# Spectral Properties and Finite Pole Assignment of Linear Neutral Systems in Banach Spaces

Authors: Xuewen Xia; Kai Liu
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We will consider a pole assignment problem for a class of linear neutral functional differential equations in Banach spaces. We will think of the neutral system studied as that of involving no time delays and reduce the study of adjoint semigroups and spectral properties of neutral equations to those of Cauchy problems. Under the
assumption that both the control and eigenspace of pole are finite dimensional, we establish the rank conditions for finite pole assignability.


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## Body

## 1. Introduction

Consider the linear system on some Banach spaceX(1.1)dy(t)dt=Ay(t),t≥0,y(0)=x∈X,
where A is the infinitesimal generator of a C0-semigroup etA, t≥0. A mild solution of (1.1) is defined as y(t)=etAx∈X for any t≥0. The null solution of (1.1) is said to be (exponentially) stable if, for any initial x∈X, the corresponding mild solution, is y(t)→0 as t→∞. It may be shown that the null solution is stable if and only if there exist positive numbers α≥1, μ>0 such that, for all t≥0, ∥etA∥≤αe-μt. If the null solution of system (1.1) is unstable,then it is important to consider stabilizability problem of its linear control system(1.2)dy(t)dt=Ay(t)+Mu(t),t≥0,y(0)=x∈X,
where M is a bounded linear operator from some Banach space U of control parameters into X. The mild solution of (1.2) is well defined for every locally integrable control function u(t), t≥0, and is given by the form(1.3)y(t)=etAx+∫0te(t-s)AMu(s)ds,t≥0.
System (1.2) is said to be feedback (exponentially) stabilizable if there exists a bounded linear operator K from X into U such that the system(1.4)dy(t)dt=(A+MK)y(t),t≥0,y(0)=x∈X,
is exponentially stable.Stability and feedback stabilization problems of the above systems and relevant nonlinear extensions, which play an important role in control theory and related topics, have been studied extensively by many researchers over the last two decades. The reader is referred to, for instance, the monograph of Luo et al.in [1] for a comprehensive statement about this topic and its applications.If we incorporate extra structure intoA, the stability and stabilizability problem would become complicated. One of the most important situations is to perturb A appropriately by a time-delay term so as that a strongly continuous family of bounded linear operators G(t) satisfying proper quasisemigroup properties completely describes the dynamics of the system studied. This idea therefore leads to the consideration of a class of linear time-delay systems(1.5)dy(t)dt=Ay(t)+∫-r0dη(θ)y(θ+t),t≥0,
where r>0, A generates a C0-semigroup etA, t≥0, and η is the Stieltjes measures given by(1.6)η(τ)=-∑i=1mχ(-∞,-ri](τ)Ai-∫τ0A0(θ)dθ,τ∈[-r,0].
Here Ai, i=1,…,m, and A0(θ), θ∈[-r,0], are properly defined linear, bounded operators from X into X (cf., Wu [2]).To our knowledge, very little paper has been done on feedback stabilization of infinite-dimensional control systems with memory. The only papers in this area are those by Yamamoto [3], Nakagiri and Yamamoto [4], Da Prato and Lunardi [5], and Jeong [6], all of which are devoted to retarded systems. In [4], the rank condition for exponential stabilizability in terms of eigenvectors and controllers was established.In the present paper, we will study the finite pole assignability problem for a class of neutral linear control system(1.7)ddt[y(t)-∫-r0dζ(θ)y(θ+t)]=Ay(t)+∫-r0dη(θ)y(θ+t)+Mu(t),t≥0,y(0)=ϕ0,y0(·)=ϕ1(·),
where ϕ=(ϕ0,ϕ1) is some initial datum to be identified later. Generally speaking, for neutral systems as above it is quite difficult to study stabilizability problem and there are few satisfactory results in this respect. The reason is that, as pointed out in the study by Salamon in [7], it is generally required a memory feedback involving derivative terms for the purposes of stabilization of (1.7) even in finite-dimensional cases. Thus we shall study in this work a weaker concept, finite pole assignability, for (1.7) by means of state feedback law which does not necessarily contain derivative terms. To this end, the whole paper is divided into five sections. After reviewing some useful notions and notations, we will establish in Section 2 a semigroup theory which enables us to reduce the neutral systems (1.7) to a class of control systems involving no delays in an appropriate infinite-dimensional space. In order to formulate systems (1.7) in the L2 product space setting, we restrict ourselves to the case that the neutral delay term on the left-hand side of (1.7) does not involve discrete delays. The associated semigroup is well defined by a solution state (y(t),yt), where yt denotes a t-segment of solutions, a situation which is different from that in-Burns et al. in [8]. The infinitesimal generator of this semigroup is explicitly described and its relationship with neutral resolvent operators is explored. In Section 3, we will establish an adjoint theory which will play an important role in the study of the usual controllability and stabilizability. Sections 4 and 5 are devoted to the investigation of spectral properties and pole assignability, respectively. Under suitable conditions such as the finite dimensionality of spectral modes, we will establish useful criteria of finite pole assignability.The real and complex number vector spaces are denoted byℝn and ℂn, n≥1, respectively. Also, ℝ+ denotes the set of all nonnegative numbers. For any λ∈ℂ1, the symbols Reλ and Imλ denote the real and imaginary parts of complex number λ, respectively. Let X and U be complex, separable Banach spaces and X*, U* their adjoint spaces with norms ∥·∥X and ∥·∥U and the dual pairings 〈·,·〉X,X* and 〈·,·〉U,U*, respectively. We use ℒ(U,X) to denote the space consisting of all bounded linear operators T from U into X with domain U. When X=U, ℒ(X,X) is denoted by ℒ(X). Each operator norm is simply denoted by ∥·∥ when there is no danger of confusion. For any operator T, we employ 𝒟(T) to denote the domain of T, and the symbols KerT and ImT will be used to denote the kernel and image of operator T, respectively. For any fixed constant r>0, we denote by Lr2=L2([-r,0];X) the space of all X-valued equivalence classes of measurable functions which are squarely integrable on [-r,0]. Let 𝒳 denote the Banach space X×Lr2 with the norm(1.8)∥ϕ∥𝒳=∥ϕ0∥X2+∥ϕ1∥Lr22,∀ϕ=(ϕ0,ϕ1)∈𝒳.
Let W1,2([-r,0];X) denote the Sobolev space of X-valued functions x(t) on [-r,0] such that x(t) and its distributional derivative belong to L2([-r,0];X).


## 2. Neutral Control Systems

Consider the following neutral linear functional differential equation on the Banach spaceX:(2.1)ddt[y(t)-∫-r0dζ(θ)y(θ+t)‍]=Ay(t)+∫-r0dη(θ)y(θ+t),t≥0,y(0)=ϕ0,y0(·)=ϕ1(·),ϕ=(ϕ0,ϕ1)∈𝒳,
where A generates a C0-semigroup etA, t≥0, y0(θ)=y(θ), θ∈[-r,0], and η, ζ are the Stieltjes measures given by(2.2)η(τ)=-∑i=1mχ(-∞,-ri](τ)Ai-∫τ0A0(θ)dθ,τ∈[-r,0],ζ(τ)=-∫τ0B0(θ)dθ,τ∈[-r,0].
Here 0<r1<r2<⋯<rm≤r, Ai∈ℒ(X), i=1,…,m, A0(·)∈L2([-r,0];ℒ(X)), and B0(·)∈L2([-r,0];ℒ(X)). Unless otherwise specified, we always use ∫-r0dη(θ)y(t+θ):y∈L2([-r,T];X)→X, T≥0, to denote the bounded, linear extension of the mapping(2.3)∫-r0dη(θ)y(t+θ)=∑i=1mAiy(t-ri)+∫-r0A0(θ)y(t+θ)dθ,y∈C([-r,T];X),
for any t≥0, and the same remark applies to ∫-r0dζ(θ)y(t+θ):y∈L2([-r,T];X)→X in an obvious way.We also wish to consider the hereditary neutral controlled system of (2.1) on X:(2.4)ddt[y(t)-∫-r0dζ(θ)y(θ+t)]=Ay(t)+∫-r0dη(θ)y(θ+t)+Mu(t),t≥0,y(0)=ϕ0,y0(·)=ϕ1(·),ϕ=(ϕ0,ϕ1)∈𝒳,u∈L2([0,∞);U),
where M∈ℒ(U,X). A mild solutiony(t,ϕ,u) of (2.4) is defined as the unique solution of the following integral equation on the Banach space X,(2.5)y(t,ϕ,u)=∫-r0dζ(θ)y(t+θ,ϕ,u)‍+etA[ϕ0-∫-r0dζ(θ)ϕ1(θ)‍]+∫0t‍e(t-s)A[∫-r0‍dη(θ)y(s+θ,ϕ,u)+∫-r0‍Adζ(θ)y(s+θ,ϕ,u)]ds+∫0t‍e(t-s)AMu(s)ds,∀t>0,
with y(0,ϕ,u)=ϕ0, y0(·,ϕ,u)=ϕ1(·), and ϕ=(ϕ0,ϕ1)∈𝒳.To ensure the uniqueness and existence of mild solutions, we further assume that, for eachi, i=1,…,m, and θ∈[-r,0], Im(B0(θ))⊂𝒟(A) such that AB0(·)∈L2([-r,0];ℒ(X)). Under these conditions, it has been shown in the study by Liu in [9] that there exists a unique mild solution y(t,ϕ,u) for (2.4) with y(0,ϕ,u)=ϕ0 and y0(·,ϕ,u)=ϕ1(·).Note that, for anyϕ∈𝒳, the mild solution y(t,ϕ) is continuous for t>0. To see this, it suffices to notice that, for any s, t>0,(2.6)∥∫-r0‍dζ(θ)y(θ+s,ϕ)dθ-∫-r0dζ(θ)y(θ+t,ϕ)dθ∥x≤∥B0∥L2([-r,0];ℒ(X))∥ys(·)-yt(·)∥L2([-r,0];X).
We define a mapping 𝒮(t) on 𝒳, t≥0, by(2.7)𝒮(t)ϕ=(y(t,ϕ),yt(·,ϕ)),t≥0,
where yt(·,ϕ)=y(t+·,ϕ) for any t≥0. It turns out that 𝒮(t), t≥0, is a strongly continuous semigroup on 𝒳.Proposition 2.1.
For anyt≥s≥0 and ϕ∈𝒳, the following relation holds:
(2.8)𝒮(t-s)(y(s,ϕ),ys(·,ϕ))=(y(t,ϕ),yt(·,ϕ)).
That is,
(2.9)𝒮(t-s)𝒮(s)ϕ=𝒮(t)ϕ.
Moreover, 𝒮(t) is a C0-semigroup of bounded linear operators on 𝒳.Proof.
The linearity of𝒮(t) is obvious. Strong continuity of 𝒮(t) on 𝒳 follows from the fact that y(t,ϕ)→ϕ0 in X as t→0+ by virtue of (2.5) and (2.6), and on the other hand, it is easy to see that yt(·,ϕ)→ϕ1 in L2([-r,0];X) as t→0+. In order to show the semigroup property (2.8), let t≥s and
(2.10)Φ(s)=𝒮(s)ϕ=(y(s,ϕ),ys(·,ϕ))∈𝒳.
Then from (2.5), it is easy to verify that
(2.11)y(t-s,Φ(s))-∫-r0dζ(θ)y(t-s+θ,Φ(s))=e(t-s)A(y(s,ϕ)-∫-r0dζ(θ)y(θ+s,ϕ))+∫st‍e(t-u)A[∫-r0dη(θ)y(u-s+θ,Φ(s))+∫-r0Adζ(θ)y(u-s+θ,Φ(s))]du=e(t-s)A[esA(ϕ0-∫-r0dζ(θ)ϕ1(θ))+∫0se(s-u)A[∫-r0dη(θ)y(u+θ,ϕ)+∫-r0Adζ(θ)y(u+θ,ϕ)]du+∫ste(t-u)A[∫-r0dη(θ)y(u-s+θ,Φ(s))+∫-r0Adζ(θ)y(u-s+θ,Φ(s))]]du=etA(ϕ0-∫-r0dζ(θ)ϕ1(θ))+∫0se(t-u)A[‍∫-r0dη(θ)y(u+θ,ϕ)+∫-r0Adζ(θ)y(u+θ,ϕ)]du+∫ste(t-u)A[∫-r0dη(θ)y(u-s+θ,Φ(s)‍)+∫-r0Adζ(θ)y(u-s+θ,Φ(s))‍]du.
On the other hand, we have for t≥s that
(2.12)y(t,ϕ)-∫-r0‍dζ(θ)y(t+θ,ϕ)=etA(ϕ0-∫-r0‍dζ(θ)ϕ1(θ))+∫0s‍e(t-u)A[∫-r0‍dη(θ)y(u+θ,ϕ)+∫-r0‍Adζ(θ)y(u+θ,ϕ)]du+∫ste(t-u)A[∫-r0dη(θ)y(u+θ,ϕ)+‍∫-r0Adζ(θ)y(u+θ,ϕ)]du‍.
Thus, by the uniqueness of solutions of (2.5) with M=0, it implies that
(2.13)y(t-s,Φ(s))=y(t,ϕ),foralmostallt≥s.
Hence, [𝒮(t-s)𝒮(s)ϕ]0=[𝒮(t)ϕ]0 for all t≥s, and so [𝒮(t-s)𝒮(s)ϕ]1=[𝒮(t)ϕ]1 in Lr2([-r,0];X). The semigroup property (2.8) is thus proved and the proof is complete.Let𝒜 be the infinitesimal generator of 𝒮(t) and denote 𝒮(t) simply by et𝒜. The next theorem explicitly describes the operator 𝒜.Theorem 2.2.
The infinitesimal generator𝒜 of et𝒜 is described by
(2.14)𝒟(𝒜)={ϕ=(ϕ0,ϕ1)∈𝒳:ϕ1∈W1,2([-r,0];X),ϕ0=ϕ1(0)∈𝒟(A)},𝒜ϕ=(Aϕ0+∫-r0dζ(θ)ϕ1′(θ)‍+∫-r0dη(θ)ϕ1(θ),ϕ1′(θ)‍),
for any ϕ=(ϕ0,ϕ1)∈𝒟(𝒜).Proof.
We denote by𝒜̃ and 𝒟(𝒜̃) the infinitesimal generator of et𝒜 and its domain, respectively. Let ϕ=(ϕ0,ϕ1)∈𝒟(𝒜̃) and
(2.15)𝒜̃ϕ=(ψ0,ψ1).
Since the second coordinate of et𝒜ϕ is the t-shift y(t+·), it follows immediately that
(2.16)y(θ)=ϕ1(θ)∈W1,2([-r,0];X),θ∈[-r,0],d+dθy(θ)=ϕ1′(θ)=ψ1(θ),inL2([-r,0];X),θ∈[-r,0],
where d+/dθ denotes the right-hand derivative. By redefining on the set of measure zero, we can suppose that y(θ)=ϕ1(θ) is absolutely continuous from [-r,0] to X by Theorem 2.2, p. 19, of [10]. Since y(0)=ϕ0, this implies that ϕ1(0)=ϕ0 and y(·):[-r,∞)→X is strongly continuous. Then the functions ∫-r0dη(θ)y(t+θ) and ∫0tAdζ(θ)y(t+θ) are strongly continuous in t≥0 such that
(2.17)limt→0+1t∫0te(t-s)A‍∫-r0‍dη(θ)y(s+θ)ds=∫-r0‍dη(θ)ϕ1(θ),limt→0+1t∫0te(t-s)A‍∫-r0‍Aζ(θ)y(s+θ)ds=∫-r0‍Adζ(θ)ϕ1(θ).
Applying (2.17) to the first coordinate of (2.15), we obtain that
(2.18)ψ0=limt→0+1t(y(t,ϕ)-ϕ0)=limt→0+1t{∫-r0dζ(θ)y(t+θ,ϕ)+etA‍(ϕ0-∫-r0dζ(θ)ϕ1(θ))+∫0te(t-s)A‍[∫-r0dη(θ)y(s+θ,ϕ)‍+∫-r0Adζ(θ)y(s+θ,ϕ)‍]ds-ϕ0}=limt→0+1t{[∫-r0‍dζ(θ)y(t+θ,ϕ)-∫-r0‍dζ(θ)ϕ1(θ)]-[etA∫-r0‍dζ(θ)ϕ1(θ)-∫-r0‍dζ(θ)ϕ1(θ)]+∫0te(t-s)A[∫-r0‍dη(θ)y(s+θ,ϕ)+∫-r0‍Adζ(θ)y(s+θ,ϕ)]ds+[etAϕ0-ϕ0]}=∫-r0‍dζ(θ)ϕ1′(θ)-∫-r0‍Adζ(θ)ϕ1(θ)+∫-r0‍dη(θ)ϕ1(θ)+∫-r0‍Adζ(θ)ϕ1(θ)+limt→0+1t[etAϕ0-ϕ0]=∫-r0‍dζ(θ)ϕ1′(θ)+∫-r0‍dη(θ)ϕ1(θ)+limt→0+1t[etAϕ0-ϕ0].
Hence, limt→0+t-1(etAϕ0-ϕ0) exists in X;  that is, ϕ0∈𝒟(A), and
(2.19)ψ0=Aϕ0+∫-r0‍dζ(θ)ϕ1′(θ)+∫-r0‍dη(θ)ϕ1(θ),
which shows that
(2.20)𝒟(𝒜̃)⊂𝒟(𝒜),𝒜ϕ=𝒜̃ϕ,forϕ∈𝒟(𝒜̃).
Next we will show the reverse inclusion. Let ϕ=(ϕ0,ϕ1)∈𝒟(𝒜); then it is easy to see that y(·,ϕ)∈W1,2([-r,T];X) for any T>0, from which (2.17) follow. Combining this with ϕ0=ϕ1(0)∈𝒟(A), we see that
(2.21)limt→0+1t(y(t,ϕ)-ϕ0)=Aϕ0+∫-r0‍dζ(θ)ϕ1′(θ)+∫-r0‍dη(θ)ϕ1(θ).
Noting that
(2.22)yt(θ,ϕ)-ϕ1(θ)t-ϕ1′(θ)=y(t+θ,ϕ)-y(θ,ϕ)t-y'(θ,ϕ)=(1t)∫0t(y'(s+θ,ϕ)-y'(θ,ϕ))ds
for θ∈[-r,0], we obtain by using Hölder inequality that
(2.23)∥1t(yt(·,ϕ)-ϕ1)-ϕ1′∥Lr22≤1t∫0t[∫-r0∥y'(s+θ,ϕ)-y'(θ,ϕ)∥X2dθ]ds.
This implies that limt→0+t-1(yt(·,ϕ)-ϕ1) exists in L2([-r,0];X) and equals ϕ1′. Therefore, we prove that 𝒟(𝒜)⊂𝒟(𝒜̃) and 𝒜ϕ=𝒜̃ϕ for ϕ∈𝒟(𝒜), and (2.14) are shown.For eachλ∈ℂ1, define the densely defined, closed linear operator Δ(λ,A,η,ζ) by(2.24)Δ(λ,A,η,ζ)=λI-A-∫-r0‍eλθdη(θ)-∫-r0‍λeλθdζ(θ).
The neutral resolvent setρ(A,η,ζ) is defined as the set of all values λ in ℂ1 for which the operator Δ(λ,A,η,ζ) has a bounded inverse on X.Proposition 2.3.
For anyλ∈ℂ1, the relation
(2.25)(λI-𝒜)(ϕ1(0),ϕ1)=(ψ0,ψ1)∈𝒳,
is equivalent to
(2.26)ϕ1(θ)=eλθϕ1(0)+∫θ0‍eλ(θ-τ)ψ1(τ)dτ∈W1,2([-r,0];X),(2.27)Δ(λ,A,η,ζ)ϕ1(0)=ψ0+∫-r0‍dη(θ)∫θ0‍eλ(θ-τ)ψ1(τ)dτ+∫-r0‍λdζ(θ)∫θ0‍eλ(θ-τ)ψ1(τ)dτ-∫-r0‍dζ(θ)ψ1(θ)dθ.
In particular, the resolvent set ρ(𝒜) is equal to ρ(A,η,ζ).Proof.
Note that the relation (2.25) is equivalent to
(2.28)ψ0=λϕ1(0)-Aϕ1(0)-∫-r0‍dη(θ)ϕ1(θ)-∫-r0‍dζ(θ)ϕ1′(θ),(2.29)ψ1(θ)=λϕ1(θ)-dϕ1(θ)dθ,θ∈[-r,0].
The variation of constants formula for the ordinary differential equation (2.29) on [-r,0] shows that
(2.30)ϕ1(θ)=eλθϕ1(0)+∫θ0‍eλ(θ-τ)ψ1(τ)dτ∈W1,2([-r,0];X).
In order to show (2.27), note that, from (2.28), we have
(2.31)(λI-A)ϕ1(0)=ψ0+∫-r0‍dζ(θ)ϕ1′(θ)+∫-r0‍dη(θ)ϕ1(θ).
Substituting (2.30) into (2.31) immediately yields the desired (2.27). The equality of resolvent sets ρ(𝒜) and ρ(A,η,ζ) is easily seen according to the equivalence of (2.25) and (2.26), (2.27). The proof is now complete.


## 3. Adjoint Systems

In the remainder of this work, unless otherwise specified, we always assume thatX is reflexive. As indicated in the study by  Hale  in [11], the adjoint theory of neutral linear functional differential equations in C([-r,0];X) is quite complicated. However, for the control equation (2.4), it is possible to construct an elementary adjoint theory for 𝒮(t).Letψ*=(ψ0*,ψ1*)∈𝒳* and define a “formal" transposed neutral system of (2.1) on X* by(3.1)ddt[y(t)-∫-r0‍dζ*(θ)y(t+θ)]=A*y(t)+∫-r0‍dη*(θ)y(t+θ),t>0,y(0)=ψ0*,y0(t)=ψ1*(t),t∈[-r,0),
where η*(θ), ζ*(θ) and A* denote the adjoint operators of η(θ), ζ(θ), and A, respectively. It is well known that A* generates a C0-semigroup S*(t) on X* which is the adjoint of S(t), t≥0. For any λ∈ℂ1, define(3.2)Δ(λ,A*,η*,ζ*)=λI-A*-∫-r0‍eλθdη*(θ)-∫-r0‍λeλθdζ*(θ).Proposition 3.1.
For anyλ∈ℂ1, the equation
(3.3)(λ¯I-𝒜*)(ϕ0*,ϕ1*)=(ψ0*,ψ1*(·))
is equivalent to
(3.4)Δ(λ¯,A*,η*,ζ*)ϕ0*=ψ0*+∫-r0‍eλ¯θψ1*(θ)dθ,(3.5)ϕ1*(θ)=∫-rθ‍eλ¯(τ-θ)ψ1*(τ)dτ-B0*(θ)ϕ0*+∫-rθ‍eλ¯(τ-θ)dη*(τ)ϕ0*+∫-rθ‍λ¯eλ¯(τ-θ)B0*(τ)dτϕ0*
almost everywhere for any ψ*=(ψ0*,ψ1*)∈𝒳*.Proof.
Note that(λ¯I-𝒜*)-1=((λI-𝒜)-1)*, so we may calculate the adjoint operator of (λI-𝒜)-1. In view of (2.26), it is not difficult to see that, for any ψ=(ψ0,ψ1)∈𝒳,
(3.6)〈(ψ0,ψ1),((λI-𝒜)-1)*(ψ0*,ψ1*)〉𝒳,𝒳*=〈(λI-𝒜)-1(ψ0,ψ1),(ψ0*,ψ1*)〉𝒳,𝒳*=〈ϕ1(0),ψ0*〉X,X*+∫-r0‍〈eλθϕ1(0)-∫0θeλ(θ-τ)ψ1(τ)dτ,ψ1*(θ)‍〉X,X*dθ=〈ϕ1(0),ψ0*〉X,X*+∫-r0‍〈eλθϕ1(0),ψ1*(θ)〉X,X*dθ-∫-r0∫0θ‍〈eλ(θ-τ)ψ1(τ),ψ1*(θ)〉X,X*dτdθ=〈ϕ1(0),ψ0*+∫-r0‍eλ¯θψ1*(θ)dθ〉X,X*+∫-r0∫θ0‍〈eλ(θ-τ)ψ1(τ),ψ1*(θ)〉X,X*dτdθ.
We reformulate the expression in (3.6), starting with the last term:
(3.7)∫-r0∫θ0〈eλ(θ-τ)ψ1(τ),ψ1*(θ)〉X,X*dτdθ=∫-r0∫-rτ‍〈eλ(θ-τ)ψ1(τ),ψ1*(θ)〉X,X*dθdτ=∫-r0‍〈ψ1(τ),∫-rτeλ¯(θ-τ)ψ1*(θ)dθ〉X,X*dτ.
Letting κ*=ψ0*+∫-r0eλ¯θψ1*(θ)dθ∈X* and applying Proposition 2.3 to such a ϕ=(ϕ1(0),ϕ1) yield that
(3.8)〈ϕ1(0),ψ0*+∫-r0eλ¯θψ1*(θ)dθ〉X,X*=〈ϕ1(0),κ*〉X,X*=〈Δ(λ,A,η,ζ)-1[∫-r0‍B0(θ)ψ1(θ)dθψ0-∫-r0‍B0(θ)ψ1(θ)dθ+∫-r0B0(τ)∫τ0‍λeλ(τ-θ)ψ1(θ)dθdτ+∫-r0dη(τ)∫τ0eλ(τ-θ)ψ1(θ)dθ],κ*〉X,X*=〈ψ0,(Δ(λ,A,η,ζ)-1)*κ*〉X,X*-∫-r0‍〈ψ1(θ),B0*(θ)(Δ(λ,A,η,ζ)-1)*κ*〉X,X*dθ+∫-r0〈ψ1(θ),∫-rθ(eλ¯(τ-θ)dη*(τ)+λ¯eλ¯(τ-θ)B0*(τ)dτ)(Δ(λ,A,η,ζ)-1)*κ*〉X,X*dθ.
If we combine these equalities and use the fact that ϕ0*=(Δ(λ,A,η,ζ)*)-1κ*, then we obtain
(3.9)〈(ψ0,ψ1),((λI-𝒜)-1)*(ψ0*,ψ1*)〉𝒳,𝒳*=〈ψ0,ϕ0*〉X,X*-∫-r0‍〈ψ1(θ),B0*(θ)ϕ0*〉X,X*dθ+∫-r0〈ψ1(θ),∫-rθ(eλ¯(τ-θ)dη*(τ)+λ¯eλ¯(τ-θ)B0*(τ)dτ)ϕ0*+∫-rθeλ¯(τ-θ)ψ1*(τ)dτ〉X,X*dθ,‍
and this proves the desired result. The proof is now complete.The following corollary which characterizes the infinitesimal generator𝒜* of the semigroup 𝒮*(t) on 𝒳* is a direct result of Proposition 3.1.Corollary 3.2.
The infinitesimal generator𝒜* of 𝒮*(t) is given by
(3.10)𝒟(𝒜*)={ϕ*=(ϕ0*,ϕ1*)∈𝒳*:ϕ0*∈𝒟(A*),ϕ1*(θ)=∫-rθdη*(τ)ϕ0*-B0*(θ)ϕ0*-φ*(θ),a.e.θ∈[-r,0]whereφ*(·)∈W1,2([-r,0];X*)∫-rθdη*(τ)ϕ0*-B0*(θ)ϕ0*-φ*(θ),},
and moreover
(3.11)𝒜*ϕ*=(A*ϕ0*+ϕ1*(0+)+B0*(0+)ϕ0*,ddθ[∫-rθdη*(τ)ϕ0*-ϕ1*(θ)-B0*(θ)ϕ0*]),ϕ*=(ϕ0*,ϕ1*)∈𝒟(𝒜*),
where ϕ1*(0+)+B0*(0+)ϕ0* is given by the limit limθ→0+(ϕ1*(θ)+B0*(θ)ϕ0*).Proof.
Letλ=0 in (3.4) and (3.5),then it follows that
(3.12)(-A*-∫-r0dη*(θ))ϕ0*=ψ0*+∫-r0ψ1*(θ)dθ,(3.13)ϕ1*(θ)=∫-rθψ1*(τ)dτ-B0*(θ)ϕ0*+∫-rθdη*(τ)ϕ0*,a.e.θ∈[-r,0].
Since the term ∫-rθdη*(τ)ϕ0* in (3.13) is left continuous at θ=0, then the sum ϕ1*(θ)+B0*(θ)ϕ0* is also left continuous at θ=0. Then we see from (3.13) that the limit limθ→0+(ϕ1*(θ)+B0*(θ)ϕ0*) exists in X and
(3.14)ϕ1*(0+)+B0*(0+)ϕ0*=∫-r0ψ1*(τ)dτ+∫-r0dη*(τ)ϕ0*.
Substituting (3.14) into (3.12) yield
(3.15)-ψ0*=A*ϕ0*+ϕ1*(0+)+B0*(0+)ϕ0*,
and letting λ=0 in (3.5) and further taking derivative with respect to θ∈[-r,0] yields
(3.16)-ψ1*(θ)=ddθ[∫-rθdη*(τ)ϕ0*-ϕ1*(θ)-B0*(θ)ϕ0*]
from which the desired results are easily obtained. The proof is now complete.Theadjoint neutral resolvent setρ(A*,η*,ζ*) is defined similarly as the set of all values λ in ℂ1 for which the operator Δ(λ,A*,η*,ζ*) has a bounded inverse on X*. Then by applying the adjoint version of Proposition 3.1, we see that ρ(A*)=ρ(A*,η*,ζ*).


## 4. Spectral Properties

In this section we investigate the spectral properties of operators𝒜 and 𝒜* by means of Δ(λ,A,η,ζ) and Δ(λ¯,A*,η*,ζ*) in preceding sections. In the remainder of this paper, we denote Δ(λ,A,η,ζ) by Δ(λ). Also recall that theneutral spectrumσ(Δ(λ,A,η,ζ)), or simply σ(Δ), is defined by σ(Δ)=ℂ1∖ρ(A,η,ζ). The spectrumσ(Δ) of Δ(λ,A,η,ζ) can be divided into three disjoint subsets in the following manner. The continuous spectrum σC(Δ) is the set of values of λ for which Δ(λ,A,η,ζ) has an unbounded inverse with dense domain in X. The residual spectrum σR(Δ) is the set of values of λ for which Δ(λ,A,η,ζ) has an inverse whose domain is not dense in X. The point spectrum σP(Δ) is the set of values of λ for which no inverse of Δ(λ,A,η,ζ) exists. Define the subset σd(Δ) of σP(Δ) by(4.1)σd(Δ)={λ:λ∈σP(Δ)anddimKerΔ(λ)isfinite}.
Throughout this paper we suppose that σR(Δ)=∅ and σd(Δ) is a denumerable nonempty set. Further we suppose on σd(Δ) that, for each pair λ1,λ2∈σd(Δ), there exists a continuous rectifiable arc C⊂ρ(Δ) joining λ1 and λ2. This condition implies that for any finite set Λ in σd(Δ) there exists a continuous rectifiable arc CΛ⊂ρ(Δ) which surrounds Λ inside and contains no other points in σ(Δ).We also denoteΔ(λ,A*,η*,ζ*) by Δ*(λ) and define the spectrum sets σ(Δ*), σP(Δ*) and σd(Δ*), in a similar way to those for Δ(λ,A,η,ζ). The proposition below shows some identical relations between the neutral point spectrum of 𝒜, 𝒜* and Δ, Δ*.Proposition 4.1.
The neutral point spectrum of𝒜 (resp., 𝒜*) satisfies that σP(𝒜)=σP(Δ) (resp., σP(𝒜*)=σP(Δ*)) and σd(𝒜)=σd(Δ) (resp., σd(𝒜*)=σd(Δ*)).Proof.
Recall that, by Proposition2.3, for any λ∈ℂ1 the relation (λI-𝒜)ϕ=ψ, ϕ∈𝒟(𝒜), ψ∈𝒳 is equivalent to the relation Δ(λ)ϕ1(0)=Gλ(ψ), ϕ0=ϕ1(0)∈𝒟(A), where Gλ(ψ) is given by
(4.2)Gλ(ψ)=ψ0+∫-r0dη(θ)∫θ0eλ(θ-τ)ψ1(τ)dτ+∫-r0λdζ(θ)∫θ0λeλ(θ-τ)ψ1(τ)dτ-∫-r0dζ(θ)ψ1(θ)dθ,ϕ1(θ)=eλθϕ1(0)+∫θ0eλ(θ-τ)ψ1(τ)dτ,θ∈[-r,0]
If we substitute ψ=0 in the above equalities, we have that Ker(λI-𝒜)={0} is equivalent to KerΔ(λ)={0}, and hence Ker(λI-𝒜)={0} if and only if KerΔ(λ)={0}. This concludes, by definition, σP(𝒜)=σP(Δ). It is easy to see that ϕ=(ϕ(0),ϕ)∈ Ker(λI-𝒜) if and only if ϕ(0)∈ KerΔ(λ) and ϕ(θ)=eλθϕ1(0). By this equivalence it is easily seen that dimKerΔ(λ)=dimKer(λI-𝒜). This shows σd(𝒜)=σd(Δ). The other equalities σP(𝒜*)=σP(Δ*) and σd(𝒜*)=σd(Δ*) can be proved similarly.In what follows we omit the symbolI for the identity operator; for example, λ-𝒜 denotes λI-𝒜. For each isolated point λ∈σ(𝒜), the spectral projection Pλ and the quasinilpotent operator Qλ are defined, respectively, by
(4.3)Pλ=12πi∫γλ(z-𝒜)-1dz,Qλ=12πi∫γλ(z-λ)(z-𝒜)-1dz,
where γλ is a small circle with center λ such that its interior and γλ contain no points of σ(𝒜). Let 𝒩λ=Pλ𝒳 be the generalized eigenspace corresponding to the eigenvalue λ of A. It is obvious that
(4.4)Qλj=12πi∫γλ(z-λ)j(z-𝒜)-1dz,j=1,2,…·
Further, if λ is a pole of (z-𝒜)-1 of order kλ, then we have
(4.5)Qλkλ=O,ImQλ⊂𝒩λ,(4.6)Ker(λ-𝒜)=𝒩λ∩KerQλ,
where O is the null operator, and the direct sum decompositions of𝒳(4.7)𝒩λ=Ker(λ-𝒜)kλ,𝒳=𝒩λ⊕Im(λ-𝒜)kλ hold  (cf. Chapter 3 in Kato [12], Chapter 8 in Tanabe [13]). The relations (4.3)–(4.7) hold for 𝒜* and each isolated λ∈σ(𝒜*). In view of Proposition 2.3, we see that
(4.8)Ker(λ-𝒜)={(ϕ0,eλ·ϕ0):Δ(λ)ϕ0=0}.Note thatλ is a pole and each Ker(λ-𝒜) and 𝒩λ are finite dimensional if λ∈σd(Δ). Let Λ⊂σd(𝒜) be a finite set of isolated spectrum. Suppose that there exists a rectifiable Jordan curve γΛ which surrounds Λ and separates Λ and ℂ1∖Λ. We define the projection PΛ on Λ by
(4.9)PΛ=12πi∫γΛ(z-𝒜)-1dz.
Then the following decomposition of 𝒳 holds:
(4.10)𝒳=𝒩Λ⊕ℛΛ,
where
(4.11)𝒩Λ=PΛ𝒳,ℛΛ=(I-PΛ)𝒳,
and I is the identity operator on 𝒳.Now we introduce the bounded operatorFλ:X*→L2([-r,0];X*) defined by
(4.12)[Fλϕ0*](θ)=[-B0*(θ)+∫-rθeλ(τ-θ)dη*(τ)+∫-rθλeλ(τ-θ)B0*(τ)dτ]ϕ0*,a.e.θ∈[-r,0]
for any λ∈ℂ1. Then by applying the adjoint version of Proposition 3.1, we have
(4.13)Ker(λ¯-𝒜*)={(ϕ0*,Fλ¯ϕ0*):Δ*(λ¯)ϕ0*=0},
where Δ*(λ¯)=Δ(λ¯,A*,η*,ζ*) and, if λ¯∈σd(𝒜*)=σd(Δ*), then
(4.14)dimKer(λ¯-𝒜*)=dimKerΔ*(λ¯)<∞.
Let λ∈σ(𝒜) be an isolated point. Then, by Kato [12],  λ¯∈σ(𝒜*) is also isolated and the eigenspace Ker(λ¯-𝒜*) and the generalized eigenspace 𝒩λ¯*=(Pλ)*𝒳*=Pλ¯*𝒳* are well defined, and
(4.15)dimKer(λ-𝒜)=dimKer(λ¯-𝒜*)≤∞,dim𝒩λ=dim𝒩λ¯*≤∞,
where Pλ¯* is the projection
(4.16)Pλ¯*=12πi∫γλ∨(z-𝒜*)-1dz
and γλ∨ is the mirror image of γλ in (4.3). Hence, we have the following result.Proposition 4.2.
For eachλ∈σd(Δ), one has λ¯∈σd(Δ*)=σd(𝒜*) and
(4.17)Ker(λ¯-𝒜*)={(ϕ0*,Fλ¯ϕ0*):ϕ0*∈KerΔ*(λ¯)},dimKer(λ¯-𝒜*)=dimKerΔ*(λ¯)<∞,dim𝒩λ¯*=dim𝒩λ<∞.


## 5. Pole Assignment

In general, for system (2.1) we have no ideas whether or not the associated generator 𝒜 in Theorem 2.2 satisfies the spectral determined growth condition(5.1)sup{Reλ:λ∈σ(𝒜)}=limt→∞ln∥et𝒜∥t.
Consequently, it is difficult for standard results, for example, those established by Hale [11], to be applied to the mild solution y(·,ϕ) of (2.1). Instead of considering the stability and stabilizability problem for the control system (2.4), we will study in this section the finite pole assignment problem for (2.4) on the product space 𝒳.We are concerned about the finite pole assignment problem of the control system (2.4): under what conditions on M can we construct a feedback law such that any finite set in σd(Δ) is shifted to any preassigned set in the complex plane?To this end, first note that, by means of𝒜, we can reformulate system (2.4) into the space 𝒳 as a control system without delay:(5.2)dY(t)dt=𝒜Y(t)+ℳu(t),t≥0,Y(0)=ϕ=(ϕ0,ϕ1)∈𝒳,u∈L2([0,∞);U),
where ℳ:U→𝒳 is defined by ℳu=(Mu,0), u∈U.Definition 5.1.
LetΛ0={λ1,…,λl}⊂σd(Δ) and Λ1={μ1,…,μl} be finite sets in the complex plane. The control system (5.2) is said to be pole assignable with respect to (Λ0,Λ1) if and only if there exists a bounded linear operator K∈ℒ(𝒳,U) such that
(5.3)σ(𝒜+ℳK)=(σ(𝒜)∖Λ0)∪Λ1.We remark that the operatorK has the form(5.4)Kϕ=K0ϕ0+∫-r0K1(θ)ϕ1(θ)dθ,ϕ=(ϕ0,ϕ1)∈𝒳,
where K0∈ℒ(X,U) and K1∈L2([-r,0];ℒ(X,U)).We will show three results which are important in the subsequent finite pole assignability.Proposition 5.2.
For arbitraryλ∈ℂ1, the following relations are equivalent:(i)
Im(λ-𝒜)+Imℳ=𝒳;(ii)
ImΔ(λ)+ImM=X.Proof.
Relation (i) holds if and only if, for anyψ∈𝒳, there exist ϕ=(ϕ0,ϕ1)∈𝒟(𝒜) with ϕ0=ϕ1(0)∈𝒟(A) and u∈U such that
(5.5)(λ-𝒜)ϕ+ℳu=ψ.
This is equivalent, in view of Proposition 2.3, to
(5.6)λϕ1(0)-Aϕ1(0)-∫-r0dζ(θ)ϕ1′(θ)-∫-r0dη(θ)ϕ1(θ)+Mu=ψ0,ϕ1(0)∈𝒟(A),(5.7)λϕ1(θ)-ddθϕ1(θ)=ψ1(θ),θ∈[-r,0].
We solve the differential equation (5.7) to obtain
(5.8)ϕ1(θ)=eλθϕ1(0)+∫θ0eλ(θ-τ)ψ1(τ)dτ,θ∈[-r,0].
Substituting (5.8) into (5.6) and using Fubini's theorem, we have
(5.9)Δ(λ)ϕ1(0)+Mu=ψ0+∫-r0dη(θ)∫θ0eλ(θ-τ)ψ1(τ)dτ+∫-r0dζ(θ)ψ1(θ)+∫-r0dζ(θ)∫θ0λeλ(θ-τ)ψ1(τ)dτ,ϕ1(0)∈𝒟(A).
Also, condition (ii) holds if and only if, for any φ0∈X, there exist κ0∈𝒟(A) and u∈U such that
(5.10)Δ(λ)κ0+Mu=φ0.
Assume that (i) holds and let φ0∈X be an arbitrarily given vector. If we put ψ=(ψ0,ψ1)=(φ0,0)∈𝒳, then by virtue of (5.9) there exist ϕ=(ϕ1(0),ϕ1)∈𝒟(𝒜) and u∈U such that
(5.11)Δ(λ)ϕ1(0)+Mu=φ0,ϕ=(ϕ1(0),eλ·ϕ1(0)).
By setting κ0=ϕ1(0), we have (5.10) so that (ii) is valid. Next, we will show the implication (ii) ⇒ (i). To this end, assume that (ii) is valid and let ψ=(ψ0,ψ1)∈𝒳. If we put
(5.12)φ0=ψ0+∫-r0dη(θ)∫θ0eλ(θ-τ)ψ1(τ)dτ+∫-r0dζ(θ)∫θ0λeλ(θ-τ)ψ1(τ)dτ+∫-r0dζ(θ)ψ1(θ),
then by virtue of (5.10) we have
(5.13)Δ(λ)κ0+Mu=φ0
for some κ0∈𝒟(A) and u∈U. For such a vector κ0, we define ϕ1(θ) by
(5.14)ϕ1(θ)=eλθκ0+∫θ0eλ(θ-τ)ψ1(τ)dτ,θ∈[-r,0].
Then the function ϕ1(θ) satisfies ϕ1(0)=κ0∈𝒟(A),ϕ=(ϕ1(0),ϕ1(·))∈𝒟(𝒜) satisfies (5.6) and (5.7), and relation (i) is therefore proved to be valid.Proposition 5.3.
Forλ∈ℂ1, the following relations are equivalent:(i)
Ker(λ¯-𝒜*)∩Kerℳ*={0};(ii)
KerΔ*(λ¯)∩KerM*={0};(iii)
Im(λ-𝒜)+Imℳ¯=𝒳, that is, Im(λ-𝒜)+Imℳ is dense in 𝒳(iv)
ImΔ(λ)+ImM¯=X, that is, ImΔ(λ)+ImM is dense in X.Proof.
We first show the equivalence of (i) and (ii). By the very definitions of adjoint operators and operatorℳ, we have that, for any ϕ*=(ϕ0*,ϕ1*)∈𝒳* and u∈U,
(5.15)〈u,M*ϕ0*〉U,U*=〈Mu,ϕ0*〉X,X*,〈u,ℳ*ϕ*〉U,U*=〈ℳu,ϕ*〉𝒳,𝒳*=〈(Mu,0),(ϕ0*,ϕ1*)〉𝒳,𝒳*=〈Mu,ϕ0*〉X,X*.
Thus, the condition ϕ*∈Kerℳ* is equivalent to the condition ϕ0*∈KerM*. Now assume that (i) holds and let ϕ0*∈KerΔ*(λ¯)∩KerM*. If we set
(5.16)ϕ*=(ϕ0*,ϕ1*)=(ϕ0*,-B0*(·)+∫-r·eλ¯(θ-·)dη*(θ)ϕ0*+∫-r·λ¯eλ¯(θ-·)B0*(θ)dθϕ0*)∈𝒳*,
then, by virtue of Proposition 3.1, we have ϕ*∈Ker(λ¯-𝒜*)∩Kerℳ*, so that, by (i), ϕ*=(ϕ0*,ϕ1*)=0 and thus ϕ0*=0. This proves the implication (i) ⇒ (ii). To show the converse implication, suppose that (ii) is true and let ϕ*=(ϕ0*,ϕ1*)∈Ker(λ¯-𝒜*)∩Kerℳ*. Then again by virtue of Proposition 3.1, we have that ϕ0*∈KerΔ*(λ¯),
(5.17)ϕ*=(ϕ0*,-B0*(·)+∫-r·eλ¯(θ-·)dη*(θ)ϕ0*+∫-r·λ¯eλ¯(θ-·)B0*(θ)dθϕ0*),
and ϕ0*∈KerM*; hence ϕ0*=0 in view of (ii). Then ϕ*=0, and thus relation (i) is shown to be true.Now we show the equivalence of (i) and (iii). Define the closed operator [λ-𝒜,ℳ]:𝒟(𝒜)×U⊂𝒳×U→𝒳 by
(5.18)[λ-𝒜,ℳ](ϕ,u)=(λ-𝒜)ϕ+ℳu,(ϕ,u)∈𝒟(𝒜)×U.
Here 𝒳×U is a complex Banach space equipped with the norm ∥(ϕ,u)∥𝒳×U=∥ϕ∥𝒳+∥u∥U for any (ϕ,u)∈𝒳×U. Then by the duality theorem, condition (iii) is equivalent to Ker[λ-𝒜,ℳ]*={0}. By calculating the adjoint operator that involves duality pairings, we can readily verify that the adjoint [λ-𝒜,ℳ]*:𝒳*→𝒳*×U* is given by
(5.19)[λ-𝒜,ℳ]*ϕ*=((λ¯I-𝒜*)ϕ*,ℳ*ϕ*),ϕ*∈𝒳*.
It then follows from (5.19) that Ker[λ-𝒜,ℳ]*={0} if and only if
(5.20)Ker(λ¯-𝒜*)∩Kerℳ*={0}.
This proves the desired equivalence of (i) and (iii). We note here that the adjoint operator Δ(λ)* is given by Δ*(λ¯). Then the equivalence of (ii) and (iv) can be shown as in the proof of the equivalence of (i) and (iii). Hence the proof is complete.Given arbitrarily sets
(5.21)Λ0={λ1,…,λl}⊂σd(Δ),Λ1={μ1,…,μl}⊂ℂl,
let U=ℂN and the controller M:ℂN→X be defined by
(5.22)Mu=∑i=1Nukbk,u=(u1,…,uN)∈ℂN,bk∈X,k=1,…,N.
For λ∈σd(Δ), it is clear that λ¯∈σd(𝒜*)=σd(Δ*) and we can thus denote the basis of the kernel KerΔ*(λ¯) by {φλj*}j=1dλ, where dλ=dimKerΔ*(λ¯).Proposition 5.4.
Assume thatM is given by (5.22). For any λ∈σd(Δ), the following conditions are equivalent:(i)
KerΔ*(λ¯)∩KerM*={0};(ii)
Rank(〈bk,φλj*〉X,X*:k=1,…,N,j=1,…,dλ)=dλ.Proof.
First we note thatKerM* is given by the orthogonal complement
(5.23)KerM*=(ImM)⊥={bk:1≤k≤N}⊥.
To show the implication (i) ⇒ (ii), let us suppose contrarily that the rank condition (ii) is not satisfied. Then there exists a nonzero vector z=(z1,…,zdλ)∈ℂdλ such that
(5.24)∑j=1dλzj〈bk,φλj*〉X,X*=0,k=1,…,N.
If we set φ*=∑j=1dλz¯jφλj*, then φ*∈KerΔ*(λ¯) is nonzero and
(5.25)〈bk,φ*〉X,X*=〈bk,∑j=1dλz¯jφλj*〉X,X*=∑j=1dλzj〈bk,φλj*〉X,X*=0,k=1,…,N.
Thus φ*∈{bk:1≤k≤N}⊥=KerM*. This implies that (i) does not hold. Next we will show the converse implication (ii) ⇒ (i). Assume that (ii) is valid and let φ*∈KerΔ*(λ¯)∩{bk:1≤k≤N}⊥. Suppose that φ* is represented as φ*=∑j=1dλzjφλj*, zj∈ℂ1, by use of the basis {φλj*} of KerΔ*(λ¯), and the condition 〈bk,φ*〉X,X*=0, k=1,…,N, are written as
(5.26)(0,…,0)T=(〈b1,∑j=1dλ‍zjφλj*〉X,X*,…,〈bN,∑j=1dλ‍zjφλj*〉X,X*)T=(∑j=1dλ‍z¯j〈b1,φλj*〉X,X*,…,∑j=1dλ‍z¯j〈bN,φλj*〉X,X*)T=Bλ(z¯1,…,z¯dλ)T,inℂN,
where Bλ=(〈bk,φλj*〉X,X* : k=1,…,N,j=1,…,dλ). Here (·)T means the transpose operation of matrices. So the rank condition (ii) implies that z¯j=zj=0, j=1,…,dλ. Thus φ*=0, which obviously shows (i).We can summarize the previous results in the following form.Theorem 5.5.
Assume thatM is given by (5.22). For any λ∈σd(Δ), the following relations (i)– (v) are equivalent:(i)
Im(λ-𝒜)+Imℳ=𝒳;(ii)
ImΔ(λ)+ImM=X;(iii)
Ker(λ¯-𝒜*)∩Kerℳ*={0};(iv)
KerΔ*(λ¯)∩KerM*={0};(v)
Rank(〈bk,φλj*〉X,X*:k=1,…,N,j=1,…,dλ)=dλ.Proof.
SinceM is given by (5.22), ImM is finite dimensional. Whereas dimPλ𝒳=dimℳλ is finite by λ∈σd(𝒜), from Theorem 5.28 by Kato in [12], the operator λ-𝒜 is Fredholm and hence by Lemma1.9 by Kato in [12], the sum Im(λ-𝒜)+Imℳ is closed. It is also clear that ImΔ(λ) is closed and so is ImΔ(λ)+ImM for λ∈σd(Δ). Then the equivalences (i)–(v) follow from Propositions 5.2, 5.3 and 5.4.For a finite setΛ0={λ1,…,λl}⊂σd(Δ), there exists, by assumption, a rectifiable Jordan curve γΛ0 which surrounds Λ0 and separates Λ0 and ℂ1∖Λ0. If we denote by PΛ0 the projection on Λ0, then we can decompose the space 𝒳 as(5.27)𝒳=𝒩Λ0⊕ℛΛ0,
where(5.28)𝒩Λ0=PΛ0𝒳,ℛΛ0=(I-PΛ0)𝒳.
As Λ0⊂σd(Δ), each λi∈Λ0 is a pole of (z-𝒜)-1 and the subspace 𝒩Λ0 is finite dimensional by Proposition 4.2. For the mappings ℳ and 𝒜, we define the operators 𝒜Λ0 and ℳΛ0 by(5.29)𝒜Λ0=PΛ0𝒜,ℳΛ0=PΛ0ℳ.
Since the operators 𝒜Λ0 and ℳΛ0 are bounded and linear in the finite-dimensional space 𝒩Λ0, the exponential operator et𝒜Λ0=𝒮Λ0(t) is well defined on 𝒩Λ0. Let ϕ∈𝒳 and u∈L2(ℝ+;ℂN). We introduce the following finite-dimensional control system on ℳΛ0 by(5.30)dY0(t)dt=𝒜Λ0Y0(t)+ℳΛ0u(t),t≥0,Y0(0)=PΛ0ϕ∈𝒩Λ0.
In view of the study by Wonham in [14], the dual observed system of (5.30) on the adjoint space (𝒩Λ0)*⊂𝒳* is given by(5.31)dZ0(t)dt=𝒜Λ0*Z0(t),t≥0,Z0(0)=(PΛ0)*ϕ*∈(𝒩Λ0)*=𝒩Λ¯0*,Y0(t)=ℳΛ0*Z0(t),t≥0,
where (PΛ0)*=PΛ¯0*,Λ¯0={λ¯1,…,λ¯l} is given by PΛ¯0*=12πi∫γΛ0∨(z-𝒜*)-1dz,γΛ0∨being mirror image of γΛ0, 𝒩Λ¯0*=PΛ¯0*𝒳*,AΛ0*=(PΛ0A)*=PΛ¯0*𝒜* is a bounded linear operator on𝒩Λ¯0*,ℳΛ0*∈ℒ(𝒳*,ℂN) is given by ℳΛ0*ψ*=ℳ*PΛ¯0*ψ*,ψ*∈𝒳*. And We denote by 𝒮Λ0*(t)the exponential operator generated by𝒜Λ0*. It is obvious that 𝒮Λ0*(t)=𝒮*(t)PΛ¯0*.In finite-dimensional control theory it is well known (cf., Wonham [14]) that the finite-dimensional control system (5.30) is controllable; that is,(5.32)⋃t>0{∫0t𝒮Λ0(t-s)ℳΛ0u(s)ds:u∈L2([0,t];ℂN)}=𝒩Λ0
if and only if the observed system (5.31) is observable; that is,(5.33)ℳΛ0*𝒮Λ0*(t)ψ*=0,t≥0,ψ*∈𝒩Λ¯0*impliesthatψ*=0.
The space 𝒩Λ0 is decomposed as𝒩Λ0=𝒩λ1⊕⋯⊕𝒩λl (direct sum) so that we have the similar direct sum 𝒩Λ¯0*=𝒩λ¯1*⊕⋯⊕𝒩λ¯l*. The projection PΛ¯0* is also decomposed as PΛ¯0*=Pλ¯1*+⋯+Pλ¯l*, and hence(5.34)ℳΛ0*𝒮Λ0*(t)ψ*=ℳ*𝒮*(t)PΛ¯0*ψ*=ℳ*𝒮*(t)Pλ¯1*ψ*+⋯+ℳ*𝒮*(t)Pλ¯l*ψ*.
Therefore, condition (5.33) is equivalent to the statement(5.35)ℳ*𝒮*(t)ψ*=0,t≥0,ψ*∈𝒩λ¯i*impliesthatψ*=0
for each i=1,…,l.Since the eigenvalueλ¯i of 𝒜* is a pole of (z-𝒜*)-1 with order kλi, we have(5.36)𝒮*(t)Pλ¯i*=12πi∫γλ¯ietz(z-𝒜*)-1dz=etλ¯i·12πi∫γλ¯iet(z-λ¯i)(z-𝒜*)-1dz=etλ¯i∑j=0kλi-1tjj!·12πi∫γλ¯i(z-λ¯i)j(z-𝒜*)-1dz=etλ¯i(Pλ¯i*+∑j=1kλi-1tjj!(Qλ¯i*)j).
Hence, forψ*∈𝒩λ¯i*the equality(5.37)ℳ*𝒮*(t)ψ*=0,t≥0,
is equivalent to(5.38)ℳ*Pλ¯i*ψ*=0,ℳ*(Qλ¯i*)jψ*=0,j=1,…,kλi-1.
Recall that, by virtue of Proposition 4.2, we have(5.39)di=dimKer(λi-𝒜)=dimKer(λ¯i-𝒜*)=dimKerΔ*(λ¯i)<∞.
We denote the basis of KerΔ*(λ¯i) by{φi1*,…,φidi*}⊂𝒟(A*). Then again by Proposition 4.2, the basis of Ker(λ¯i-𝒜*)is given by(5.40){(φi1*,Fλ¯iφi1*),…,(φidi*,Fλ¯iφidi*)}⊂𝒳*.We setΦij*=(φij*,Fλ¯iφidi*), then(5.41)〈bk,φij*〉X,X*=〈(bk,0),Φij*〉𝒳,𝒳*
holds for each k=1,…,N and j=1,…,di. Indeed, we have(5.42)〈(bk,0),Φij*〉𝒳,𝒳*=〈(bk,0),(φij*,Fλ¯iφidi*)〉𝒳,𝒳*=〈bk,φij*〉X,X*.In order to prove the finite pole assignability theorem we need the following proposition on the rank condition.Proposition 5.6.
Assume thatM is given by (5.22). Then the following two statements are equivalent (i)
The equalities(5.43)ℳ*Pλ¯i*ψ*=0,ℳ*(Qλ¯i*)jψ*=0,j=1,…,kλi,ψ*∈𝒩λ¯i*,
imply that  ψ*=0.(ii)
The rank condition(5.44)Rank(〈bk,φij*〉X,X*:k=1,…,N,j=1,…,di)=di
holds.Proof.
SinceM is given by (5.22), it follows from standard calculations that the adjoint operator M0*∈ℒ(X*,ℂN) is given by
(5.45)M0*ψ0*=(〈b1,ψ0*〉¯,…,〈bN,ψ0*〉¯),ψ0*∈X*,
and ℳ*∈ℒ(𝒳*,ℂN) is given by ℳ*ψ*=M0*ψ0*for ψ*=(ψ0*,ψ1*)∈𝒳*. Hence, the equalities (5.43) can be rewritten as
(5.46)〈bk,[Pλ¯i*ψ*]0〉X,X*=〈(bk,0),Pλ¯i*ψ*〉𝒳,𝒳*=0,k=1,…,N,(5.47)〈bk,[(Qλ¯i*)jψ*]0〉X,X*=〈(bk,0),(Qλ¯i*)jψ*〉𝒳,𝒳*=0,k=1,…,N,j=1,…,kλi.
We first show the implication (i) ⇒ (ii). Suppose to the contrary that the rank condition (5.44), or equivalently  by the rank condition  (5.41),
(5.48)Rank(〈(bk,0),Φij*〉𝒳,𝒳*:k=1,…,N,j=1,…,di)=di
is not satisfied. Then there exists a nonzero vector z=(z1,…,zdi)∈ℂN such that
(5.49)∑j=1dizj〈(bk,0),Φij*〉𝒳,𝒳*=〈(bk,0),∑j=1diz¯jΦij*〉𝒳,𝒳*=0,k=1,…,N.
If we set Φ*=∑j=1diz¯jΦij*, then Φ*∈Ker(λ¯i-𝒜*)⊂𝒩λ¯i*is nonzero and by (5.49)
(5.50)〈(bk,0),Φ*〉𝒳,𝒳*=0,k=1,…,N.
Since Pλ¯i*Φ*=Φ*, equalities (5.46) hold owing to (5.50). The relation Ker(λ¯i-𝒜*)=𝒩λ¯i*∩KerQλ¯i*(cf., (4.6)) yields Qλ¯i*Φ*=0, and hence (Qλ¯i*)jΦ*=0  for eachj=1,2,…. This implies equality (5.47). Hence (i) does not hold, which is a contradiction.Next we show the converse implication (ii)⇒ (i). Let ψ*∈𝒩λ¯i*. Assume that the rank condition (5.44), or equivalently (5.48), and equalities (5.46) hold. Since (Qλ¯i*)kλi=O andImQλ¯i*⊂ℳλ¯i*by (4.5), then φ1*≡(Qλ¯i*)kλi-1ψ*∈KerQλ¯i*, so that φ1*∈Ker(λ¯i-𝒜*) by (4.6). Thenφ1* is written as
(5.51)φ1*=∑j=1dizjΦij*,zj∈ℂ1,j=1,…,di.
Hence, it follows from (5.41), (5.45), and the last equality in (5.47) that
(5.52)ℳ*(Qλ¯i*)kλi-1ψ*=M*[∑j=1dizjΦij*]0=(〈(b1,0),∑j=1dizjΦij*〉𝒳,𝒳*¯,…,〈(bN,0),∑j=1dizjΦij*〉𝒳,𝒳*¯)=(〈b1,∑j=1dizjφij*〉X,X*¯,…,〈bN,∑j=1dizjφij*〉X,X*¯)=(0,…,0)∈ℂN,
and thus
(5.53)Bi(z1,…,zdi)T=(0,…,0)T,
where Bi=(〈bk,φij*〉X,X*) and ·T denotes the transpose of the vector. Since the rank condition (5.44) is satisfied, (5.53) implies that z1=⋯=zdi=0. That is, φ1*=0. Hence φ2*≡(Qλ¯i*)kλi-2ψ* is an element of KerQλ¯i*, so that φ2*∈Ker(λ¯i-A*)by (4.6). We can repeat this procedure via (5.44) to obtain
(5.54)φ2*=0,(Qλ¯i*)kλi-3ψ*=0,…,Qλ¯i*ψ*=0,Pλ¯i*ψ*=ψ*=0.
Therefore (i) is shown.Recall notation (5.21), and for each λi, i=1,…,l, let {φij*}j=1di⊂X* be the basis of the null space KerΔ*(λ¯), where di=dimKerΔ*(λ¯). We may further obtain the following result by virtue of Proposition 5.6.Theorem 5.7.
Assume thatM is given by (5.22) in system (5.2). Let a finite set Λ0={λ1,…,λl}⊂σd(Δ) be given. For each λi∈Λ0, let Bi, i=1,…,l, be N×di matrices given by
(5.55)Bi=(〈bk,φij*〉X,X*:k=1,…N,j=1,…,di).
Then the control system (5.2) is pole assignable with respect to (Λ0,Λ1) for any finite set Λ1={μ1,…,μl} in ℂ1 if and only if the rank conditions
(5.56)RankBi=di,for  eachi=1,…,l,
are satisfied.Proof.
Given thatΛ1={μ1,…,μl}⊂ℂ1, by Theorem 5.5 and Proposition 5.6 we have the equivalences of (5.32), (5.33), (5.35), (5.43), (5.44) and (5.56). Condition (5.32) means that the finite-dimensional control system (5.30) on 𝒩Λ0 is controllable. Then by Wonham's pole assignment theorem [14], (5.30) on𝒩Λ0 is controllable if and only if there exists a linear operator K0∈ℒ(𝒩Λ0,ℂN) such that
(5.57)σ(AΛ0+BΛ0K0)=Λ1,on𝒩Λ0.
Define the operator K∈ℒ(𝒳,ℂN) by
(5.58)Kϕ={K0ϕ,ϕ∈𝒩Λ0,0,ϕ∈ℛΛ0.
It is clear that ℳK=O on ℛΛ0 and ℳK=ℳΛ0K0 on 𝒩Λ0. Hence 𝒜+ℳK=𝒜 on ℛΛ0 and 𝒜+ℳK=𝒜Λ0+ℳΛ0K0 on ℳΛ0, which implies that σ(𝒜+ℳK)=σ(𝒜)∖Λ0 on ℛΛ0 and σ(𝒜+ℳK)=Λ1 on 𝒩Λ0 by (5.57). Thus, we obtain the conclusion (5.3) by the direct sum decomposition (4.10). This completes the proof of the theorem.



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*Source: 948764-2010-08-05.xml*