., [6] for equations involving only bounded operators), and
even the unboundedness of the given operators are allowed [37, 39, 40, 42 – 45,
47, 49, 50, 53]. Sylvester’s equation in this chapter is of finite dimension, so that
there arises no difficulty caused by the complexity of infinite dimension. Its
infinite-dimensional version and the properties are discussed later in Chapters 4,](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-19-320.jpg)
![4 Theory of Stabilization for Linear Boundary Control Systems
6, and 7. Given a positive integer s and vectors ξk ∈ Hs, 1 ⩽ k ⩽ N, let us
consider the following Sylvester’s equation in Hn:
XL−MX = −ΞW, Ξ ∈ L (CN
; Hs), where
Ξz =
N
∑
k=1
zkξk for z = (z1 ... zN)T
∈ CN
.
(1.5)
Here, M denotes a given operator in L (Hs), and ξk vectors to be designed in Hs.
A possible solution X would belong to L (Hn; Hs). An approach via Sylvester’s
equations is found, e.g., in [7, 10], in which, by setting n = s, a condition for the
existence of the bounded inverse X−1 ∈ L (Hn) is sought. Choosing an M such
that σ(M) ⊂ C+, it is then proved that
L+(X−1
Ξ)W = X−1
MX, σ(X−1
MX) = σ(M) ⊂ C+,
the left-hand side of which means a desired perturbed operator. The procedure of
its derivation is, however, rather complicated, and the choice of the ξk is unclear.
In fact, X−1 might not exist sometimes for some ξk.
The approach in this chapter is new and rather different. Let us characterize
the operator L in (1.5). There is a set of generalized eigenpairs {λi, φij} with the
following properties:
(i) σ(L) = {λi; 1 ⩽ i ⩽ ν (⩽ n)}, λi ̸= λj for i ̸= j; and
(ii) Lφij = λiφij +∑kj αi
jkφik, 1 ⩽ i ⩽ ν, 1 ⩽ j ⩽ mi.
Let Pλi
be the projector in Hn corresponding to the eigenvalue λi. Then, we see
that Pλi
u = ∑
mi
j=1 uijφij for u ∈ Hn. The restriction of L onto the invariant
subspace Pλi
Hn is, in the basis {φi1, ..., φimi }, is represented by the mi × mi
upper triangular matrix Λi, where
Λi|(j,k) =
αi
k j, j k,
λi, j = k,
0, j k.
(1.6)
If we set Λi = λi +Ni, the matrix Ni is nilpotent, that is, Nmi
i = 0. The minimum
integer n such that ker Nn
i = ker Nn+1
i , denoted as li, is called the ascent of λi −L.
It is well known that the ascent li coincides with the order of the pole λi of the
resolvent (λ −L)−1. Laurent’s expansion of (λ −L)−1 in a neighborhood of the
pole λi ∈ σ(L) is expressed as
(λ −L)−1
=
li
∑
j=1
K−j
(λ −λi)j
+
∞
∑
j=0
(λ −λi)j
Kj, where
li ⩽ mi, Kj =
1
2πi
∫
|ζ−λi|=δ
(ζ −L)−1
(ζ −λi)j+1
dζ, j = 0,±1,±2,....
(1.7)](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-20-320.jpg)
![Preliminary results - Stabilization of linear systems of finite dimension 5
Note that K−1 = Pλi
. The set {φij; 1 ⩽ i ⩽ ν, 1 ⩽ j ⩽ mi} forms a basis for Hn.
Each x ∈ Hn is uniquely expressed as x = ∑i,j xijφij. Let T be a bijection, defined
as Tx =
(
x11 x12 ... xνmν
)T
. Then, L is identified with the upper triangular matrix
Λ;
TLT−1
= Λ = diag
(
Λ1 Λ2 ... Λν
)
. (1.8)
Let us turn to the operator M in (1.5). Let
{
ηij; 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ ℓi
}
be an orthonormal basis for Hs. Then necessarily s = ∑n
i=1 ℓi ⩾ n. Every vector
v ∈ Hs is expressed as
v =
n
∑
i=1
ℓi
∑
j=1
vijηij, where vij =
⟨
v, ηij
⟩
s
.
Let {µi}n
1=1 be a set of positive numbers such that 0 µ1 ··· µn, and set
Mv =
n
∑
i=1
ℓi
∑
j=1
µivijηij (1.9)
for v = ∑i, j vijηij. It is apparent that (i) σ(M) = {µi}n
i=1; and (ii) (µi −M)ηij = 0,
1 ⩽ i ⩽ n, 1 ⩽ j ⩽ ℓi. The operator M is self-adjoint, and potive-definite,
⟨Mv, v⟩s =
n
∑
i=1
ℓi
∑
j=1
µi|vij|2
⩾ µ1∥v∥2
s .
Let Qµi be the projector in Hs corresponding to the eigenvalue µi ∈ σ(M), say
Qµi v = ∑
ℓi
j=1 vijηij for v = ∑i,j vijηij. We put an additional condition on M:
σ(L)∩σ(M) = ∅. (1.10)
Assuming (1.10), we derive our first result as Proposition 1.1. Since the proof is
carried out in exactly the same manner as in [37, 44, 45, 50], it is omitted.
Proposition 1.1. Suppose that the condition (1.10) is satisfied. Then,
Sylvester’s equation (1.5) admits a unique operator solution X ∈ L (Hn; Hs).
The solution X is expressed as
Xu =
1
2πi
∫
C
(λ −M)−1
ΞW(λ −L)−1
udλ
= ∑
λ ∈σ(M)
Qλ ΞW(λ −L)−1
u
=
n
∑
i=1
Qµi ΞW(µi −L)−1
u,
(1.11)](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-21-320.jpg)
![6 Theory of Stabilization for Linear Boundary Control Systems
where C denotes a Jordan contour encircling σ(M) in its inside, with σ(L)
outside C. The above first expression is the so called Rosenblum formula [6].
The main results are stated as Theorem 2.1 and Proposition 2.2 in the next
section, where a more explicit and concrete expression than ever before of a set
of stabilizing actuators gk in (1.3) is obtained. As we see in the next section, an
advantage of considering the operator X ∈ L (Hn; Hs) with s ⩾ n is that the
bounded inverse (X∗X)−1 is ensured under a reasonable assumption on the
operator Ξ. A numerical example is also given. Finally, Proposition 2.3 is
stated, where our feedback scheme exactly coincides with the standard pole
assignment theory [70] in the case where we can choose N = 1.
1.2 Main Results
We assume that σ(L) ∩ C− ̸= ∅, so that the semigroup e−tL, t ⩾ 0, is
unstable. We construct suitable actuators gk ∈ Hn in (1.3) such that e−t(L−GW)
has a preassigned decay rate, say −µ1 (see (1.9)). The operator
(
W WL ... WLn−1
)T
belongs to L (Hn; CnN). The observability condition
on the pair (W, L) means that the above operator is injective, in other words,
ker
(
W WL ... WLn−1
)T
= {0}. Throughout the section, the separation
condition (1.10) is assumed in Sylvester’s equation (1.5). Then, we obtain one
of the main results:
Theorem 2.1. Assume that the conditions
ker
(
W WL ... WLn−1
)T
= {0}, and
ker Qµi Ξ = {0}, 1 ⩽ i ⩽ n
(2.1)
are satisfied. Then, kerX = {0}.
Proof. Let Xu = 0. In view of Proposition 1.1, we see that
Qµi ΞW(µi −L)−1
u = 0, 1 ⩽ i ⩽ n.
Since ker Qµi Ξ = {0}, 1 ⩽ i ⩽ n, by (2.1), we obtain
W(µi −L)−1
u = 0, 1 ⩽ i ⩽ n, or
⟨
(µi −L)−1
u, wk
⟩
n
= 0, 1 ⩽ k ⩽ N, 1 ⩽ i ⩽ n.
(2.2)
Set fk(λ; u) =
⟨
(λ −L)−1u, wk
⟩
n
. By recalling that T(λ − L)−1T−1 =
(λ − Λ)−1 (see (1.8)), fk(λ; u) is rewritten as
⟨
(λ − Λ)−1Tu,
(
T−1
)∗
wk
⟩
Cn .
Each element of the n × n matrix (λ − Λ)−1 is a rational function of λ; its
denominator consists of a polynomial of order n; and the numerator at most of
order n − 1. This means that each fk(λ; u) is a rational function of λ, the](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-22-320.jpg)
![Preliminary results - Stabilization of linear systems of finite dimension 9
In addition, there is no generalized eigenspace for any λ ∈ σ(X −1M ).
Remark: By Proposition 2.2, we obtain a decay estimate
exp
(
−t
(
L+(X∗
X)−1
X∗
ΞW
))
= e−t(X −1M )
⩽ const e−µ1t
, t ⩾ 0.
(2.10)
In fact, the last assertion of the proposition ensures that no algebraic growth in
time arises in the semigroup, regarding the smallest eigenvalue. Thus, a set of
actuators gk = −(X∗X)−1X∗ξk, 1 ⩽ k ⩽ N, in other words, G = −(X∗X)−1X∗Ξ
explicitly gives a desired set of actuators in (1.3).
Proof of Proposition 2.2. Since X is positive-definite, we can find a non-
unique bijection U ∈ L (Hn) such that
X = X∗
X = U ∗
U , (2.11)
the so called Cholesky factorization. Let us define
M ′
= (U ∗
)−1
M U −1
= (U −1
)∗
M U −1
.
Then, M ′ ∈ L (Hn) is a self-adjoint operator, enjoying some properties similar
to those of X −1M . In fact, let λ ∈ σ(X −1M ), or (λX −M )u = 0 for some
u ̸= 0. Then, since
0 = (λU ∗
U −M )u = U ∗
(
λ −(U ∗
)−1
M U −1
)
U u
= U ∗
(
λ −M ′
)
U u = 0,
we see that λ belongs to σ(M ′). The converse relation is also correct, which
means that
σ(X −1
M ) = σ(M ′
) ⊂ R1
. (2.12)
Inequality (2.9) is achieved by applying the well known min-max principle [11]
to M ′, or more directly by the following observation: Let λ ∈ σ(X −1M ), and
(λX −M )u = 0 for some u ̸= 0. Then
λ ∥Xu∥2
s = λ ⟨X u, u⟩n = ⟨M u, u⟩n = ⟨MXu, Xu⟩s ⩾ µ1∥Xu∥2
s ,
from which (2.9) immediately follows, since Xu ̸= 0.
Next let us show that there is no generalized eigenspace for
λ ∈ σ(X −1M ). Let (λ − X −1M )2u = 0 for some u ̸= 0. Setting
v = (λ −X −1M )u, we calculate
0 = X (λ −X −1
M )2
u = (λX −M )v
= (λU ∗
U −M )v = U ∗
(
λ −(U ∗
)−1
M U −1
)
U v
= U ∗
(
λ −M ′
)
w = 0, w = U v,](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-25-320.jpg)
![12 Theory of Stabilization for Linear Boundary Control Systems
The other eigenvalues are those of the matrix,
1
γ
(
|β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3 +γa |β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3
|α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 |α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 +γb
)
.
(2.14)
To see that these eigenvalues are generally greater than µ1, let us consider a
numerical example: Let (µ1 µ2 µ3) = (2 3 4), a = 0, and b = −1. Then,
α =
(
1
2
1
3
1
4
)T
, β =
(
1
3
1
4
1
5
)T
, |α|2
=
61
144
, |β|2
=
769
3600
,
⟨α, β⟩3 =
3
10
, ⟨α, 1⟩3 =
13
12
, ⟨β, 1⟩3 =
47
60
,
γ = |α|2
|β|2
−⟨α, β⟩3
2
=
253
518400
.
One of the eigenvalues a+⟨α, 1⟩3 /|α|2 is 156/61 2 (= µ1). The matrix (2.14)
is then
1
253
(
−1860 −1860
3540 3287
)
, the eigenvalues of which are denoted as ζ1 and
ζ2. Then, µ1 = 2 ζ1 156/61 ζ2, and thus λ∗ = ζ1 2.
We close this section with the following remark: There is a case where λ∗
coincides with µ1. Following [52], let us consider (1.3) in the space Hn = Cn
(see (1.8)). All operators L, G, and W are then matrices of respective size. Let
σ(L) consist only of simple eigenvalues, so that mi = 1, 1 ⩽ i ⩽ n, and n = ν.
Thus we can choose N = 1, ℓi = 1, 1 ⩽ i ⩽ n, and thus s = n. The operator in
(2.10) is written as L + (X∗X)−1X∗ΞW, where Ξu = uξ for u ∈ C1, and W =
⟨·, w⟩n, w =
(
w1 w2 ... wn
)T
∈ Cn. The observability condition then turns out to
be wi ̸= 0, 1 ⩽ i ⩽ n. Let us consider Sylvester’s equation (1.5) in Hs = Cn. By
setting ξ =
(
1 1 ... 1
)T
∈ Cn, the solution X to (1.5) is an n×n matrix, and has
a bounded inverse:
X = ΦW̃, (2.15)
where
Φ =
(
1
µi −λj
;
i ↓ 1, ..., n
j → 1, ..., n
)
, and W̃ = diag
(
w1 w2 ... wn
)
.
Thus, L + (X∗X)−1X∗ΞW = L + X−1ξ wT. It is shown [52] that, given a set
{µi}1⩽i⩽n, there is a unique g ∈ Cn such that σ
(
L−gwT
)
= {µi}1⩽i⩽n, and that
g is concretely expressed as](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-28-320.jpg)
![bottle decanted, was, from time to time, denominated a FELLOW-
COMMONER.
1867. Collins, The Public Schools, p. 26. Thomas Middleton
petitions King Charles, on his restoration, to grant his royal letters
to the Winchester electors in favour of his son’s admittance “as a
child in Winchester College, where he has now spent three years
as FELLOW-COMMONER.”
1891. Harry Fludyer at Cambridge, 38. She said she had heard
from her cousin, who is, I think, a FELLOW-COMMONER, or something
of that sort, at Downing College, that Harry is one of the most
popular men at Cambridge.
Feoffee, subs. (Manchester Grammar:
obsolete).—The original name for the
trustees in whose hands the foundation
estate was placed by Hugh Bexwycke. [A.S.
feo = fee or inheritance.]
Ferk. See Firk.
Ferula, subs. (Stonyhurst).—See Tolly.
Festive, adj. (Charterhouse).—Said of a boy
who has not learned his duty to his
superiors and seniors.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-31-320.jpg)
![Fighting-green, subs. (Westminster).—The
old battle-ground in the western cloister.
Figures, subs. (Stonyhurst).—The Second
Form: formerly Great Figures. See Little
Figures.
Fin, intj. (Christ’s Hospital).—A form of
negative. Ex. “FIN the small court” = “I won’t
have, c.” [Lat. fend.] See Fains.
Find, subs. (Harrow).—A mess of, usually, two
upper boys which takes breakfast and tea in
the rooms of one or other of the set: a
privilege of the Sixth Form. Whence FIND-FAG
= a fag who lays the table for the upper
boys. [Find (dial.) = to supply; to supply
with provisions.] Also as verb.
1867. Collins, The Public Schools, p. 316. Immediately a certain
number of rolls (FINDS they were called—etymology unknown) were
ordered at the baker’s, and were rebaked every morning until they
were pretty nearly as hard as pebbles. At nine o’clock on the
morning fixed for the rolling in, the members of the hall ranged
themselves on the long table which ran along one side of the
room, each with his pile of these rolls before him, and a fag to pick
them up.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-33-320.jpg)
![Finder, subs. (Oxford).—A waiter.
Finjy! intj. (Winchester).—An exclamation
excusing one from participation in an
unpleasant or unacceptable task, which he
who says the word last has to undertake. Cf.
Fains.
Firk (or Ferk), verb (Winchester).—To
proceed; to hasten; to expel; to send; to
drive away. [O.E. fercian.] Also TO FIRK UP and
TO FIRK DOWN.
1283. William of Palerne. Thei bisiliche fondede (tried) fast to FERKE
him forthward.
c. 1400. Troy Book. I you helpe shall the flese for to fecche, and
FERKE it away.
[?] MS. Lincoln, Morte Arthure, f. 79. The Kyng FERKES furthe on a
faire stede.
1599. Shakspeare, Henry V., iv. 4. Pistol. I’ll fer him, and FIRK him,
and ferret him, discuss the same in French unto him. Boy. I do not
know the French for fer and ferret and FIRK.
1611. Barry, Ram Alley [Dodsley, Old Plays (Reed), v. 466]. Nay, I
will FIRK my silly novice, as he was never FIRK’D Since mid-wives
bound his noddle.
1640. Brome, Antipodes. As tumblers do ... by FIRKING up their
breeches.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-34-320.jpg)
![1795. Sewell, Hist. of Quakers. At this the judge said, “Take him
away: prevaricator! I’ll FERK him.”
Five, The, subs. (Charterhouse).—The Five
bell.
Flannels, subs. (Harrow).—The members of
either School Eleven.
1899. Public School Mag., Dec., p. 446. Up to the present the
eleven have won four matches and lost one, while Monro,
Cookson, Wyckoff, and Borwick have all received their FLANNELS.
Flat, subs. (Royal High School, Edin.).—An
objectionable person; a “bounder.” [A
misuse of flat = fool.]
Fleshy, subs. (Winchester).—A thick cut out
of the middle of a shoulder of mutton. See
Dispar.
Flies. Squashed flies, subs. (Durham:
obsolete).—Biscuits with currants.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-35-320.jpg)
![For, phr. (Tonbridge).—A form of ridicule: e.g.
“first eleven FOR one” would be used in
jeering at a boy who had recently obtained
his colours.
Forakers (or Foricus), subs. (Winchester).—
The water-closet. [Formerly foricus, and
probably a corruption of foricas, an English
plural of the Latin forica.]
Force. Out by force, phr. (Stonyhurst).—Of a
football when it goes out from two opposite
players at the same time.
Founders, subs. (Winchester).—Boys who
proved their descent from the Founder, and
were afterwards elected (by rote among the
Electors) as such. Only two were admitted
each year, and only two were sent to New
College, but these two were put at the head
of the Roll (q.v.) whatever their previous
position in Sixth Book (q.v.) might have
been. They were not obliged to leave at the
age of eighteen, as the other boys were,
but were allowed to remain till they were](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-38-320.jpg)
![with the founder’s family; and gradually the customs obtained of
electing two only of these favoured candidates at the head of the
roll for admission, and filling up the remaining vacancies by a
process of successive nominations by each of the six electors, the
Warden of New College having the first turn, until the number of
vacancies was supplied.
Founder’s-Ob., subs. (Winchester).—The
anniversary of the Founder’s death.
Four-holed Middlings, subs. phr.
(Winchester: obsolete).—Ordinary walking
shoes. Cf. Beeswaxers.
Fourth, subs. (Cambridge).—A rear or jakes.
[Origin uncertain; said to have been first
used at St. John’s or Trinity, where the
closets are situated in the Fourth Court.
Whatever its derivation, the term is now the
only one in use at Cambridge, and is
frequently heard outside the university.] The
verbal phrase is TO KEEP A FOURTH.
Fourth Book, subs. (Winchester: obsolete).—
All the boys below Junior Part the Fifth. See
Books.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-40-320.jpg)
![and the shore, produce an effect not easily forgotten. A
pyrotechnic illumination of the College arms concludes the
ceremonies, and is the signal for the crews to land and march in
jubilant disorder back to College.
Fox-and-dowdy, subs. (King Edward’s,
Birm.: obsolete).—See Action.
Fragment, subs. (Winchester: obsolete).—A
dinner for six (served in College Hall, after
the ordinary dinner), ordered by a Fellow in
favour of a particular boy, who was at
liberty to invite five others to join him. A
fragment was supposed to consist of three
dishes.—Winchester Word-Book [1891].
Free, adj. (Oxford).—Impudent; self-
possessed.
1864. Tennyson, Northern Farmer (Old Style), line 25. But parson a
coomes an’ a goos, an’ a says it eäsy an’ FREEÄ.
Freed, adv. (Stonyhurst).—Of an extra
recreation: given for some special reason.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-43-320.jpg)
![retaining wall to the bottom of an incline up
which coal-stores, c., could be brought
into the playground. The new science and
art rooms have covered the site, and MONKEY-
DEN has superseded the terrors of this local
Tarpeian Rock. The FRESH-HERRING is always
told that he must bring beeswax and
turpentine for the purpose of polishing his
desk, and he not infrequently comes armed
with this or some other form of furniture-
polish, to the glee of the “stuffer-up.”
Freshman (or Fresher), subs. (University).—
A University man during his first year. In
Dublin University he is a JUNIOR FRESHMAN
during his first year, and a SENIOR FRESHMAN the
second year. At Oxford the title lasts for the
first term. See Soph.
1596. Nashe, Saffron Walden, in Works, iii. 8. When he was but yet
a FRESHMAN in Cambridge.
1611. Middleton, Roaring Girl, Act iii. sc. 3. S. Alex. Then he’s a
graduate. S. Davy. Say they trust him not. S. Alex. Then is he held
a FRESHMAN and a sot.
1650. Howell, Familiar Letters [Nares]. I am but a FRESHMAN yet in
France, therefore I can send you no news, but that all is here
quiet, and ’tis no ordinary news, that the French should be quiet.
1671. Cotgrave, Wit’s Interpreter, p. 221. First, if thou art a
FRESHMAN, and art bent To bear love’s arms, and follow Cupid’s tent.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-45-320.jpg)
![Freshman’s Church, subs. phr. (Cambridge).
—The Pitt Press. [From its ecclesiastical
architecture.]
Freshman’s Landmark, subs. phr.
(Cambridge).—King’s College Chapel. [From
the situation.]
Freshmanship, subs. (old).—Of the quality or
state of being a freshman.
1605. Jonson, Volpone, or the Fox, iv. 3. Well, wise Sir Pol., since
you have practised thus, Upon my FRESHMANSHIP, I’ll try your salt-
head With what proof it is against a counter-plot.
Froust, subs. (Harrow).—1. Extra sleep
allowed on Sunday mornings and whole
holidays. Also (2) an easy-chair. Hence
Frouster.
Frout, adj. (Winchester).—Angry; vexed.
Fudge, subs. 1. (Christ’s Hospital).—To copy;
to crib; to dodge or escape: also see quot.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-47-320.jpg)
![ag, subs. 1. (Christ’s Hospital).—See
quot. Gag-eater = a term of reproach.
1813. Lamb, Christ’s Hospital, in Works, p. 324 (ed. 1852). L. has
recorded the repugnance of the school to GAGS, or the fat of fresh
beef boiled; and sets it down to some superstition.... A GAG-EATER in
our time was equivalent to a ghoul, ... and held in equal
estimation.
2. (Winchester: obsolete).—An exercise
(said to have been invented by Dr. Gabell)
which consists in writing Latin criticisms on
some celebrated piece, in a book sent in
about once a month. In the Parts below
Sixth Book and Senior Part, the GAGS
consisted in historical analysis. [An
abbreviation of “gathering.”]
c. 1840. Mansfield, School-Life at Winchester College, p. 108. From
time to time, also, they had to write ... an analysis of some
historical work; these productions were called GATHERINGS (or GAGS).
Gain. See Election.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-51-320.jpg)
![Gaits (Geits, Gytes, or Gites), subs. (Royal
High School, Edin.).—The first, or lowest
class. See Cats.
Gallery, subs. (Winchester).—A Commoner
bedroom. [From a tradition of GALLERIES in
Commoners.] Hence GALLERY NYMPH = a
housemaid.
Gang, subs. (Felsted: obsolete).—A particular
friend. From the ordinary meaning of the
word, applied first to the two friends, then
to each of them. Used only of “acute”
friendship. Also as verb = to carry on such a
friendship with another.
Garden, The (Stonyhurst).—The playgrounds,
built on the site of part of the old garden,
long kept this name. “The boys went to the
GARDEN” = “into the playground”: obsolete.
Gater, subs. (Winchester: obsolete).—A
plunge head foremost into a POT (q.v.).](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-52-320.jpg)
![To be at gates, verb. phr. (Winchester:
obsolete).—To assemble in Seventh
Chamber passage, preparatory to going Hills
or Cathedral.
Gaudeamus, subs. (general).—A feast; a
drinking bout; any sort of merry-making.
[German students’, but now general. From
the first word of the mediæval (students’)
ditty.]
Gaudy (or Gaudy-day), subs. (general).—A
feast or entertainment: specifically, the
annual dinner of the Fellows of a college in
memory of founders or benefactors; or a
festival of the Inns of Court. [Lat. gaudere
= to rejoice.]
1540. Palsgrave, Acolastus [Halliwell]. We maye make our
tryumphe, kepe our GAUDYES, or let us sette the cocke on the hope,
and make good chere within dores. Ibid., I have good cause to set
the cocke on the hope, and make GAUDYE chere.
1608. Shakspeare, Antony and Cleopatra, iii. 11. Come, Let’s have
one other GAUDY night; call to me All my sad captains; fill our
bowls; once more Let’s mock the midnight bell.
1636. Suckling, Goblins [Dodsley, Old Plays (Reed), x. 143]. A foolish
utensil of state, Which, like old plate upon a GAUDY day, ’s brought
forth to make a show, and that is all.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-54-320.jpg)
![Genuine, subs. (Winchester).—Praise. Also as
verb = to praise. [It is suggested (but see
quot.) that the derivation may be from
genuina, the “jaw-tooth,” praise being
nothing but “jaw”: cf. Parsius, i. 115.]
1891. Wrench, Winchester Word-Book, s.v. Genuine.... He was
awfully quilled and GENUINED my task. Possibly from calling a thing
genuine. Cf. to blackguard, to lord, c. But fifty years ago it was a
subs. only. [See Appendix.]
Gip (or Gyp), subs. (Cambridge).—A college
servant.
1891. Harry Fludyer at Cambridge, 8. My GYP said he thought he
knew some one who’d give me eighteen shillings for it.
Girdlestoneites (Charterhouse).—A
boarding-house. [From a master’s name.]
See Out-houses.
Glope, verb (Winchester: obsolete).—To spit.
Go. To go down, verb (University).—To leave
school or college: by special EXEAT (q.v.) or at
vacation. Whence TO BE SENT DOWN = to be
under discipline; to be rusticated.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-56-320.jpg)
![1628. Earle, Microcosmography. His companion is ordinarily some
stale fellow that has been notorious for an ingle to GOLD HATBANDS,
whom hee admires at first, afterwards scornes.
1889. Gentleman’s Mag., June, p. 598. Noblemen at the
universities, since known as “tufts,” because of the gold tuft or
tassel to their cap, were then known as GOLD HATBANDS.
Golgotha, subs. (old University).—The Dons’
gallery at Cambridge; also a certain part of
the theatre at Oxford. [That is, “the place of
skulls” (cf. Luke xxiii. 33 and Matt. xxvii.
33); whence the pun, Dons being the heads
of houses.]
1730. Jas. Miller, Humours of Oxford, Act ii., p. 23 (2nd ed.).
Sirrah, I’ll have you put in the black-book, rusticated—expelled—I’ll
have you coram nobis at GOLGOTHA, where you’ll be bedevilled,
Muck-worm, you will.
1785. Grose, Vulg. Tongue, s.v.
1791. G. Huddesford, Salmagundi (Note on, p. 150). Golgotha, “The
place of a Skull,” a name ludicrously affixed to the Place in which
the Heads of Colleges assemble.
1808. J. T. Conybeare in C. K. Sharp’s Correspondence (1888), i. 324.
The subject then of the ensuing section is Oxford News ... we will
begin by GOLGOTHA.... Cole has already obtained the Headship of
Exeter, and Mr. Griffiths ... is to have that of University.
Gomer, subs. 1. (Winchester).—A large
pewter dish used in College. [Probably from](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-59-320.jpg)
![its holding a homer or omer in measure: see
quots.]
1610-31. Donne. Not satisfied with his GOMER of manna.
d. 1656. Hall, Satires, Bk. v. He that gave a GOMER to each.
1778. Inventory of Kitchen and Hall. Twenty-four GOMERS (amongst
dishes and brass pots).
2. A new hat: specifically, a beaver when
first introduced: but see quot., Peals, and
Appendix.
1867. Collins, The Public Schools, 68. Top-boots are no longer
considered, by young gentlemen of twelve, “your only wear” to go
home in, although the term for them—GOMERS (i.e. go-homers)—
still survives in the Winchester vocabulary.
Good-breakfast, subs. (Stonyhurst).—A
breakfast given to those Distinguished (q.v.)
every term: also called Distinction-breakfast.
Cf. Do and Good-supper.
Good-creatures, subs. (Charterhouse).—
Meat, vegetables, and pudding. [From a
quaint old-fashioned “Scholars’
grace”—“Lord, bless to us these thy GOOD-
CREATURES,” c.]](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-60-320.jpg)
![Gownboy-arch (Charterhouse).—An arch
near the east end of the chapel, formerly
the doorway from Scholars’ Court into
Gown-boys. The earliest Old Carthusian
name inscribed on it bears date 1778.
Gownboy-cricket, subs. (Charterhouse).—
Cricket in which there are twenty bowlers to
one batsman, with no fielders.
Gownboys (Charterhouse).—A boarding-
house. [Because on migration to Godalming
in 1872 nearly all the old Gownboys (q.v.)
were received there.]
Gowner, subs. (Winchester).—The Goal (q.v.)
at football stood with his legs stretched out,
and a gown, rolled up into a ball, at each
foot. When the ball was kicked over either
of these gowns, without goal’s touching it,
this counted two for the party who kicked it.
—Mansfield (c. 1840). Also see Goal and
Schitt. Now obsolete.](https://image.slidesharecdn.com/14645-250621062536-6596644a/85/Theory-of-stabilization-for-linear-boundary-control-systems-1st-Edition-Nambu-63-320.jpg)
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- 5. Theory of Stabilization for Linear Boundary Control Systems
- 7. Theory of Stabilization for Linear Boundary Control Systems Fermented Meat Products Health Aspects Edited by Nevijo Zdolec University of Zagreb, Faculty of Veterinary Medicine Department of Hygiene Technology and Food Safety Heinzelova 55 10000 Zagreb, Croatia Takao Nambu Professor Emeritus Department of Applied Mathematics Kobe University Kobe, Japan
- 8. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20160708 International Standard Book Number-13: 978-1-4987-5847-5 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, 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 stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.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 pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy 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. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 9. To my family, Mariko, Ryutaro, and Hiromu.
- 11. Preface This monograph studies the stabilization theory for linear systems governed by partial differential equations of parabolic type in a unified manner. As long as controlled plants are relatively small, such as electric circuits and mechanical oscillations/rotations of rigid bodies, ordinary differential equations, abbreviated as ode(s), are suitable mathematical models to describe them. When the controlled plants are, e.g., chemical reactors, wings of aircrafts, or other flexible systems such as robotics arms, plates, bridges, and cranes, however, effects of space variables are essential and non-neglegeble terms. For the set up of mathematical models describing these plants, partial differential equations, abbreviated as pde(s), are a more suitable language. It is generally expected that control laws based on more accurate pde models would work effectively in actual applications. The origin of control theory is said to be the paper, “On governors” by J.C. Maxwell (1868). For many years, control theory has been studied mainly for systems governed by odes in which controlled plants are relatively small. Control theory for pdes began in 60’s of the 20th century, and the study of stabilization in mid 70’s to cope with much larger systems. Fundamental concepts of control such as controllability, observability, optimality, and stabilizability are the same as in those of odes, and translated by the language of pdes. The essence of pdes consists in their infinite-dimensional properties, so that control problems of pdes face serious difficulties in respective aspects, which have never been experienced in the world of odes: However, these difficulties provide us rich and challenging fields of study both from mathematical and engineering viewpoints. Among other control problems of pdes such as optimal control problems, etc., we concentrate ourselves on the topic of stabilization problems. Stabilization problems of pdes have a new aspect of pdes in the framework of synthesis (or design) of a desirable spectrum by involving the concept of vii
- 12. viii Theory of Stabilization for Linear Boundary Control Systems observation/control, and are connected not only with functional analysis but also non-harmonic analysis and classical Fourier analysis, etc. The monograph consists of eight chapters which strongly reflects the author’s works over thirty years except for Chapter 2: Some were taught in graduate courses at Kobe University. The organization of the monograph is stated as follows: It begins with the linear tabilization problem of finite dimension in Chapter 1. Finite-dimensional models constitute pseudo-internal structures of pdes. Although the problem is entirely solved by W. M. Wonham in 1967 [70], we develop a much easier new approach, which has never appeared even among the community of finite-dimensional control theory: It is based on Sylvester’s equation. Infinite-dimensional versions of the equation appear in later chapters as an essential tool for stabilization problems throughout the monograph. Chapter 2 is a brief introduction of basic results on standard elliptic differential operators L and related Sobolev spaces necessary for our control problems: These results are well known among the pdes community, but proofs of some results are stated for the readers’ convenience. As for results requiring much preparation we only provide some references instead of proofs. In Chapters 3 through 7, the main topics discussed are, where stabilization problems of linear parabolic systems are successfully solved in the boundary observation/boundary feedback scheme. The elliptic operator L is derived from a pair of standard (but general enough) differential operators (L , τ), and forms the coefficient of our control systems, where L denotes a uniformly elliptic differential operator and τ a boundary operator. The operator L is sectorial, and thus −L turns out to be an infinitesimal generator of an analytic semigroup. One of important issues is certainly the existence or non-existence of Riesz bases associated with L: When an associated Riesz basis exists, a sequence of finite-dimensional approximation models of the original pde is quantitatively justified, so that the control laws based on the approximated finite-dimensional models effectively works. There is an attempt to draw out a class of elliptic operators with Riesz bases (see the footnote in the beginning of Chapter 4). However, is the class of pdes admitting associated Riesz bases general enough or much narrower than expected? We do not have a satisfactory solution to the question yet. Based on these observations, our feedback laws are constructed so that they are applied to a general class of pdes, without assuming Riesz bases. There are two kinds of feedback schemes: One is a static feedback scheme, and the other a dynamic feedback scheme. In Chapter 3, the stabilization problem and related problems are discussed in the static feedback scheme, in which the outputs of the system are directly fed back into the system through the actuators. While the scheme has difficulties in engineering implementations, it works as an auxiliary means in the dynamic feedback schemes. In Chapter 4, we establish stabilization in the scheme of boundary observation/boundary feedback. The feedback scheme is the dynamic feedback scheme, in which the outputs on the boundary are fed back into the system through another
- 13. Preface ix differential equation described in another abstract space. This differential equation is called a dynamic compensator, the concept of which originates from D. G. Luenberger’s paper [33] in 1966 for linear odes. In his paper, two kinds of compensators are proposed: One is an identity compensator, and the other a compensator of general type. We formulate the latter compensator in the feedback loop to cope with the stabilization problem, and finally reduce the compensator to a finite-dimensional one. All arguments are algebraic, and do not depend on the kind of boundary operators τ. In Chapter 5, the problem is discussed from another viewpoint when the system admits a Riesz basis. Since a finite-dimensional approximation to the pde is available as a strongly effective means, an identity compensator is installed in the feedback loop. Most stabilization results in the literature are based on identity compensators, but have difficulty in terms of mathematical generality. In Chapters 4 and 5, observability and controllability conditions on sensors and actuators, respectively, are assumed on the pseudo-internal substructure of finite dimension. We then ask in Chapter 6 the following: What can we claim when the observability and controllability conditions are lost? Output stabilization is one of the answers: Assuming an associated Riesz basis, we propose sufficient conditions on output stabilization. A related problem is also discussed, which leads to a new problem, that is, the problem of pole allocation with constraints. To show mathematical generality of our stabilization scheme, we generalize in Chapter 7 the class of operators L, in which −L is a generator of eventually differentiable semigroups: A class of delay-differential equations generates such operators L. In our general stabilization scheme, we solve an inverse problem associated with the infinite-dimensional Sylvester’s equation. The problem forms a so called ill-posed problem lacking of continuity property. Finally in Chapter 8, we propose a numerical approximation algorhism to the inverse problem, the solution of which is mathematically ensured. The algorhism consists of a simple idea, but needs tedious calculations. Although the algorhism has some restrictions at present, it is expected that it would work in more general settings of the parameters. Numerical approximation itself is a problem independent of our stabilization problem. However, the latter certainly leads to a development of new problems in numerical analysis. The author hopes that willing readers could open a new area in effective numerical algorhisms. The author in his graduate school days had an opportunity to read papers by Y. Sakawa, by H. O. Fattorini, and by S. Agmon and L. Nirenberg ([2, 17, 18, 57]) among others, and learned about the close relationships lying in differential equations, functional analysis, and the theory of functions. Inspired by these results, he had a hope to contribute to deep results of such nature, since then. He
- 14. x Theory of Stabilization for Linear Boundary Control Systems is not certain now, but would be happy, if the monograph coould reflect his hope even a little. Takao Nambu December, 2015 Kobe
- 15. Preface.....................................................................................................................vii 1. Preliminary results—Stabilization of linear systems of finite dimension....1 1.1 Introduction .............................................................................................1 1.2 Main results .............................................................................................6 1.3 Observability: Reduction to substructures ............................................14 1.4 The case of a single observation............................................................17 2 Preliminary results: Basic theory of elliptic operators................................27 2.1 Introduction ...........................................................................................27 2.2 Brief survey of Sobolev spaces .............................................................28 2.3 Elliptic boundary valule problems.........................................................36 2.3.1 The Dirichlet boundary ...............................................................36 2.3.2 The Robin boundary....................................................................41 2.3.3 The case of a general boundary...................................................43 2.3.4 On the domain of fractional powers Lc q with Robinboundary ....48 2.4 Analytic semigroup.....................................................................51 3 Stabilization of linear systems of infinite dimension: Static feedback.......55 3.1 Introduction ...........................................................................................55 3.2 Decomposition of the system ................................................................59 3.3 Remark on the choice of the decay rate.................................................64 3.4 Stability enhancement ...........................................................................69 3.5 Some generalization ..............................................................................78 4 Stabilization of linear systems of infinite dimension: Dynamic feedback...........................................................................................................93 4.1 Introduction.................................................................................93 4.2 Boundary Control Systems .......................................................107 4.3 Stabilization...............................................................................113 4.4 Another Construction of Stabilizing Compensators .................134 4.5 Alternative Framework of Stabilization ..............................................140 Contents
- 16. 4.6 The Robin Boundary and Fractional Powers.......................................152 4.7 Some Related Topics ...........................................................................159 4.7.1 On the growth rate of s(B)........................................................159 4.7.2 On fractional powers of elliptic operators characterized by feedback boundary conditions..............................................163 5 Stabilization of linear systems with Riesz Bases: Dynamic feedback ......171 5.1 Introduction .........................................................................................171 5.2 Boundary Control Systems..................................................................173 5.3 Another Model of Identity Compensators...........................................187 6 Output stabilization : lack of the observability and/or the controllability conditions..............................................................................193 6.1 Introduction .........................................................................................193 6.2 Output stabilization .............................................................................196 6.3 Application to boundary control systems............................................202 6.3.1 Algebraic approach to boundary control systems.....................205 6.3.2 Some generalization..................................................................209 6.4 Operator L admitting generalized eigenvectors...................................211 6.5 Some functionals .................................................................................212 7 Stabilization of a class of linear control systems generating C0 -semigroups................................................................................................223 7.1 Introduction .........................................................................................223 7.2 Basic properties of the semigroup .......................................................225 7.3 Stabilization ........................................................................................234 8 A Computational Algorhism for an Infinite-Dimensional Sylvester’s Equation .....................................................................................247 8.1 Introduction .........................................................................................247 8.2 An algorhism .......................................................................................251 References.............................................................................................................265 Index......................................................................................................................271 xii Theory of Stabilization for Linear Boundary Control Systems
- 17. Chapter 1 Preliminary results - Stabilization of linear systems of finite dimension 1.1 Introduction We develop in this chapter the basic problem arising from stabilization problems of finite-dimension. Since the celebrated pole assignment theory [70] (see also [56, 68]) for linear control systems of finite dimension appeared, the theory has been applied to various stabilization problems both of finite dimension and infinite dimension such as the one with boundary output/boundary input scheme (see, e.g., [12, 13, 28, 37 – 40, 42 – 45, 47 – 50, 53, 58, 59] and the references therein). The symbol Hn, n = 1, 2, ..., hereafter will denote a finite-dimensional Hilbert space with dim Hn = n, equipped with inner product ⟨·, ·⟩n and norm ∥·∥. The symbol ∥·∥ is also used for the L (Hn)-norm. Let L, G, and W be operators in L (Hn), L (CN; Hn), and L (Hn; CN), respectively. Here and hereafter, the symbol L (R; S), R and S being linear spaces of finite or infinite dimension, means the set of all linear bounded operators mapping R into S. The set L (R; S) forms a linear space. When R = S, L (R; R) is abbreviated simply as L (R). Given L, W, and any set of n complex numbers, Z = {ζi}1⩽i⩽n, the problem is to seek a suitable G such that σ(L − GW) = Z, where σ(L − GW) means the spectrum of the operator
- 18. 2 Theory of Stabilization for Linear Boundary Control Systems L−GW. Or alternatively, given L and G, its algebraic counterpart is to seek a W such that σ(L − GW) = Z. Stimulated by the result of [70], various approaches and algorhisms for computation of G or W have been proposed since then (see, e.g., [7, 10, 14]). However, each approach needs much preparation and a deep background in linear algebra to achieve stabilization and determine the necessary parameters. Explicit realizations of G or W sometimes seem complicated. One for this is no doubt the complexity of the process in determining G or W exactly satisfying the relation, σ(L−GW) = Z. Let us describe our control system: Our system, consisting of a state u(·) ∈ Hn, output y = Wu ∈ CN, and input f ∈ CN, is described by a linear differential equation in Hn, du dt +Lu = Gf, y = Wu, u(0) = u0 ∈ Hn. (1.1) Here, Gf = N ∑ k=1 fkgk for f = (f1 ... fN)T ∈ CN , Wu = ( ⟨u, w1⟩n ... ⟨u, wN⟩n )T for u ∈ Hn, (1.2) (...)T denoting the transpose of vectors or matrices throughout the monogtaph. The vectors wk ∈ Hn denote given weights of the observation (output); and gk ∈ Hn are actuators to be constructed. By setting f = y in (1.1), the control system yields a feedback system, du dt +(L−GW)u = 0, u(0) = u0 ∈ Hn. (1.3) According to the choice of a basis for Hn, the operators L, G, and W are identified with matrices of respective size. We hereafter employ the above symbols somewhat different from those familiar in the control theory community of finite dimension, in which state of the system, for example, would be often represented as x(·); output Cx; input u; and equation dx dt = Ax+Bu = (A+BC)x, u = Cx. The reason for employing present symbols is that they are consistent with those in systems of infinite dimension discussed in later chapters. Let us assume that σ(L) ∩ C− ̸= ∅, so that the system (1.1) with f = 0 is unstable. Given a µ 0, the stabilization problem for the finite dimensional control system (1.3) is to seek a G or W such that e−t(L−GW) ⩽ const e−µt , t ⩾ 0. (1.4) The pole assignment theory [70] plays a fundamental role in the above problem, and has been applied so far to various linear systems. The theory is concretely
- 19. Preliminary results - Stabilization of linear systems of finite dimension 3 stated as follows: Let Z = {ζi}1⩽i⩽n be any set of n complex numbers, where some ζi may coincide. Then, there exists an operator G such that σ(L − GW) = Z, if and only if the pair (W, L) is observable. Thus, if the set Z is chosen such that minζ∈Z Re ζ, say µ (= Re ζ1) is positive, and if there is no generalized eigenspace of L−GW corresponding to ζ1, we obtain the decay estimate (1.4). Now we ask: Do we need all information on σ(L − GW) for stabilization? In fact, to obtain the decay estimate (1.4), it is not necessary to designate all elements of the set Z: What is really necessary is the number, µ = minζi∈Z Re ζi, say = Re ζ1, and the spectral property that ζ1 does not allow any generalized eigenspace; the latter is the requirement that no factor of algebraic growth in time is added to the right-hand side of (1.4). In fact, when an algebraic growth is added, the decay property becomes a little worse, and the gain constant (⩾ 1) in (1.4) increases. The above operator L − GW also appears, as a pseudo-substructure, in the stabilization problems of infinite dimensional linear systems such as parabolic systems and/or retarded systems (see, e.g., [16]): These systems are decomposed into two, and understood as composite systems consisting of two states; one belonging to a finite dimensional subspace, and the other to an infinite dimensional one. It is impossible, however, to manage the infinite dimensional substructures. Thus, no matter how precisely the finite dimensional spectrum σ(L − GW) could be assigned, it does not exactly dominate the whole structure of infinite dimension. In other words, the assigned spectrum of finite dimension is not necessarily a subset of the spectrum of the infinite-dimensional feedback control system. In view of the above observations, our aim in this chapter is to develop a new approach much simpler than those in existing literature, which allows us to construct a desired operator G or a set of actuators gk ensuring the decay (1.4) in a simpler and more explicit manner (see (2.10) just below Lemma 2.2). The result is, however, not as sharp as in [70] in the sense that it does not generally provide the precise location of the assigned eigenvalues. From the above viewpoint of infinite-dimensional control theory, however, the result would be meaningful enough, and satisfactory for stabilization. We note that our result exactly coincides with the standard pole assignment theory in the case where we can choose N = 1 (see Proposition 2.3 in Section 2). The results of this chapter are based on those discussed in [48, 51, 52]. Our approach is based on Sylvester’s equation of finite dimension. Sylvester’s equation in infinite-dimensional spaces has also been studied extensively (see, e.g., [6] for equations involving only bounded operators), and even the unboundedness of the given operators are allowed [37, 39, 40, 42 – 45, 47, 49, 50, 53]. Sylvester’s equation in this chapter is of finite dimension, so that there arises no difficulty caused by the complexity of infinite dimension. Its infinite-dimensional version and the properties are discussed later in Chapters 4,
- 20. 4 Theory of Stabilization for Linear Boundary Control Systems 6, and 7. Given a positive integer s and vectors ξk ∈ Hs, 1 ⩽ k ⩽ N, let us consider the following Sylvester’s equation in Hn: XL−MX = −ΞW, Ξ ∈ L (CN ; Hs), where Ξz = N ∑ k=1 zkξk for z = (z1 ... zN)T ∈ CN . (1.5) Here, M denotes a given operator in L (Hs), and ξk vectors to be designed in Hs. A possible solution X would belong to L (Hn; Hs). An approach via Sylvester’s equations is found, e.g., in [7, 10], in which, by setting n = s, a condition for the existence of the bounded inverse X−1 ∈ L (Hn) is sought. Choosing an M such that σ(M) ⊂ C+, it is then proved that L+(X−1 Ξ)W = X−1 MX, σ(X−1 MX) = σ(M) ⊂ C+, the left-hand side of which means a desired perturbed operator. The procedure of its derivation is, however, rather complicated, and the choice of the ξk is unclear. In fact, X−1 might not exist sometimes for some ξk. The approach in this chapter is new and rather different. Let us characterize the operator L in (1.5). There is a set of generalized eigenpairs {λi, φij} with the following properties: (i) σ(L) = {λi; 1 ⩽ i ⩽ ν (⩽ n)}, λi ̸= λj for i ̸= j; and (ii) Lφij = λiφij +∑kj αi jkφik, 1 ⩽ i ⩽ ν, 1 ⩽ j ⩽ mi. Let Pλi be the projector in Hn corresponding to the eigenvalue λi. Then, we see that Pλi u = ∑ mi j=1 uijφij for u ∈ Hn. The restriction of L onto the invariant subspace Pλi Hn is, in the basis {φi1, ..., φimi }, is represented by the mi × mi upper triangular matrix Λi, where Λi|(j,k) = αi k j, j k, λi, j = k, 0, j k. (1.6) If we set Λi = λi +Ni, the matrix Ni is nilpotent, that is, Nmi i = 0. The minimum integer n such that ker Nn i = ker Nn+1 i , denoted as li, is called the ascent of λi −L. It is well known that the ascent li coincides with the order of the pole λi of the resolvent (λ −L)−1. Laurent’s expansion of (λ −L)−1 in a neighborhood of the pole λi ∈ σ(L) is expressed as (λ −L)−1 = li ∑ j=1 K−j (λ −λi)j + ∞ ∑ j=0 (λ −λi)j Kj, where li ⩽ mi, Kj = 1 2πi ∫ |ζ−λi|=δ (ζ −L)−1 (ζ −λi)j+1 dζ, j = 0,±1,±2,.... (1.7)
- 21. Preliminary results - Stabilization of linear systems of finite dimension 5 Note that K−1 = Pλi . The set {φij; 1 ⩽ i ⩽ ν, 1 ⩽ j ⩽ mi} forms a basis for Hn. Each x ∈ Hn is uniquely expressed as x = ∑i,j xijφij. Let T be a bijection, defined as Tx = ( x11 x12 ... xνmν )T . Then, L is identified with the upper triangular matrix Λ; TLT−1 = Λ = diag ( Λ1 Λ2 ... Λν ) . (1.8) Let us turn to the operator M in (1.5). Let { ηij; 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ ℓi } be an orthonormal basis for Hs. Then necessarily s = ∑n i=1 ℓi ⩾ n. Every vector v ∈ Hs is expressed as v = n ∑ i=1 ℓi ∑ j=1 vijηij, where vij = ⟨ v, ηij ⟩ s . Let {µi}n 1=1 be a set of positive numbers such that 0 µ1 ··· µn, and set Mv = n ∑ i=1 ℓi ∑ j=1 µivijηij (1.9) for v = ∑i, j vijηij. It is apparent that (i) σ(M) = {µi}n i=1; and (ii) (µi −M)ηij = 0, 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ ℓi. The operator M is self-adjoint, and potive-definite, ⟨Mv, v⟩s = n ∑ i=1 ℓi ∑ j=1 µi|vij|2 ⩾ µ1∥v∥2 s . Let Qµi be the projector in Hs corresponding to the eigenvalue µi ∈ σ(M), say Qµi v = ∑ ℓi j=1 vijηij for v = ∑i,j vijηij. We put an additional condition on M: σ(L)∩σ(M) = ∅. (1.10) Assuming (1.10), we derive our first result as Proposition 1.1. Since the proof is carried out in exactly the same manner as in [37, 44, 45, 50], it is omitted. Proposition 1.1. Suppose that the condition (1.10) is satisfied. Then, Sylvester’s equation (1.5) admits a unique operator solution X ∈ L (Hn; Hs). The solution X is expressed as Xu = 1 2πi ∫ C (λ −M)−1 ΞW(λ −L)−1 udλ = ∑ λ ∈σ(M) Qλ ΞW(λ −L)−1 u = n ∑ i=1 Qµi ΞW(µi −L)−1 u, (1.11)
- 22. 6 Theory of Stabilization for Linear Boundary Control Systems where C denotes a Jordan contour encircling σ(M) in its inside, with σ(L) outside C. The above first expression is the so called Rosenblum formula [6]. The main results are stated as Theorem 2.1 and Proposition 2.2 in the next section, where a more explicit and concrete expression than ever before of a set of stabilizing actuators gk in (1.3) is obtained. As we see in the next section, an advantage of considering the operator X ∈ L (Hn; Hs) with s ⩾ n is that the bounded inverse (X∗X)−1 is ensured under a reasonable assumption on the operator Ξ. A numerical example is also given. Finally, Proposition 2.3 is stated, where our feedback scheme exactly coincides with the standard pole assignment theory [70] in the case where we can choose N = 1. 1.2 Main Results We assume that σ(L) ∩ C− ̸= ∅, so that the semigroup e−tL, t ⩾ 0, is unstable. We construct suitable actuators gk ∈ Hn in (1.3) such that e−t(L−GW) has a preassigned decay rate, say −µ1 (see (1.9)). The operator ( W WL ... WLn−1 )T belongs to L (Hn; CnN). The observability condition on the pair (W, L) means that the above operator is injective, in other words, ker ( W WL ... WLn−1 )T = {0}. Throughout the section, the separation condition (1.10) is assumed in Sylvester’s equation (1.5). Then, we obtain one of the main results: Theorem 2.1. Assume that the conditions ker ( W WL ... WLn−1 )T = {0}, and ker Qµi Ξ = {0}, 1 ⩽ i ⩽ n (2.1) are satisfied. Then, kerX = {0}. Proof. Let Xu = 0. In view of Proposition 1.1, we see that Qµi ΞW(µi −L)−1 u = 0, 1 ⩽ i ⩽ n. Since ker Qµi Ξ = {0}, 1 ⩽ i ⩽ n, by (2.1), we obtain W(µi −L)−1 u = 0, 1 ⩽ i ⩽ n, or ⟨ (µi −L)−1 u, wk ⟩ n = 0, 1 ⩽ k ⩽ N, 1 ⩽ i ⩽ n. (2.2) Set fk(λ; u) = ⟨ (λ −L)−1u, wk ⟩ n . By recalling that T(λ − L)−1T−1 = (λ − Λ)−1 (see (1.8)), fk(λ; u) is rewritten as ⟨ (λ − Λ)−1Tu, ( T−1 )∗ wk ⟩ Cn . Each element of the n × n matrix (λ − Λ)−1 is a rational function of λ; its denominator consists of a polynomial of order n; and the numerator at most of order n − 1. This means that each fk(λ; u) is a rational function of λ, the
- 23. Preliminary results - Stabilization of linear systems of finite dimension 7 denominator of which is a polynomial of order n, and the numerator of order n − 1. Since the numerator of fk has at least n distinct zeros µi, 1 ⩽ i ⩽ n, by (2.2), we conclude that fk(λ; u) = ⟨ (λ −L)−1 u, wk ⟩ n = 0, λ ∈ ρ(L), 1 ⩽ k ⩽ N. (2.3) Let c be a number such that −c ∈ ρ(L), and set Lc = L+c. In view of the identity (λ −L)−1 = Lc(λ −L)−1 Lc −1 = −Lc −1 +(λ +c)(λ −L)−1 Lc −1 , let us introduce a series of rational functions fl k(λ; u), l = 0, 1, ..., as f0 k (λ; u) = fk(λ; u), fl+1 k (λ; u) = fl k(λ; u) λ +c , l = 0, 1, .... (2.4) It is easily seen that fl k(λ; u) = ⟨ (λ −L)−1 L−l c u, wk ⟩ n − l ∑ i=1 1 (λ +c)i ⟨ L −(l+1−i) c u, wk ⟩ n (2.5) and fl k(λ; u) = 0, λ ∈ ρ(L){−c}, 1 ⩽ k ⩽ N, l ⩾ 0. In view of Laurent’s expansion (1.7) of (λ − L)−1 in a neighborhood of λi, we obtain the relation 0 = fk(λ; u) = li ∑ j=1 ⟨ K−ju, wk ⟩ n (λ −λi)j + ∞ ∑ j=0 (λ −λi)j ⟨ Kju, wk ⟩ n , 1 ⩽ k ⩽ N, in a neighborhood of λi. Calculation of the residue of fk(λ; u) at λi implies that ⟨K−1u, wk⟩n = ⟨ Pλi u, wk ⟩ n = 0, 1 ⩽ i ⩽ ν, 1 ⩽ k ⩽ N, or WPλi u = 0, 1 ⩽ i ⩽ ν. (2.6) As for fl k(λ; u), ℓ ⩾ 1, we have a similar expression in a neighborhood of λi, fl k(λ; u) = li ∑ j=1 ⟨ A−jL−l c u, wk ⟩ n (λ −λi)j + ∞ ∑ j=0 (λ −λi)j ⟨ AjLc −l u, wk ⟩ n − l ∑ i=1 1 (λ +c)i ⟨ L −(l+1−i) c u, wk ⟩ n = 0
- 24. 8 Theory of Stabilization for Linear Boundary Control Systems by (2.5). Note that K−1L−l c u = Pλi L−l c u = L−l c Pλi u. Calculation of the residue of fl k(λ; u) at λi similarly implies that ⟨ K−1L−l c u, wk ⟩ n = ⟨ L−l c Pλi u, wk ⟩ n = 0, 1 ⩽ i ⩽ ν, 1 ⩽ k ⩽ N, or WL−l c Pλi u = 0, 1 ⩽ i ⩽ ν, l ⩾ 1. Combining these with the above relation (2.6), we see that ( W WL−1 c ... WL −(n−1) c )T Pλi u = 0, 1 ⩽ i ⩽ ν. (2.7) It is clear that ker ( W WL ... WLn−1 )T = ker ( W WLc ... WLn−1 c )T , where Lc = L+c. Thus, by the first condition of (2.1), it is easily seen that ker ( W WL−1 c ... WL −(n−1) c )T = ker ( W WL ... WLn−1 )T = {0}. Thus, (2.7) immediately implies that Pλi u = 0 for 1 ⩽ i ⩽ ν, and finally that u = 0. By Theorem 2.1, there is a positive constant such that ∥Xu∥s ⩾ const ∥u∥, ∀u ∈ Hn. The derivation of the above positive lower bound of ∥Xu∥s is due to a specific nature of finite-dimensional spaces. The operator X∗X ∈ L (Hn) is self-adjoint, and positive-definite. In fact, by the relation const ∥u∥2 ⩽ ∥Xu∥2 s = ⟨Xu, Xu⟩s = ⟨X∗ Xu, u⟩n ⩽ ∥X∗ Xu∥∥u∥, we see that ∥X∗Xu∥ ⩾ const ∥u∥. Thus the bounded inverse (X∗X)−1 ∈ L (Hn) exists. We go back to Sylvester’s equation (1.5). Setting X∗X = X ∈ L (Hn) and X∗MX = M ∈ L (Hn), we obtain the relation, L−(X∗ X)−1 X∗ MX = −(X∗ X)−1 X∗ ΞW or L+ N ∑ k=1 ⟨·, wk⟩n X −1 X∗ ξk = X −1 M . (2.8) Both operators X and M are self-adjoint, but X −1M is not. The following assertion is the second of our main results, and leads to a stabilization result: Proposition 2.2. Assume that (2.1) is satisfied. Then, σ(X −1M ) is contained in R1 +. Actually, λ∗ = min σ(X −1 M ) ⩾ µ1. (2.9)
- 25. Preliminary results - Stabilization of linear systems of finite dimension 9 In addition, there is no generalized eigenspace for any λ ∈ σ(X −1M ). Remark: By Proposition 2.2, we obtain a decay estimate exp ( −t ( L+(X∗ X)−1 X∗ ΞW )) = e−t(X −1M ) ⩽ const e−µ1t , t ⩾ 0. (2.10) In fact, the last assertion of the proposition ensures that no algebraic growth in time arises in the semigroup, regarding the smallest eigenvalue. Thus, a set of actuators gk = −(X∗X)−1X∗ξk, 1 ⩽ k ⩽ N, in other words, G = −(X∗X)−1X∗Ξ explicitly gives a desired set of actuators in (1.3). Proof of Proposition 2.2. Since X is positive-definite, we can find a non- unique bijection U ∈ L (Hn) such that X = X∗ X = U ∗ U , (2.11) the so called Cholesky factorization. Let us define M ′ = (U ∗ )−1 M U −1 = (U −1 )∗ M U −1 . Then, M ′ ∈ L (Hn) is a self-adjoint operator, enjoying some properties similar to those of X −1M . In fact, let λ ∈ σ(X −1M ), or (λX −M )u = 0 for some u ̸= 0. Then, since 0 = (λU ∗ U −M )u = U ∗ ( λ −(U ∗ )−1 M U −1 ) U u = U ∗ ( λ −M ′ ) U u = 0, we see that λ belongs to σ(M ′). The converse relation is also correct, which means that σ(X −1 M ) = σ(M ′ ) ⊂ R1 . (2.12) Inequality (2.9) is achieved by applying the well known min-max principle [11] to M ′, or more directly by the following observation: Let λ ∈ σ(X −1M ), and (λX −M )u = 0 for some u ̸= 0. Then λ ∥Xu∥2 s = λ ⟨X u, u⟩n = ⟨M u, u⟩n = ⟨MXu, Xu⟩s ⩾ µ1∥Xu∥2 s , from which (2.9) immediately follows, since Xu ̸= 0. Next let us show that there is no generalized eigenspace for λ ∈ σ(X −1M ). Let (λ − X −1M )2u = 0 for some u ̸= 0. Setting v = (λ −X −1M )u, we calculate 0 = X (λ −X −1 M )2 u = (λX −M )v = (λU ∗ U −M )v = U ∗ ( λ −(U ∗ )−1 M U −1 ) U v = U ∗ ( λ −M ′ ) w = 0, w = U v,
- 26. 10 Theory of Stabilization for Linear Boundary Control Systems or (λ −M ′)w = 0. On the other hand, since w = U v = U (λ −X −1 M )u = U (λ −U −1 (U ∗ )−1 M )u = (λ −(U ∗ )−1 M U −1 )U u = (λ −M ′ )U u, we see that 0 = ( λ −M ′ ) w = ( λ −M ′ )2 U u, U u ̸= 0. But, M ′ is self-adjoint, so that there is no generalized eigenspace for λ ∈ σ(M ′). Thus, U u turns out to be an eigenvector of M ′ for λ, and 0 = U ∗ (λ −M ′ )U u = U ∗ (λ −(U ∗ )−1 M U −1 )U u = (λU ∗ U −M )u = (λX −M )u. This means that u is an eigenvector of X −1M for λ. The following example shows that λ∗ = min σ(X −1M ) does not generally coincide with the prescribed µ1. Example: Let n = 3, and set H3 = C3, so that L is a 3×3 matrix. Let L = diag ( a a b ) , where a, b ⩽ 0 and a ̸= b. Since n = 3, ν = 2, m1 = 2, and m2 = 1, we choose N = 2, s = 6, H6 = C6, and ℓ1 = ℓ2 = ℓ3 = 2. As for the operator W ∈ L (C3; C2), let us consider the case, for example, where w1 = (1 0 1)T and w2 = (0 1 0)T. The operator W is a 2 × 3 matrix given by ( 1 0 1 0 1 0 ) . The pair (W, L) is then observable, and the first condition of (2.1) is satisfied. To consider Sylvester’s equation (1.5), let {ηij; 1 ⩽ i ⩽ 3, j = 1, 2} be a standard basis for C6 such that η11 = (1 0 0 ... 0)T, η12 = (0 1 0 ... 0)T, η21 = (0 0 1 ... 0)T, ..., and η32 = (0 ... 0 1)T. Set M = diag (µ1 µ1 µ2 µ2 µ3 µ3) for 0 µ1 µ2 µ3. In the operator Ξ given by Ξu = u1ξ1 +u2ξ2 for u = (u1 u2)T ∈ C2 , set ξ1 = (1 0 1 0 1 0)T and ξ2 = (0 1 0 1 0 1)T. Then, we see that ker Qµi Ξ = {0}, 1 ⩽ i ⩽ 3, and the second condition of (2.1) is satisfied. The unique solution X ∈ L (C3; C6) to Sylvester’s equation (1.5) is a 6×3 matrix described as (u = (u11 u12 u21)T ∈ C3)
- 27. Preliminary results - Stabilization of linear systems of finite dimension 11 Xu = ⟨ (µ1 −L)−1u, w1 ⟩ 3 ⟨ (µ1 −L)−1u, w2 ⟩ 3 ⟨ (µ2 −L)−1u, w1 ⟩ 3 ⟨ (µ2 −L)−1u, w2 ⟩ 3 ⟨ (µ3 −L)−1u, w1 ⟩ 3 ⟨ (µ3 −L)−1u, w2 ⟩ 3 = 1 µ1 −a 0 1 µ1 −b 0 1 µ1 −a 0 1 µ2 −a 0 1 µ2 −b 0 1 µ2 −a 0 1 µ3 −a 0 1 µ3 −b 0 1 µ3 −a 0 u11 u12 u21 , (2.13) where ⟨·, ·⟩3 denotes the inner product in C3. Setting, for computational convenience, α = ( 1 µ1 −a 1 µ2 −a 1 µ3 −a )T , β = ( 1 µ1 −b 1 µ2 −b 1 µ3 −b )T , and 1 = (1 1 1)T , we see that (X∗ X)−1 = 1 γ |β|2 0 −⟨α, β⟩3 0 |β|2 −⟨α, β⟩3 2 /|α|2 0 −⟨α, β⟩3 0 |α|2 , where γ = |α|2 |β|2 − ⟨α, β⟩3 2 . By noting that X∗ξ1 = ( ⟨α, 1⟩3 0 ⟨β, 1⟩3 )T and X∗ξ2 = ( 0 ⟨α, 1⟩3 0 )T , the matrix L + (X∗X)−1X∗ΞW is concretely described as diag ( a a b ) + 1 γ × |β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3 0 |β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3 0 ⟨α, 1⟩3 ( |β|2 − ⟨α, β⟩3 2 |α|2 ) 0 |α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 0 |α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 . It is apparent that one of the eigenvalues of this matrix is the (2,2)-element: a+ ⟨α, 1⟩3 γ ( |β|2 − ⟨α, β⟩3 2 |α|2 ) = a+ ⟨α, 1⟩3 |α|2 , and is certainly greater than µ1. Note that 0 λ∗ − µ1 ⩽ 1 |α|2 ( µ2 − µ1 (µ2 −a)2 + µ3 − µ1 (µ3 −a)2 ) → 0, µ2, µ3 → ∞.
- 28. 12 Theory of Stabilization for Linear Boundary Control Systems The other eigenvalues are those of the matrix, 1 γ ( |β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3 +γa |β|2 ⟨α, 1⟩3 −⟨α, β⟩3 ⟨β, 1⟩3 |α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 |α|2 ⟨β, 1⟩3 −⟨α, β⟩3 ⟨α, 1⟩3 +γb ) . (2.14) To see that these eigenvalues are generally greater than µ1, let us consider a numerical example: Let (µ1 µ2 µ3) = (2 3 4), a = 0, and b = −1. Then, α = ( 1 2 1 3 1 4 )T , β = ( 1 3 1 4 1 5 )T , |α|2 = 61 144 , |β|2 = 769 3600 , ⟨α, β⟩3 = 3 10 , ⟨α, 1⟩3 = 13 12 , ⟨β, 1⟩3 = 47 60 , γ = |α|2 |β|2 −⟨α, β⟩3 2 = 253 518400 . One of the eigenvalues a+⟨α, 1⟩3 /|α|2 is 156/61 2 (= µ1). The matrix (2.14) is then 1 253 ( −1860 −1860 3540 3287 ) , the eigenvalues of which are denoted as ζ1 and ζ2. Then, µ1 = 2 ζ1 156/61 ζ2, and thus λ∗ = ζ1 2. We close this section with the following remark: There is a case where λ∗ coincides with µ1. Following [52], let us consider (1.3) in the space Hn = Cn (see (1.8)). All operators L, G, and W are then matrices of respective size. Let σ(L) consist only of simple eigenvalues, so that mi = 1, 1 ⩽ i ⩽ n, and n = ν. Thus we can choose N = 1, ℓi = 1, 1 ⩽ i ⩽ n, and thus s = n. The operator in (2.10) is written as L + (X∗X)−1X∗ΞW, where Ξu = uξ for u ∈ C1, and W = ⟨·, w⟩n, w = ( w1 w2 ... wn )T ∈ Cn. The observability condition then turns out to be wi ̸= 0, 1 ⩽ i ⩽ n. Let us consider Sylvester’s equation (1.5) in Hs = Cn. By setting ξ = ( 1 1 ... 1 )T ∈ Cn, the solution X to (1.5) is an n×n matrix, and has a bounded inverse: X = ΦW̃, (2.15) where Φ = ( 1 µi −λj ; i ↓ 1, ..., n j → 1, ..., n ) , and W̃ = diag ( w1 w2 ... wn ) . Thus, L + (X∗X)−1X∗ΞW = L + X−1ξ wT. It is shown [52] that, given a set {µi}1⩽i⩽n, there is a unique g ∈ Cn such that σ ( L−gwT ) = {µi}1⩽i⩽n, and that g is concretely expressed as
- 29. Preliminary results - Stabilization of linear systems of finite dimension 13 g = g1 g2 g3 . . . gn = 1 ∆ 1 w1 ∆1 f(λ1) − 1 w2 ∆2 f(λ2) 1 w3 ∆3 f(λ3) . . . (−1)n−1 1 wn ∆n f(λn) , (2.16) where f(λ) = n ∏ i=1 (λ − µi), and ∆ = ∏ 1⩽ij⩽n (λi −λj), ∆k = ∏ 1⩽ij⩽n, i, j̸=k (λi −λj), 1 ⩽ k ⩽ n. The proof will be given later in Section 4. Proposition 2.3. Suppose in Proposition 2.2 that σ(L) consists only of simple eigenvalues. Set ξ = (1 1 ... 1)T as above. Then X−1ξ = −g, and thus λ∗ = µ1. In fact, we have σ ( L+(X∗X)−1X∗ΞW ) = {µi}1⩽i⩽n. Proof. The relation, X−1ξ = −g is rewritten as −∆ 1 1 1 . . . 1 = ΦW̃ 1 w1 ∆1 f(λ1) − 1 w2 ∆2 f(λ2) 1 w3 ∆3 f(λ3) . . . (−1)n−1 1 wn ∆n f(λn) = Φ ∆1 f(λ1) −∆2 f(λ2) ∆3 f(λ3) . . . (−1)n−1∆n f(λn) . In other words, we show that − n ∑ j=1 (−1)j−1∆j f(λj) µi −λj = n ∑ j=1 (−1)j−1 ∆j ( =λn−1 j +··· ) z }| { ∏ 1⩽ℓ⩽n, ℓ̸=i (λj − µℓ) = ∆, 1 ⩽ i ⩽ n. (2.17) The left-hand side of (2.17), a polynomial of λi, 1 ⩽ i ⩽ n, is in particular a polynomial of λ1 of order n − 1, and the coefficient of λn−1 1 is ∆1 = ∏2⩽ij⩽n(λi − λj). For j k, let us compare the jth and the kth terms. The following lemma is elementary:
- 30. Another Random Scribd Document with Unrelated Content
- 31. bottle decanted, was, from time to time, denominated a FELLOW- COMMONER. 1867. Collins, The Public Schools, p. 26. Thomas Middleton petitions King Charles, on his restoration, to grant his royal letters to the Winchester electors in favour of his son’s admittance “as a child in Winchester College, where he has now spent three years as FELLOW-COMMONER.” 1891. Harry Fludyer at Cambridge, 38. She said she had heard from her cousin, who is, I think, a FELLOW-COMMONER, or something of that sort, at Downing College, that Harry is one of the most popular men at Cambridge. Feoffee, subs. (Manchester Grammar: obsolete).—The original name for the trustees in whose hands the foundation estate was placed by Hugh Bexwycke. [A.S. feo = fee or inheritance.] Ferk. See Firk. Ferula, subs. (Stonyhurst).—See Tolly. Festive, adj. (Charterhouse).—Said of a boy who has not learned his duty to his superiors and seniors.
- 32. Fez, subs. (Harrow).—The equivalent of the Cap (q.v.) for cricket: the FEZ being given to the House Eleven for distinction at football. Field, verb. 1. (Winchester).—To take care of; to support: in swimming. 2. (Harrow).—See Lick. 3. (Eton).—See Wall. The Field, subs. (Sherborne).—See Fields. Fields, subs. (Sherborne: obsolete).—The playing-ground: seventeenth century. The modern term is “The Field,” though there are five separate grounds. Fifteens, subs. (Winchester).—A football match. See Six-and-six. Fifty, The, subs. (Tonbridge).—The chief football ground; the next immediately below it is the Middle Fifty, then the Lower Fifty, and the Fourth Fifty. Cf. Hundred, which is now obsolete.
- 33. Fighting-green, subs. (Westminster).—The old battle-ground in the western cloister. Figures, subs. (Stonyhurst).—The Second Form: formerly Great Figures. See Little Figures. Fin, intj. (Christ’s Hospital).—A form of negative. Ex. “FIN the small court” = “I won’t have, c.” [Lat. fend.] See Fains. Find, subs. (Harrow).—A mess of, usually, two upper boys which takes breakfast and tea in the rooms of one or other of the set: a privilege of the Sixth Form. Whence FIND-FAG = a fag who lays the table for the upper boys. [Find (dial.) = to supply; to supply with provisions.] Also as verb. 1867. Collins, The Public Schools, p. 316. Immediately a certain number of rolls (FINDS they were called—etymology unknown) were ordered at the baker’s, and were rebaked every morning until they were pretty nearly as hard as pebbles. At nine o’clock on the morning fixed for the rolling in, the members of the hall ranged themselves on the long table which ran along one side of the room, each with his pile of these rolls before him, and a fag to pick them up.
- 34. Finder, subs. (Oxford).—A waiter. Finjy! intj. (Winchester).—An exclamation excusing one from participation in an unpleasant or unacceptable task, which he who says the word last has to undertake. Cf. Fains. Firk (or Ferk), verb (Winchester).—To proceed; to hasten; to expel; to send; to drive away. [O.E. fercian.] Also TO FIRK UP and TO FIRK DOWN. 1283. William of Palerne. Thei bisiliche fondede (tried) fast to FERKE him forthward. c. 1400. Troy Book. I you helpe shall the flese for to fecche, and FERKE it away. [?] MS. Lincoln, Morte Arthure, f. 79. The Kyng FERKES furthe on a faire stede. 1599. Shakspeare, Henry V., iv. 4. Pistol. I’ll fer him, and FIRK him, and ferret him, discuss the same in French unto him. Boy. I do not know the French for fer and ferret and FIRK. 1611. Barry, Ram Alley [Dodsley, Old Plays (Reed), v. 466]. Nay, I will FIRK my silly novice, as he was never FIRK’D Since mid-wives bound his noddle. 1640. Brome, Antipodes. As tumblers do ... by FIRKING up their breeches.
- 35. 1795. Sewell, Hist. of Quakers. At this the judge said, “Take him away: prevaricator! I’ll FERK him.” Five, The, subs. (Charterhouse).—The Five bell. Flannels, subs. (Harrow).—The members of either School Eleven. 1899. Public School Mag., Dec., p. 446. Up to the present the eleven have won four matches and lost one, while Monro, Cookson, Wyckoff, and Borwick have all received their FLANNELS. Flat, subs. (Royal High School, Edin.).—An objectionable person; a “bounder.” [A misuse of flat = fool.] Fleshy, subs. (Winchester).—A thick cut out of the middle of a shoulder of mutton. See Dispar. Flies. Squashed flies, subs. (Durham: obsolete).—Biscuits with currants.
- 36. Floor, verb (general).—To pluck; to plough. Also = to master; to prove oneself superior to the occasion: e.g. TO FLOOR A PAPER, LESSON, EXAMINATION, EXAMINER, c. Cf. Bowl; Throw. 1852. Bristed, Five Years in an English University, p. 12. Somehow I nearly FLOORED the paper. 1853. Bradley, Verdant Green, iv. Mr. Filcher thoroughly understood the science of “FLOORING” a freshman. 1861. Hughes, Tom Brown at Oxford. I’ve FLOORED my Little Go. 1891. Harry Fludyer at Cambridge, 98. These blessed exams. are getting awfully close now, but I think I shall FLOOR mine. Fluke, verb (general).—To shirk. 1864. Eton School-Days, ch. xvi. p. 203. “By Jove! I think I shall FLUKE doing Verses; I should like to see Paddy drive tandem through College,” said Butler Burke. Flyer, subs. (Winchester).—A half-volley at football. A MADE-FLYER is when the bound of the ball is gained from a previous kick, by the same side, against canvas or any other obstacle, or is dropped, as in a “drop-kick.” This is now confused with a “kick-up.”— Wrench.
- 37. Flying-man, subs. (Eton).—The boy who stands behind the “bully,” and either runs down, or kicks hard, as may be required. 1864. Eton School-Days, ch. xxiii. p. 255. He possessed good wind, and was a very good “kick-off,” and he could “bully” a ball as well as any one. He was a little too heavy for FLYING-MAN, but he made a decent “sidepost,” and now and then he officiated as “corner.” Fobs, subs. (Durham: obsolete).—Boiled bread and milk. Footer, subs. (Harrow).—(1) Football; (2) a player of football according to Rugby rules; and (3) the ball itself. 1890. Great Public Schools, p. 96. Directly after the goose match (Michaelmas Day) FOOTER proper begins, and is the principal game played at the school during the Christmas term. The game as played at Harrow differs considerably from the game as played at Eton and other schools, and has distinct rules of its own; it may be said to be more like the Association game than any other. 1896. Felstedian, Nov., p. 139. H. H. H. who wants to have a “second” FOOTER shirt. Footer-hill, The (Harrow).—The hill from the football-fields and DUCKER (q.v.).
- 38. For, phr. (Tonbridge).—A form of ridicule: e.g. “first eleven FOR one” would be used in jeering at a boy who had recently obtained his colours. Forakers (or Foricus), subs. (Winchester).— The water-closet. [Formerly foricus, and probably a corruption of foricas, an English plural of the Latin forica.] Force. Out by force, phr. (Stonyhurst).—Of a football when it goes out from two opposite players at the same time. Founders, subs. (Winchester).—Boys who proved their descent from the Founder, and were afterwards elected (by rote among the Electors) as such. Only two were admitted each year, and only two were sent to New College, but these two were put at the head of the Roll (q.v.) whatever their previous position in Sixth Book (q.v.) might have been. They were not obliged to leave at the age of eighteen, as the other boys were, but were allowed to remain till they were
- 39. twenty-five. They were supposed to have particularly thick skulls.—Mansfield (c. 1840). Founder’s-Com., subs. (Winchester).—The four days on which there were festivals in commemoration of the Founder, when there was Amen-chapel (q.v.); the Fellows and Masters gave a dinner in Common-room, and the Founders (q.v.) received a sovereign each.—Mansfield (c. 1840). Founder’s-day, subs. (Harrow).—The 3rd of October, the anniversary of the death of John Lyon: usually kept on the nearest Thursday to the date in question. Founder’s-kin, subs. (various).—Those, who at Winchester, Harrow, c., could show descent from William of Wykeham or John Lyon, c., as the case might be, and who were entitled to priority of election on the foundation. 1867. Collins, The Public Schools, p. 32. The preference assigned to FOUNDER’S-KIN in the election soon brought into the field, as may be supposed, young Wykehams and Williamses from all quarters, with others who proved more or less satisfactorily their connection
- 40. with the founder’s family; and gradually the customs obtained of electing two only of these favoured candidates at the head of the roll for admission, and filling up the remaining vacancies by a process of successive nominations by each of the six electors, the Warden of New College having the first turn, until the number of vacancies was supplied. Founder’s-Ob., subs. (Winchester).—The anniversary of the Founder’s death. Four-holed Middlings, subs. phr. (Winchester: obsolete).—Ordinary walking shoes. Cf. Beeswaxers. Fourth, subs. (Cambridge).—A rear or jakes. [Origin uncertain; said to have been first used at St. John’s or Trinity, where the closets are situated in the Fourth Court. Whatever its derivation, the term is now the only one in use at Cambridge, and is frequently heard outside the university.] The verbal phrase is TO KEEP A FOURTH. Fourth Book, subs. (Winchester: obsolete).— All the boys below Junior Part the Fifth. See Books.
- 41. Fourth Former (Harrow).—The oldest form room in the Old Schools: now used for morning prayer by those who go to the Old Schools, and also as the head-master’s torture-chamber. Fourth of June (Eton).—See quot. 1865. Etoniana, p. 166. Since the glories of Montem have departed, the fourth of June procession has taken its place as the great yearly festival of Etonians. It was instituted in commemoration of a visit of King George III., and is held on his birthday. It is the great trysting day of Eton, when her sons gather from far and wide, young and old, great and small,—no matter who or what, so long as they are old Etonians; that magic bond binds them all together as brothers, and levels for the time all distinctions of age or rank. The proceedings begin with the ‘speeches’ delivered in the upper school at twelve o’clock before the provost, fellows, masters, and a large audience of the boys’ friends. Selections from classical authors, ancient or modern, are recited by the Sixth-form boys, who are dressed for the occasion in black swallow-tail coats, white ties, black knee-breeches and buckles, silk stockings, and pumps. Then follows the provost’s luncheon, given in the college hall to the distinguished visitors, while similar entertainments on a smaller scale are going on in the various tutors’ and dames’ houses. At 3 o’clock there is full choral service in chapel. At 6 o’clock all hands adjourn to the Brocas, a large open meadow, to witness the great event of the day,—the procession of the Boats to Surly Hall, a public-house of that name, on the right bank of the river, some three and a half miles from Windsor. The boats are divided into two classes—Upper and Lower. The Upper division consists of the Monarch ten-oar, the Victory, and the Prince of Wales, or, as it is more usually called, the Third
- 42. Upper. The Lower boats are the Britannia, Dreadnought, Thetis, and St. George; sometimes, when the number of aspirants to a place is larger than usual, an eighth boat called the Defiance is added. The collegers have also for some years put on a four-oar— latterly expanded into an eight—which follows in the procession. The flotilla is preceded by the Eton racing eight-oar, manned by the picked crew who are to contend at Putney or Henley. Each boat has its distinctive uniform. Formerly these were very fanciful —Greek pirates, or galley slaves in silver chains, astonishing the quiet reaches of the Thames for the day. The crews of the Upper boats now wear dark blue jackets and trousers, and straw hats with ribbons, displaying the name of the boat in gold letters. The coxswains are dressed in an admiral’s uniform, with gold fittings, sword, and cocked-hat. The captain of each boat has an anchor and crown embroidered in gold on the left sleeve of his jacket. In the Lower boats, the crews wear trousers of white jean, and all ornaments and embroidery are in silver. Each boat carries a large silk flag in the stern. The procession is headed by a quaint old- fashioned boat (an Eton racing boat of primitive days) rowed by watermen and conveying a military band. The Westminster eight always receives an invitation to this celebration, and occasionally makes its appearance on the river, adding very much to the interest of the procession.... Opposite to Surly Hall, a liberal display of good things ... awaits the arrival of the crews—the Sixth Form alone being accommodated with a tent. After a few toasts, and as much champagne as can be fairly disposed of in a short time, the captain of the boat gives the word for all to re-embark, and the flotilla returns to Eton in the same order.... Singing, shouting, racing, and bumping, all go on together in the most harmonious confusion.... The boats, after their return through Windsor Bridge, turn and row two or three times round an eyot in the middle of the stream above the bridge. During this time a grand display of fireworks takes place on the eyot. The ringing of the fine old bells in the Curfew Tower, the cheering of the crews, and the brilliant coloured fires which strike across the water, and light up the dense masses of spectators along the bridge, the rafts,
- 43. and the shore, produce an effect not easily forgotten. A pyrotechnic illumination of the College arms concludes the ceremonies, and is the signal for the crews to land and march in jubilant disorder back to College. Fox-and-dowdy, subs. (King Edward’s, Birm.: obsolete).—See Action. Fragment, subs. (Winchester: obsolete).—A dinner for six (served in College Hall, after the ordinary dinner), ordered by a Fellow in favour of a particular boy, who was at liberty to invite five others to join him. A fragment was supposed to consist of three dishes.—Winchester Word-Book [1891]. Free, adj. (Oxford).—Impudent; self- possessed. 1864. Tennyson, Northern Farmer (Old Style), line 25. But parson a coomes an’ a goos, an’ a says it eäsy an’ FREEÄ. Freed, adv. (Stonyhurst).—Of an extra recreation: given for some special reason.
- 44. Fresh, adj. (University).—Said of an undergraduate in his first term. 1803. Gradus ad Cantabrigiam, s.v. 1866. Trevelyan, Horace at Athens. When you and I were FRESH. Fresher. See Freshman. Freshers. The Freshers, subs. (Cambridge). That part of the Cam which lies between the Mill and Byron’s Pool. So called because it is frequented by FRESHMEN (q.v.). Fresh-herring, subs. (King Edward’s, Birm.). —A boy newly admitted to the school. Such a one is seized on his first or first few visits to the playground, and conveyed to a corner —a MONKEY DEN—where he is more or less forcibly SQUABBED (q.v.) against the wall by as many persecutors as can get at him. The incongruity of fresh-herrings in a monkey- den does not seem to be remarked. But twenty-five to thirty years ago FRESH-HERRINGS were hurled over the Precipice. This was a drop of some six or eight feet from the general level of the playground over a
- 45. retaining wall to the bottom of an incline up which coal-stores, c., could be brought into the playground. The new science and art rooms have covered the site, and MONKEY- DEN has superseded the terrors of this local Tarpeian Rock. The FRESH-HERRING is always told that he must bring beeswax and turpentine for the purpose of polishing his desk, and he not infrequently comes armed with this or some other form of furniture- polish, to the glee of the “stuffer-up.” Freshman (or Fresher), subs. (University).— A University man during his first year. In Dublin University he is a JUNIOR FRESHMAN during his first year, and a SENIOR FRESHMAN the second year. At Oxford the title lasts for the first term. See Soph. 1596. Nashe, Saffron Walden, in Works, iii. 8. When he was but yet a FRESHMAN in Cambridge. 1611. Middleton, Roaring Girl, Act iii. sc. 3. S. Alex. Then he’s a graduate. S. Davy. Say they trust him not. S. Alex. Then is he held a FRESHMAN and a sot. 1650. Howell, Familiar Letters [Nares]. I am but a FRESHMAN yet in France, therefore I can send you no news, but that all is here quiet, and ’tis no ordinary news, that the French should be quiet. 1671. Cotgrave, Wit’s Interpreter, p. 221. First, if thou art a FRESHMAN, and art bent To bear love’s arms, and follow Cupid’s tent.
- 46. 1767. Colman, Oxonian in Town, ii. 3. And now I find you as dull and melancholy as a FRESHMAN at college after a jobation. 1841. Lever, Charles O’Malley, ch. xiv. “This is his third year,” said the Doctor, “and he is only a FRESHMAN, having lost every examination.” 1853. Bradley, Verdant Green, iii. Mr. Green saw at a glance that all the passengers were Oxford men, dressed in every variety of Oxford fashion, and exhibiting a pleasing diversity of Oxford manners. Their private remarks on the two new-comers were, like stage “asides,” perfectly audible. “Decided case of governor!” said one. “Undoubted ditto of FRESHMAN!” observed another. 1891. Harry Fludyer at Cambridge, 55. A lot of FRESHMEN got together after Hall (it was a Saints’ day, and they’d been drinking audit) and went and made hay in Marling’s rooms. 1891. Sporting Life, Mar. 20. The mile, bar accidents, will be a gift to B. C. Allen, of Corpus, who has more than maintained the reputation he gained as a FRESHER. 1895. Felstedian, Dec., 178. The new trousers and immaculate brown boots of the “FRESHER” are suffering terribly from the slush. 1898. Stonyhurst Mag., Dec., p. 149, “Life at Oxford.” Three Seniors were entertaining some fifteen or more FRESHERS. Adj. (University).—Of, or pertaining to, a FRESHMAN, or a first year student. Freshman’s Bible, subs. phr. (University).— The University Calendar.
- 47. Freshman’s Church, subs. phr. (Cambridge). —The Pitt Press. [From its ecclesiastical architecture.] Freshman’s Landmark, subs. phr. (Cambridge).—King’s College Chapel. [From the situation.] Freshmanship, subs. (old).—Of the quality or state of being a freshman. 1605. Jonson, Volpone, or the Fox, iv. 3. Well, wise Sir Pol., since you have practised thus, Upon my FRESHMANSHIP, I’ll try your salt- head With what proof it is against a counter-plot. Froust, subs. (Harrow).—1. Extra sleep allowed on Sunday mornings and whole holidays. Also (2) an easy-chair. Hence Frouster. Frout, adj. (Winchester).—Angry; vexed. Fudge, subs. 1. (Christ’s Hospital).—To copy; to crib; to dodge or escape: also see quot.
- 48. 1870-95. More Gleanings from The Blue. The Latin Grammar was a strange book to the new boy; he says he was “relieved from embarrassment by the readiness with which my schoolfellows in the class above assisted in explaining,” c. c.; so a “FUDGE” is not a modern invention, though it is expressed by a polite periphrasis. 1877. The Blue-Coat Boys, p. 97. Fudge, to prompt a fellow in class, or prompt oneself in class artificially. Thence to tell: e.g. “FUDGE me what the time is.” 2. (common).—To advance the hand unfairly at marbles. Fug, subs. (Harrow).—1. A small soft football. Also (2) the game as played with such a ball in a yard, house, c. See Appendix. Verb. 1. (Shrewsbury).—To stay in a stuffy room. 2. (Harrow).—To stop indoors. Fug-footer, subs. (Harrow).—A species of football played in passages with a FUG (q.v.) See ante. Fuggy, subs. (general).—A hot roll. Adj. (Shrewsbury).—Stuffy.
- 49. Fug-shop, subs. (Charterhouse).—The carpenter’s shop. Functior (or Functure), subs. (Winchester).— An iron bracket candlestick, used for the night-light in College Chambers. c. 1840. Mansfield, School-Life at Winchester, p. 68. Beside the window yawned the great fireplace, with its dogs, on which rested the faggots and bars for the reception of the array of boilers. Above it was a rushlight, fixed in a circular iron pan fastened to a staple in the wall; it was called the FUNCTIOR. 1891. Wrench, Winchester Word-Book, s.v. Functure. The word looks like fulctura, an earlier form of fulture, meaning a prop or stay, with phonetic change of l into n. Funking-Monday, subs. (Christ’s Hospital).— See quot. 1887. The Blue, Nov. Yet it is not from ignorance of vulgar slang that the author’s elegance springs, for he unbends once so far as to say that the Monday after the holidays is called “FUNKING-MONDAY.” Funking-room, subs. (medical).—The room at the Royal College of Surgeons where students collect on the last evening of their final during the addition of their marks, and whence each is summoned by an official announcing failure or success.
- 50. 1841. Punch, i. p. 225, col. 2. On the top of a staircase he enters a room, wherein the partners of his misery are collected. It is a long, narrow apartment, commonly known as the FUNKING-ROOM. Funkster, subs. (Winchester).—A coward. Furk. See Firk.
- 51. ag, subs. 1. (Christ’s Hospital).—See quot. Gag-eater = a term of reproach. 1813. Lamb, Christ’s Hospital, in Works, p. 324 (ed. 1852). L. has recorded the repugnance of the school to GAGS, or the fat of fresh beef boiled; and sets it down to some superstition.... A GAG-EATER in our time was equivalent to a ghoul, ... and held in equal estimation. 2. (Winchester: obsolete).—An exercise (said to have been invented by Dr. Gabell) which consists in writing Latin criticisms on some celebrated piece, in a book sent in about once a month. In the Parts below Sixth Book and Senior Part, the GAGS consisted in historical analysis. [An abbreviation of “gathering.”] c. 1840. Mansfield, School-Life at Winchester College, p. 108. From time to time, also, they had to write ... an analysis of some historical work; these productions were called GATHERINGS (or GAGS). Gain. See Election.
- 52. Gaits (Geits, Gytes, or Gites), subs. (Royal High School, Edin.).—The first, or lowest class. See Cats. Gallery, subs. (Winchester).—A Commoner bedroom. [From a tradition of GALLERIES in Commoners.] Hence GALLERY NYMPH = a housemaid. Gang, subs. (Felsted: obsolete).—A particular friend. From the ordinary meaning of the word, applied first to the two friends, then to each of them. Used only of “acute” friendship. Also as verb = to carry on such a friendship with another. Garden, The (Stonyhurst).—The playgrounds, built on the site of part of the old garden, long kept this name. “The boys went to the GARDEN” = “into the playground”: obsolete. Gater, subs. (Winchester: obsolete).—A plunge head foremost into a POT (q.v.).
- 53. Gates, subs. (University).—The being forbidden to pass outside the gate of a college. Hence as verb = to confine wholly or during certain hours within the college gate for some infraction of discipline. To BREAK GATES = to stay out of college after hours. Gate-bill (old) = the record of an undergraduate’s failure to be within the precincts of his college by a specified time at night. 1803. Gradus ad Cant., p. 128. To avoid GATE-BILLS he will be out at night as late as he pleases ... climb over the college wall, and fee his gyp well. 1835. The Snobiad (Whibley, Cap and Gown, p. 141). Two proctors kindly holding either arm Staunch the dark blood and GATE him for the term. 1853. Bradley, Verdant Green, I. ch. xii. He won’t hurt you much, Giglamps! Gate and chapel you! 1861. Hughes, Tom Brown at Oxford, ch. xii. Now you’ll both be GATED probably, and the whole crew will be thrown out of gear. 1865. Cornhill Mag., p. 227. He is requested to confine himself to college after a specified hour, which is familiarly termed being GATED. 1870. Morning Advertiser, May 23. The two least culpable of the party have been GATED. 1881. Lang, Xxxii. Ballades, “Of Midsummer Term.” When freshmen are careless of GATES.
- 54. To be at gates, verb. phr. (Winchester: obsolete).—To assemble in Seventh Chamber passage, preparatory to going Hills or Cathedral. Gaudeamus, subs. (general).—A feast; a drinking bout; any sort of merry-making. [German students’, but now general. From the first word of the mediæval (students’) ditty.] Gaudy (or Gaudy-day), subs. (general).—A feast or entertainment: specifically, the annual dinner of the Fellows of a college in memory of founders or benefactors; or a festival of the Inns of Court. [Lat. gaudere = to rejoice.] 1540. Palsgrave, Acolastus [Halliwell]. We maye make our tryumphe, kepe our GAUDYES, or let us sette the cocke on the hope, and make good chere within dores. Ibid., I have good cause to set the cocke on the hope, and make GAUDYE chere. 1608. Shakspeare, Antony and Cleopatra, iii. 11. Come, Let’s have one other GAUDY night; call to me All my sad captains; fill our bowls; once more Let’s mock the midnight bell. 1636. Suckling, Goblins [Dodsley, Old Plays (Reed), x. 143]. A foolish utensil of state, Which, like old plate upon a GAUDY day, ’s brought forth to make a show, and that is all.
- 55. 1724. E. Coles, Eng. Dict. Gaudy days, college or Inns of Court festivals. 1754. B. Martin, Eng. Dict., 2nd ed. Gaudies, double commons, such as they have on GAUDY or grand DAYS in colleges. 1760. Foote, Minor, Act i. Dine at twelve, and regale, upon a GAUDY DAY, with buns and beer at Islington. 1803. Gradus ad Cantab., p. 122. Cut lectures ... give GAUDIES and spreads. 1820. Lamb, Elia (Oxford in the Vacation). Methought I a little grudged at the coalition of the better Jude with Simon—clubbing, as it were, their sanctities together, to make up one poor GAUDY-DAY between them. 1822. Nares, Glossary, s.v. Gaudy day or Night. A time of festivity and rejoicing. The expression is yet fully retained in the University of Oxford. Blount, in his Glossographia, speaks of a foolish derivation of the word from a judge Gaudy, said to have been the institutor of such days. But such days were held in all times, and did not want a judge to invent them. 1822. Scott, Fortunes of Nigel, ch. xxiii. We had a carouse to your honour ... we fought, too, to finish off the GAUDY. 1878. Besant and Rice, By Celia’s Arbour, ch. xxxiii. Champagne ... goes equally well with a simple luncheon of cold chicken, and with the most elaborate GAUDY. General’s-day (Stonyhurst).—See Day. Gentlemen-Philosopher, subs. (Stonyhurst: obsolete).—See Philosopher.
- 56. Genuine, subs. (Winchester).—Praise. Also as verb = to praise. [It is suggested (but see quot.) that the derivation may be from genuina, the “jaw-tooth,” praise being nothing but “jaw”: cf. Parsius, i. 115.] 1891. Wrench, Winchester Word-Book, s.v. Genuine.... He was awfully quilled and GENUINED my task. Possibly from calling a thing genuine. Cf. to blackguard, to lord, c. But fifty years ago it was a subs. only. [See Appendix.] Gip (or Gyp), subs. (Cambridge).—A college servant. 1891. Harry Fludyer at Cambridge, 8. My GYP said he thought he knew some one who’d give me eighteen shillings for it. Girdlestoneites (Charterhouse).—A boarding-house. [From a master’s name.] See Out-houses. Glope, verb (Winchester: obsolete).—To spit. Go. To go down, verb (University).—To leave school or college: by special EXEAT (q.v.) or at vacation. Whence TO BE SENT DOWN = to be under discipline; to be rusticated.
- 57. 1863. H. Kingsley, Austin Elliot, i. 179. How dare you say “deuce” in my presence? You can GO DOWN, my Lord. 1886. Dickens, Dict. of Cambridge, 3. No undergraduate should GO down without obtaining his EXEAT. 1891. Harry Fludyer at Cambridge, 53. I’m thankful to say this Term’s nearly over now.... We shall be able to GO down next week ... which is a blessing. 1898. Stonyhurst Mag., Dec., p. 149, “Life at Oxford.” You will think, then, that most of us do no work. Well, a good many do precious little. Still there is this check. All who do not pass their examinations within a certain time must “GO DOWN,” i.e. they must leave. It wholly depends upon ourselves, then, how much work we do; and it is naturally a much more difficult matter to “read” in this way than when one has regular schools and studies. Goal, subs. (Winchester).—(1) At football the boy who stands at the centre of each end, acting as umpire; and (2) the score of three points made when the ball is kicked between his legs, or over his head, without his touching it. See Schitt. c. 1840. Mansfield, School-Life at Winchester College, p. 138. Midway between each of the two ends of the line was stationed another boy, as umpire (GOAL he was called), who stood with his legs wide apart, and a gown rolled up at each foot: if the ball was kicked directly over his head, or between his legs, without his touching it, it was a GOAL, and scored three for the party that kicked it.
- 58. God, subs. 1. (Eton).—A Sixth Form boy. See Appendix. 1881. Pascoe, Life in our Public Schools. A GOD at Eton is probably in a more exalted position, and receives more reverence than will ever afterwards fall to his lot. 2. (Westminster).—The juniors who, at the Westminster Play (q.v.), occupy a back gallery. A proposal was made in 1792 to exclude them from the performance on the grand nights, which, however, was successfully resisted. Whence GOD-KEEPER = a Third Election boy, who acts as deputy monitor, and keeps the gallery deities in order. 1867. Collins, The Public Schools, p. 155. A rushing noise is heard as of a party of inebriated whirlwinds coming up College, and the Di Superi (in vulgar parlance THE GODS) make their appearance. Now is the time to see the GOD-KEEPER in his glory, in kid gloves, cane, and commanding voice: “Here, Jones, go up closer. Room for three or four more in that corner. Tumble-up, Davis.” Going-out Saturday, subs. (Charterhouse). —See Exeat 2. Gold Hatband, subs. (old University).—A nobleman undergraduate; a TUFT (q.v.).
- 59. 1628. Earle, Microcosmography. His companion is ordinarily some stale fellow that has been notorious for an ingle to GOLD HATBANDS, whom hee admires at first, afterwards scornes. 1889. Gentleman’s Mag., June, p. 598. Noblemen at the universities, since known as “tufts,” because of the gold tuft or tassel to their cap, were then known as GOLD HATBANDS. Golgotha, subs. (old University).—The Dons’ gallery at Cambridge; also a certain part of the theatre at Oxford. [That is, “the place of skulls” (cf. Luke xxiii. 33 and Matt. xxvii. 33); whence the pun, Dons being the heads of houses.] 1730. Jas. Miller, Humours of Oxford, Act ii., p. 23 (2nd ed.). Sirrah, I’ll have you put in the black-book, rusticated—expelled—I’ll have you coram nobis at GOLGOTHA, where you’ll be bedevilled, Muck-worm, you will. 1785. Grose, Vulg. Tongue, s.v. 1791. G. Huddesford, Salmagundi (Note on, p. 150). Golgotha, “The place of a Skull,” a name ludicrously affixed to the Place in which the Heads of Colleges assemble. 1808. J. T. Conybeare in C. K. Sharp’s Correspondence (1888), i. 324. The subject then of the ensuing section is Oxford News ... we will begin by GOLGOTHA.... Cole has already obtained the Headship of Exeter, and Mr. Griffiths ... is to have that of University. Gomer, subs. 1. (Winchester).—A large pewter dish used in College. [Probably from
- 60. its holding a homer or omer in measure: see quots.] 1610-31. Donne. Not satisfied with his GOMER of manna. d. 1656. Hall, Satires, Bk. v. He that gave a GOMER to each. 1778. Inventory of Kitchen and Hall. Twenty-four GOMERS (amongst dishes and brass pots). 2. A new hat: specifically, a beaver when first introduced: but see quot., Peals, and Appendix. 1867. Collins, The Public Schools, 68. Top-boots are no longer considered, by young gentlemen of twelve, “your only wear” to go home in, although the term for them—GOMERS (i.e. go-homers)— still survives in the Winchester vocabulary. Good-breakfast, subs. (Stonyhurst).—A breakfast given to those Distinguished (q.v.) every term: also called Distinction-breakfast. Cf. Do and Good-supper. Good-creatures, subs. (Charterhouse).— Meat, vegetables, and pudding. [From a quaint old-fashioned “Scholars’ grace”—“Lord, bless to us these thy GOOD- CREATURES,” c.]
- 61. Good-day, subs. (Stonyhurst).—A free day given at the end of the school year to those distinguished in mathematics. There is also a “Rhetoric GOOD DAY,” given to the Rhetoricians (q.v.), and a “Certificate GOOD DAY,” given to candidates for the Higher Certificate Examination. Good-Four-o’clock, subs. (Stonyhurst: obsolete).—A repast similar in character to a Good-supper and a Good-breakfast (both of which see). Good-supper, subs. (Stonyhurst).—A supper given for a special reason: e.g. the Choir- supper (that given to members of the Choir); the Actors’-supper (that given to the participants in Shrovetide-plays); the Eleven- supper (to the Cricket eleven after an “out” match), c. Cf. Do and Good-breakfast. Goose-match, subs. (Harrow).—A cricket match played between the School Eleven and a team of Old Harrovians on Michaelmas Day, or as near to it as possible.
- 62. The Eleven opposing the School are called “the geese.” See Appendix. Gosh, subs. (Winchester).—To spit. Gown, subs. 1. (Winchester: obsolete).— Coarse brown paper. 2. (University).—The schools as distinguished from the TOWN (q.v.): e.g. Town and Gown. 1847. Thackeray, Punch’s Prize Novelists, “Codlingsby,” p. 232. From the Addenbroke’s hospital to the Blenheim turnpike, all Cambridge was in an uproar—the College gates closed—the shops barricaded —the shop-boys away in support of their brother townsmen—the battle raged, and the GOWN had the worst of the fight. 1853. Bradley, Verdant Green, II., ch. iii. When GOWN was absent, Town was miserable. 1891. Pall Mall Gaz., 30th May, p. 4, c. 3. Town and GOWN joined in harmony. Gownboy, subs. (Charterhouse).—A scholar on the foundation: they wore at the Charterhouse black Eton jackets, black trousers, shoes called Gowsers (q.v.), and gowns. This distinctive garb was abolished in 1872.
- 63. Gownboy-arch (Charterhouse).—An arch near the east end of the chapel, formerly the doorway from Scholars’ Court into Gown-boys. The earliest Old Carthusian name inscribed on it bears date 1778. Gownboy-cricket, subs. (Charterhouse).— Cricket in which there are twenty bowlers to one batsman, with no fielders. Gownboys (Charterhouse).—A boarding- house. [Because on migration to Godalming in 1872 nearly all the old Gownboys (q.v.) were received there.] Gowner, subs. (Winchester).—The Goal (q.v.) at football stood with his legs stretched out, and a gown, rolled up into a ball, at each foot. When the ball was kicked over either of these gowns, without goal’s touching it, this counted two for the party who kicked it. —Mansfield (c. 1840). Also see Goal and Schitt. Now obsolete.
- 64. Gownsman (also Gown), subs. (University). —A student. 1800. C. K. Sharpe, in Correspondence (1888), i. 96. A battle between the GOWNSMEN and townspeople ... in spite of the Vice- Chancellor and Proctors. 1850. F. E. Smedley, Frank Fairlegh, ch. xxv. The ancient town of Cambridge, no longer animated by the countless throngs of GOWNSMEN, frowned in its unaccustomed solitude. 1853. Bradley, Verdant Green, III. By the time Mr. Bouncer finished these words, the coach appropriately drew up at the “Mitre,” and the passengers tumbled off amid a knot of GOWNSMEN collected on the pavement to receive them. 1861. Hughes, Tom Brown at Oxford. The townsmen ... were met by the GOWNSMEN with settled steady pluck. Gowsers, subs. (Charterhouse: obsolete).— Shoes. Grammar, subs. 1. (Stonyhurst).—The Lower Fourth Form. 2. (Harrow).—See Upper School. Grand-matches, subs. (Stonyhurst).—The three final matches of the Stonyhurst- football (q.v.) season, played always on the Thursday before Shrove-tide, and on the
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