![The Twin Semigroup Approach Towards
Periodic Neutral Delay Equations
Sjoerd Verduyn Lunel
Abstract In the first part of this chapter we review the recently developed theory of
twin semigroups and norming dual pairs in the light of neutral delay equations. In the
second part we extend the perturbation theory for twin semigroups to include time-
dependent perturbations. Finally we apply this newly developed theory to neutral
periodic delay equations.
1 Introduction
Consider a function x defined on the half-line [0, ∞) with values in Rn
and assume
that the derivative ẋ depends on the history of x and ẋ. More precisely, we assume that
there exists h > 0 such that ẋ(t) depends on x(τ) and ẋ(τ) for t − h ≤ τ ≤ t. Given
these restrictions we would like to consider a general linear differential equation.
To formulate precisely what type of equations we consider, we first define the
segment xt : [−h, 0] → Rn
by
xt (θ) := x(t + θ), for − h ≤ θ ≤ 0. (1)
Let η and ζ be n × n-matrix-valued functions of bounded variation defined on [0, ∞)
such that η(0) = ζ(0) = 0, η and ζ are continuous from the right on (0, h), η(t) =
η(h)andζ(t) = ζ(h)fort ≥ h.Wecallsuchfunctionsη andζ ofnormalizedbounded
variation. Furthermore assume that η(t) is continuous at t = 0. (See Appendix A for
the precise definition and basic properties of such functions.)
The class of equations that we will study can now be written as
d
dt
x(t) −
h
0
dη(θ)x(t − θ)
=
h
0
dζ(θ)x(t − θ). (2)
S. Verduyn Lunel (B)
Mathematical Institute, Utrecht University, Utrecht, The Netherlands
e-mail: S.M.VerduynLunel@uu.nl
© CISM International Centre for Mechanical Sciences 2023
D. Breda (ed.), Controlling Delayed Dynamics, CISM International Centre
for Mechanical Sciences 604, https://doi.org/10.1007/978-3-031-01129-0_1
1](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-14-320.jpg)
![2 S. Verduyn Lunel
To single out a unique solution we have to provide an initial condition at a certain
time s. The initial condition should specify the values of x on the interval of length
h preceding time s. Let y satisfy (2) for t ≥ s and the initial condition
y(s + θ) = ϕ(θ), −h ≤ θ ≤ 0,
where ϕ ∈ B ([−h, 0]; Rn
), the Banach space of bounded Borel measurable func-
tions provided with the supremum norm (see Sect.A for the precise definition and
basic properties). Then x defined for t ≥ 0 by x(t) = y(s + t), satisfies (2) for t ≥ 0
and the initial condition
x(θ) = ϕ(θ), −h ≤ θ ≤ 0. (3)
Equation (2) is time invariant and called autonomous. So we can, without loss of
generality, restrict our attention to an initial condition imposed at time zero. This in
contrast to time periodic equations which we will consider in Sect.8.
Equation (2) is called a neutral functional differential equation (NFDE). A
solution of the initial-value problem (2)–(3) on the half-line [0, ∞) is a function
x ∈ B ([0, ∞); Rn
) such that
(i) (3) holds;
(ii) on (0, ∞), the function x is absolutely continuous and (2) holds;
(iii) the following limit exists
lim
t↓0
1
t
x(t) −
h
0
dη(θ)x(t − θ) − ϕ(0) −
h
0
dη(θ) ϕ(−θ)
and equals
h
0 dζ(θ) ϕ(−θ).
We end the introduction with an outline of this chapter. In Sect.2 we derive a
representation of the solution of a NFDE by direct methods. The main result is given
in Theorem 2.4. In Sect.3 we introduce the notions of norming dual pair and twin
semigroup following Diekmann and Verduyn Lunel (2021). In Sect.4 we introduce
a concrete norming dual pair that will be used in Sect.5 to represent the solution
semigroup corresponding to a NFDE as a twin semigroup. In Sect.6 we use the
twin semigroup approach towards NFDE to prove a variation-of-constants formula,
see Theorem 6.4. In Sect.7 we develop the perturbation theory for bounded time-
dependent perturbations of twin semigroups. The main result is given in Theorem
7.5. In Sect.8 we consider periodic NFDE as an application of the perturbation
theory developed in Sect.7 and prove that periodic NFDE define a twin evolutionary
system. Finally in Appendix A we review some basic properties of functions of
bounded variations and complex Borel measures.](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-15-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 3
2 Introduction to NFDE
This section is concerned with the existence, uniqueness and representation of a
solution of the initial-value problem (2)–(3). For 0 ≤ t ≤ h, we can combine the two
separate pieces of information given in (2) and (3) and write
d
dt
x(t) −
h
0
dη(θ)x(t − θ)
=
t
0
dζ(θ)x(t − θ) +
h
t
dζ(θ)ϕ(t − θ). (4)
By integration and changing the order of integration we can write (4) as
x(t) −
h
0
dη(θ)x(t − θ) =
t
0
ζ(θ)x(t − θ) dθ + g(t), (5)
where
g(t) := ϕ(0) −
h
0
dη(θ)ϕ(−θ) +
t
0
h
s
dζ(θ)ϕ(s − θ)
ds. (6)
Next we write (5) as follows
x(t) =
t
0
dη(θ)x(t − θ) +
t
0
ζ(θ)x(t − θ) dθ + f (t), (7)
where, using (6),
f (t) := g(t) +
h
t
dη(θ)ϕ(t − θ)
= ϕ(0) +
h
0
[ζ(t + σ) − ζ(σ)] ϕ(−σ) dσ
+
h
0
d [η(t + σ) − η(σ)] ϕ(−σ). (8)
Here we have used that
t
0
h
s
dζ(θ)ϕ(s − θ)
ds =
h
0
[ζ(t + σ) − ζ(σ)] ϕ(−σ) dσ
and that h
t
dη(θ)ϕ(t − θ) =
h
0
dη(t + σ)ϕ(−σ).
It follows from Theorem A.2 that the function f defined by (8) is a bounded Borel
measurable function on [0, ∞) that is constant on [h, ∞).](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-16-320.jpg)
![4 S. Verduyn Lunel
Define the function μ by
μ(θ) := η(θ) +
θ
0
ζ(s) ds, 0 ≤ θ ≤ h, (9)
and μ(θ) = μ(h) for θ ≥ h, then μ is a n × n-matrix-valued function of normalized
bounded variation. Note that, since η(θ) is continuous at θ = 0, we have that μ(θ) is
continuous at θ = 0.
Theconvolutionproductofan × n-matrix-valuedfunctionofnormalizedbounded
variation μ and a bounded Borel measurable function f is defined by
(μ ∗ f ) (t) :=
t
0
dμ(θ) f (t − θ), t ≥ 0. (10)
From Theorem A.1, it follows that μ ∗ f is a bounded Borel measurable function on
[0, ∞).
Using the convolution product defined by (10), the initial-value problem (2)–(3),
i.e., (7), can be rewritten as a renewal equation for x
x = μ ∗ x + f, (11)
where μ is given by (9) and f , given by (8), can be rewritten as
f (t) = ϕ(0) +
h
0
d [μ(t + σ) − μ(σ)] ϕ(−σ). (12)
Therefore to prove existence and uniqueness of solutions of the initial-value problem
(2)–(3), it suffices to prove existence and uniqueness of solutions of the renewal
equation (11).
The convolution product of two n × n-matrix-valued functions of normalized
bounded variation μ and ν, defined by
(μ ∗ ν) (t) :=
t
0
dμ(θ)ν(t − θ), t ≥ 0, (13)
is again a function of bounded variation (see Appendix A and, in particular, Theorem
A.3).
The resolvent kernel ρ of a renewal equation (11) with kernel μ and convolution
product (13) is defined as the matrix solution of the resolvent equation
ρ = ρ ∗ μ + μ = μ ∗ ρ + μ. (14)
The key property of the resolvent concerns the representation of the solution of the
renewal equation (11) as
x = f + ρ ∗ f. (15)](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-17-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 5
Indeed taking to convolution with ρ on the left and right of (11) yields
ρ ∗ x = (ρ ∗ μ) ∗ x + ρ ∗ f = (ρ − μ) ∗ x + ρ ∗ f.
Hence μ ∗ x = ρ ∗ f and substituting this relation into (11) yields (15).
We now discuss the existence and uniqueness of the solution of (14) under the
assumption that μ is a n × n-matrix-valued function of normalized bounded varia-
tion. It follows from Appendix A and in particular Theorem A.1 that functions of
normalized bounded variation are in one-to-one correspondence to complex Borel
measures. This allows us to use measure theory to prove existence and uniqueness
of the solution of (14). We start with some preparations.
Let E denote the Borel σ-algebra on [0, ∞). The Banach space of complex
Borel measures of bounded total variation is denoted by M ([0, ∞)) (see (82)). Let
Mloc ([0, ∞)) denote the vector space of local measures, i.e., set functions that are
defined on relatively compact Borel measurable subsets of [0, ∞) and that locally
behave like bounded measures: for every T 0 the set function μT defined by
μT (E) := μ (E ∩ [0, T ]) , E ∈ E,
belongs to M ([0, ∞)). The elements of Mloc ([0, ∞)) are called Radon measures.
Since the restriction to [0, T ] of μ ∗ ν depends only on the restrictions of μ and ν to
[0, T ], we can unambiguously extend the convolution product to Mloc ([0, ∞)) (see
(84)).
We continue with the existence of the resolvent ρ of a complex Borel measure μ
supported on [0, ∞). For details see Diekmann and Verduyn Lunel (2021, Theorem
A.7) and for further information and details see Grippenberg et. al. (1990).
Theorem 2.1 Suppose that μ ∈ Mloc
[0, ∞); Rn×n
. There exists a unique measure
ρ ∈ Mloc
[0, ∞); Rn×n
satisfying either one of the following identities
ρ − μ ∗ ρ = μ = ρ − ρ ∗ μ (16)
if and only if det [I − μ({0})] = 0. Furthermore, if μ((0, t]) is continuous as t = 0,
then ρ((0, t]) is continuous at t = 0 as well.
The following theorem summarizes some relevant results for renewal equations
(Diekmann and Verduyn Lunel 2021, Theorem A.9).
Theorem 2.2 Let μ ∈ Mloc
[0, ∞), Rn×n
with det [I − μ({0}] = 0.
(i) For every f ∈ Bloc ([0, ∞), Rn
), the renewal equation (15) has a unique solution
x ∈ Bloc ([0, ∞), Rn
) given by
x = f + ρ ∗ f,
where ρ satisfies (16). Furthermore, if f is locally absolutely continuous, then
the solution x is locally absolutely continuous as well.](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-18-320.jpg)
![6 S. Verduyn Lunel
(ii) If the kernel μ has no discrete part and if f ∈ C ([0, ∞), Rn
), then x ∈
C ([0, ∞), Rn
).
We now summarize the conclusions obtained so far in this section in the following
theorem.
Theorem 2.3 Let η and ζ be of normalized bounded variation. Let ϕ ∈
B ([−h, 0]; Rn
) be given. Define μ by (9). If det [I − μ(0)] = 0, then the NFDE
(2) provided with the initial condition (3) admits a unique solution. For t ≥ 0 this
solution coincides with the unique solution of the renewal equation (11) and the
solution has the representation (15) where ρ satisfies the resolvent equation (14) and
f is given by (8).
Representation (15) will be used to derive a representation of the solution of
(2)–(3) directly in terms of the initial data x0 = ϕ. We first need a definition. The
fundamental solution of the delay equation (2)–(3) on [−h, ∞) is defined by the
n × n-matrix-valued function
X(t) :=
I + ρ((0, t]) for t ≥ 0,
0 for − h ≤ t 0,
(17)
where ρ is the resolvent of μ given by Theorem 2.1. Since t → μ((0, t]) is continuous
at t = 0, it follows from Theorem 2.1 that ρ((0, t]) is continuous at t = 0. Therefore
we can conclude that X(t) has a jump at t = 0.
By construction, the fundamental matrix solution X(t) satisfies (2) with initial
data
X0(θ) =
I for θ = 0,
0 for − h ≤ θ 0.
(18)
Using the fundamental matrix solution X(t) given by (17) and Fubini’s theorem,
we can rewrite the representation formula (15) in terms of the forcing function f
given by (8) directly in terms of the initial condition ϕ.
We summarize the result in a theorem.
Theorem 2.4 The solution of (2)–(3) is given explicitly by
x(t; ϕ) = X(t)ϕ(0) +
h
0
d
t
−h
dX(τ) (μ(t − τ + σ) − μ(σ))
ϕ(−σ). (19)
Or, equivalently, in terms of the resolvent ρ we have
x(t; ϕ) = (I + ρ((0, t])) ϕ(0) +
h
0
d [μ(t + σ) − μ(σ)])ϕ(−σ)
+
h
0
d
t
0
ρ(dτ) (μ(t − τ + σ) − μ(σ))
ϕ(−σ). (20)
Here μ is given by (9).](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-19-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 7
3 Norming Dual Pairs and Twin Semigroups
The system of equations (2)–(3) defines an infinite-dimensional dynamical system
on the state space B ([−h, 0]; Rn
), but for the qualitative study of such a dynamical
system we need an adjoint theory in place (see Hale and Verduyn Lunel 1993). In
the classical theory of delay equations this is the main reason to work with the state
space C ([−h, 0]; Rn
) despite the fact that the initial data of the fundamental solution
(see (18)) does not belong to this space. From the Riesz representation theorem it
follows that the dual space of C ([−h, 0]; Rn
) has a nice characterization as the space
of functions of normalized bounded variation.
The state space B ([−h, 0]; Rn
) includes the initial data of the fundamental solu-
tion but its dual space does not have a nice characterization. So although the state
space B ([−h, 0]; Rn
) is a more natural space to consider, it has not yet been used
because its dual space is too large to provide a useful adjoint theory. A beautiful idea
to repair this discrepancy is to use the notion of a dual pair (see Aliprantis and Border
2006) made precise in Kunze (2011) for infinite-dimensional dynamical systems in
the following way.
Two Banach spaces Y and Y are called a norming dual pair (cf. Kunze (2011))
if a bilinear map
· , · : Y × Y → R
exists such that, for some M ∈ [1, ∞),
| y , y| ≤ My y
and, moreover,
y := sup | y , y| | y ∈ Y , y ≤ 1
y := sup | y , y| | y ∈ Y, y ≤ 1 .
So we can consider Y as a closed subspace of Y ∗
, the dual of Y , and Y as a closed
subspace of Y∗
and both subspaces are necessarily weak∗
dense since they separate
points.
The collection of linear functionals Y defines a weak topology on Y, denoted by
σ(Y, Y ). The corresponding locally convex topological vector space is denoted by
(Y, σ(Y, Y )). A crucial point in our approach is that the dual space (Y, σ(Y, Y ))
is (isometrically isomorphic to) Y (Rudin 1991, Theorem 3.10). So if a linear func-
tional on Y is continuous with respect to the topology induced by Y , it can be
(uniquely) represented by an element of Y .
The next key idea to study infinite-dimensional dynamical systems on a norming
dual pair is the notion of a twin operator introduced in Diekmann and Verduyn Lunel
(2021).](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-20-320.jpg)
![10 S. Verduyn Lunel
y = S(λ)(λy − z)
3. y, z ∈ Y and for all t 0
t
0
S(τ)z dτ = S(t)y − y.
Here it should be noted that item 3. includes the assertions
• the integral
t
0 S(τ)z dτ defines an element of Y (even though at first it only defines
an element of Y ∗
);
• the integral
t
0 S(τ)z dτ does not depend on the choice of z ∈ Cy in case C is
multi-valued.
Lemma 3.3 For all t 0 and y ∈ Y, we have
t
0 S(τ)y dτ ∈ D (C) and
S(t)y − y ∈ C
t
0
S(τ)y dτ.
4 The Norming Dual Pair (B, N BV)
In the study of delay differential equations, the natural dual pair is given by
Y = B
[−1, 0], Rn
and Y = N BV
[0, 1], Rn
(24)
with the pairing
y , y =
[0,1]
y (dσ) · y(−σ) (25)
(see Appendix A for the definition of N BV ). Here Y is provided with the supremum
norm and Y with the total variation norm (see (83)). See Diekmann and Verduyn
Lunel 2021.
In the study of renewal equations, the natural dual pair is given by
Y = N BV
[−1, 0], Rn
and Y = B
[0, 1], Rn
with the pairing
y , y =
[−1,0]
y(dσ) · y (−σ).
Returning to (24)–(25), we first make two trivial, yet useful, observations: fix
1 ≤ i ≤ n and −1 ≤ θ ≤ 0,](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-23-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 11
[0,1]
y (dσ) · y(−σ) = yi (θ),
if yj (σ) = 0, 0 ≤ σ ≤ 1, j = i, and yi (σ) = 0 for 0 ≤ σ −θ and yi (σ) = 1 for
σ ≥ −θ, and similarly
[0,1]
y (dσ) · y(−σ) = yi (−θ),
if yj (−σ) = 0, 0 ≤ σ ≤ 1, j = i, and yi (−σ) = 1 for 0 ≤ σ ≤ −θ and yi (−σ) = 0
for σ −θ.
The point is that, consequently, in case of (24)–(25), convergence in both
(Y, σ(Y, Y )) and (Y , σ(Y , Y)) entails pointwise convergence (in, respectively,
B ([−1, 0], Rn
) and N BV ([0, 1], Rn
)).
In the first case, the dominated convergence theorem implies that, conversely, a
bounded pointwise convergent sequence in B ([−1, 0], Rn
) converges in
(Y, σ(Y, Y )). For N BV ([0, 1], Rn
), this is not so clear. It is true that the pointwise
limit of a sequence of functions of bounded variation is again of bounded variation
(Helly’s theorem), but there is no dominated convergence theorem for measures.
The following theorem is proved in Diekmann and Verduyn Lunel (2021, Theorem
B.1).
Theorem 4.1 The dual pair given by (24) and (25) is a norming dual pair, i.e.,
y = sup | y , y| | y ∈ Y , y ≤ 1
y = sup | y , y| | y ∈ Y, y ≤ 1 .
Furthermore
(i) (Y, σ(Y, Y )) is sequentially complete;
(ii) a linear map (Y, σ(Y, Y )) → R is continuous if it is sequentially continuous.
5 The Twin Semigroup Approach to NFDE
Consider the norming dual pair (Y, Y ) with Y and Y as given in Sect.4 by (24).
By solving (2)–(3), see Theorem 2.3, we can define a Y-valued function u :
[0, ∞) → Y by
u(t; ϕ) := xt ( · ; ϕ), t ≥ 0, (26)
where xt is defined by (1), and bounded linear operators S(t) : Y → Y by
S(t)ϕ = u(t; ϕ). (27)](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-24-320.jpg)
![12 S. Verduyn Lunel
The initial condition (2) translates into
S(0)ϕ = u(0; ϕ) = ϕ
and (27) reflects that we define a dynamical system on Y by translating along the
function ϕ extended according to (2). Below we show that {S(t)} is a twin semigroup
and we characterize its generator C. But first we present some heuristics.
In order to motivate an abstract ODE for the Y-valued function u, we first observe
that the infinitesimal formulation of the translation rule (26) amounts to the PDE
∂u
∂t
−
∂u
∂θ
= 0.
We need to combine this with (2), in terms of u(t)(0) = x(t), and we have to specify
the domain of definition of the derivative with respect to θ. The latter is actually rather
subtle. An absolutely continuous function has almost everywhere a derivative and
when the function is Lipschitz continuous this derivative is bounded. Thus a Lipschitz
function specifies a unique L∞
-equivalence class by the process of differentiation.
But not a unique element of Y. In fact the set
Cψ = ψ
∈ Y | ψ(θ) = ψ(−1) +
θ
−1
ψ
(σ) dσ,
ψ
(0) −
h
0
dη(θ)ψ
(−θ) =
h
0
dζ(θ)ψ(−θ)
(28)
is, for a given Lipschitz continuous function ψ, very large indeed. Note that the
boundary condition
ψ
(0) −
h
0
dη(θ)ψ
(−θ) =
h
0
dζ(θ)ψ(−θ)
takes care of (2). We define C as a multi-valued, unbounded, operator on Y by (28)
with domain given by
D (C) = Lip
[−1, 0], Rn
. (29)
We claim that (2)–(3) and (26) correspond to an abstract differential equation
du
dt
∈ Cu.
To substantiate this claim, we shall verify that {S(t)}t≥0 defined by (27) is a twin
semigroup and, finally, that C is the corresponding generator in the sense of (23)
where S(λ) is given by (21).
From the representation (19) of the solution of (2)–(3) we can derive an explicit
representation of the semigroup {S(t)}t≥0 defined by (27).](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-25-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 13
Theorem 5.1 The semigroup {S(t)}t≥0 defined by (27) is given by
(S(t)ϕ) (θ) =
h
0
Kt (θ, dσ) ϕ(−σ) (30)
with for σ 0 and −h ≤ θ ≤ 0 the kernel Kt (θ, σ) defined by
Kt (θ, σ) := H(σ + t + θ) + H(t + θ)ρ(t + θ)
+ H(t + θ)
t+θ
0
dX(ξ) (μ(t + θ + σ − ξ) − μ(σ)) , (31)
and Kt (θ, 0) = 0. Here ρ denotes the resolvent of μ with μ defined in (9), X denotes
the fundamental solution given by (17), and H is the standard Heaviside function.
Proof For t + θ 0 the second and third terms in the expression for Kt do not
contribute, and the first term yields
(S(t)ϕ) (θ) = ϕ(t + θ)
which is in accordance with (27) because of (3).
Now assume that t + θ ≥ 0. Clearly the first term contributes a unit jump at σ = 0
and H(t + θ) = 1. The second term has, as a function of σ, a jump of magnitude
ρ(t + θ) at σ = 0, an absolutely continuous part with derivative given by
t+θ
0
dX(ξ) (ζ(t + θ + σ − ξ) − ζ(σ)) ,
and a part of bounded variation given by
t+θ
0
dX(ξ) (η(t + θ + σ − ξ) − η(σ)) .
The jumps yield the first term at the right hand side of (19) (see also (20)) evaluated
at t + θ, the absolutely continuous part yields the second, and the bounded variation
part the third term.
Note that Kt is bounded, in the sense (cf. Kunze 2009, Definition 3.2) that for
fixed θ in [−1, 0] the function σ → Kt (θ, σ) is of normalized bounded variation,
while for fixed σ ∈ [0, 1] the function θ → Kt (θ, σ) is bounded and measurable.
The next corollary is a general property of kernel operators.
Corollary 5.2 The operator S(t) extends to a twin operator.
Proof The proof directly follows from the observation that we can represent the
action of y S(t) explicitly as](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-26-320.jpg)
![16 S. Verduyn Lunel
λ ¯
f (λ) = ϕ(0) +
∞
0
λe−λt
t
0
h
s
dζ(θ)ϕ(s − θ) ds
dt
+ λ
∞
0
e−λt
h
t
dη(θ)ϕ(t − θ) dt
= ϕ(0) +
∞
0
e−λt
h
t
dζ(θ)ϕ(t − θ) dt
+ λ
∞
0
e−λt
h
t
dη(θ)ϕ(t − θ) dt
= ϕ(0) +
h
0
dζ(θ)
θ
0
e−λt
ϕ(t − θ) dt
+ λ
h
0
dη(θ)
θ
0
e−λt
ϕ(t − θ) dt
= ϕ(0) +
h
0
dζ(θ)e−λθ
0
−θ
e−λσ
ϕ(σ) dσ
λ
h
0
dη(θ)e−λθ
0
−θ
e−λσ
ϕ(σ) dσ
= H(λ; ϕ).
Therefore it follows from (33) that indeed ψ(0) = x̄(λ; ϕ) and this completes the
proof of the identity (34).
In Diekmann and Verduyn Lunel (2021), we proved Theorems 5.1, 5.3 and 5.4 for
retarded functional differential equations, and gave an alternative proof of Theorem
5.3 in the neutral case using a relative bounded perturbation argument, see Diekmann
and Verduyn Lunel (2021, Theorem 11.1).
6 The Variation-of-Constants Formula for NFDE
It is a direct consequence of (29) that
X = D (C) = C
[−1, 0], Rn
.
Clearly Cψ ∩ X is either empty or a singleton, cf. (28), and for the set to be nonempty
we need that ψ ∈ C1
and
ψ
(0) −
h
0
dη(θ)ψ
(−θ) =
h
0
dζ(θ)ψ(−θ).
So the generator A of the restriction {T (t)}t≥0 of {S(t)}t≥0 to X is given by](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-29-320.jpg)
![18 S. Verduyn Lunel
y , u(t) = y S(t)u0 +
t
0
y S(t − τ)q f (τ) dτ, (37)
where the standard assumptions are
(i) (Y, Y ) is a norming dual pair;
(ii) q ∈ Y;
(iii) f : [0, T ] → R is bounded and measurable;
(iv) {S(t)} is a twin semigroup,
and where u0 (corresponding to ϕ in (36)) is an arbitrary element of Y.
The definition of the first term at the right hand side of (37) is no problem at all,
it contributes S(t)u0 to u(t). The second term defines an element of Y ∗
, but it is not
clear that this element is, without additional assumptions, represented by an element
of Y. The following lemma provides a sufficient condition.
Lemma 6.1 In addition to (i)–(iv) assume that
Y, σ(Y, Y ) is sequentially complete. (38)
Then
y →
t
0
y S(t − τ)q f (τ) dτ (39)
is represented by an element of Y, to be denoted as
t
0 S(t − τ)q f (τ) dτ.
Proof There exists a sequence of step functions fm such that | fm| ≤ | f | and fm → f
pointwise. Lemma 3.3 shows that
t
0
S(t − τ)q fm(τ) dτ
belongs to Y (in fact even to D (C)). Since (see Definition 3.1(ii))
y S(t − τ)q fm(τ)
≤ Meω(t−τ)
qy sup
σ
| f (σ)|,
the dominated convergence theorem implies that for every y ∈ Y
lim
m→∞
t
0
y S(t − τ)q fm(τ) dτ =
t
0
y S(t − τ)q f (τ) dτ.
The sequential completeness next guarantees that the limit too is represented by an
element of Y.
In Diekmann and Verduyn Lunel (2021) we have developed a perturbation theory
to study neutral equations directly as an unbounded perturbation of a retarted equa-
tion. In order to do this, we have to replace f (τ) dτ by F(dτ) with F of bounded](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-31-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 19
variation. In this setting the approximation by step functions used in the proof of
Lemma 6.1 no longer works. This observation motivates to look for an alternative
sufficient condition to replace (38). This is taken care of in the following lemma.
Lemma 6.2 In addition to (i)–(iv) assume that
a linear map
Y , σ(Y , Y) → R is continuous
if it is sequentially continuous. (40)
Then the assertion of Lemma 6.1 holds.
Proof Again we are going to make use of the dominated convergence theorem.
Consider a sequence {ym} in Y such that for every y ∈ Y the sequence ym, y
converges to zero in R. Then for all relevant t and τ we have
lim
m→∞
ym S(t − τ)q = 0
and consequently
lim
m→∞
t
0
ym S(t − τ)q f (τ) dτ = 0.
So the linear map (39) is, in the sense described in (40), sequentially continuous and
therefore, by the assumption, continuous. Since
Y , σ(Y , Y)
= Y,
we conclude that (39) is represented by an element of Y.
We are going to use the above results to show that a certain type of perturbation of
a twin semigroup {S(t)} yields again a twin semigroup. In order to do this we need
a dual version of (37), i.e., we want to define an element u (t) of Y by way of the
action on Y expressed in the formula
u (t), y = u0 S(t)y +
t
0
q S(t − τ)y f (τ) dτ, (41)
where the standard assumptions are as before with (ii) replaced by (ii)
, i.e.,
(i) (Y, Y ) is a norming dual pair;
(ii)
q ∈ Y ;
(iii) f : [0, T ] → R is bounded and measurable;
(iv) {S(t)} is a twin semigroup,
and where u0 is an arbitrary element of Y . This implies that
y →
t
0
q S(t − τ)y f (τ) dτ (42)](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-32-320.jpg)
![20 S. Verduyn Lunel
is represented by an element of Y , to be denoted as
t
0 q S(t − τ) f (τ) dτ.
Applying the two lemmas above, with the role of Y and Y interchanged, we find
that this is indeed the case if either
Y , σ(Y , Y) is sequentially complete (43)
or
a linear map
Y, σ(Y, Y ) → R is continuous
if it is sequentially continuous. (44)
Therefore to develop a perturbation theory for twin semigroups we need both (39)
and (42) to be represented by elements from, respectively, Y and Y . This motivates
the following definition.
Definition 6.3 We say that a norming dual pair (Y, Y ) is suitable for twin pertur-
bation if
(a) at least one of (38) and (40) holds; and
(b) at least one of (43) and (44) holds
Recall from Theorem 4.1 that for the norming dual pair (B, N BV ) introduced
in Sect.4 we have that (38) and (44) are satisfied. This shows that the norming dual
pair (B, N BV ) is suitable for twin perturbation.
We are now ready to give a rigorous proof of the variation-of-constants formula
for NFDE.
Theorem 6.4 The solution of the inhomogeneous NFDE (35) can be
represented by the variation-of-constants formula (36), i.e.,
xt = S(t)ϕ +
t
0
S(t − τ)q f (τ) dτ,
where S(t) is the twin semigroup given by (30).
Proof It follows from Theorem 4.1 that
Y = B([−1, 0]; Rn
) and Y = N BV ([0, 1]; Rn
)
is a norming dual pair suitable for twin perturbation. Therefore the claim follows by
applying Lemma 6.1 with respect to the norming dual pair (B, N BV ) and Lemma
6.2 with respect to the norming dual pair (N BV, B).
In the treatment of renewal equations in Diekmann and Verduyn Lunel (2021)
we assumed (43) and (40). In fact for delay differential equations we take as normal
dual pair (Y, Y ) with Y = B([−1, 0]) and Y = N BV ([0, 1]), while for renewal
equations we take (Y, Y ) with Y = N BV ([−1, 0]) and Y = B([0, 1]).](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-33-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 21
7 Bounded Time-Dependent Perturbation of a Twin
Semigroup
In this section we assume
• (Y, Y ) is a norming dual pair that is suitable for twin perturbation, cf. Definition
6.3;
• {S0(t)} is a twin semigroup on (Y, Y ) with generator C0;
• For j = 1, . . . , n the elements qj ∈ Y and t → qj (t) ∈ Y are given.
Definition 7.1 A two-parameter family U = {U(t, s)}t≥s of twin operators on a
norming dual pair (Y, Y ) is called a twin evolutionary system if
(i) U(s, s) = I and U(t, s) = U(t,r)U(r, s) for s ≤ r ≤ t
(ii) there exist constants M ≥ 1 and ω0 ∈ R such that for all y ∈ Y, y ∈ Y
|y U(t, s)y| ≤ Meω0(t−s)
y y , t ≥ s;
(iii) Let the set ⊂ R2
be defined by = {(t, s) | −∞ s ≤ t ∞}. For all
y ∈ Y, y ∈ Y the function
(t, s) → y U(t, s)y ∈ R
is measurable.
Our aim is to define constructively a twin evolutionary system {U(t, s)} corre-
sponding to the abstract multi-valued differential equation
du
dt
∈ C(t)u, t ≥ s, u(s) given, (45)
with
D (C(t)) = D (C0) , C(t)y = C0 y +
n
j=1
qj (t), yqj . (46)
The first step is to introduce a n × n-matrix-valued function k(t, s) on R × R via
k(t, s) = 0 for −∞ t ≤ s ∞ and
ki j (t, s) := qi (t)S0(t − s)qj , −∞ s ≤ t ∞. (47)
Note that for each pair c1, c2 with −∞ c1 c2 ∞ and for each
f ∈ L1
([c1, c2]; Rn
), we have
sup
f ≤1
c2
c1
c2
c1
k(t, s) f (s) ds
dt ∞.](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-34-320.jpg)
![22 S. Verduyn Lunel
Here f denotes the norm of f as function belonging to L1
([c1, c2]; Rn
) and the
map
f →
t
c1
k(t, s) f (s) ds, c1 ≤ t ≤ c2,
defines a bounded linear operator on L1
([c1, c2]; Rn
) which we shall denote by K.
The linear space of lower triangular kernel functions on [c1, c2] × [c1, c2] of type
L1
loc endowed with the norm
k1 := sup
f ≤1
c2
c1
c2
c1
k(t, s) f (s) ds
dt
= ess sup
s∈[c1,c2]
c2
c1
k(t, s) dt (48)
is a Banach space (see Theorem 9.2.4 and Proposition 9.2.7 of Grippenberg et. al.
1990) which we will denote by L1
+
[c1, c2] × [c1, c2]; Rn×n
.
Now let k be a lower triangular kernel function of type L1
loc. We call an n × n-
matrix-function r(t, s) a resolvent kernel function of k if r(t, s) is a lower triangular
kernel function of type L1
loc and
r(t, s) = k(t, s) +
t
s
r(t, a)k(a, s) da, −∞ s ≤ t ∞, (49)
= k(t, s) +
t
s
k(t, a)r(a, s) da, −∞ s ≤ t ∞. (50)
Define the integral operator R similar as the operator K but with the kernel k(t, s)
replaced by r(t, s), i.e.,
(R f ) (t) :=
t
c1
r(t, s) f (s) ds, c1 ≤ t ≤ c2.
Using the integral operators K and R, it follows from the identity (50) that for
c1 t c2 we have
(K R f )(t) =
t
c1
k(t, s)(R f )(s) ds
=
t
c1
k(t, s)
s
0
r(s, τ) f (τ) dτ
ds
=
t
c1
t
τ
k(t, s)r(s, τ) ds
f (τ) dτ
=
t
c1
(r(t, τ) − k(t, τ)) f (τ) dτ
= (R f )(t) − (K f )(t), c1 ≤ t ≤ c2.](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-35-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 23
It follows that K R = R − K. Similarly, using (49), we have RK = R − K. This
yields K R = RK and
(I − K)(I + R) = (I + R)(I − K) = I, (51)
where I is the identity operator on L1
([c1, c2]; Rn
). Thus I − K is an invertible
operator on L1
([c1, c2]; Rn
), and its inverse is given by I + R.
Theorem 7.2 If k(t, s) is a lower triangular kernel function of type L1
loc, then k(t, s)
has a unique resolvent kernel function r(t, s) of type L1
loc. In particular, the integral
equation x = K x + f has a unique solution given by x = f + R f .
Proof The proof will be done in three steps. Throughout k(t, s) is a lower triangular
kernel function of type L1
loc.
Step 1. First note that if k1 and k2 are lower triangular kernel functions on R × R,
then the same holds true for the functions
(t, s) →
t
s
k1(t, a)k2(a, s) da and (t, s) →
t
s
k2(t, a)k1(a, s) da.
Furthermore, from the discussion in the paragraph preceding the present theorem it
follows that a resolvent kernel function of type L1
loc is unique whenever it exists.
Step 2. Because of uniqueness of the resolvent kernel of type L1
loc, it suffices to prove
existence of a resolvent kernel on [c1, c2] for every c1, c2 ∈ (0, ∞) with c1 c2.
Assume first that k1 ≤ 1 with k1 given by (48), then the map
r(t, s) →
t
s
k(t, a)r(a, s) da + k(t, s)
is a contraction on L1
+
[c1, c2] × [c1, c2]; Rn×n
. This shows that (50) (and, using
(51), similarly (49)) has a unique solution, and this solution is a resolvent kernel of
type L1
loc.
Step 3. Since k(t, s) is a lower triangular kernel function of type L1
loc, we define a
scaled lower triangular kernel function of type L1
loc by
k(t, s) := e−γ(t−s)
k(t, s).
Since the norm of
k is defined by (see (48))
k1 := ess sup
s∈[c1,c2]
c2
c1
k(t, s) dt = ess sup
s∈[c1,c2]
c2
c1
e−γ(t−s)
k(t, s) dt,
we can choose γ so large that
k1 1. From Step 2, it follows that the equation
r(t, s) =
k(t, s) +
t
s
k(t, a)
r(a, s) da](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-36-320.jpg)
![24 S. Verduyn Lunel
has a unique solution
r ∈ L1
+
[c1, c2] × [c1, c2]; Rn×n
. Therefore, we have
r(t, s) = e−γ(t−s)
k(t, s) +
t
s
e−γ(t−a)
k(t, a)
r(a, s) da,
and hence
eγ(t−s)
r(t, s) = k(t, s) +
t
s
k(t, a)eγ(a−s)
r(a, s) da.
Thus
r(t, s) = k(t, s) +
t
s
k(t, a)r(a, s) da,
where r(t, s) = eγ(t−s)
r(t, s). This completes the proof.
Proposition 7.3 If k(t, s) is a lower triangular kernel function that satisfies the esti-
mate k(t, s) ≤ m(t) for 0 ≤ s ≤ t and r(t, s) denotes the corresponding resolvent
kernel function, then
r(t, s) ≤ m(t) exp
t
s
m(σ) dσ
, 0 ≤ s ≤ t ∞.
Proof From the estimate k(t, s) ≤ m(t) for 0 ≤ s ≤ t we obtain the following
integral inequality for the function u(t, s) := r(t, s) on 0 ≤ s ≤ t:
u(t, s) ≤ m(t) + m(t)
t
s
u(a, s) da, 0 ≤ s ≤ t ∞. (52)
Now fix s ∈ [0, ∞), and put
q(t) := exp
−
t
s
m(σ) dσ
t
s
u(a, s) da. t ≥ s. (53)
Differentiation of q with respect to t yields
dq
dt
(t) = −m(t)q(t) + exp
−
t
s
m(σ) dσ
u(t, s)
=
u(t, s) − m(t)
t
s
u(a, s) da
exp
−
t
s
m(σ) dσ
≤ m(t) exp
−
t
s
m(σ) dσ
,
where we have used (52). Integration from s to t yields the inequality](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-37-320.jpg)
![28 S. Verduyn Lunel
so we conclude from (61) that this identity does indeed hold.
Theorem 7.5 Equation (61) in combination with (60), or Eq.(65) in combination
with (64), defines a twin evolutionary system {U(t, s)} corresponding to the abstract
differential equation (45).
Proof Fix t ≥ s. Since (Y, Y ) is suitable for twin perturbation, we can use (61) and
either Lemma 6.1 or Lemma 6.2 to deduce that U(t, s) maps Y into Y. Similarly
we can use (65) and the observation concerning (42) to deduce that U(t, s) maps Y
into Y . So U(t, s) is a twin operator.
Next we use Lemma 7.4 to derive the property
U(t, s) = U(t,r)U(r, s), t ≥ r ≥ s, (67)
To verify (67), we start from (61) and use Lemma 7.4 to write
U(t, s)y = S0(t − r)
S0(r − s)y +
r
s
S0(r − τ)q · v(τ, s)y dτ
+
t
r
S0(t − τ)q · v(τ,r)U(r, s)y dτ
= S0(t − r)U(r, s)y +
t
r
S0(t − τ)q · v(τ,r)U(r, s)y dτ
= U(t,r)U(r, s)y.
Both the property S(s, s) = I and the measurability, for all y ∈ Y, y ∈ Y , of t →
y S(t)y follow from (61) and the corresponding properties of {S0(t)}.
Finally, the exponential estimate for y S0(t)y yields exponential estimates for
both the kernel k(t, s) and the forcing function t → q (t)S0(t − s)y, t ≥ s, in the
renewal equation (59). Therefore, using Proposition 7.3 we obtain an exponential
estimate for the resolventr(t, s) and hence via (60) an exponential bound for v(t, s)y.
Finally, using (61) we obtain an exponential bound for y U(t, s)y for t ≥ s.
This completes the proof of Theorem 7.5.
8 A Perturbation Approach Towards Periodic NFDE
We shall be dealing with linear periodic neutral functional differential equations of
the following type:
⎧
⎪
⎨
⎪
⎩
d
dt
x(t) −
h
0
[dη(τ)]x(t − τ)
=
h
0
[dτ ζ(t, τ)]x(t − τ), t ≥ s,
x(s + θ) = ϕ(θ), −h ≤ θ ≤ 0.
(68)](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-41-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 29
Here dτ denotes integration with respect to the τ variable and ϕ is a given function
in B ([−h, 0], Rn
). Throughout we assume that for each t ∈ R the functions η and
ζ(t, ·) are n × n matrices of which the entries are real functions of bounded variation
on [0, h] and continuous from the left on (0, h), and η(0) = ζ(t, 0) = 0. Moreover,
it is assumed that there is a nondecreasing bounded function m ∈ L1
loc[−h, ∞) such
that
Var[−h,0] ζ(t, ·) ≤ m(t), t ≥ 0.
Theorem 8.1 Under the above conditions, Eq.(68) defines a well-posed dynam-
ical system, that is, Eq.(68) has a unique solution x on [0, ∞) such that xt ∈
B ([−h, 0], Rn
) for t ≥ 0.
The above theorem is an extension of Theorem 6.1.1 in Hale and Verduyn Lunel
(1993) to the neutral case. In this section we shall derive Theorem 8.1 as a corollary
of Theorem 7.5 using the perturbation approach developed in the previous section.
Consider as the unperturbed problem the special case ζ = 0 in (68). Let y denote
the solution of the autonomous NFDE
⎧
⎪
⎨
⎪
⎩
d
dt
y(t) −
h
0
[dη(τ)]y(t − τ)
= 0, t ≥ 0,
y(θ) = ϕ(θ), −h ≤ θ ≤ 0.
(69)
From the theory developed in Sect.2, it follows that the solution y of (69) satisfies
the autonomous renewal equation
y(t) −
t
0
dη(θ)y(t − θ) = f0(t), t ≥ s, (70)
where
f0(t) := ϕ(0) −
h
0
dη(θ)ϕ(−θ) +
h
t
dη(θ)ϕ(t − θ), t ≥ s. (71)
The solution of (70) is given by
y(t) = f0(t) +
t
0
dρ0(θ) f0(t − θ), t ≥ 0, (72)
where ρ0 denotes the resolvent of η, i.e., it satisfies the resolvent equation
ρ0 = η ∗ ρ0 + η, (73)
see Theorem 2.2. Denote by X(t) = I + ρ0(t) the fundamental matrix solution of
(69) so that we can write the solution y given by (72) as](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-42-320.jpg)
![32 S. Verduyn Lunel
Finally apply to (61) the element of Y that corresponds to the Dirac measure in
−θ ∈ [0, 1] to obtain
(U(t, s)ϕ) (θ) = y(t − s + θ) +
t
s
X(t − τ + θ) · v(τ, s)ϕ dτ
= x(t − s + θ; ϕ),
where in the last identity we have used (81).
Thus we conclude that the the perturbation approach based on the abstract
variation-of-constants formula developed in the previous section precisely yields
the twin evolutionary system defined by translation along the solution of (68).
We summarize this result in a theorem.
Theorem 8.2 Under the above conditions, translation along the solution of equation
(68) defines a twin evolutionary system {U(t, s)}t≥s given by (61).
A Review of Functions of Bounded Variation
In this appendix E denotes the Borel σ-algebra on [0, ∞). For E ∈ E, we call a
sequence of disjoint sets {E j } in E a partition of E if ∪∞
j=1 E j = E. A complex
Borel measure is a map μ : E → C such that μ(∅) = 0 and
μ(E) =
∞
j=1
μ(E j ),
for every partition {E j } of E with the series converging absolutely. In the following
we will often omit the adjective ‘bounded’. The total variation measure |μ| of a
complex Borel measure μ is given by
|μ|(E) = sup
⎧
⎨
⎩
n
j=0
|μ(E j )| | n ∈ N, {E j } a partition of E in E
⎫
⎬
⎭
.
The vector space of complex Borel measures of bounded total variation is denoted
by M ([0, ∞)). Provided with the total variation norm given by
μT V = |μ| ([0, ∞)) , (82)
the vector space M ([0, ∞)) becomes a Banach space.
Let f : [0, ∞) → C be a given function. For a partition {E j } of [0, t] with E j =
[tj−1, tj ) and 0 = t0 t1 · · · tn = t we define the function Tf : [0, ∞) →
[0, ∞] by](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-45-320.jpg)
![The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 33
Tf (t) := sup
n
j=1
| f (tj ) − f (tj−1)|,
wherethesupremumistakenovern ∈ Nandallsuchpartitionsof[0, t].Theextended
real function Tf is called the total variation function of f . Note that if 0 ≤ a b,
then Tf (b) − Tf (a) ≥ 0 and hence Tf is an increasing function.
If limt→∞ Tf (t) is finite, then we call f a function of bounded variation. We
denote the space of all such functions by BV . The space N BV ([0, ∞)) of normalized
functions of bounded variation is defined by
N BV ([0, ∞)) := { f ∈ BV | f is continuous from the right on (0, ∞)
and f (0) = 0 }.
Provided with the norm
f T V := lim
t→∞
Tf (t) (83)
the space N BV ([0, ∞)) becomes a Banach space. More generally, we define for
−∞ a b ∞, the vector space N BV ([a, b]) to be the space of functions
f : [a, b] → C such that f (a) = 0, f is continuous from the right on the open
interval (a, b), and whose total variation on [a, b], given by Tf (b) − Tf (a) = Tf (b),
is finite. Provided with the norm f T V := Tf (b), the space N BV ([a, b]) becomes a
Banach space. We extend the domain of definition of a function of bounded variation
by defining f (t) = 0 for t 0 if f ∈ N BV ([0, ∞)) and f (t) = 0 for t a and
f (t) = f (b) for t b if f ∈ N BV ([a, b]).
The following fundamental result (see Folland 1999, Theorem 3.29) provides the
correspondence between functions of bounded variation and complex Borel mea-
sures.
Theorem A.1 Let μ be a complex Borel measure on R. If f : [0, ∞) → C is defined
by f (t) = μ((0, t]), then f ∈ N BV ([0, ∞)). Conversely, if f ∈ N BV ([0, ∞)) is
given, then there is a unique complex Borel measure μ f such that μ f ((0, t]) = f (t).
Moreover |μ f | = μTf
.
Given a function f ∈ N BV ([a, b]) with corresponding measure μ f , we define
the Lebesgue-Stieltjes integral
g d f or
g(x) f (dx) to be
g dμ f . Thus, a
Lebesgue-Stieltjes integral is a special Lebesgue integral and the theory for the
Lebesgue integral applies to the Lebesgue-Stieltjes integral. We embed L1
([0, ∞))
into M ([0, ∞)) by identifying f ∈ L1
([0, ∞)) with the measure μ defined by
μ(E) =
E
f (x) dx or, in short, μ(dx) = f (x) dx.
In this section we collect some results about the convolution of a measure and a
function and the convolution of two measures needed to study renewal equations.](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-46-320.jpg)
![34 S. Verduyn Lunel
For details we refer to Appendix A of Diekmann and Verduyn Lunel (2021) and for
further results we refer to Folland (1999); Grippenberg et. al. (1990).
Let B ([0, ∞)) denote the vector space of all bounded, Borel measurable func-
tions f : [0, ∞) → R. Provided with the supremum norm (denoted by · ), the
space B ([0, ∞)) becomes a Banach space. With B ([a, b]) we denote the Banach
space of all bounded, Borel measurable functions f : [a, b] → R provided with the
supremum norm.
The half-line convolution μ ∗ f of a measure μ ∈ M([0, ∞)) and a Borel mea-
surable function f ∈ B ([0, ∞)) is the function
(μ ∗ f )(t) :=
[0,t]
μ(ds) f (t − s)
defined for those values of t for which [0, t] s → f (t − s) is |μ|-integrable.
The following result can be found in Grippenberg et. al. (1990, Theorem 3.6.1(ii)).
Theorem A.2 If f ∈ B ([0, ∞)) and μ ∈ M ([0, ∞)), then the convolution of f
and μ satisfies μ ∗ f ∈ B ([0, ∞)) and
μ ∗ f ≤ μT V f .
The half-line convolution μ ∗ ν of two measures μ, ν ∈ M ([0, ∞)) is defined by
the complex Borel measure that to each Borel set E ∈ E assigns the value
(μ ∗ ν)(E) :=
[0,∞)
μ(ds)ν ((E − s)+) , (84)
where (E − s)+ := {e − s | e ∈ E} ∩ [0, ∞) (cf. Grippenberg et. al. 1990, Defini-
tion 4.1.1)).
If χE is the characteristic function of the set E, then
ν((E − s)+) =
[0,∞)
χE (σ + s)ν(dσ),
where [0, ∞) σ → χE (σ + s) is the characteristic function of (E − s)+. It fol-
lows from Theorem A.2 that s → ν(E − s)+) belongs to B ([0, ∞)) and hence the
definition of the convolution of two measures μ ∗ ν : E → C given in (84) makes
sense. Furthermore, using Fubini’s Theorem, we have the following useful identity](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-47-320.jpg)
![40 S. Verduyn Lunel
operators, and show that the period map of a periodic neutral delay equation has
a characteristic matrix function.
We end the introduction with an outline of this chapter. In Sect.2 we introduce
and discuss the basic properties of Jordan chains. In Sect.3 we introduce the notion
of a characteristic matrix function for a class of bounded operators, and prove that
the characteristic matrix function completely determines the spectral properties of
the associated bounded operator. In Sect.4 we show that the period map associated
with a periodic neutral delay equation has a characteristic matrix function. In Sect.5
we show that in case the period is equal to the delay, then we can compute the
characteristic matrix function rather explicitly. Finally, in Sect.6, we consider a class
of periodic delay equations for which the period is two times the delay. We construct
new examples for which we can compute the characteristic matrix function explicitly.
In particular, we construct an example for which the period map has a finite spectrum.
In the literature such examples are only known in case the period is equal to the delay,
and were unknown in case the period is two times the delay.
2 Equivalence and Jordan Chains
Let X, Y, X
, Y
be complex Banach spaces, and suppose that L : U → L(X, Y)
and M : U → L(X
, Y
) are operator-valued functions, analytic on the open subset
U ⊂ C. The two operator-valued functions L and M are called equivalent on U
(see Sect.2.4 in Bart et al. (1979)) if there exist analytic operator-valued functions
E : U → L(X
, X) and F : U → L(Y, Y
), whose values are invertible operators,
such that,
M(z) = F(z)L(z)E(z), z ∈ U. (1)
Let L : U → L(X, Y) be an analytic operator-valued function. A point λ0 ∈ U is
called a root of L if there exists a vector x0 ∈ X, x0 = 0, such that,
L(λ0)x0 = 0.
An ordered set (x0, x1, . . . , xk−1) of vectors in X is called a Jordan chain for L at
λ0 if x0 = 0 and
L(z)[x0 + (z − λ0)x1 + · · · + (z − λ0)k−1
xk−1] = O((z − λ0)k
). (2)
The number k is called the length of the chain and the maximal length of the chain
starting with x0 is called the rank of x0. The analytic function
k−1
l=0
(z − λ0)l
xl](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-53-320.jpg)
![The Project Gutenberg eBook of The Palace and
Park
This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: The Palace and Park
Author: Samuel Phillips
Edward Forbes
R. G. Latham
Richard Owen
George Scharf
F. K. J. Shenton
Release date: June 30, 2019 [eBook #59843]
Language: English
Credits: Produced by Chris Curnow, Harry Lamé and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-59-320.jpg)
![continue and confirm the good it had already effected in the nation and
beyond it.
It will be remembered that the destination of the Great Exhibition
building occupied much public attention towards the close of 1851, and
that a universal regret prevailed at the threatened loss of a structure
which had accomplished so much for the improvement of the national
taste, and which was evidently capable, under intelligent direction, of
effecting so very much more. A special commission even had been
appointed for the purpose of reporting on the different useful purposes to
which the building could be applied, and upon the cost necessary to carry
them out. Further discussion on the subject, however, was rendered
unnecessary by the declaration of the Home Secretary, on the 25th of
March, 1852, that Government had determined not to interfere in any
way with the building, which accordingly remained, according to previous
agreement, in the hands of Messrs. Fox and Henderson, the builders and
contractors. Notwithstanding the announcement of the Home Secretary, a
last public effort towards rescuing the Crystal Palace for its original site in
Hyde Park, was made by Mr. Heywood in the House of Commons, on the
29th of April. But Government again declined the responsibility of
purchasing the structure, and Mr. Heywood’s motion was, by a large
majority, lost.
It was at this juncture that Mr. Leech,[1] a private gentleman, conceived
the idea of rescuing the edifice from destruction, and of rebuilding it on
some appropriate spot, by the organisation of a private company. On
communicating this view to his partner, Mr. Farquhar, he received from
him a ready and cordial approval. They then submitted their project to Mr.
Francis Fuller, who entering into their views, undertook and arranged, on
their joint behalf, a conditional purchase from Messrs. Fox and
Henderson, of the Palace as it stood. In the belief that a building, so
destined, would, if erected on a metropolitan line of railway, greatly
conduce to the interests of the line, and that communication by railway
was essential for the conveyance thither of great masses from London,
Mr. Farquhar next suggested to Mr. Leo Schuster, a Director of the
Brighton Railway, that a site for the new Palace should be selected on the
Brighton line. Mr. Schuster, highly approving of the conception, obtained
the hearty concurrence of Mr. Laing, the Chairman of the Brighton Board,
and of his brother Directors, for aiding as far as possible in the
prosecution of the work. And, accordingly, these five gentlemen, and their](https://image.slidesharecdn.com/23098801-250601153211-61a69919/85/Controlling-Delayed-Dynamics-Advances-In-Theory-Methods-And-Applications-Dimitri-Breda-88-320.jpg)
Controlling Delayed Dynamics Advances In Theory Methods And Applications Dimitri Breda
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- 5. International Centre for Mechanical Sciences CISM International Centre for Mechanical Sciences Courses and Lectures 604 Dimitri Breda Editor Controlling Delayed Dynamics Advances in Theory, Methods and Applications
- 6. CISM International Centre for Mechanical Sciences Courses and Lectures Volume 604 Managing Editor Paolo Serafini, CISM—International Centre for Mechanical Sciences, Udine, Italy Series Editors Elisabeth Guazzelli, Laboratoire Matière et Systèmes Complexes, Université Paris Diderot, Paris, France Alfredo Soldati, Institute of Fluid Mechanics and Heat Transfer, Technische Universität Wien, Vienna, Austria Wolfgang A. Wall, Institute for Computational Mechanics, Technische Universität München, Munich, Germany Antonio De Simone, BioRobotics Institute, Sant’Anna School of Advanced Studies, Pisa, Italy
- 7. For more than 40 years the book series edited by CISM, “International Centre for Mechanical Sciences: Courses and Lectures”, has presented groundbreaking developments in mechanics and computational engineering methods. It covers such fields as solid and fluid mechanics, mechanics of materials, micro- and nanomechanics, biomechanics, and mechatronics. The papers are written by international authorities in the field. The books are at graduate level but may include some introductory material.
- 8. Dimitri Breda Editor Controlling Delayed Dynamics Advances in Theory, Methods and Applications
- 9. Editor Dimitri Breda CDLab - Computational Dynamics Laboratory University of Udine Udine, Italy ISSN 0254-1971 ISSN 2309-3706 (electronic) CISM International Centre for Mechanical Sciences ISBN 978-3-031-00981-5 ISBN 978-3-031-01129-0 (eBook) https://doi.org/10.1007/978-3-031-01129-0 © CISM International Centre for Mechanical Sciences 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. Preface Delays are ubiquitous in engineering and natural sciences: communication delays in control devices or the incubation period during an epidemic spread are just a couple of noteworthy examples. The inclusion of past history in the time evolution adds nontrivial complexities with respect to ordinary systems, balancing the advantage of dealing with more realistic models. Equations involving delays generate infinite- dimensional dynamical systems, asking for advanced tools and methods in the back- ground mathematical analysis, the numerical treatment and the development, design and optimization of control strategies. Eventually, understanding fundamental issues like stability is crucial, especially for varying or uncertain parameters. These premises motivated the organization of an international course at CISM in 2019, and this book collects contributions of the lecturers about analytical, numerical and application aspects of time-delay systems, under the paradigm of control theory. The aim is at discussing recent advances in these different contexts, also highlighting the interdisciplinary connections. Chapter “The Twin Semigroup Approach Towards Periodic Neutral Delay Equa- tions” deals with twin semigroups and norming dual pairs for neutral delay equa- tions, including time-dependent perturbations in view of periodic problems. Then in chapter “Characteristic Matrix Functions and Periodic Delay Equations”, charac- teristic matrix functions are introduced to analyze spectral properties, focusing on monodromy operators of neutral periodic delay equations. Chapters “Pseudospectral Methods for the Stability Analysis of Delay Equa- tions. Part I: The Infinitesimal Generator Approach and “Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach” concern the use of pseudospectral collocation techniques to reduce to finite dimension the dynamical analysis of both delay differential and renewal equations. Discretizations of the infinitesimal generator of the relevant semigroup (Chapter “Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach”) and of the semigroup itself (Chapter “Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach”) are described in view of analyzing local stability and performing bifurcation analysis. v
- 11. vi Preface The focus moves then to the characteristic roots of linear time-invariant time- delay systems in view of stability. Frequency-sweeping techniques are illustrated in chapter “Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications”, while frequency- domain approaches are presented in chapter “Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods”, linking maximal multiplicity to dominancy, also in view of low-complexity controllers. Chapter “Bifurcation Analysis of Systems With Delays: Methods and Their Use in Applications” presents a dynamical systems point of view to study problems with possiblystate-dependentdelays.ByusingthemostrecentreleaseofDDE-BIFTOOL, the numerical continuation of steady states and periodic orbits, their bifurcations and relevant normal forms are addressed, also through the analysis of two longer case studies. Chapters“DesignofStructuredControllersforLinearTime-DelaySystems”gives an overview of control design methods, grounded in matrix theory and numerical linear algebra and relying on a direct optimization of stability, robustness and perfor- mance indicators as a function of controller or design parameters. Then Chapter “A Scalable Controller Synthesis Method for the Robust Control of Networked Systems” concentrates on a scalable controller synthesis method in the framework of H∞-norm control for networked systems. Finally, chapters “Regenerative Machine Tool Vibrations” and “Dynamics of Human Balancing” discuss models of, respectively, machine tool vibrations and human balancing tasks. In the former, the phenomenon called surface regeneration is analyzed in terms of the delay differential equations governing the vibrations, and stability diagrams are constructed. In the latter, the central role played by the reaction time is addressed by discussing stabilizability issues in terms of the critical delay for different feedback concepts. After two years of the global pandemic, the time has eventually come to put an end to this volume: after all, some delay is not completely out of place given the subject. Once more, let me thank the lecturers: I am sure that their stimulating contributions to the course have been much appreciated by the 46 attendees from 13 different countries, to which I gratefully add myself. The priceless help and kind presence of CISM administrative staff are also acknowledged with true pleasure. This book is the result of the lecturers’ effort, Sjoerd Verduyn Lunel, Silviu-Iulian Niculescu, Bernd Krauskopf, Wim Michiels and Tamás Insperger (and of that of their co-authors, whom I thank as well). I am tremendously grateful to them, as well as to many other colleagues for all I could learn about delay systems. Since the list would be excessively long, let me just give credit to Gabor Stépán for having inspired the course behind this volume, and to Rossana Vermiglio and Stefano Maset for having patiently introduced me to this research field, for which I trust this volume will be a valid resource. Udine, Italy Dimitri Breda
- 12. Contents The Twin Semigroup Approach Towards Periodic Neutral Delay Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sjoerd Verduyn Lunel Characteristic Matrix Functions and Periodic Delay Equations . . . . . . . . 37 Sjoerd Verduyn Lunel Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach . . . . . . . . . . . . . 65 Dimitri Breda Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach . . . . . . . . . . . . . . . . . 95 Dimitri Breda Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Silviu-Iulian Niculescu, Xu-Guang Li and Arben Çela Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Silviu-Iulian Niculescu and Islam Boussaada Bifurcation Analysis of Systems With Delays: Methods and Their Use in Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Bernd Krauskopf and Jan Sieber Design of Structured Controllers for Linear Time-Delay Systems . . . . . . 247 Wim Michiels A Scalable Controller Synthesis Method for the Robust Control of Networked Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Pieter Appeltans and Wim Michiels vii
- 13. viii Contents Regenerative Machine Tool Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Tamás Insperger and Gabor Stépán Dynamics of Human Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Tamás Insperger, Gabor Stépán and John Milton
- 14. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations Sjoerd Verduyn Lunel Abstract In the first part of this chapter we review the recently developed theory of twin semigroups and norming dual pairs in the light of neutral delay equations. In the second part we extend the perturbation theory for twin semigroups to include time- dependent perturbations. Finally we apply this newly developed theory to neutral periodic delay equations. 1 Introduction Consider a function x defined on the half-line [0, ∞) with values in Rn and assume that the derivative ẋ depends on the history of x and ẋ. More precisely, we assume that there exists h > 0 such that ẋ(t) depends on x(τ) and ẋ(τ) for t − h ≤ τ ≤ t. Given these restrictions we would like to consider a general linear differential equation. To formulate precisely what type of equations we consider, we first define the segment xt : [−h, 0] → Rn by xt (θ) := x(t + θ), for − h ≤ θ ≤ 0. (1) Let η and ζ be n × n-matrix-valued functions of bounded variation defined on [0, ∞) such that η(0) = ζ(0) = 0, η and ζ are continuous from the right on (0, h), η(t) = η(h)andζ(t) = ζ(h)fort ≥ h.Wecallsuchfunctionsη andζ ofnormalizedbounded variation. Furthermore assume that η(t) is continuous at t = 0. (See Appendix A for the precise definition and basic properties of such functions.) The class of equations that we will study can now be written as d dt x(t) − h 0 dη(θ)x(t − θ) = h 0 dζ(θ)x(t − θ). (2) S. Verduyn Lunel (B) Mathematical Institute, Utrecht University, Utrecht, The Netherlands e-mail: S.M.VerduynLunel@uu.nl © CISM International Centre for Mechanical Sciences 2023 D. Breda (ed.), Controlling Delayed Dynamics, CISM International Centre for Mechanical Sciences 604, https://doi.org/10.1007/978-3-031-01129-0_1 1
- 15. 2 S. Verduyn Lunel To single out a unique solution we have to provide an initial condition at a certain time s. The initial condition should specify the values of x on the interval of length h preceding time s. Let y satisfy (2) for t ≥ s and the initial condition y(s + θ) = ϕ(θ), −h ≤ θ ≤ 0, where ϕ ∈ B ([−h, 0]; Rn ), the Banach space of bounded Borel measurable func- tions provided with the supremum norm (see Sect.A for the precise definition and basic properties). Then x defined for t ≥ 0 by x(t) = y(s + t), satisfies (2) for t ≥ 0 and the initial condition x(θ) = ϕ(θ), −h ≤ θ ≤ 0. (3) Equation (2) is time invariant and called autonomous. So we can, without loss of generality, restrict our attention to an initial condition imposed at time zero. This in contrast to time periodic equations which we will consider in Sect.8. Equation (2) is called a neutral functional differential equation (NFDE). A solution of the initial-value problem (2)–(3) on the half-line [0, ∞) is a function x ∈ B ([0, ∞); Rn ) such that (i) (3) holds; (ii) on (0, ∞), the function x is absolutely continuous and (2) holds; (iii) the following limit exists lim t↓0 1 t x(t) − h 0 dη(θ)x(t − θ) − ϕ(0) − h 0 dη(θ) ϕ(−θ) and equals h 0 dζ(θ) ϕ(−θ). We end the introduction with an outline of this chapter. In Sect.2 we derive a representation of the solution of a NFDE by direct methods. The main result is given in Theorem 2.4. In Sect.3 we introduce the notions of norming dual pair and twin semigroup following Diekmann and Verduyn Lunel (2021). In Sect.4 we introduce a concrete norming dual pair that will be used in Sect.5 to represent the solution semigroup corresponding to a NFDE as a twin semigroup. In Sect.6 we use the twin semigroup approach towards NFDE to prove a variation-of-constants formula, see Theorem 6.4. In Sect.7 we develop the perturbation theory for bounded time- dependent perturbations of twin semigroups. The main result is given in Theorem 7.5. In Sect.8 we consider periodic NFDE as an application of the perturbation theory developed in Sect.7 and prove that periodic NFDE define a twin evolutionary system. Finally in Appendix A we review some basic properties of functions of bounded variations and complex Borel measures.
- 16. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 3 2 Introduction to NFDE This section is concerned with the existence, uniqueness and representation of a solution of the initial-value problem (2)–(3). For 0 ≤ t ≤ h, we can combine the two separate pieces of information given in (2) and (3) and write d dt x(t) − h 0 dη(θ)x(t − θ) = t 0 dζ(θ)x(t − θ) + h t dζ(θ)ϕ(t − θ). (4) By integration and changing the order of integration we can write (4) as x(t) − h 0 dη(θ)x(t − θ) = t 0 ζ(θ)x(t − θ) dθ + g(t), (5) where g(t) := ϕ(0) − h 0 dη(θ)ϕ(−θ) + t 0 h s dζ(θ)ϕ(s − θ) ds. (6) Next we write (5) as follows x(t) = t 0 dη(θ)x(t − θ) + t 0 ζ(θ)x(t − θ) dθ + f (t), (7) where, using (6), f (t) := g(t) + h t dη(θ)ϕ(t − θ) = ϕ(0) + h 0 [ζ(t + σ) − ζ(σ)] ϕ(−σ) dσ + h 0 d [η(t + σ) − η(σ)] ϕ(−σ). (8) Here we have used that t 0 h s dζ(θ)ϕ(s − θ) ds = h 0 [ζ(t + σ) − ζ(σ)] ϕ(−σ) dσ and that h t dη(θ)ϕ(t − θ) = h 0 dη(t + σ)ϕ(−σ). It follows from Theorem A.2 that the function f defined by (8) is a bounded Borel measurable function on [0, ∞) that is constant on [h, ∞).
- 17. 4 S. Verduyn Lunel Define the function μ by μ(θ) := η(θ) + θ 0 ζ(s) ds, 0 ≤ θ ≤ h, (9) and μ(θ) = μ(h) for θ ≥ h, then μ is a n × n-matrix-valued function of normalized bounded variation. Note that, since η(θ) is continuous at θ = 0, we have that μ(θ) is continuous at θ = 0. Theconvolutionproductofan × n-matrix-valuedfunctionofnormalizedbounded variation μ and a bounded Borel measurable function f is defined by (μ ∗ f ) (t) := t 0 dμ(θ) f (t − θ), t ≥ 0. (10) From Theorem A.1, it follows that μ ∗ f is a bounded Borel measurable function on [0, ∞). Using the convolution product defined by (10), the initial-value problem (2)–(3), i.e., (7), can be rewritten as a renewal equation for x x = μ ∗ x + f, (11) where μ is given by (9) and f , given by (8), can be rewritten as f (t) = ϕ(0) + h 0 d [μ(t + σ) − μ(σ)] ϕ(−σ). (12) Therefore to prove existence and uniqueness of solutions of the initial-value problem (2)–(3), it suffices to prove existence and uniqueness of solutions of the renewal equation (11). The convolution product of two n × n-matrix-valued functions of normalized bounded variation μ and ν, defined by (μ ∗ ν) (t) := t 0 dμ(θ)ν(t − θ), t ≥ 0, (13) is again a function of bounded variation (see Appendix A and, in particular, Theorem A.3). The resolvent kernel ρ of a renewal equation (11) with kernel μ and convolution product (13) is defined as the matrix solution of the resolvent equation ρ = ρ ∗ μ + μ = μ ∗ ρ + μ. (14) The key property of the resolvent concerns the representation of the solution of the renewal equation (11) as x = f + ρ ∗ f. (15)
- 18. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 5 Indeed taking to convolution with ρ on the left and right of (11) yields ρ ∗ x = (ρ ∗ μ) ∗ x + ρ ∗ f = (ρ − μ) ∗ x + ρ ∗ f. Hence μ ∗ x = ρ ∗ f and substituting this relation into (11) yields (15). We now discuss the existence and uniqueness of the solution of (14) under the assumption that μ is a n × n-matrix-valued function of normalized bounded varia- tion. It follows from Appendix A and in particular Theorem A.1 that functions of normalized bounded variation are in one-to-one correspondence to complex Borel measures. This allows us to use measure theory to prove existence and uniqueness of the solution of (14). We start with some preparations. Let E denote the Borel σ-algebra on [0, ∞). The Banach space of complex Borel measures of bounded total variation is denoted by M ([0, ∞)) (see (82)). Let Mloc ([0, ∞)) denote the vector space of local measures, i.e., set functions that are defined on relatively compact Borel measurable subsets of [0, ∞) and that locally behave like bounded measures: for every T 0 the set function μT defined by μT (E) := μ (E ∩ [0, T ]) , E ∈ E, belongs to M ([0, ∞)). The elements of Mloc ([0, ∞)) are called Radon measures. Since the restriction to [0, T ] of μ ∗ ν depends only on the restrictions of μ and ν to [0, T ], we can unambiguously extend the convolution product to Mloc ([0, ∞)) (see (84)). We continue with the existence of the resolvent ρ of a complex Borel measure μ supported on [0, ∞). For details see Diekmann and Verduyn Lunel (2021, Theorem A.7) and for further information and details see Grippenberg et. al. (1990). Theorem 2.1 Suppose that μ ∈ Mloc [0, ∞); Rn×n . There exists a unique measure ρ ∈ Mloc [0, ∞); Rn×n satisfying either one of the following identities ρ − μ ∗ ρ = μ = ρ − ρ ∗ μ (16) if and only if det [I − μ({0})] = 0. Furthermore, if μ((0, t]) is continuous as t = 0, then ρ((0, t]) is continuous at t = 0 as well. The following theorem summarizes some relevant results for renewal equations (Diekmann and Verduyn Lunel 2021, Theorem A.9). Theorem 2.2 Let μ ∈ Mloc [0, ∞), Rn×n with det [I − μ({0}] = 0. (i) For every f ∈ Bloc ([0, ∞), Rn ), the renewal equation (15) has a unique solution x ∈ Bloc ([0, ∞), Rn ) given by x = f + ρ ∗ f, where ρ satisfies (16). Furthermore, if f is locally absolutely continuous, then the solution x is locally absolutely continuous as well.
- 19. 6 S. Verduyn Lunel (ii) If the kernel μ has no discrete part and if f ∈ C ([0, ∞), Rn ), then x ∈ C ([0, ∞), Rn ). We now summarize the conclusions obtained so far in this section in the following theorem. Theorem 2.3 Let η and ζ be of normalized bounded variation. Let ϕ ∈ B ([−h, 0]; Rn ) be given. Define μ by (9). If det [I − μ(0)] = 0, then the NFDE (2) provided with the initial condition (3) admits a unique solution. For t ≥ 0 this solution coincides with the unique solution of the renewal equation (11) and the solution has the representation (15) where ρ satisfies the resolvent equation (14) and f is given by (8). Representation (15) will be used to derive a representation of the solution of (2)–(3) directly in terms of the initial data x0 = ϕ. We first need a definition. The fundamental solution of the delay equation (2)–(3) on [−h, ∞) is defined by the n × n-matrix-valued function X(t) := I + ρ((0, t]) for t ≥ 0, 0 for − h ≤ t 0, (17) where ρ is the resolvent of μ given by Theorem 2.1. Since t → μ((0, t]) is continuous at t = 0, it follows from Theorem 2.1 that ρ((0, t]) is continuous at t = 0. Therefore we can conclude that X(t) has a jump at t = 0. By construction, the fundamental matrix solution X(t) satisfies (2) with initial data X0(θ) = I for θ = 0, 0 for − h ≤ θ 0. (18) Using the fundamental matrix solution X(t) given by (17) and Fubini’s theorem, we can rewrite the representation formula (15) in terms of the forcing function f given by (8) directly in terms of the initial condition ϕ. We summarize the result in a theorem. Theorem 2.4 The solution of (2)–(3) is given explicitly by x(t; ϕ) = X(t)ϕ(0) + h 0 d t −h dX(τ) (μ(t − τ + σ) − μ(σ)) ϕ(−σ). (19) Or, equivalently, in terms of the resolvent ρ we have x(t; ϕ) = (I + ρ((0, t])) ϕ(0) + h 0 d [μ(t + σ) − μ(σ)])ϕ(−σ) + h 0 d t 0 ρ(dτ) (μ(t − τ + σ) − μ(σ)) ϕ(−σ). (20) Here μ is given by (9).
- 20. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 7 3 Norming Dual Pairs and Twin Semigroups The system of equations (2)–(3) defines an infinite-dimensional dynamical system on the state space B ([−h, 0]; Rn ), but for the qualitative study of such a dynamical system we need an adjoint theory in place (see Hale and Verduyn Lunel 1993). In the classical theory of delay equations this is the main reason to work with the state space C ([−h, 0]; Rn ) despite the fact that the initial data of the fundamental solution (see (18)) does not belong to this space. From the Riesz representation theorem it follows that the dual space of C ([−h, 0]; Rn ) has a nice characterization as the space of functions of normalized bounded variation. The state space B ([−h, 0]; Rn ) includes the initial data of the fundamental solu- tion but its dual space does not have a nice characterization. So although the state space B ([−h, 0]; Rn ) is a more natural space to consider, it has not yet been used because its dual space is too large to provide a useful adjoint theory. A beautiful idea to repair this discrepancy is to use the notion of a dual pair (see Aliprantis and Border 2006) made precise in Kunze (2011) for infinite-dimensional dynamical systems in the following way. Two Banach spaces Y and Y are called a norming dual pair (cf. Kunze (2011)) if a bilinear map · , · : Y × Y → R exists such that, for some M ∈ [1, ∞), | y , y| ≤ My y and, moreover, y := sup | y , y| | y ∈ Y , y ≤ 1 y := sup | y , y| | y ∈ Y, y ≤ 1 . So we can consider Y as a closed subspace of Y ∗ , the dual of Y , and Y as a closed subspace of Y∗ and both subspaces are necessarily weak∗ dense since they separate points. The collection of linear functionals Y defines a weak topology on Y, denoted by σ(Y, Y ). The corresponding locally convex topological vector space is denoted by (Y, σ(Y, Y )). A crucial point in our approach is that the dual space (Y, σ(Y, Y )) is (isometrically isomorphic to) Y (Rudin 1991, Theorem 3.10). So if a linear func- tional on Y is continuous with respect to the topology induced by Y , it can be (uniquely) represented by an element of Y . The next key idea to study infinite-dimensional dynamical systems on a norming dual pair is the notion of a twin operator introduced in Diekmann and Verduyn Lunel (2021).
- 21. 8 S. Verduyn Lunel A twin operator L on a norming dual pair (Y, Y ) is a bounded bilinear map from Y × Y to R that defines both a bounded linear map from Y to Y and a bounded linear map from Y to Y . More precisely, L : Y × Y → R (y , y) → y Ly is such that (i) for some C 0 the inequality |y Ly| ≤ Cy y holds for all y ∈ Y and y ∈ Y ; (ii) forgiven y ∈ Y themap y → y Ly iscontinuousasamapfrom(Y , σ(Y , Y)) to R and hence there exists Ly ∈ Y such that y , Ly = y Ly for all y ∈ Y ; (iii) forgiven y ∈ Y themap y → y Ly iscontinuousasamapfrom(Y, σ(Y, Y )) to R and hence there exists y L ∈ Y such that y L, y = y Ly for all y ∈ Y. So all three maps are denoted by the symbol L, but to indicate on which space L acts we write, inspired by Feller (1953) which, in turn, is inspired by matrix notation, either y Ly, Ly or y L. As a concrete example, consider the identity operator. It maps (y , y) to y , y, y to y and y to y . If our starting point is a bounded linear operator L : Y → Y then there exists an associated twin operator if and only if the adjoint of L leaves the embedding of Y into Y∗ invariant. We express this in words by saying that L extends to a twin operator. Likewise, if our starting point is an operator L : Y → Y then L extends to a twin operator if and only if the adjoint of L leaves the embedding of Y into Y ∗ invariant. So a twin operator on a norming dual pair is reminiscent of the combination of a bounded linear operator on a reflexive Banach space and its adjoint, whence the adjective “twin”. The composition of bounded bilinear maps is, in general, not defined. But for twin operators it is! Indeed, if L1 and L2 are both twin operators on the norming dual pair (Y, Y ), we define the composition L1 L2 by y L1 L2 y := y L1, L2 y. Note that this definition entails that L1 L2 acts on Y by first applying L2 and next L1, whereas L1 L2 acts on Y by first applying L1 and next L2.
- 22. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 9 Definition 3.1 A family {S(t)}t≥0 of twin operators on a norming dual pair (Y, Y ) is called a twin semigroup if (i) S(0) = I, and S(t + s) = S(t)S(s) for t, s ≥ 0; (ii) there exist constants M ≥ 1 and ω ∈ R such that |y S(t)y| ≤ Meωt y y ; (iii) for all y ∈ Y, y ∈ Y the function t → y S(t)y is measurable; (iv) for Re λ ω (with ω as introduced in ii)) there exists a twin operator S(λ) such that y S(λ)y = ∞ 0 e−λt y S(t)y dt. (21) Note that the combination of ii) and iii) allows us to conclude that the right hand side of (21) defines a bounded bilinear map, but not that it defines a twin operator. Hence iv) is indeed an additional assumption. Wecall S(λ)definedon{λ | Re λ ω}theLaplacetransform of{S(t)}.Itactually suffices to assume that the assertion of iv) holds for λ = λ0 with Re λ0 ω. This assumption allows us to introduce the multi-valued operator C = λ0 I − S(λ0)−1 (22) on Y and next define the function λ → S(λ) by S(λ) = (λI − C)−1 (23) on an open neighbourhood of λ0. In Definition 2.6 of Kunze (2009) an operator C is called the generator of the semigroup provided the Laplace transform is injective and hence C is single-valued. In Diekmann and Verduyn Lunel (2021) we adopted a more pliant position and call C the generator even when it is multi-valued and we refer to this paper for additional information. Focusing on {S(t)}t≥0 as a semigroup of bounded linear operators on Y, we now list some basic results from Kunze (2011). Lemma 3.2 The following statements are equivalent 1. y ∈ D (C) and z ∈ Cy; 2. there exist λ ∈ C with Re λ ω, here ω is as introduced in ii) of Definition 3.1, and y, z ∈ Y such that
- 23. 10 S. Verduyn Lunel y = S(λ)(λy − z) 3. y, z ∈ Y and for all t 0 t 0 S(τ)z dτ = S(t)y − y. Here it should be noted that item 3. includes the assertions • the integral t 0 S(τ)z dτ defines an element of Y (even though at first it only defines an element of Y ∗ ); • the integral t 0 S(τ)z dτ does not depend on the choice of z ∈ Cy in case C is multi-valued. Lemma 3.3 For all t 0 and y ∈ Y, we have t 0 S(τ)y dτ ∈ D (C) and S(t)y − y ∈ C t 0 S(τ)y dτ. 4 The Norming Dual Pair (B, N BV) In the study of delay differential equations, the natural dual pair is given by Y = B [−1, 0], Rn and Y = N BV [0, 1], Rn (24) with the pairing y , y = [0,1] y (dσ) · y(−σ) (25) (see Appendix A for the definition of N BV ). Here Y is provided with the supremum norm and Y with the total variation norm (see (83)). See Diekmann and Verduyn Lunel 2021. In the study of renewal equations, the natural dual pair is given by Y = N BV [−1, 0], Rn and Y = B [0, 1], Rn with the pairing y , y = [−1,0] y(dσ) · y (−σ). Returning to (24)–(25), we first make two trivial, yet useful, observations: fix 1 ≤ i ≤ n and −1 ≤ θ ≤ 0,
- 24. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 11 [0,1] y (dσ) · y(−σ) = yi (θ), if yj (σ) = 0, 0 ≤ σ ≤ 1, j = i, and yi (σ) = 0 for 0 ≤ σ −θ and yi (σ) = 1 for σ ≥ −θ, and similarly [0,1] y (dσ) · y(−σ) = yi (−θ), if yj (−σ) = 0, 0 ≤ σ ≤ 1, j = i, and yi (−σ) = 1 for 0 ≤ σ ≤ −θ and yi (−σ) = 0 for σ −θ. The point is that, consequently, in case of (24)–(25), convergence in both (Y, σ(Y, Y )) and (Y , σ(Y , Y)) entails pointwise convergence (in, respectively, B ([−1, 0], Rn ) and N BV ([0, 1], Rn )). In the first case, the dominated convergence theorem implies that, conversely, a bounded pointwise convergent sequence in B ([−1, 0], Rn ) converges in (Y, σ(Y, Y )). For N BV ([0, 1], Rn ), this is not so clear. It is true that the pointwise limit of a sequence of functions of bounded variation is again of bounded variation (Helly’s theorem), but there is no dominated convergence theorem for measures. The following theorem is proved in Diekmann and Verduyn Lunel (2021, Theorem B.1). Theorem 4.1 The dual pair given by (24) and (25) is a norming dual pair, i.e., y = sup | y , y| | y ∈ Y , y ≤ 1 y = sup | y , y| | y ∈ Y, y ≤ 1 . Furthermore (i) (Y, σ(Y, Y )) is sequentially complete; (ii) a linear map (Y, σ(Y, Y )) → R is continuous if it is sequentially continuous. 5 The Twin Semigroup Approach to NFDE Consider the norming dual pair (Y, Y ) with Y and Y as given in Sect.4 by (24). By solving (2)–(3), see Theorem 2.3, we can define a Y-valued function u : [0, ∞) → Y by u(t; ϕ) := xt ( · ; ϕ), t ≥ 0, (26) where xt is defined by (1), and bounded linear operators S(t) : Y → Y by S(t)ϕ = u(t; ϕ). (27)
- 25. 12 S. Verduyn Lunel The initial condition (2) translates into S(0)ϕ = u(0; ϕ) = ϕ and (27) reflects that we define a dynamical system on Y by translating along the function ϕ extended according to (2). Below we show that {S(t)} is a twin semigroup and we characterize its generator C. But first we present some heuristics. In order to motivate an abstract ODE for the Y-valued function u, we first observe that the infinitesimal formulation of the translation rule (26) amounts to the PDE ∂u ∂t − ∂u ∂θ = 0. We need to combine this with (2), in terms of u(t)(0) = x(t), and we have to specify the domain of definition of the derivative with respect to θ. The latter is actually rather subtle. An absolutely continuous function has almost everywhere a derivative and when the function is Lipschitz continuous this derivative is bounded. Thus a Lipschitz function specifies a unique L∞ -equivalence class by the process of differentiation. But not a unique element of Y. In fact the set Cψ = ψ ∈ Y | ψ(θ) = ψ(−1) + θ −1 ψ (σ) dσ, ψ (0) − h 0 dη(θ)ψ (−θ) = h 0 dζ(θ)ψ(−θ) (28) is, for a given Lipschitz continuous function ψ, very large indeed. Note that the boundary condition ψ (0) − h 0 dη(θ)ψ (−θ) = h 0 dζ(θ)ψ(−θ) takes care of (2). We define C as a multi-valued, unbounded, operator on Y by (28) with domain given by D (C) = Lip [−1, 0], Rn . (29) We claim that (2)–(3) and (26) correspond to an abstract differential equation du dt ∈ Cu. To substantiate this claim, we shall verify that {S(t)}t≥0 defined by (27) is a twin semigroup and, finally, that C is the corresponding generator in the sense of (23) where S(λ) is given by (21). From the representation (19) of the solution of (2)–(3) we can derive an explicit representation of the semigroup {S(t)}t≥0 defined by (27).
- 26. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 13 Theorem 5.1 The semigroup {S(t)}t≥0 defined by (27) is given by (S(t)ϕ) (θ) = h 0 Kt (θ, dσ) ϕ(−σ) (30) with for σ 0 and −h ≤ θ ≤ 0 the kernel Kt (θ, σ) defined by Kt (θ, σ) := H(σ + t + θ) + H(t + θ)ρ(t + θ) + H(t + θ) t+θ 0 dX(ξ) (μ(t + θ + σ − ξ) − μ(σ)) , (31) and Kt (θ, 0) = 0. Here ρ denotes the resolvent of μ with μ defined in (9), X denotes the fundamental solution given by (17), and H is the standard Heaviside function. Proof For t + θ 0 the second and third terms in the expression for Kt do not contribute, and the first term yields (S(t)ϕ) (θ) = ϕ(t + θ) which is in accordance with (27) because of (3). Now assume that t + θ ≥ 0. Clearly the first term contributes a unit jump at σ = 0 and H(t + θ) = 1. The second term has, as a function of σ, a jump of magnitude ρ(t + θ) at σ = 0, an absolutely continuous part with derivative given by t+θ 0 dX(ξ) (ζ(t + θ + σ − ξ) − ζ(σ)) , and a part of bounded variation given by t+θ 0 dX(ξ) (η(t + θ + σ − ξ) − η(σ)) . The jumps yield the first term at the right hand side of (19) (see also (20)) evaluated at t + θ, the absolutely continuous part yields the second, and the bounded variation part the third term. Note that Kt is bounded, in the sense (cf. Kunze 2009, Definition 3.2) that for fixed θ in [−1, 0] the function σ → Kt (θ, σ) is of normalized bounded variation, while for fixed σ ∈ [0, 1] the function θ → Kt (θ, σ) is bounded and measurable. The next corollary is a general property of kernel operators. Corollary 5.2 The operator S(t) extends to a twin operator. Proof The proof directly follows from the observation that we can represent the action of y S(t) explicitly as
- 27. 14 S. Verduyn Lunel y S(t) (σ) = h 0 y (dτ) Kt (−τ, σ). Theorem 5.3 The semigroup {S(t)}t≥0 defined by (30) is a twin semigroup. Proof With reference to Definition 3.1 we note that S(0) = I follows directly from (30)–(31), while the semigroup property follows from the uniqueness of solutions to (2)–(3) and the fact that S(t) corresponds to translation along the solution. The exponential estimates (ii) are well-established in the theory of NFDE, see Sect.9.3 of Hale and Verduyn Lunel (1993) or the proof of Proposition 7.3 below. Property (iii), the measurability of t → y S(t)y, is a direct consequence of the way Kt (θ, σ), defined in (31), depends on t. It remains to verify that the Laplace transform defines a twin operator. By Fubini’s Theorem, the Laplace transform is a kernel operator with kernel ∞ 0 e−λt Kt (θ, σ) dt. Theorem 5.4 The operator C defined by (28) and (29) is the generator (in the sense of (23)) of {S(t)}t≥0 defined by (30). Proof Assume ϕ ∈ (λI − C)ψ. Then there exists ψ ∈ Y which is a.e. a derivative of ψ such that λψ − ψ = ϕ, −1 ≤ θ 0, satisfying the boundary condition λψ(0) − h 0 dη(θ)ψ (−θ) − h 0 dζ(θ)ψ(−θ) = ϕ(0). Solving the differential equation yields that ψ(θ) = eλθ ψ(0) + eλθ 0 θ e−λσ ϕ(σ) dσ (32) and accordingly the boundary condition for θ = 0 boils down to ψ(0) = (λ)−1 H(λ; ϕ), (33) where
- 28. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 15 H(λ; ϕ) := ϕ(0) + λ h 0 dη(σ)e−λσ 0 −σ e−λτ ϕ(τ) dτ + h 0 dζ(σ)e−λσ 0 −σ e−λτ ϕ(τ) dτ. This requires that det (λ) = 0 with (λ) = λ I − h 0 dη(σ)e−λσ + h 0 dζ(σ)e−λσ . Our claim is that the identity (λI − C)−1 ϕ = ∞ 0 e−λt S(t)ϕ dt (34) or, equivalently, ψ(θ) = ∞ 0 e−λt (S(t)ϕ) (θ) dt holds. To verify this, we first note that ∞ 0 e−λt (S(t)ϕ) (θ) dt = ∞ 0 e−λt x(t + θ; ϕ) dt = −θ 0 e−λt ϕ(t + θ) dt + ∞ −θ e−λt x(t + θ) dt = eλθ 0 θ e−λσ ϕ(σ) dσ + eλθ x̄(λ; ϕ), where x̄(λ; ϕ) := ∞ 0 e−λt x(t; ϕ) dt, with x(t; ϕ) the solution of (2)–(3) given by (19). So, since (32) holds, to prove (34) it remains to check that ψ(0) = x̄(λ; ϕ). By taking the Laplace transform on both sides of (11) we deduce that x̄(λ; ϕ) = 1 − ∞ 0 e−λt dμ(t) −1 ¯ f (λ) = (λ)−1 λ ¯ f (λ), where ¯ f (λ) := ∞ 0 e−λt f (t) dt. Therefore, using the representation of f in (12), it follows that
- 29. 16 S. Verduyn Lunel λ ¯ f (λ) = ϕ(0) + ∞ 0 λe−λt t 0 h s dζ(θ)ϕ(s − θ) ds dt + λ ∞ 0 e−λt h t dη(θ)ϕ(t − θ) dt = ϕ(0) + ∞ 0 e−λt h t dζ(θ)ϕ(t − θ) dt + λ ∞ 0 e−λt h t dη(θ)ϕ(t − θ) dt = ϕ(0) + h 0 dζ(θ) θ 0 e−λt ϕ(t − θ) dt + λ h 0 dη(θ) θ 0 e−λt ϕ(t − θ) dt = ϕ(0) + h 0 dζ(θ)e−λθ 0 −θ e−λσ ϕ(σ) dσ λ h 0 dη(θ)e−λθ 0 −θ e−λσ ϕ(σ) dσ = H(λ; ϕ). Therefore it follows from (33) that indeed ψ(0) = x̄(λ; ϕ) and this completes the proof of the identity (34). In Diekmann and Verduyn Lunel (2021), we proved Theorems 5.1, 5.3 and 5.4 for retarded functional differential equations, and gave an alternative proof of Theorem 5.3 in the neutral case using a relative bounded perturbation argument, see Diekmann and Verduyn Lunel (2021, Theorem 11.1). 6 The Variation-of-Constants Formula for NFDE It is a direct consequence of (29) that X = D (C) = C [−1, 0], Rn . Clearly Cψ ∩ X is either empty or a singleton, cf. (28), and for the set to be nonempty we need that ψ ∈ C1 and ψ (0) − h 0 dη(θ)ψ (−θ) = h 0 dζ(θ)ψ(−θ). So the generator A of the restriction {T (t)}t≥0 of {S(t)}t≥0 to X is given by
- 30. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 17 D (A) = ψ ∈ C1 | ψ (0) − h 0 dη(θ)ψ (−θ) = h 0 dζ(θ)ψ(−θ) Aψ = ψ in complete agreement with the standard theory. As S(t) maps Y into X for t ≥ 1, one might wonder whether we gained anything at all by the extension from X to Y? Already in the pioneering work of Jack Hale he emphasized that if one adds a forcing term to (2), one needs q(θ) := 1 for θ = 0, 0 for − 1 ≤ θ 0, to describe the solution by way of the variation-of-constants formula. Indeed, the solution of d dt x(t) − h 0 dη(θ)x(t − θ) = h 0 dζ(θ)x(t − θ) + f (t), t ≥ 0, x(θ) = ϕ(θ), −1 ≤ θ ≤ 0, (35) is explicitly given by xt = S(t)ϕ + t 0 S(t − τ)q f (τ) dτ, (36) where S(t) is given by (30) and xt is as defined in (1). This formally follows directly from the fact that the inhomogeneous NFDE (35) corresponds to the initial value problem du dt ∈ Cu + q f, u(0) = ϕ, where as before u(t) = xt . Note that the solution with initial condition q is the so-called fundamental solution, cf. (18) and (17). The integration theory presented next provides a precise underpinning of the integral in (36) and the remainder of this section is devoted to a proof of (36). In the original approach of Hale, the hidden argument θ in (36) is inserted and thus the integral reduces to the integration of an Rn -valued function. Note that evaluation in a point corresponds to the application of a Dirac functional, so our approach yields, in a sense, a theoretical underpinning of Hale’s approach. As a final remark, we emphasize that the variation-of-constants formula (36) is the key first step towards a local stability and bifurcation theory for nonlinear problems, as shown in detail in Diekmann et. al. (1995) for retarded functional differential equations. For neutral functional differential equations this is work in progress. Motivated by (36), we want to define an element u(t) of Y by way of the action on Y expressed in the formula
- 31. 18 S. Verduyn Lunel y , u(t) = y S(t)u0 + t 0 y S(t − τ)q f (τ) dτ, (37) where the standard assumptions are (i) (Y, Y ) is a norming dual pair; (ii) q ∈ Y; (iii) f : [0, T ] → R is bounded and measurable; (iv) {S(t)} is a twin semigroup, and where u0 (corresponding to ϕ in (36)) is an arbitrary element of Y. The definition of the first term at the right hand side of (37) is no problem at all, it contributes S(t)u0 to u(t). The second term defines an element of Y ∗ , but it is not clear that this element is, without additional assumptions, represented by an element of Y. The following lemma provides a sufficient condition. Lemma 6.1 In addition to (i)–(iv) assume that Y, σ(Y, Y ) is sequentially complete. (38) Then y → t 0 y S(t − τ)q f (τ) dτ (39) is represented by an element of Y, to be denoted as t 0 S(t − τ)q f (τ) dτ. Proof There exists a sequence of step functions fm such that | fm| ≤ | f | and fm → f pointwise. Lemma 3.3 shows that t 0 S(t − τ)q fm(τ) dτ belongs to Y (in fact even to D (C)). Since (see Definition 3.1(ii)) y S(t − τ)q fm(τ) ≤ Meω(t−τ) qy sup σ | f (σ)|, the dominated convergence theorem implies that for every y ∈ Y lim m→∞ t 0 y S(t − τ)q fm(τ) dτ = t 0 y S(t − τ)q f (τ) dτ. The sequential completeness next guarantees that the limit too is represented by an element of Y. In Diekmann and Verduyn Lunel (2021) we have developed a perturbation theory to study neutral equations directly as an unbounded perturbation of a retarted equa- tion. In order to do this, we have to replace f (τ) dτ by F(dτ) with F of bounded
- 32. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 19 variation. In this setting the approximation by step functions used in the proof of Lemma 6.1 no longer works. This observation motivates to look for an alternative sufficient condition to replace (38). This is taken care of in the following lemma. Lemma 6.2 In addition to (i)–(iv) assume that a linear map Y , σ(Y , Y) → R is continuous if it is sequentially continuous. (40) Then the assertion of Lemma 6.1 holds. Proof Again we are going to make use of the dominated convergence theorem. Consider a sequence {ym} in Y such that for every y ∈ Y the sequence ym, y converges to zero in R. Then for all relevant t and τ we have lim m→∞ ym S(t − τ)q = 0 and consequently lim m→∞ t 0 ym S(t − τ)q f (τ) dτ = 0. So the linear map (39) is, in the sense described in (40), sequentially continuous and therefore, by the assumption, continuous. Since Y , σ(Y , Y) = Y, we conclude that (39) is represented by an element of Y. We are going to use the above results to show that a certain type of perturbation of a twin semigroup {S(t)} yields again a twin semigroup. In order to do this we need a dual version of (37), i.e., we want to define an element u (t) of Y by way of the action on Y expressed in the formula u (t), y = u0 S(t)y + t 0 q S(t − τ)y f (τ) dτ, (41) where the standard assumptions are as before with (ii) replaced by (ii) , i.e., (i) (Y, Y ) is a norming dual pair; (ii) q ∈ Y ; (iii) f : [0, T ] → R is bounded and measurable; (iv) {S(t)} is a twin semigroup, and where u0 is an arbitrary element of Y . This implies that y → t 0 q S(t − τ)y f (τ) dτ (42)
- 33. 20 S. Verduyn Lunel is represented by an element of Y , to be denoted as t 0 q S(t − τ) f (τ) dτ. Applying the two lemmas above, with the role of Y and Y interchanged, we find that this is indeed the case if either Y , σ(Y , Y) is sequentially complete (43) or a linear map Y, σ(Y, Y ) → R is continuous if it is sequentially continuous. (44) Therefore to develop a perturbation theory for twin semigroups we need both (39) and (42) to be represented by elements from, respectively, Y and Y . This motivates the following definition. Definition 6.3 We say that a norming dual pair (Y, Y ) is suitable for twin pertur- bation if (a) at least one of (38) and (40) holds; and (b) at least one of (43) and (44) holds Recall from Theorem 4.1 that for the norming dual pair (B, N BV ) introduced in Sect.4 we have that (38) and (44) are satisfied. This shows that the norming dual pair (B, N BV ) is suitable for twin perturbation. We are now ready to give a rigorous proof of the variation-of-constants formula for NFDE. Theorem 6.4 The solution of the inhomogeneous NFDE (35) can be represented by the variation-of-constants formula (36), i.e., xt = S(t)ϕ + t 0 S(t − τ)q f (τ) dτ, where S(t) is the twin semigroup given by (30). Proof It follows from Theorem 4.1 that Y = B([−1, 0]; Rn ) and Y = N BV ([0, 1]; Rn ) is a norming dual pair suitable for twin perturbation. Therefore the claim follows by applying Lemma 6.1 with respect to the norming dual pair (B, N BV ) and Lemma 6.2 with respect to the norming dual pair (N BV, B). In the treatment of renewal equations in Diekmann and Verduyn Lunel (2021) we assumed (43) and (40). In fact for delay differential equations we take as normal dual pair (Y, Y ) with Y = B([−1, 0]) and Y = N BV ([0, 1]), while for renewal equations we take (Y, Y ) with Y = N BV ([−1, 0]) and Y = B([0, 1]).
- 34. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 21 7 Bounded Time-Dependent Perturbation of a Twin Semigroup In this section we assume • (Y, Y ) is a norming dual pair that is suitable for twin perturbation, cf. Definition 6.3; • {S0(t)} is a twin semigroup on (Y, Y ) with generator C0; • For j = 1, . . . , n the elements qj ∈ Y and t → qj (t) ∈ Y are given. Definition 7.1 A two-parameter family U = {U(t, s)}t≥s of twin operators on a norming dual pair (Y, Y ) is called a twin evolutionary system if (i) U(s, s) = I and U(t, s) = U(t,r)U(r, s) for s ≤ r ≤ t (ii) there exist constants M ≥ 1 and ω0 ∈ R such that for all y ∈ Y, y ∈ Y |y U(t, s)y| ≤ Meω0(t−s) y y , t ≥ s; (iii) Let the set ⊂ R2 be defined by = {(t, s) | −∞ s ≤ t ∞}. For all y ∈ Y, y ∈ Y the function (t, s) → y U(t, s)y ∈ R is measurable. Our aim is to define constructively a twin evolutionary system {U(t, s)} corre- sponding to the abstract multi-valued differential equation du dt ∈ C(t)u, t ≥ s, u(s) given, (45) with D (C(t)) = D (C0) , C(t)y = C0 y + n j=1 qj (t), yqj . (46) The first step is to introduce a n × n-matrix-valued function k(t, s) on R × R via k(t, s) = 0 for −∞ t ≤ s ∞ and ki j (t, s) := qi (t)S0(t − s)qj , −∞ s ≤ t ∞. (47) Note that for each pair c1, c2 with −∞ c1 c2 ∞ and for each f ∈ L1 ([c1, c2]; Rn ), we have sup f ≤1 c2 c1 c2 c1 k(t, s) f (s) ds dt ∞.
- 35. 22 S. Verduyn Lunel Here f denotes the norm of f as function belonging to L1 ([c1, c2]; Rn ) and the map f → t c1 k(t, s) f (s) ds, c1 ≤ t ≤ c2, defines a bounded linear operator on L1 ([c1, c2]; Rn ) which we shall denote by K. The linear space of lower triangular kernel functions on [c1, c2] × [c1, c2] of type L1 loc endowed with the norm k1 := sup f ≤1 c2 c1 c2 c1 k(t, s) f (s) ds dt = ess sup s∈[c1,c2] c2 c1 k(t, s) dt (48) is a Banach space (see Theorem 9.2.4 and Proposition 9.2.7 of Grippenberg et. al. 1990) which we will denote by L1 + [c1, c2] × [c1, c2]; Rn×n . Now let k be a lower triangular kernel function of type L1 loc. We call an n × n- matrix-function r(t, s) a resolvent kernel function of k if r(t, s) is a lower triangular kernel function of type L1 loc and r(t, s) = k(t, s) + t s r(t, a)k(a, s) da, −∞ s ≤ t ∞, (49) = k(t, s) + t s k(t, a)r(a, s) da, −∞ s ≤ t ∞. (50) Define the integral operator R similar as the operator K but with the kernel k(t, s) replaced by r(t, s), i.e., (R f ) (t) := t c1 r(t, s) f (s) ds, c1 ≤ t ≤ c2. Using the integral operators K and R, it follows from the identity (50) that for c1 t c2 we have (K R f )(t) = t c1 k(t, s)(R f )(s) ds = t c1 k(t, s) s 0 r(s, τ) f (τ) dτ ds = t c1 t τ k(t, s)r(s, τ) ds f (τ) dτ = t c1 (r(t, τ) − k(t, τ)) f (τ) dτ = (R f )(t) − (K f )(t), c1 ≤ t ≤ c2.
- 36. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 23 It follows that K R = R − K. Similarly, using (49), we have RK = R − K. This yields K R = RK and (I − K)(I + R) = (I + R)(I − K) = I, (51) where I is the identity operator on L1 ([c1, c2]; Rn ). Thus I − K is an invertible operator on L1 ([c1, c2]; Rn ), and its inverse is given by I + R. Theorem 7.2 If k(t, s) is a lower triangular kernel function of type L1 loc, then k(t, s) has a unique resolvent kernel function r(t, s) of type L1 loc. In particular, the integral equation x = K x + f has a unique solution given by x = f + R f . Proof The proof will be done in three steps. Throughout k(t, s) is a lower triangular kernel function of type L1 loc. Step 1. First note that if k1 and k2 are lower triangular kernel functions on R × R, then the same holds true for the functions (t, s) → t s k1(t, a)k2(a, s) da and (t, s) → t s k2(t, a)k1(a, s) da. Furthermore, from the discussion in the paragraph preceding the present theorem it follows that a resolvent kernel function of type L1 loc is unique whenever it exists. Step 2. Because of uniqueness of the resolvent kernel of type L1 loc, it suffices to prove existence of a resolvent kernel on [c1, c2] for every c1, c2 ∈ (0, ∞) with c1 c2. Assume first that k1 ≤ 1 with k1 given by (48), then the map r(t, s) → t s k(t, a)r(a, s) da + k(t, s) is a contraction on L1 + [c1, c2] × [c1, c2]; Rn×n . This shows that (50) (and, using (51), similarly (49)) has a unique solution, and this solution is a resolvent kernel of type L1 loc. Step 3. Since k(t, s) is a lower triangular kernel function of type L1 loc, we define a scaled lower triangular kernel function of type L1 loc by k(t, s) := e−γ(t−s) k(t, s). Since the norm of k is defined by (see (48)) k1 := ess sup s∈[c1,c2] c2 c1 k(t, s) dt = ess sup s∈[c1,c2] c2 c1 e−γ(t−s) k(t, s) dt, we can choose γ so large that k1 1. From Step 2, it follows that the equation r(t, s) = k(t, s) + t s k(t, a) r(a, s) da
- 37. 24 S. Verduyn Lunel has a unique solution r ∈ L1 + [c1, c2] × [c1, c2]; Rn×n . Therefore, we have r(t, s) = e−γ(t−s) k(t, s) + t s e−γ(t−a) k(t, a) r(a, s) da, and hence eγ(t−s) r(t, s) = k(t, s) + t s k(t, a)eγ(a−s) r(a, s) da. Thus r(t, s) = k(t, s) + t s k(t, a)r(a, s) da, where r(t, s) = eγ(t−s) r(t, s). This completes the proof. Proposition 7.3 If k(t, s) is a lower triangular kernel function that satisfies the esti- mate k(t, s) ≤ m(t) for 0 ≤ s ≤ t and r(t, s) denotes the corresponding resolvent kernel function, then r(t, s) ≤ m(t) exp t s m(σ) dσ , 0 ≤ s ≤ t ∞. Proof From the estimate k(t, s) ≤ m(t) for 0 ≤ s ≤ t we obtain the following integral inequality for the function u(t, s) := r(t, s) on 0 ≤ s ≤ t: u(t, s) ≤ m(t) + m(t) t s u(a, s) da, 0 ≤ s ≤ t ∞. (52) Now fix s ∈ [0, ∞), and put q(t) := exp − t s m(σ) dσ t s u(a, s) da. t ≥ s. (53) Differentiation of q with respect to t yields dq dt (t) = −m(t)q(t) + exp − t s m(σ) dσ u(t, s) = u(t, s) − m(t) t s u(a, s) da exp − t s m(σ) dσ ≤ m(t) exp − t s m(σ) dσ , where we have used (52). Integration from s to t yields the inequality
- 38. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 25 q(t) ≤ t s m(a) exp − a s m(σ) dσ da = 1 − exp − t s m(σ) dσ . Together with the definition of q in (53) we arrive at m(t) t s u(a, s) da = m(t) exp t s m(σ) dσ q(t) ≤ −m(t) + m(t) exp t s m(σ) dσ . Substitution into (52) yields u(t, s) ≤ m(t) exp t s m(σ) dσ , 0 ≤ s ≤ t ∞, which completes the proof. In the context of the variation-of-constants spirit (46) motivates us to presuppose that U(t, s) and S0(t) should be related to each other by the equation U(t, s) = S0(t − s) + t s S0(t − τ)B(τ)U(τ, s) dτ, t ≥ s, (54) where B(t)y := n j=1 qj (t), yqj , t ≥ s. (55) By letting B(t) act on (54) we obtain, for a given initial point y ∈ Y, a finite dimen- sional renewal equation. To derive this renewal equation, we first write (55) as B(t)y = q (t), y · q, t ≥ s, (56) where t → q (t) is the n-vector-valued function with Y -valued components qj (t) and q is the n-vector-valued with Y-valued components qj . Here we use · to denote the inner product in Rn . We can factor (a rank factorization) B as B = B2 B1 with B1 : Y → Rn and B2 : Rn → Y defined by B1(t)y := q (t), y, B2x := n j=1 xj qj , t ≥ s. (57) Now let (54) act on y ∈ Y and use (56) to obtain
- 39. 26 S. Verduyn Lunel U(t, s)y = S0(t − s)y + t s S0(t − τ)q (τ)U(τ, s)y · q dτ, t ≥ s. (58) Next act on both sides of (58) with the operator B1(t) as defined in (57) to arrive at v(t, s)y = q (t)S0(t − s)y + t s k(t, τ)v(τ, s)y dτ, t ≥ s, (59) where v(t, s)y := B1(t)U(t, s)y = q (t)U(t, s)y, t ≥ s, and the lower triangular kernel function k(t, s) is given by (47). Using Theorem 7.2 we can express the solution of (59) in terms of the resolvent r(t, s) of the kernel k(t, s) and the forcing function t → q (t)S0(t − s)y by the formula v(t, s)y = q (t)S0(t − s)y + t s r(t, τ)q (τ)S0(τ − s)y dτ, t ≥ s. (60) And now that the function v(t, s)y, representing q (t)U(t, s)y, can be considered as known, Eq.(54) becomes an explicit formula for U(t, s): U(t, s) = S0(t − s) + t s S0(t − τ)q · v(τ, s) dτ, t ≥ s. (61) Please note that, with this definition of U(t, s), we do indeed have that v(t, s)y = q (t)U(t, s)y (compare (61) to (59)). Formula (61) is well suited for proving, on the basis of Lemma 6.1 or Lemma 6.2, that U(t, s) maps Y into Y. But not for proving that U(t, s) maps Y into Y . So even though this may seem superfluous, we now provide an alternative dual constructive definition starting from the following equation U(t, s) = S0(t − s) + t s U(t, τ)B(τ)S0(τ − s) dτ, t ≥ s, (62) which is the variant of (54) in which the roles of U(t, s) and S0(t) are interchanged. Let (62) act (from the right) on y ∈ Y and next let the resulting identity act on the vector q. Using (56) this yields the equation y w(t, s) = y S0(t − s)q + t s y w(t, τ)k(τ, s) dτ, t ≥ s, (63) where y w(t, s) := y U(t, s)q and k(t, s) is given by (47). The formula
- 40. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 27 y w(t, s) = y S0(t − s)q + t s y S0(t − τ)q r(τ, s) dτ, t ≥ s, (64) expresses the solution of (63) in terms of the forcing function in (63) and the resolvent r(t, s) of the kernel k(t, s). Next use (56) to rewrite (62) in the form U(t, s) = S0(t − s) + t s w(t, τ) · q S0(τ − s) dτ, t ≥ s. (65) Please note that indeed y w(t, s) = y U(t, s)q (compare (65) to (63)). Of course we should now verify that the integrals in (61) and (65) do indeed define the same object. Writing the integral in (61) as w0 ∗ v and the integral in (65) as w ∗ v0, equality follows from (60) written in the form v = v0 + r ∗ v0 and (64) written in the form w = w0 + w0 ∗ r since w0 ∗ v = w0 ∗ (v0 + r ∗ v0) = w0 ∗ v0 + w0 ∗ r ∗ v0 = (w0 + w0 ∗ r) ∗ v0 = w ∗ v0. Before we can prove Theorem 7.5 below we first need an auxiliary result. Lemma 7.4 The solution v(t, s)y of (59) has the property v(t, s)y = v(t,r)U(r, s)y, t ≥ r ≥ s. (66) Proof From (59) it follows that v(t, s)y = q (t)S0(t − r)S0(r − s)y + r s k(t, τ)v(τ, s)y dτ + t r k(t, τ)v(τ, s)y dσ, t ≥ r ≥ s, and, by uniqueness, (66) follows provided the following identity holds q (t)S0(t − r)S0(r − s)y + r s k(t, τ)v(τ, s)y dτ = q (t)S0(t − r)U(r, s)y. Recall from (47) that k(t, s) = q S0(t − s)q = q S0(t − r)S0(r − s)q, t ≥ r ≥ s,
- 41. 28 S. Verduyn Lunel so we conclude from (61) that this identity does indeed hold. Theorem 7.5 Equation (61) in combination with (60), or Eq.(65) in combination with (64), defines a twin evolutionary system {U(t, s)} corresponding to the abstract differential equation (45). Proof Fix t ≥ s. Since (Y, Y ) is suitable for twin perturbation, we can use (61) and either Lemma 6.1 or Lemma 6.2 to deduce that U(t, s) maps Y into Y. Similarly we can use (65) and the observation concerning (42) to deduce that U(t, s) maps Y into Y . So U(t, s) is a twin operator. Next we use Lemma 7.4 to derive the property U(t, s) = U(t,r)U(r, s), t ≥ r ≥ s, (67) To verify (67), we start from (61) and use Lemma 7.4 to write U(t, s)y = S0(t − r) S0(r − s)y + r s S0(r − τ)q · v(τ, s)y dτ + t r S0(t − τ)q · v(τ,r)U(r, s)y dτ = S0(t − r)U(r, s)y + t r S0(t − τ)q · v(τ,r)U(r, s)y dτ = U(t,r)U(r, s)y. Both the property S(s, s) = I and the measurability, for all y ∈ Y, y ∈ Y , of t → y S(t)y follow from (61) and the corresponding properties of {S0(t)}. Finally, the exponential estimate for y S0(t)y yields exponential estimates for both the kernel k(t, s) and the forcing function t → q (t)S0(t − s)y, t ≥ s, in the renewal equation (59). Therefore, using Proposition 7.3 we obtain an exponential estimate for the resolventr(t, s) and hence via (60) an exponential bound for v(t, s)y. Finally, using (61) we obtain an exponential bound for y U(t, s)y for t ≥ s. This completes the proof of Theorem 7.5. 8 A Perturbation Approach Towards Periodic NFDE We shall be dealing with linear periodic neutral functional differential equations of the following type: ⎧ ⎪ ⎨ ⎪ ⎩ d dt x(t) − h 0 [dη(τ)]x(t − τ) = h 0 [dτ ζ(t, τ)]x(t − τ), t ≥ s, x(s + θ) = ϕ(θ), −h ≤ θ ≤ 0. (68)
- 42. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 29 Here dτ denotes integration with respect to the τ variable and ϕ is a given function in B ([−h, 0], Rn ). Throughout we assume that for each t ∈ R the functions η and ζ(t, ·) are n × n matrices of which the entries are real functions of bounded variation on [0, h] and continuous from the left on (0, h), and η(0) = ζ(t, 0) = 0. Moreover, it is assumed that there is a nondecreasing bounded function m ∈ L1 loc[−h, ∞) such that Var[−h,0] ζ(t, ·) ≤ m(t), t ≥ 0. Theorem 8.1 Under the above conditions, Eq.(68) defines a well-posed dynam- ical system, that is, Eq.(68) has a unique solution x on [0, ∞) such that xt ∈ B ([−h, 0], Rn ) for t ≥ 0. The above theorem is an extension of Theorem 6.1.1 in Hale and Verduyn Lunel (1993) to the neutral case. In this section we shall derive Theorem 8.1 as a corollary of Theorem 7.5 using the perturbation approach developed in the previous section. Consider as the unperturbed problem the special case ζ = 0 in (68). Let y denote the solution of the autonomous NFDE ⎧ ⎪ ⎨ ⎪ ⎩ d dt y(t) − h 0 [dη(τ)]y(t − τ) = 0, t ≥ 0, y(θ) = ϕ(θ), −h ≤ θ ≤ 0. (69) From the theory developed in Sect.2, it follows that the solution y of (69) satisfies the autonomous renewal equation y(t) − t 0 dη(θ)y(t − θ) = f0(t), t ≥ s, (70) where f0(t) := ϕ(0) − h 0 dη(θ)ϕ(−θ) + h t dη(θ)ϕ(t − θ), t ≥ s. (71) The solution of (70) is given by y(t) = f0(t) + t 0 dρ0(θ) f0(t − θ), t ≥ 0, (72) where ρ0 denotes the resolvent of η, i.e., it satisfies the resolvent equation ρ0 = η ∗ ρ0 + η, (73) see Theorem 2.2. Denote by X(t) = I + ρ0(t) the fundamental matrix solution of (69) so that we can write the solution y given by (72) as
- 43. 30 S. Verduyn Lunel y(t) = t 0 dX(τ) f0(t − τ), t ≥ 0. (74) It follows from Theorem 5.3 that the semigroup {S0(t)} defined by translation along the solution of (69), i.e., (S0(t)ϕ) (θ) = y(t + θ; ϕ), −h ≤ θ ≤ 0, t ≥ 0, is a twin semigroup. Define for i = 1, . . . , n elements qi ∈ Y and functions t → qi (t) ∈ Y by qi (θ) := 0 for − h ≤ θ 0, ei for θ = 0, (75) where ei is the i-th unit vector in Rn and the maps t → qi (t) are defined by qi (t) (θ) := ζi (t, θ), −h ≤ θ ≤ 0, t ≥ 0, (76) where ζi is the i-th row of the n × n-matrix-valued function ζ. For the matrix kernel k(t, s) introduced in (47) we have, using (75) and (76), the representation ki j (t, s) = qi (t)S0(t − s)qj = t−s 0 dτ ζi (t, τ)X j (t − s − θ), t ≥ s, (77) where X j (t) is the j-th column of the fundamental matrix solution X(t). Furthermore for ϕ ∈ Y, using (76), q (t)S0(t − s)ϕ = q (t)y(t − s; ϕ) = h 0 dζ(t, θ)y(t − s − θ; ϕ), t ≥ s. Let v(t, s)ϕ be the solution to the renewal equation (59), i.e., v(t, s)ϕ = q (t)S0(t − s)ϕ + t s k(t, τ)v(τ, s)ϕ dτ, t ≥ s, where the kernel k(t, s) is given by (77). We claim that the solution v(t, s)ϕ is given by v(t, s)ϕ = h 0 dζ(t, θ)x(t − θ; ϕ), t ≥ s, (78) where x(·; ϕ) is the solution of (68).
- 44. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 31 Define v̄(t, s)ϕ := h 0 dζ(t, θ)x(t − θ; ϕ). (79) To prove that v(t, s) = v̄(t, s) it suffices to show that v̄(t, s)ϕ is also a solution of the renewal equation (59). Let x(·; ϕ) be the solution of (68). Similar as before we can rewrite equation (68) to obtain that x is a solution of the renewal equation x(t) − t 0 dη(θ)x(t − θ) = t s v̄(σ, s)ϕ dσ + f0(t), (80) where f0 is given by (71). Note that the left hand side of (80) can be written as x − η ∗ x. Using the resolvent equation (73) we obtain (1 + ρ0) ∗ (x − η ∗ x) = x − η ∗ x + ρ0 ∗ x − ρ0 ∗ η ∗ x = x − η ∗ x + ρ0 ∗ x − (ρ0 − η) ∗ x = x. Thus if we take on both sides of (80) the convolution with the fundamental solution X(t) = I + ρ0(t) of (69) then x(t) = y(t; ϕ) + t s dX(t − τ) τ s v̄(σ, s)ϕ dσ = y(t; ϕ) + t s X(t − τ)v̄(τ, s)ϕ dτ, (81) where y is given by (74). Finally take the convolution with q (t) on both sides of (81) to arrive at v̄(t, s)ϕ = q (t)y(t; ϕ) + t s q (t)X(t − τ)v̄(τ, s)ϕ dτ = q (t)S0(t − s)ϕ + t s h 0 dζ(t, θ)X(t − τ − θ) v̄(τ, s)ϕ dτ = q (t)S0(t − s)ϕ + t s k(t, τ)v̄(τ, s)ϕ dτ, where we have used (77) and (78). Therefore v̄(t, s)ϕ given by (79) satisfies the identity v̄(t, s)ϕ = q (t)S0(t − s)ϕ + t s k(t, τ)v̄(τ, s)ϕ dτ. This shows that v̄(t, s)ϕ is a solution to the renewal equation (59) and completes the proof of the claim (78).
- 45. 32 S. Verduyn Lunel Finally apply to (61) the element of Y that corresponds to the Dirac measure in −θ ∈ [0, 1] to obtain (U(t, s)ϕ) (θ) = y(t − s + θ) + t s X(t − τ + θ) · v(τ, s)ϕ dτ = x(t − s + θ; ϕ), where in the last identity we have used (81). Thus we conclude that the the perturbation approach based on the abstract variation-of-constants formula developed in the previous section precisely yields the twin evolutionary system defined by translation along the solution of (68). We summarize this result in a theorem. Theorem 8.2 Under the above conditions, translation along the solution of equation (68) defines a twin evolutionary system {U(t, s)}t≥s given by (61). A Review of Functions of Bounded Variation In this appendix E denotes the Borel σ-algebra on [0, ∞). For E ∈ E, we call a sequence of disjoint sets {E j } in E a partition of E if ∪∞ j=1 E j = E. A complex Borel measure is a map μ : E → C such that μ(∅) = 0 and μ(E) = ∞ j=1 μ(E j ), for every partition {E j } of E with the series converging absolutely. In the following we will often omit the adjective ‘bounded’. The total variation measure |μ| of a complex Borel measure μ is given by |μ|(E) = sup ⎧ ⎨ ⎩ n j=0 |μ(E j )| | n ∈ N, {E j } a partition of E in E ⎫ ⎬ ⎭ . The vector space of complex Borel measures of bounded total variation is denoted by M ([0, ∞)). Provided with the total variation norm given by μT V = |μ| ([0, ∞)) , (82) the vector space M ([0, ∞)) becomes a Banach space. Let f : [0, ∞) → C be a given function. For a partition {E j } of [0, t] with E j = [tj−1, tj ) and 0 = t0 t1 · · · tn = t we define the function Tf : [0, ∞) → [0, ∞] by
- 46. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 33 Tf (t) := sup n j=1 | f (tj ) − f (tj−1)|, wherethesupremumistakenovern ∈ Nandallsuchpartitionsof[0, t].Theextended real function Tf is called the total variation function of f . Note that if 0 ≤ a b, then Tf (b) − Tf (a) ≥ 0 and hence Tf is an increasing function. If limt→∞ Tf (t) is finite, then we call f a function of bounded variation. We denote the space of all such functions by BV . The space N BV ([0, ∞)) of normalized functions of bounded variation is defined by N BV ([0, ∞)) := { f ∈ BV | f is continuous from the right on (0, ∞) and f (0) = 0 }. Provided with the norm f T V := lim t→∞ Tf (t) (83) the space N BV ([0, ∞)) becomes a Banach space. More generally, we define for −∞ a b ∞, the vector space N BV ([a, b]) to be the space of functions f : [a, b] → C such that f (a) = 0, f is continuous from the right on the open interval (a, b), and whose total variation on [a, b], given by Tf (b) − Tf (a) = Tf (b), is finite. Provided with the norm f T V := Tf (b), the space N BV ([a, b]) becomes a Banach space. We extend the domain of definition of a function of bounded variation by defining f (t) = 0 for t 0 if f ∈ N BV ([0, ∞)) and f (t) = 0 for t a and f (t) = f (b) for t b if f ∈ N BV ([a, b]). The following fundamental result (see Folland 1999, Theorem 3.29) provides the correspondence between functions of bounded variation and complex Borel mea- sures. Theorem A.1 Let μ be a complex Borel measure on R. If f : [0, ∞) → C is defined by f (t) = μ((0, t]), then f ∈ N BV ([0, ∞)). Conversely, if f ∈ N BV ([0, ∞)) is given, then there is a unique complex Borel measure μ f such that μ f ((0, t]) = f (t). Moreover |μ f | = μTf . Given a function f ∈ N BV ([a, b]) with corresponding measure μ f , we define the Lebesgue-Stieltjes integral g d f or g(x) f (dx) to be g dμ f . Thus, a Lebesgue-Stieltjes integral is a special Lebesgue integral and the theory for the Lebesgue integral applies to the Lebesgue-Stieltjes integral. We embed L1 ([0, ∞)) into M ([0, ∞)) by identifying f ∈ L1 ([0, ∞)) with the measure μ defined by μ(E) = E f (x) dx or, in short, μ(dx) = f (x) dx. In this section we collect some results about the convolution of a measure and a function and the convolution of two measures needed to study renewal equations.
- 47. 34 S. Verduyn Lunel For details we refer to Appendix A of Diekmann and Verduyn Lunel (2021) and for further results we refer to Folland (1999); Grippenberg et. al. (1990). Let B ([0, ∞)) denote the vector space of all bounded, Borel measurable func- tions f : [0, ∞) → R. Provided with the supremum norm (denoted by · ), the space B ([0, ∞)) becomes a Banach space. With B ([a, b]) we denote the Banach space of all bounded, Borel measurable functions f : [a, b] → R provided with the supremum norm. The half-line convolution μ ∗ f of a measure μ ∈ M([0, ∞)) and a Borel mea- surable function f ∈ B ([0, ∞)) is the function (μ ∗ f )(t) := [0,t] μ(ds) f (t − s) defined for those values of t for which [0, t] s → f (t − s) is |μ|-integrable. The following result can be found in Grippenberg et. al. (1990, Theorem 3.6.1(ii)). Theorem A.2 If f ∈ B ([0, ∞)) and μ ∈ M ([0, ∞)), then the convolution of f and μ satisfies μ ∗ f ∈ B ([0, ∞)) and μ ∗ f ≤ μT V f . The half-line convolution μ ∗ ν of two measures μ, ν ∈ M ([0, ∞)) is defined by the complex Borel measure that to each Borel set E ∈ E assigns the value (μ ∗ ν)(E) := [0,∞) μ(ds)ν ((E − s)+) , (84) where (E − s)+ := {e − s | e ∈ E} ∩ [0, ∞) (cf. Grippenberg et. al. 1990, Defini- tion 4.1.1)). If χE is the characteristic function of the set E, then ν((E − s)+) = [0,∞) χE (σ + s)ν(dσ), where [0, ∞) σ → χE (σ + s) is the characteristic function of (E − s)+. It fol- lows from Theorem A.2 that s → ν(E − s)+) belongs to B ([0, ∞)) and hence the definition of the convolution of two measures μ ∗ ν : E → C given in (84) makes sense. Furthermore, using Fubini’s Theorem, we have the following useful identity
- 48. The Twin Semigroup Approach Towards Periodic Neutral Delay Equations 35 μ ∗ ν(E) = [0,∞) μ(ds)ν ((E − s)+) = [0,∞) [0,∞) χE (σ + s)μ(ds)ν(dσ) = [0,∞) μ ((E − s)+) ν(ds). The following result can be found in Grippenberg et. al. (1990, Theorem 4.1.2(ii)). Theorem A.3 Let μ, ν ∈ M ([0, ∞)) and let the convolution μ ∗ ν be defined by (84). (i) The convolution μ ∗ ν belongs to M ([0, ∞)) and μ ∗ νT V ≤ μT V νT V . (ii) For any bounded Borel function h ∈ B ([0, ∞)), we have [0,∞) h(t) (μ ∗ ν) (dt) = [0,∞) [0,∞) h(t + s) μ(dt)ν(ds). Using the one-to-one correspondence between complex Borel measures and func- tions of bounded variation, see Theorem A.1, we can combine the above results to obtain the following theorem (see Diekmann and Verduyn Lunel 2021, Theorem A.5). Theorem A.4 If f ∈ N BV ([0, ∞)) and μ ∈ M([0, ∞)), then the convolution of μ and f satisfies μ ∗ f ∈ N BV ([0, ∞)) and μ ∗ f T V ≤ μT V f T V . We also need the following result (see Diekmann and Verduyn Lunel 2021, The- orem A.6). Theorem A.5 Let μ ∈ M ([0, ∞)) and let f : [0, ∞) → C be a bounded continu- ous function. If μ has no discrete part, then μ ∗ f is a bounded continuous function and μ ∗ f ≤ μT V f .
- 49. 36 S. Verduyn Lunel References Aliprantis, C. D., Border, K. C. (2006). Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd Edn. Berlin: Springer. Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M., Walther, H. O. (1995). Delay Equations: Functional-, Complex-, and Nonlinear Analysis. New York: Springer. Diekmann, O., Verduyn Lunel, S. M. (2021). Twin semigroups and delay equations. Journal of Differential Equations, 286, 332–410. Feller, W. (1953). Semigroups of transformations in general weak topologies. Annals of Mathemat- ics, 57, 287–308. Hale, J. K., Verduyn Lunel, S. M. (1993). Introduction to Functional Differential Equations. New York: Springer. Folland, G. B. (1999). Real Analysis, 2nd Edn. Wiley-Interscience. Gripenberg, G., Londen, S.-O., Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge: Cambridge University Press. Kunze, M. (2009). Continuity and equicontinuity of semigroups on norming dual pairs. Semigroup Forum, 79, 540–560. Kunze, M. (2011). A Pettis-type integral and applications to transition semigroups. Czechoslovak Mathematical Journal, 61, 437–459. Rudin, W. (1991). Functional Analysis (2nd ed.). New York: McGraw-Hill.
- 50. Characteristic Matrix Functions and Periodic Delay Equations Sjoerd Verduyn Lunel Abstract In the first part of this chapter we recall the notion of a characteristic matrix function for classes of operators as introduced in Kaashoek and Verduyn Lunel (2023). The characteristic matrix function completely describes the spectral properties of the corresponding operator. In the second part we show that the period map or monodromy operator associated with a periodic neutral delay equation has a characteristic matrix function. We end this chapter with a number of illustrative examples of periodic neutral delay equations for which we can compute the charac- teristic matrix function explicitly. 1 Introduction Let X denote a complex Banach space, and let A : D (A) → X be a linear operator with domain D (A) a subspace of X. A complex number λ belongs to the resolvent set ρ(A) of A if and only if the resolvent (zI − A)−1 exists and is bounded, i.e., (i) λI − A is one-to-one; (ii) Im λI − A = X; (iii) (zI − A)−1 is bounded. Note that for closed operators, (iii) is superfluous, since it is a direct consequence of the other assumptions by the closed graph theorem. The spectrum σ(A) is by definition the complement of ρ(A) in C. The point spectrum σp(A) is the set of those λ ∈ C for which λI − A is not one-to-one, i.e., Aϕ = λϕ for some ϕ = 0. One then calls λ an eigenvalue and ϕ an eigenvector corresponding to λ. The null space Ker (λI − A) is called the eigenspace and its dimension the geo- metric multiplicity of λ. The generalized eigenspace Mλ = Mλ(A) is the smallest closed linear subspace that contains all Ker (λI − A)j for j = 1, 2, . . . and its dimen- sion M(A; λ) is called the algebraic multiplicity of λ. If, in addition, λ is an isolated S. Verduyn Lunel (B) Mathematical Institute, Utrecht University, Utrecht, The Netherlands e-mail: S.M.VerduynLunel@uu.nl © CISM International Centre for Mechanical Sciences 2023 D. Breda (ed.), Controlling Delayed Dynamics, CISM International Centre for Mechanical Sciences 604, https://doi.org/10.1007/978-3-031-01129-0_2 37
- 51. 38 S. Verduyn Lunel point in σ(T ) and M(A; λ) is finite, then λ is called an eigenvalue of finite type. When M(A; λ) = 1 we say that λ is a simple eigenvalue. A class of operators for which the eigenvalues are of finite type is formed by the compact operators. Other classes appear later in this chapter. If λ is an eigenvalue of finite type, the operator T = A |Mλ is a bounded operator from a finite dimensional space into itself. So the situation is reduced to the finite dimensional case, which we shall, therefore, discuss first. Let T : Cm → Cm be a bounded linear operator. The eigenvalues of T are pre- cisely given by the roots of the characteristic polynomial C(z) := det (zI − T ) . Over the scalar field C the characteristic polynomial can be factorized into a product of m linear factors C(z) = m j=1 (z − λj ), where λj ∈ σ(T ). Define the multiplicity m(λj , zI − T ) of λj to be the number of times the factor (z − λj ) appears, or, in other words, the order of λj as a zero of the characteristic polynomial C. The characteristic polynomial is an annihilating polynomial of T , i.e., C(T ) = 0. The minimal polynomial Cm of T is defined to be an annihilating polynomial of T that divides any other annihilating polynomial. Necessarily, Cm is of the form Cm(z) = l j=1 (z − λj )kj , where σ(T ) = {λ1, . . . , λl}, and for j = 1, . . . ,l, the number kj is positive and called the ascent of λj . Define Mj := Ker (λj I − T )kj . This is a T -invariant subspace, i.e., T Mj ⊆ Mj , and we can define the part of T in Mj , i.e., Tj = T |Mj : Mj → Mj . This yields Cm = M1 ⊕ · · · ⊕ Ml. The operator T decomposes accordingly T = l ⊕ j=1 Tj . This decomposition is unique (up to the order of summands). The action of T can be broken down to the study of the action of Tj . To continue the decomposition one first studies the structure of the subspaces Mj more closely. Let λ ∈ σ(T ). A vector x is called a generalized eigenvector of order r if
- 52. Characteristic Matrix Functions and Periodic Delay Equations 39 (λI − T )r x = 0 while (λI − T )r−1 x = 0. Suppose xr−1 is a generalized eigenvector of order r; then there are vectors (xr−2, . . . , x1, x0) for which x0 = 0 and T x0 = λx0, T x1 = λx1 + x0, . . . T xr−1 = λxr−1 + xr−2 and hence xj ∈ Ker (λI − T )j+1 . Such a sequence is called a Jordan chain. Obvi- ously, the length of the Jordan chain is less than or equal to kλ, the ascent of λ, and a Jordan chain consists of linearly independent elements. As a consequence of this con- struction, the matrix representation of Tj with respect to the basis (xr−2, . . . , x1, x0) is given by a Jordan block of order r corresponding to λ. See Diekmann et al. (1995, Chap. IV) and the next section for more information about Jordan chains for analytic matrix-valued functions. Next consider the case that T is an operator defined on an infinite dimensional complex Banach space X, then, in general, T no longer has a matrix representation and we cannot define the characteristic polynomial of T by det (zI − T ). Neverthe- less there is a large class of operators for which one has a characteristic function whose zeros determine the spectrum of the corresponding operator. For example, this is true for the infinitestimal generator of solution semigroup corresponding to autonomous delay equations, see Diekmann et al. (1995, Chap. I). As it turned out the abstract notion of a characteristic matrix function, introduced in Kaashoek and Verduyn Lunel (1992) for unbounded operators, can be used to explain this connec- tion. As a consequence it was possible to extend the finite dimensional theory to specific classes of unbounded operators. To briefly explain the connection between unbounded operators A : D (A) → X and analytic matrix functions, as developed in Kaashoek and Verduyn Lunel (1992), let : → L(Cn ) be an analytic n × n matrix function with ⊂ C. Wecallacharacteristicmatrixfunctionfor A onifthereexistanalyticoperator functions E and F, E : → L(Cn ⊕ X) and F : → L(Cn ⊕ X), whose values are invertible operators, such that (z) 0 0 I = F(z) ICn 0 0 zI − A E(z), z ∈ . The characteristic matrix function completely determines the spectral properties of the unbounded operator A. See Kaashoek and Verduyn Lunel (1992) and Diekmann et al. (1995, Chap. IV) for details. In this chapter we will follow recent work, Kaashoek and Verduyn Lunel (2023), and extend the notion of a characteristic matrix function to classes of bounded
- 53. 40 S. Verduyn Lunel operators, and show that the period map of a periodic neutral delay equation has a characteristic matrix function. We end the introduction with an outline of this chapter. In Sect.2 we introduce and discuss the basic properties of Jordan chains. In Sect.3 we introduce the notion of a characteristic matrix function for a class of bounded operators, and prove that the characteristic matrix function completely determines the spectral properties of the associated bounded operator. In Sect.4 we show that the period map associated with a periodic neutral delay equation has a characteristic matrix function. In Sect.5 we show that in case the period is equal to the delay, then we can compute the characteristic matrix function rather explicitly. Finally, in Sect.6, we consider a class of periodic delay equations for which the period is two times the delay. We construct new examples for which we can compute the characteristic matrix function explicitly. In particular, we construct an example for which the period map has a finite spectrum. In the literature such examples are only known in case the period is equal to the delay, and were unknown in case the period is two times the delay. 2 Equivalence and Jordan Chains Let X, Y, X , Y be complex Banach spaces, and suppose that L : U → L(X, Y) and M : U → L(X , Y ) are operator-valued functions, analytic on the open subset U ⊂ C. The two operator-valued functions L and M are called equivalent on U (see Sect.2.4 in Bart et al. (1979)) if there exist analytic operator-valued functions E : U → L(X , X) and F : U → L(Y, Y ), whose values are invertible operators, such that, M(z) = F(z)L(z)E(z), z ∈ U. (1) Let L : U → L(X, Y) be an analytic operator-valued function. A point λ0 ∈ U is called a root of L if there exists a vector x0 ∈ X, x0 = 0, such that, L(λ0)x0 = 0. An ordered set (x0, x1, . . . , xk−1) of vectors in X is called a Jordan chain for L at λ0 if x0 = 0 and L(z)[x0 + (z − λ0)x1 + · · · + (z − λ0)k−1 xk−1] = O((z − λ0)k ). (2) The number k is called the length of the chain and the maximal length of the chain starting with x0 is called the rank of x0. The analytic function k−1 l=0 (z − λ0)l xl
- 54. Characteristic Matrix Functions and Periodic Delay Equations 41 in (2) is called a root function of L corresponding to λ0. Proposition 2.1 If two analytic operator functions L and M are equivalent on U, then there is a one-to-one correspondence between their Jordan chains. Proof The equivalence relation (1) is symmetric, and thus it suffices to show that Jordan chains for L yield Jordan chains for M. If (x0, . . . , xk−1) is a Jordan chain for L at λ0, then E(z)−1 (x0 + (z − λ0)x1 + · · · + (z − λ0)k−1 xk−1) = y0 + (z − λ0)y1 + · · · + (z − λ0)k−1 yk−1 + h.o.t. and (y0, . . . , yk−1) is a Jordan chain for M at λ0. Here h.o.t. stands for the higher order terms. Furthermore, the equivalence yields that the null spaces Ker L(λ0) and Ker M(λ0) are isomorphic and this proves the proposition. Let ⊂ C and : → L(Cn ) denote an entire n × n matrix function. If the determinant of is not identically zero, then we define m(λ, ) to be the order of λ as a zero of det and k(λ, ) is the order of λ as pole of the matrix function (·)−1 . Let λ0 be an isolated root of , then the Jordan chains for at λ0 have finite length, and we can organize the chains as follows. Choose an eigenvector, say x1,0, with maximal rank, say r1. Next, choose a Jordan chain (x1,0, . . . , x1,r1−1) of length r1 and let N1 be the complement in Ker (λ0) of the subspace spanned by x1,0. In N1 we choose an eigenvector x2,0 of maximal rank, say r2, and let (x2,0, . . . , x2,r2−1) be a corresponding Jordan chain of length r2. We continue as follows, let N2 be the complement in N1 of the subspace spanned by x2,0 and replace N1 by N2 in the above described procedure. In this way, we obtain a basis {x1,0, . . . , xp,0} of Ker (λ0) and a corresponding canonical system of Jordan chains x1,0, . . . , x1,r1−1, x2,0, . . . , x2,r2−1, xp,0, . . . , xp,rp−1 for at λ0. It is easy to see that the rank of any eigenvector x0 corresponding to the root λ0 is always equal to one of the rj for 1 ≤ j ≤ p. Thus, the integers r1, . . . ,rp do not depend on the particular choices made in the procedure described above and are called the zero-multiplicities of at λ0. Their sum r1 + · · · + rp is called the algebraic multiplicity of at λ0 and will be denoted by M((λ0)).
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- 59. The Project Gutenberg eBook of The Palace and Park This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: The Palace and Park Author: Samuel Phillips Edward Forbes R. G. Latham Richard Owen George Scharf F. K. J. Shenton Release date: June 30, 2019 [eBook #59843] Language: English Credits: Produced by Chris Curnow, Harry Lamé and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)
- 60. *** START OF THE PROJECT GUTENBERG EBOOK THE PALACE AND PARK ***
- 61. Please see the Transcriber’s Notes at the end of this text. The cover image has been created for this e-text and is in the public domain.
- 62. VIEW OF BUILDING FROM THE NORTH. THE PALACE AND PARK: ITS N AT U R A L H I S T O R Y, AND ITS
- 63. 1. PALACE AND PARK; 2. PORTRAIT GALLERY; 4. EXTINCT ANIMALS; 5. POMPEIAN COURT; PORTRAIT GALLERY, TOGETHER WITH A DESCRIPTION OF THE POMPEIAN COURT. IN THE UNDERMENTIONED GUIDES: 3. ETHNOLOGY NATURAL HISTORY. CRYSTAL PALACE LIBRARY, C R Y S TA L P A L A C E , S Y D E N H A M . 1859.
- 64. GUIDE TO THE CRYSTAL PALACE AND ITS Park and Gardens. By SAMUEL PHILLIPS. A NEWLY ARRANGED AND ENTIRELY REVISED EDITION, By F. K. J. SHENTON. WITH NEW PLANS AND ILLUSTRATIONS, AND AN INDEX OF PRINCIPAL OBJECTS.
- 65. CRYSTAL PALACE LIBRARY; C R Y S TA L P A L A C E , S Y D E N H A M . 1859. L O N D O N : R O B E R T K . B U R T, P R I N T E R , H O L B O R N H I L L .
- 66. PART I. PRELIMINARY AND INTRODUCTION. Note.—This Division of the Guide-Book contains the Index to Principal Objects; and the Company’s Official Announcements; with the Refreshment Tariff; an Introduction to the General Guide- book; and an Account of the Building.
- 67. ADVERTISEMENT. The arrangement of the present edition has been made with the view of simplifying, as much as possible, the reference to particular objects; as well as of enabling the visitor to regularly explore with advantage every portion of the Crystal Palace. The splendid Botanical Collection, now thoroughly acclimatised, and very complete, is described as fully as the nature of the book will permit. PREFACE TO ORIGINAL EDITION. The following pages are presented to the public as a brief but connected and carefully prepared account of the exterior and interior of the Crystal Palace. It is believed that no important or interesting object in connexion with the Exhibition is without its record in this little volume; although, in so vast a collection of works of architecture, sculpture, and industrial manufacture, it is clearly impossible to compress within the limits of a General Hand-book all the information which is necessary to satisfy the visitor desirous of precise and accurate knowledge of the numberless objects offered to his contemplation.
- 68. A general and comprehensive view of the Crystal Palace will unquestionably be obtained by the perusal of the present manual. The Hand-books of the respective departments will supply all the detailed information necessary to fill in the broad and rapidly drawn outlines. In them, Literature will faithfully serve as the handmaiden to Art, and complete the great auxiliary work of education which it is the first aim of the Crystal Palace to effect. These Hand-books are published at prices varying from three-pence to eighteen-pence, according to the size of the volume. The lowest possible price has been affixed to one and all. It may be fearlessly asserted that books containing the same amount of entertainment, information, and instruction, it would be difficult to purchase at a more reasonable rate elsewhere.
- 69. PRINTED BY R. K. BURT, HOLBORN HILL, CITY. Large map. CONVEYANCE BY ROAD AND RAIL. The trains start punctually from the London Bridge and Pimlico Stations at the times advertised in the official bills to be found in various parts of the building; but special trains are put on always as occasion may require.
- 70. The shortest route from London, by carriage, will be found marked on the accompanying map. The ordinary entrances from the road are at the South and Central Transepts. Entrances are also provided opposite Sydenham Church, and at the bottom of the Park, below the Grand Lake and Extinct Animals. Omnibuses leave Gracechurch Street for the Crystal Palace at intervals from 10 in the morning. An omnibus also leaves the Paddington Station at a quarter to 11 A.M. Also one from the Kings and Key, Fleet Street, at 12 o’clock, and one from the Green Man, Oxford Street, at the same time. Omnibuses leave the City for Camberwell every 10 minutes. Conveyance can also be procured from Peckham and Clapham. On fête days omnibuses run at frequent intervals, at times according to the season. Crystal Palace and Lower Norwood to Oxford Street, viâ Norwood, Brixton Road, Elephant and Castle, Westminster Road, Whitehall, Waterloo Place, and Regent Street—(c) green; (m) Norwood. From Crystal Palace, week days only, 8.30, 10.25, A.M., 2.20, 5.15, 6.30, 7.10, 8.45, P.M. From King’s Head, Norwood, week days, 8.55, 9.55, 10.50, A.M., 12.55, 1.50, 2.50, 4.5, 5.35, 7.35, 9.15, P.M. Sundays, 9.40, 10.50, A.M., 1.0, 1.40, 2.10, 4.20, 5.50, 7.20, 8.10, 9.35, P.M. From New Church, Tulse Hill, week days, 9.5, 10.5, 11.0, A.M., 1.5, 2.0, 3.0, 4.15, 5.45, 7.15, 9.25, P.M.; Sundays, 9.50, 11.0 A.M., 1.10, 1.50, 2.20, 4.30, 6.0, 7.30, 8.20, 9.45, P.M. From Oxford Street, Regent Circus, week days, 10.0, 11.20, A.M., 12.10, 2.20, 3.10, 4.10, 5.30, 7.0, 8.30, 10.45, P.M.; Sundays, 10.20, 10.50, A.M., 12.0, 2.10, 3.0, 3.30, 5.30, 7.0, 9.30, 10.50, P.M. From Charing Cross, week days, 10.15, 11.35, A.M., 12.25, 2.35, 3.25, 4.25, 5.45, 7.15, 8.45, 11.0, P.M.; Sundays, 10.35, 11.5, A.M., 12.15, 2.25, 3.15, 3.45, 5.45, 7.15, 9.45, 11.5, P.M. Fares, Upper Norwood to Kennington Gate, Charing Cross, or Oxford Street, 1s.; Cemetery, Lower Norwood, and Oxford Street, 1s.; ditto, ditto, Charing Cross, 9d.; ditto, ditto, Kennington Gate, 6d. A complete system of omnibus conveyance has been established by the London General Omnibus Company between the following districts and the Crystal Palace Railway Station at London Bridge:—Hammersmith, Putney, Brompton, Paddington, Bayswater, St. John’s Wood, Holloway, Hornsey Road, Islington, Kingsland, Hoxton, Newington Causeway, and Kent Road.
- 71. Crystal Palace Company. OFFICIAL ANNOUNCEMENTS. SEASON TICKETS. FIRST CLASS. Two Guineas each for Adults, One Guinea for Children under twelve. To admit the holder on all occasions whatever, excepting the three performances of the Handel Festival. SECOND CLASS. One Guinea each for Adults, Half a Guinea each for Children under twelve. To admit the holder on all occasions whatever, excepting the three performances of the Handel Festival, and when the price of admission is Five Shillings, or upwards, on payment of Half a Crown. The Tickets may be obtained at— The Crystal Palace; The Offices of the London and Brighton Railway Company, London Bridge, and Regent Circus, Piccadilly; and at the Stations on the Palace Railways, and various Lines in connection therewith. The Central Ticket Office, 2, Exeter Hall; And of the following Agents to the Company:— Addison Hollier, Regent-street; W. Austen, Hall-keeper, St. James’s Hall; Cramer, Beale, Co., 201, Regent-street; Dando, Todhunter, Smith, 22, Gresham-street, Bank; Duff Hodgson, Oxford-street; Gray Warren, Croydon; M. Hammond Nephew, 27, Lombard-street; Keith, Prowse, Co., 48, Cheapside; Letts, Son, Co., 8, Royal Exchange; Mead Powell, Railway Arcade, London Bridge; J. Mitchell, 33, Old Bond-street; W. R. Sams, 1, St. James’s-street; W. R. Stephens, 36, Throgmorton- street; Charles Westerton, 20, St. George’s-place, Knightsbridge.
- 72. Remittances for Season Tickets to be by Post-office Orders on the General Post-office, payable to George Grove. RATES OF ADMISSION, RAILWAY ARRANGEMENTS, ETC. Ordinary Rates of Admission.—These remain as before, viz.:— On Mondays, Tuesdays, Wednesdays, Thursdays, and Fridays (unless on special occasions) One Shilling. On Saturdays, Half-a-Crown, unless on special occasions, and excepting those in August, September, and October, when the Price of Admission may be reduced to One Shilling. Children under 12 Years of Age, Half-Price. Books, containing 25 admissions for ordinary Shilling days, till the 30th of April, 1860, are issued at the following rates:— Shilling Days, 25 for £1 2 6 Half-crown Days, 25 for 2 10 0 GRAND MILITARY MUSICAL FÊTE. To commemorate the suppression of the Indian Rebellion. This Fête, which will bring together a larger number of wind instruments than has been before heard together in this country, will take place in the new Orchestra of the Great Handel Festival, on May 2nd. FLOWER SHOWS. There will be Flower Shows at the following dates:— s. d. Wednesday, May 18th Admission 7 6 „ June 8th „ 7 6 „ Sept. 7th „ 2 6 Thursday, Sept. 8th „ 1 0 Wednesday, c., Nov. 9th and 10th „ 1 0 Tickets for the first two Shows will be issued prior to the day of the Show, at the reduced rate of Five Shillings, on the written order of a
- 73. Season Ticket-holder. OPERA CONCERTS. The Directors have made arrangements with Mr. Gye for a series of Six Grand Concerts, to be supported by the artistes of the Royal Italian Opera, Covent Garden. These Concerts are fixed to take place on— Wednesday, May 11th. Wednesday, July 6th. „ May 25th. „ „ 13th. „ June 15th. „ „ 20th. SATURDAY PROMENADES AND SECOND SERIES OF CONCERTS. During the period embraced by the Concerts of the Royal Italian Opera Company the Saturday Promenades will be continued as during last Season, admission Half-a-crown. After the conclusion of that series, it is proposed to combine the Concert and Promenade on the Saturdays, commencing with the 23rd July, for a Second Series. For these Concerts the Directors are happy to announce that they have entered into arrangements for the services of some of the most celebrated Artistes, Continental and English, amongst whom will be found several who are highly popular with the public, and who have not yet appeared at the Crystal Palace. The admission to these Concerts will be to Non-Season Ticket-holders Five Shillings. OTHER MUSICAL ENTERTAINMENTS. Other Concerts will take place during the Season; and of these due notice will be given. In the meantime the Directors may state that they will be favoured with the co-operation of Mr. Henry Leslie’s Choir: Also that some Grand Performances of Classical Music, on an extensive scale, by the Vocal Association, under the able baton of Mr. Benedict, embracing several novelties, may be looked forward to. It is further announced with pleasure that the Metropolitan Schools Choral Society, numbering among its ranks many thousands of the Children of the National Schools, whose
- 74. singing last year, conducted by Mr. G. W. Martin, elicited such warm approval, will hold another celebration on Saturday, 11th June; as also will the members of the Tonic Sol-Fa Association, under the same able conduct as before. Another great meeting of the Metropolitan Charity Children is anticipated. THE SATURDAY WINTER CONCERT Will be resumed in November, as during the last Season. Every opportunity will be taken to widen the range and increase the attractions of these Concerts, and to add to the convenience of the visitors who attend them. With the latter intention, in obedience to a desire very generally expressed, it has been determined that a limited number of Reserved Seats will be provided at each Concert. OPEN AIR MUSIC. Performances of Music by a Band of Wind Instruments in the open air having, during former seasons, afforded much gratification to the Public, it is proposed to resume these performances during the coming Summer months, at frequent intervals, and at such times of the afternoon as will be most convenient for the largest number of Visitors. LECTURES. The Lectures delivered by Mr. Pepper during the past Autumn and Winter will be resumed at the end of the Summer Season, and no exertions will be spared to make them efficient and interesting. THE GREAT HANDEL FESTIVAL. The dates of each performance will be as follows:— Monday June 20 “Messiah.” Wednesday, June 22 “Dettingen Te Deum:” Selections from “Saul,” “Samson,” “Belshazzar,” “Judas Maccabeus,” and other Works. Friday June 24 “Israel in Egypt.”
- 75. The Great Orchestra is 216 feet wide, with a central depth of 100 feet; and will contain on the occasion nearly 4,000 performers. PARK, GROUNDS, AND OUT-DOOR AMUSEMENTS. These will continue to receive the attention of the Directors. The Cricket Ground is rising into public favour, and is becoming the resort of several clubs of importance; a Rifle Ground, a Bowling Green, and a Gymnasium of approved construction, are now added to it. CRYSTAL PALACE ART-UNION. The detailed plan and arrangements of this Institution are set forth in the official statement issued by the Council, which may be obtained on application at the Company’s proper offices. EXCURSIONS. Benevolent Societies, Schools, and other large bodies may visit the Palace at the following reduced rates:—applying only to Shilling Days and Third-class Carriages. s. d. s. d. For a number of Excursionists over 250 and under 500 1 3 pr. head instead of 1 6 Exceeding 500 and under 750 1 2 „ „ 1 6 Exceeding 750 and under 1000 1 1 „ „ 1 6 Exceeding 1000 1 0 „ „ 1 6 Children, half-price. Parties wishing to arrange for Refreshments, must apply at the Palace, to Mr. F. Strange, who is prepared to make a reduction in favour of large parties, according to the kind of Refreshment desired. ⁂ When the Excursion consists mainly, or in part, of Children, it is requested that the persons in charge of them will prevent their touching any works of Fine Art in the Courts, or gathering leaves or flowers in or out of the building. Considerable damage has frequently been thus done
- 76. THE SALOON by children, and serious noise and annoyance is caused by their running along the galleries, or playing boisterously—a practice which it is desirable to stop. BATH CHAIRS. Wheel-chairs for invalids and others, may be hired in the building on the following terms:— Within the Palace, with Assistants 1s. 6d. per hour. In the Grounds „ 2s. 6d. „ Without Assistants, 6d. less. Perambulators 0s. 6d. „ Double Perambulators 1s. 0d. „ Lifting Chairs for carrying Invalids up the stairs from the Railway Station, or to the Galleries, 1s. The principal stand is near the entrance to the building, from the railways. Visitors can also be conveyed by these chairs to any hotel or residence in Sydenham or Norwood. Crystal Palace, May, 1859. THE REFRESHMENT DEPARTMENT. The various Saloons and Dining Rooms allotted for the Refreshment Department are all situated at the South End of the Palace, but branch stations for light refreshments will be found in various convenient positions throughout the building, and on special occasions requiring it, in the grounds. Mr. Frederick Strange is the lessee of the whole department. is entered at the right-hand corner of the extreme South End of the Palace, and is richly carpeted and decorated, and fitted with
- 77. THE DINING ROOM SOUTH WING DINING ROOM. THE TERRACE DINING ROOM every elegant convenience. The very highest class of entertainment is served here to due notice and order. Hot Dinners—Soups, Fish, Entrées, c., c.—to order at a few minutes’ notice. Price as per detailed Carte. The authorised charge for attendance is 3d. each person. is on the left of the Saloon. s. d. Dinner from the Hot Joint 2 0 Sweets, c., according to daily Bill of Fare. The authorised charge for attendance is 2d. each person. The South Wing Dining Room is entered at the left-hand corner of the extreme South End of the Palace, as the Saloon is at the right-hand. It is the most spacious dining hall of the kind in England, and is constructed entirely of glass and iron. The end and the long façade next the gardens are fitted for the whole extent with magnificent plate glass (which can be opened at convenience), commanding, from the dinner tables, a perfect view of the Terraces, Fountains, the Gardens, and the great prospect of rich landscape beyond. The dishes are served direct from the kitchen by a special covered tramway. is entered from the garden end of the South Transept, near to the entrance from the Railways. The front, toward the garden, is glass, giving a view of the terraces and grounds. Cold dinners only are served in this room. s. d. Cold Meat or Veal Pie, with Cheese and Bread 1 6 Chicken, with Ham and Tongue, and ditto 2 6 Lobster Salad, per dish 2 6 Jelly or Pudding 0 6 Ice (Nesselrode) Pudding 1 0
- 78. THE THIRD CLASS ROOMS The authorised charge for attendance is 1d. each person. are situated near the Railway Colonnade, in the lower story of the South Wing, and near the staircase at the end of the Machinery Department. s. d. Plate of Meat 0 6 Bread 0 1 Bread and Cheese 0 3 Porter (per Quart) 0 4 Ale „ 0 6 Ale „ 0 8 Coffee or Tea (per cup) 0 3 Roll and Butter 0 2 Biscuit 0 1 Bun 0 1 Bath Bun 0 2 Soda Water, c. 0 3 GENERAL TARIFF. s. d. Ices, Cream or Water 0 6 Coffee, or Tea (per Cup) 0 4 French Chocolate 0 6 Sandwich 0 6 Pork Pie 1 0 Pale Ale or Double Stout (Tankard) 0 6 Pale Ale or Double Stout (Glass) 0 3 Soda Water, Lemonade, c. 0 4 Confectionery at the usual prices. No charge for attendance is authorised on light refreshments. Note.—The Full Wine List will be found on all the tables, and at all the Stations.
- 79. ⁂ In case of any complaint against Waiters, Visitors are requested to report the circumstance, together with the number of the Waiter, at the Office of Mr. Strange. Waiters are not allowed to receive any gratuity. INDEX TO THE PRINCIPAL OBJECTS OF INTEREST IN THE CRYSTAL PALACE, ITS GARDENS AND PARK, AT SYDENHAM, As described or named in the General Guide-Book. A. PAGE Aboo Simbel, Tomb from, 28 Agricultural Implements, The, 142 Alhambra Court, The, 38 Amazonian Natives, 94 Anoplotheria, The, 165 Aquaria, Fresh Water, 96 Aquaria, Sea Water, 96-100 Araucaria Cookii, 131 Arcades for Waterfalls, 160 Archery Ground, The, 158 Arctic Illustrations, The, 95 Arundel Society Exhibition, 82 Assyrian Court, The, 43 Atrium of Greek Court, The, 33 Augsburg Cathedral, Bronze Doors from, 54 Australian Natives of Cape York, 92 Aviaries, The, 116
- 80. Aegina Marbles, The, 118 B. Bavaria, Colossal Head of, 109 Beni Hassan, Tomb from, 27 Bernini, Virgin and Christ, 78 Birkin Church, Norman Doorway from, 56 Birmingham Court, the, 84 Boilers and Furnaces, The, 13 Bosjesmen, The, 97 Botany of the Palace, The, 120 Botocudos, The, 93 Bramante, Doors from the Cancellaria at Rome, 79 Byzantine Court, The, 47 Byzantine Mosaic Ornament, 52 Byzantine Portraits of Justinian, Theodora, Charles the Bald, and Nicephorus Botoniates, 52 C. Campanile, Venice, Bronze Castings from the, 78 Canadian Court, 108 Cantilupe Shrine, The, 81 Caribs, The, 91 Cellini, Benvenuto, the Nymph of Fontainebleau, 72 Cellini, Benvenuto, Perseus, 119 Ceramic Court, The, 102 Certosa at Pavia, Sculptures and Architectural specimens from the, 71, 72, 73, 80 Chameleons, The, 117 Charles I., Statue of, 105 Chatham, Earl of, Statue of, 106 Chinese Chamber of Curiosities, 141 Choragic Monument of Lysicrates, 112 Cimabue, Paintings from Convent of Assisi in Italy, 55 Cloisters, Romanesque, 53 Cloisters, from Guisborough Abbey, 61
- 81. Coliseum at Rome, Large Model of, 37 Colleone, Equestrian Statue of, 82 Cologne Cathedral, Architectural Details from, 57 Concert Room, The, 121 Cotton Spinning Machine, 144 Cricket Ground, The, 160 Crosses, Irish and Manx, 55, 81 Crystal Palace, Account of the Building, 10 Crystal Palace, Measurements of the, 14 D. Danakils, The, 97 Dicynodons, The, 163 Donatello, Bas-Reliefs and Sculptures by, 72, 73, 81 Doria Palace, Doorways from the, 72, 73 Duquesne, Admiral, Colossal Statue, 111 E. Eardsley Church, Font from, 55 Effigy of Richard Cœur de Lion, 53 Egyptian Court, The, 24 Egyptian Frieze, 26 Egyptian Pictures, 26 Egyptian Figures, The Great, 118 Elgin Marbles, The, 34 Elizabethan Court, The, 74 Elks, The Irish, 165 Ely Cathedral, Door of Bishop West’s Chapel, 61 Ely Cathedral, The Prior’s Door, 53 Engineering and Architectural models, 139 Entrance, The, 21 Extinct Animals, The, 163 F. Fancy Manufactures, 103
- 82. Farnese Hercules, 111 Farnese Flora, 111 Fine Arts Court, Introduction to, 23 Fontevrault Abbey, Effigies from, 54 Forum at Rome, The, 37 Fortification, Mr. Fergusson’s System of, 139 Fountains, the System of, 172 Fountains, The Bronze, 114 Fountains, The Crystal, 21 Fountains, From Heisterbach, 53 Fountains, of Renaissance period, 70 Fountains, The Tartarughe, 77 Francis I., Equestrian Statue, 107 Franconia, Colossal Statue, 109 Frescos, Indian, 140 G. Galleries, The, 133 Gardens, The, 150 Gardens, The Italian Flower, 150 Gardens, The English Landscape, 157 Gattemelata, Bronze Equestrian Statue by Donatello, 82 Geerts, Charles, Ecclesiastical Sculpture by, 56 Geological Illustrations, 160 Ghiberti, Lorenzo, Bronze Gates from the Baptistery at Florence, 72 Glass Manufactures, Foreign, 101 Gold Fish, The, 114 Gothic Sepulchral Monuments, 64 Goujon, Jean, Carved doors from St. Maclou, 71, 72, 73 Goujon, Jean, Caryatides, from the Louvre, 72 Greek Court, The, 31 Greenlander, The, 95 Gutenberg Monument, The, 107
- 83. H. Hawton Church, The Easter Sepulchre from, 63 Hildesheim Cathedral, Doors from, 54 Hildesheim Cathedral, Bronze Column from, 81 Hot-Water Apparatus, 16 Hotel Bourgtheroulde, Restorations from, 70 Hylæosaurus, The, 164 I. Ichthyosaurus, The, 164 Iguanodons, The, 164 Indian Court, The, 140 Indians, American, 93 Inventions, Court of, 84 Italian Court, The, 76 Italian Court, Vestibule, 79 K. Kaffres, Zulu, 97 Karnak, Temple of, 28 Kilpeck Door (Norman), 54 Krafft, Adam, Ecclesiastical Sculpture by, 57, 58 L. Labyrinthodons, The, 163 Landscape view from the Terraces, 148 Laocoon, The, 32 Lessing, Portrait Statue of, 107 Lepidosiren, The, 115 Library and Reading Room, The, 109 Lichfield Cathedral, Door from, 64 Lincoln Cathedral, John O’Gaunt’s Window, 64 Lincoln Cathedral, Architectural Details from, 62 Lizards, The, 117 Lombardo, Pietro, Bronze Altar of La Madona della Scarpa, 80
- 84. M. Machinery in motion, 144 Mammoth Tree, 119 Marine Aquaria, 96-100 Mayence Cathedral, Monument from, 57 Medal Press, The, 83 Mediæval Court, The English, 53 Mediæval Court, The German, 56 Mediæval Court, The French and Italian, 67 Megalosaurus, The, 164 Megatherium, The, 165 Mexicans, 94, 99 Michael Angelo, Statues by, 77, 78 Michael Angelo, The Medici Tombs, 78 Monuments of art, Court of, 81 Monuments in front of Mediæval Courts, 117, 118 Mosasaurus, The, 164 Museum, Industrial and Technological Collection, 135 N. Natural History Illustrations, 90 Naval Museum, The, 139 Nave, The, 103 Niobe Sculptures, The, 34 Notre Dame of Paris, Arches and Iron Doors from, 67 Nuremberg Doorway, The, 56 O. Orchestra, Great Festival, 111 Orchestra, Concert, 112 P. Palæotherium, The, 164 Pantheon at Rome, The, 37 Papuans, The, 92 Park and Gardens, The, 147
- 85. Parthenon, Large Model of, 33 Parthenon, Frieze, 33 Perugino, Painted Ceiling from Perugia, 73 Philoe Portico, The, 27 Photographs in Galleries, Architectural, 138, 142 Picture Gallery, The, 134 Pilon, Germain, The Graces and other Statues, 73 Pisano, Giovanni, and Nino, Statues by, 67 Pipes in Gardens, System of, 155 Plesiosaurus, The, 164 Pocklington Cross, The, 81 Pompeian Court, The, 85 Portrait Gallery, Commencement of, 33 Portrait Gallery, The, 138 Pterodactyles, Great, 165 Ptolemaic Architecture, 26 Q. Quail, Californian, 115 Quercia, Jacopo della, Monument from Lucca Cathedral, 73 R. Raffaelle, Frescos from the Loggie of the Vatican, 77, 78 Raffaelle, Jonah and the Whale, 78 Raffaelle, Painted Ceiling from the “Camera Della Segnatura” of the Vatican, 78 Rameses the Great, Figures of, 27 Rathain Church, Old Window from, 55 Renaissance Court, The, 68 Robbia, Lucca della, Bas-Reliefs by, 72 Robbia Family, The, Frieze from Pistoia, 70 Rochester Cathedral, Doorway from, 62 Roman Court, The, 35 Romanesque (Byzantine) Court, The, 47 Rosary, The, 156 Rosetta Stone, The, 28
- 86. Rubens, Colossal Statue of, 111 S. Samoiedes, The, 95 Sansovino, Bronze Statues from the Campanile Loggia at Venice, 76 Sansovino, Bronze Door from St. Mark’s, Venice, 78 Screen of the Kings and Queens, 103-4 Sheffield Court, The, 85 Shobdon Side-Door and Chancel Arch, 54 Site of the Crystal Palace, The, 147 Somnauth Gates, The, 141 Stationery Court, The, 82 St. John Lateran, Arcade from, 54 T. Teleosaurus, The, 166 Terraces, The, 154 Testament, The King of Prussia’s, 110 Tibetans, The, 99 Toro Farnese, The, 112 Torrigiano, Monument of the Countess of Richmond from Westminster, 75 Towers, The Great Water-Towers, 168 Transepts, The, 105, 111, 114 Tropical Department, The, 114 Tuam Cathedral, Details and Examples from, 55 V. Vecchietta of Sienna, Bronze Effigy by, 73 Venus of Milo, The, 32 Veit Stoss, Ecclesiastical Sculpture by, 56, 58 Vestibule to English Mediæval Court, 66 W. Water Colour Copies of Great Masters, 79
- 87. T Well and Water Supply, The, 170 Wells Cathedral, Sculpture and Details from, 62, 63, 64 Winchester Cathedral, Portion of the Altar Screen, 64 Winchester Cathedral, Black Norman Font from, 55 Worcester Cathedral, Prince Arthur’s Door from, 61 INTRODUCTION. he map of the routes to the Crystal Palace will enable the visitor to ascertain the shortest and least troublesome way of reaching the Palace from the various parts of the great metropolis and its environs. The railway communication is by the London and Brighton, and the West End Railways, which serve as the great main lines for the conveyance of visitors by rail from London to the Palace doors. We will presume that the visitor has taken his railway ticket, which, for his convenience, includes admission within the Palace, and that his twenty minutes’ journey has commenced. Before he alights, and whilst his mind is still unoccupied by the wonders that are to meet his eye, we take the opportunity to relate, as briefly as we can, the History of the Crystal Palace, from the day upon which the Royal Commissioners assembled within its transparent walls to declare their great and successful mission ended, until the 10th of June, 1854, when reconstructed, and renewed and beautified in all its proportions, it again opened its wide doors to
- 88. continue and confirm the good it had already effected in the nation and beyond it. It will be remembered that the destination of the Great Exhibition building occupied much public attention towards the close of 1851, and that a universal regret prevailed at the threatened loss of a structure which had accomplished so much for the improvement of the national taste, and which was evidently capable, under intelligent direction, of effecting so very much more. A special commission even had been appointed for the purpose of reporting on the different useful purposes to which the building could be applied, and upon the cost necessary to carry them out. Further discussion on the subject, however, was rendered unnecessary by the declaration of the Home Secretary, on the 25th of March, 1852, that Government had determined not to interfere in any way with the building, which accordingly remained, according to previous agreement, in the hands of Messrs. Fox and Henderson, the builders and contractors. Notwithstanding the announcement of the Home Secretary, a last public effort towards rescuing the Crystal Palace for its original site in Hyde Park, was made by Mr. Heywood in the House of Commons, on the 29th of April. But Government again declined the responsibility of purchasing the structure, and Mr. Heywood’s motion was, by a large majority, lost. It was at this juncture that Mr. Leech,[1] a private gentleman, conceived the idea of rescuing the edifice from destruction, and of rebuilding it on some appropriate spot, by the organisation of a private company. On communicating this view to his partner, Mr. Farquhar, he received from him a ready and cordial approval. They then submitted their project to Mr. Francis Fuller, who entering into their views, undertook and arranged, on their joint behalf, a conditional purchase from Messrs. Fox and Henderson, of the Palace as it stood. In the belief that a building, so destined, would, if erected on a metropolitan line of railway, greatly conduce to the interests of the line, and that communication by railway was essential for the conveyance thither of great masses from London, Mr. Farquhar next suggested to Mr. Leo Schuster, a Director of the Brighton Railway, that a site for the new Palace should be selected on the Brighton line. Mr. Schuster, highly approving of the conception, obtained the hearty concurrence of Mr. Laing, the Chairman of the Brighton Board, and of his brother Directors, for aiding as far as possible in the prosecution of the work. And, accordingly, these five gentlemen, and their
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