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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2010, Article ID 948764, 27 pages doi:10.1155/2010/948764

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

Xuewen Xia1 and Kai Liu2

1 2

Faculty of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL, UK

Correspondence should be addressed to Xuewen Xia, xxw1234567@163.com Received 28 March 2010; Accepted 13 May 2010 Academic Editor: Viorel Barbu Copyright q 2010 X. Xia and K. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We will consider a pole assignment problem for a class of linear neutral functional di?erential 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 ?nite dimensional, we establish the rank conditions for ?nite pole assignability.

1. Introduction

Consider the linear system on some Banach space X dy t dt Ay t , t ≥ 0, y 0 x ∈ X, 1.1

where A is the in?nitesimal generator of a C0 -semigroup etA , t ≥ 0. A mild solution of 1.1 is de?ned as y t etA x ∈ 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 dy t dt Ay t Mu t , t ≥ 0, y 0 x ∈ X, 1.2

2

Abstract and Applied Analysis

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 de?ned for every locally integrable control function u t , t ≥ 0, and is given by the form

t 0

y t

etA x

e t?s A Mu s ds,

t ≥ 0.

1.3

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 dy t dt t ≥ 0, y 0 x ∈ X,

A

MK y t ,

1.4

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 into A, 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 dy t dt

0

Ay t

?r

dη θ y θ

t ,

t ≥ 0,

1.5

where r > 0, A generates a C0 -semigroup etA , t ≥ 0, and η is the Stieltjes measures given by

m i 1 0

η τ

?

χ ?∞,?ri τ Ai ?

A0 θ dθ,

τ

τ ∈ ?r, 0 .

1.6

Here Ai , i 1, . . . , m, and A0 θ , θ ∈ ?r, 0 , are properly de?ned linear, bounded operators from X into X cf., Wu 2 . To our knowledge, very little paper has been done on feedback stabilization of in?nite-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.

Abstract and Applied Analysis

3

In the present paper, we will study the ?nite pole assignability problem for a class of neutral linear control system d y t ? dt

0 ?r 0

dζ θ y θ

t y 0

Ay t φ0 ,

?r

dη θ y θ φ1 · ,

t

Mu t ,

t ≥ 0,

1.7

y0 ·

where φ φ0 , φ1 is some initial datum to be identi?ed later. Generally speaking, for neutral systems as above it is quite di?cult 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 ?nite-dimensional cases. Thus we shall study in this work a weaker concept, ?nite 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 ?ve 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 in?nite-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 de?ned by a solution state y t , yt , where yt denotes a t-segment of solutions, a situation which is di?erent from that in-Burns et al. in 8 . The in?nitesimal 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 ?nite dimensionality of spectral modes, we will establish useful criteria of ?nite pole assignability. The real and complex number vector spaces are denoted by Rn and Cn , n ≥ 1, respectively. Also, R denotes the set of all nonnegative numbers. For any λ ∈ C1 , the symbols Re λ and Im λ 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 Ker T and Im T will be used to denote the kernel and image of operator L2 ?r, 0 ; X the space of T , respectively. For any ?xed constant r > 0, we denote by L2 r all X -valued equivalence classes of measurable functions which are squarely integrable on ?r, 0 . Let X denote the Banach space X × L2 r with the norm φ

X

φ0

2 X

φ1

2 L2 r

,

?φ

φ0 , φ1 ∈ X.

1.8

Let W 1,2 ?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 .

4

Abstract and Applied Analysis

2. Neutral Control Systems

Consider the following neutral linear functional di?erential equation on the Banach space X : d dt y t ?

0 ?r 0

dζ θ y θ φ0 ,

t

Ay t φ1 · , φ

?r

dη θ y θ φ0 , φ1 ∈ X,

t,

t ≥ 0,

2.1

y 0

y0 ·

where A generates a C0 -semigroup etA , t ≥ 0, y0 θ measures given by η τ ? ?

m i 1 0

y θ , θ ∈ ?r, 0 , and η, ζ are the Stieltjes

χ ?∞,?ri τ Ai ?

0

A0 θ dθ,

τ

τ ∈ ?r, 0 , 2.2

ζ τ

B0 θ dθ,

τ

τ ∈ ?r, 0 .

Here 0 < r1 < r2 < · · · < rm ≤ r , Ai ∈ L X , i 1, . . . , m, A0 · ∈ L2 ?r, 0 ; L X , and 0 2 B0 · ∈ L ?r, 0 ; L X . Unless otherwise speci?ed, we always use ?r dη θ y t θ : y ∈ L2 ?r, T ; X → X , T ≥ 0, to denote the bounded, linear extension of the mapping

0 ?r m

dη θ y t

θ

i 1

Ai y t ? ri

0 ?r

A0 θ y t

θ dθ,

y ∈ C ?r, T ; X ,

2.3

for any t ≥ 0, and the same remark applies to ?r dζ 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 : d y t ? dt y 0

0 ?r 0

0

dζ θ y θ y0 ·

t

Ay t φ

?r

dη θ y θ

t

Mu t ,

t ≥ 0,

2.4

φ0 ,

φ1 · ,

φ0 , φ1 ∈ X,

u ∈ L2 0, ∞ ; U ,

where M ∈ L U, X . A mild solution y t, φ, u of 2.4 is de?ned as the unique solution of the following integral equation on the Banach space X ,

0

y t, φ, u

?r

dζ θ y t

t 0 t 0

θ, φ, u

0 ?r

etA φ0 ? θ, φ, u

0 ?r

dζ θ φ1 θ

0

e t?s A

dη θ y s

?r

A dζ θ y s

θ, φ, u

ds

2.5

e t?s A Mu s ds,

?t > 0, φ0 , φ1 ∈ X.

with y 0, φ, u

φ0 , y0 ·, φ, u

φ1 · , and φ

Abstract and Applied Analysis

5

To ensure the uniqueness and existence of mild solutions, we further assume that, for each i, 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 φ1 · . mild solution y t, φ, u for 2.4 with y 0, φ, u φ0 and y0 ·, φ, u Note that, for any φ ∈ X, the mild solution y t, φ is continuous for t > 0. To see this, it su?ces to notice that, for any s, t > 0,

0 ?r

dζ θ y θ ≤ B0

s, φ dθ ?

0 ?r

dζ θ y θ

t, φ dθ

x

2.6

L2 ?r,0 ;L X

ys · ? yt ·

L2 ?r,0 ;X

.

We de?ne a mapping S t on X, t ≥ 0, by

S t φ

y t, φ , yt ·, φ ,

t ≥ 0,

2.7

y t where yt ·, φ semigroup on X.

·, φ for any t ≥ 0. It turns out that S t , t ≥ 0, is a strongly continuous

Proposition 2.1. For any t ≥ s ≥ 0 and φ ∈ X, the following relation holds:

S t ? s y s, φ , ys ·, φ

y t, φ , yt ·, φ .

2.8

That is,

S t?s S s φ

S t φ.

2.9

Moreover, S t is a C0 -semigroup of bounded linear operators on X. Proof. The linearity of S 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

Φ s

S s φ

y s, φ , ys ·, φ

∈ X.

2.10

6 Then from 2.5 , it is easy to verify that y t ? s, Φ s ?

0 ?r

Abstract and Applied Analysis

dζ θ y t ? s

0 ?r

θ, Φ s s, φ θ, Φ s

0 ?r

e t?s A y s, φ ?

t s

dζ θ y θ

e t?u A

0 ?r

dη θ y u ? s

0 ?r

A dζ θ y u ? s

θ, Φ s

du

e t?s A esA φ0 ?

s 0 t s

dζ θ φ1 θ

0 ?r 0

e s?u A

dη θ y u

θ, φ θ, Φ s

0 ?r

?r

A dζ θ y u

0 ?r

θ, φ

du θ, Φ s

e t?u A

0 ?r

0 ?r

dη θ y u ? s

s

A dζ θ y u ? s

0

du

etA φ0 ?

t s

dζ θ φ1 θ

0 0 ?r

e t?u A

dη θ y u

0

θ, φ

?r

A dζ θ y u

θ, φ

du

e t?u A

dη θ y u ? s

θ, Φ s

?r

A dζ θ y u ? s

θ, Φ s

du. 2.11

On the other hand, we have for t ≥ s that y t, φ ?

0 ?r

dζ θ y t

0 ?r

θ, φ

etA φ0 ?

s 0 t s

dζ θ φ1 θ

0 ?r 0 ?r 0

2.12 A dζ θ y u θ, φ du

e t?u A e t?u A

dη θ y u

θ, φ

?r 0

dη θ y u

θ, φ

?r

A dζ θ y u

θ, φ

du.

Thus, by the uniqueness of solutions of 2.5 with M y t ? s, Φ s y t, φ ,

0, it implies that 2.13

1

for almost all t ≥ 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 ? r, 0 ; X . The semigroup property 2.8 is thus proved and the proof is complete. L2 r

in

Abstract and Applied Analysis

7

Let A be the in?nitesimal generator of S t and denote S t simply by etA . The next theorem explicitly describes the operator A. Theorem 2.2. The in?nitesimal generator A of etA is described by

D A Aφ

φ Aφ0

φ0 , φ1 ∈ X : φ1 ∈ W 1,2 ?r, 0 ; X , φ0

0 ?r 0

φ1 0 ∈ D A ,

, 2.14

dζ θ φ1 θ

?r

dη θ φ1 θ , φ1 θ

for any φ

φ0 , φ1 ∈ D A .

Proof. We denote by A and D A the in?nitesimal generator of etA and its domain, respectively. Let φ φ0 , φ1 ∈ D A and

Aφ

ψ0 , ψ1 .

2.15

Since the second coordinate of etA φ is the t-shift y t

· , it follows immediately that

y θ 7pt d y θ dθ

φ1 θ ∈ W 1,2 ?r, 0 ; X , φ1 θ ψ1 θ ,

θ ∈ ?r, 0 , 2.16

in L2 ?r, 0 ; X , θ ∈ ?r, 0 ,

where d /dθ denotes the right-hand derivative. By rede?ning 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. φ0 and y · : ?r, ∞ → X is strongly 19, of 10 . Since y 0 φ0 , this implies that φ1 0 0 t continuous. Then the functions ?r dη θ y t θ and 0 A dζ θ y t θ are strongly continuous in t ≥ 0 such that

t→0

lim

1 t

t 0 t 0

e t?s A e t?s A

0 ?r 0 ?r

0

dη θ y s Aζ θ y s

θ ds θ ds

?r 0 ?r

dη θ φ1 θ , 2.17 A dζ θ φ1 θ .

1 lim t→0 t

8

Abstract and Applied Analysis

Applying 2.17 to the ?rst coordinate of 2.15 , we obtain that ψ0 lim lim 1 y t, φ ? φ0 t 1 t

t 0 0 ?r

t→0

t→0

dζ θ y t

0 ?r 0 ?r

θ, φ

etA φ0 ?

0

0 ?r

dζ θ φ1 θ ds ? φ0

e t?s A 1 t

0 ?r

dη θ y s

θ, φ

0 ?r

?r

A dζ θ y s

θ, φ

t→0

lim

dζ θ y t

θ, φ ?

0 ?r

dζ θ φ1 θ

? etA

t 0 0 ?r 0 ?r 0 ?r

dζ θ φ1 θ ?

0 ?r 0 ?r

dζ θ φ1 θ

0

e t?s A

dη θ y s

θ, φ

0

?r

A dζ θ y s

θ, φ

ds

etA φ0 ? φ0

dζ θ φ1 θ ? A dζ θ φ1 θ dζ θ φ1 θ

A dζ θ φ1 θ 1 tA e φ0 ? φ0 t

?r

dη θ φ1 θ

t→0 0 ?r

lim

dη θ φ1 θ

t→0

lim

1 tA e φ0 ? φ0 . t 2.18

Hence, limt → 0 t?1 etA φ0 ? φ0 exists in X ; that is, φ0 ∈ D A , and

0 0

ψ0

Aφ0

?r

dζ θ φ1 θ

?r

dη θ φ1 θ ,

2.19

which shows that D A ?D A , Aφ Aφ, for φ ∈ D A . 2.20

Next we will show the reverse inclusion. Let φ φ0 , φ1 ∈ D A ; then it is easy to see that y ·, φ ∈ W 1,2 ?r, T ; X for any T > 0, from which 2.17 follow. Combining this with φ0 φ1 0 ∈ D A , we see that 1 y t, φ ? φ0 t

0 0

t→0

lim

Aφ0

?r

dζ θ φ1 θ

?r

dη θ φ1 θ .

2.21

Abstract and Applied Analysis Noting that yt θ, φ ? φ1 θ ? φ1 θ t y t 1 t θ, φ ? y θ, φ ? y θ, φ t

t 0

9

y s

θ, φ ? y θ, φ ds

2.22

for θ ∈ ?r, 0 , we obtain by using Holder inequality that ¨ 1 yt ·, φ ? φ1 ? φ1 t

2 L2 r

≤

1 t

t 0

0 ?r

y s

θ, φ ? y θ, φ

2 X dθ

ds.

2.23

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 A and Aφ Aφ for φ ∈ D A , and 2.14 are shown. For each λ ∈ C1 , de?ne the densely de?ned, closed linear operator Δ λ, A, η, ζ by

0 ?r 0 ?r

Δ λ, A, η, ζ

λI ? A ?

eλθ dη θ ?

λeλθ dζ θ .

2.24

The neutral resolvent set ρ A, η, ζ is de?ned 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 any λ ∈ C1 , the relation λI ? A φ1 0 , φ1 ψ0 , ψ1 ∈ X, 2.25

is equivalent to

0 θ

φ1 θ Δ λ, A, η, ζ φ1 0

eλθ φ1 0

0

eλ θ?τ ψ1 τ dτ ∈ W 1,2 ?r, 0 ; X ,

0

2.26 eλ θ?τ ψ1 τ dτ 2.27

ψ0 ?

0 ?r

?r

dη θ

θ

eλ θ?τ ψ1 τ dτ

0 ?r

0

λ dζ θ

θ

dζ θ ψ1 θ dθ.

In particular, the resolvent set ρ A is equal to ρ A, η, ζ .

10 Proof. Note that the relation 2.25 is equivalent to

0 ?r

Abstract and Applied Analysis

ψ0

λφ1 0 ? Aφ1 0 ? ψ1 θ

dη θ φ1 θ ?

0 ?r

dζ θ φ1 θ ,

2.28 2.29

λφ1 θ ?

dφ1 θ , dθ

θ ∈ ?r, 0 .

The variation of constants formula for the ordinary di?erential equation 2.29 on ?r, 0 shows that eλθ φ1 0

0 θ

φ1 θ

eλ θ?τ ψ1 τ dτ ∈ W 1,2 ?r, 0 ; X .

2.30

In order to show 2.27 , note that, from 2.28 , we have

0 0

λI ? A φ1 0

ψ0

?r

dζ θ φ1 θ

?r

dη θ φ1 θ .

2.31

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 speci?ed, we always assume that X is re?exive. As indicated in the study by Hale in 11 , the adjoint theory of neutral linear functional di?erential 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 . ? ? ψ0 , ψ1 ∈ X? and de?ne a “formal” transposed neutral system of 2.1 on X ? Let ψ ? by d y t ? dt

0 ?r 0 ?r

dζ? θ y t

θ y 0

A? y t

? ψ0 ,

dη? θ y t

? ψ1

θ ,

t > 0,

3.1

y0 t

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 , de?ne Δ λ, A? , η? , ζ? λI ? A? ?

0 ?r

eλθ dη? θ ?

0 ?r

λeλθ dζ? θ .

3.2

Abstract and Applied Analysis Proposition 3.1. For any λ ∈ C1 , the equation

11

λI ? A?

? ? , φ1 φ0

? ? , ψ1 · ψ0

3.3

is equivalent to

? Δ λ, A? , η? , ζ? φ0 θ ?r θ ?r

? ψ0

0 ?r

? eλθ ψ1 θ dθ, θ ?r

3.4

? ? τ dτφ0 λeλ τ ?θ B0

? θ φ1

? ? ? eλ τ ?θ ψ1 τ dτ ? B0 θ φ0

? eλ τ ?θ dη? τ φ0

3.5

almost everywhere for any ψ ?

? ? ψ0 , ψ1 ∈ X? .

Proof. Note that λI ? A? ?1 λI ? A ?1 ? , so we may calculate the adjoint operator of ?1 ψ0 , ψ1 ∈ X, λI ? A . In view of 2.26 , it is not di?cult to see that, for any ψ

ψ0 , ψ1 , λI ? A λI ? A

? φ1 0 , ψ0 ?1

?1 ?

? ? , ψ1 ψ0

X,X?

? ? , ψ1 ψ0 , ψ1 , ψ0 0

X,X? θ 0 ? eλ θ?τ ψ1 τ dτ, ψ1 θ X,X ? 0 ?r θ 0

X,X ?

?r 0

eλθ φ1 0 ?

dθ

? φ1 0 , ψ0

X,X ? 0 ?r

?r

? eλθ φ1 0 , ψ1 θ 0 X,X ?

X,X

dθ ? ?

? eλ θ?τ ψ1 τ , ψ1 θ

X,X ?

dτ dθ

? φ1 0 , ψ0

? eλθ ψ1 θ dθ

0 θ

?r

? eλ θ?τ ψ1 τ , ψ1 θ

X,X ?

dτ dθ. 3.6

We reformulate the expression in 3.6 , starting with the last term:

0 ?r

0 θ

? eλ θ?τ ψ1 τ , ψ1 θ

0 X,X

τ ?r

dτ dθ ?

?r 0 ?r

? eλ θ?τ ψ1 τ , ψ1 θ τ

X,X ?

dθ dτ 3.7 dτ.

ψ1 τ ,

?r

e

λ θ ?τ

? ψ1

θ dθ

X,X ?

12 Letting κ? yield that

? ψ0 0 ?r

Abstract and Applied Analysis

? eλθ ψ1 θ dθ ∈ X ? and applying Proposition 2.3 to such a φ

φ1 0 , φ1

? φ1 0 , ψ0

0 ?r

? eλθ ψ1 θ dθ X,X ? X,X ? ?1

φ1 0 , κ?

Δ λ, A, η, ζ

ψ0 ?

0 ?r

0 ?r

0

0

B0 θ ψ1 θ dθ

0

?r

B0 τ

τ

λeλ τ ?θ ψ1 θ dθdτ

dη τ

τ

eλ τ ?θ ψ1 θ dθ , κ?

X,X ? 0 ?r ? ψ1 θ , B0 θ ?1 ? ?

ψ0 , Δ λ, A, η, ζ

0 ?r θ

?1 ? ?

κ

X,X ?

?

Δ λ, A, η, ζ

κ

X,X ?

dθ dθ.

ψ1 θ ,

?r

eλ τ ?θ dη? τ

? λeλ τ ?θ B0 τ dτ

Δ λ, A, η, ζ

?1 ? ?

κ

X,X ?

3.8

? If we combine these equalities and use the fact that φ0 ?1 ? ? ? , ψ1 ψ0

Δ λ, A, η, ζ

? ?1 ?

κ , then we obtain

ψ0 , ψ1 ,

λI ? A

X,X?

? ψ0 , φ0 0 ?r

X,X ?

?

0 ?r θ

? ? θ φ0 ψ1 θ , B0

X,X ?

dθ

θ ?r ? eλ τ ?θ ψ1 τ dτ X,X ?

ψ1 θ ,

?r

eλ τ ?θ dη? τ

? ? λeλ τ ?θ B0 τ dτ φ0

dθ, 3.9

and this proves the desired result. The proof is now complete. The following corollary which characterizes the in?nitesimal generator A? of the semigroup S? t on X? is a direct result of Proposition 3.1. Corollary 3.2. The in?nitesimal generator A? of S? t is given by D A? φ?

? ? ? ? φ0 , φ1 ∈ D A? , φ1 θ ∈ X? : φ0 θ ?r ? ? ? dη? τ φ0 ? B0 ? ?? θ , θ φ0

a.e. θ ∈ ?r, 0

where ?? · ∈ W 1,2 ?r, 0 ; X ?

, 3.10

Abstract and Applied Analysis and moreover A ? φ?

? A? φ0 ? φ1 0 ? B0 0 ? , φ0

13

d dθ

θ ?r

? ? ? ? dη? τ φ0 ? φ1 θ ? B0 θ φ0

,

?

3.11

φ

? 0 where φ1 ? B0 0 ? ? φ0 is given by the limit limθ → 0 φ1 θ

?

? ? φ0 , φ1

∈D A ,

? ? B0 θ φ0 .

Proof. Let λ

0 in 3.4 and 3.5 ,then it follows that ?A? ?

? φ1 θ θ θ ?r 0 ?r

dη? θ

? φ0 θ ?r

? ψ0

0 ?r

? ψ1 θ dθ,

3.12 3.13

? ? ? ψ1 τ dτ ? B0 θ φ0

? dη? τ φ0 ,

a.e. θ ∈ ?r, 0 .

? ? ? ? in 3.13 is left continuous at θ 0, then the sum φ1 θ B0 θ φ0 is Since the term ?r dη? τ φ0 ? ? ? 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 ? φ1 0 ? B0 0 ? φ0 0 ?r ? ψ1 τ dτ 0 ?r ? dη? τ φ0 .

3.14

Substituting 3.14 into 3.12 yield

? ?ψ0 ? A? φ0 ? φ1 0 ? B0 0 ? , φ0

3.15

and letting λ

0 in 3.5 and further taking derivative with respect to θ ∈ ?r, 0 yields

? ?ψ1 θ

d dθ

θ ?r

? ? ? ? dη? τ φ0 ? φ1 θ ? B0 θ φ0

3.16

from which the desired results are easily obtained. The proof is now complete. The adjoint neutral resolvent set ρ A? , η? , ζ? is de?ned similarly as the set of all values λ in C for which the operator Δ λ, A? , η? , ζ? has a bounded inverse on X ? . Then by applying ρ A? , η ? , ζ ? . the adjoint version of Proposition 3.1, we see that ρ A?

1

4. Spectral Properties

In this section we investigate the spectral properties of operators A 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 the neutral spectrum σ Δ λ, A, η, ζ , or simply σ Δ , is de?ned by σ Δ C1 \ ρ A, η, ζ . The spectrum σ 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

14

Abstract and Applied Analysis

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. De?ne the subset σd Δ of σP Δ by σd Δ {λ : λ ∈ σP Δ and dim Ker Δ λ is ?nite}. 4.1

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 recti?able arc C ? ρ Δ joining λ1 and λ2 . This condition implies that for any ?nite set Λ in σd Δ there exists a continuous recti?able arc CΛ ? ρ Δ which surrounds Λ inside and contains no other points in σ Δ . We also denote Δ λ, A? , η? , ζ? by Δ? λ and de?ne 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 of A (resp., A? ) satis?es that σP A σP Δ? ) and σd A σd Δ (resp., σd A? σd Δ? ). σP A? σP Δ (resp.,

Proof. Recall that, by Proposition 2.3, for any λ ∈ C1 the relation λI ? A φ ψ , φ ∈ D A , Gλ ψ , φ0 φ1 0 ∈ D A , where Gλ ψ is ψ ∈ X is equivalent to the relation Δ λ φ1 0 given by

0 0

Gλ ψ

ψ0

0 ?r

?r

dη θ

θ 0

eλ θ?τ ψ1 τ dτ

0 ?r

λdζ θ

θ 0 θ

λeλ θ?τ ψ1 τ dτ ?

dζ θ ψ1 θ dθ,

4.2

φ1 θ

eλθ φ1 0

eλ 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, σP Δ . It is easy to see that φ φ 0 , φ ∈ Ker λI ? A if and only if by de?nition, σP A φ 0 ∈ KerΔ λ and φ θ eλθ φ1 0 . By this equivalence it is easily seen that dim Ker Δ λ σd Δ . The other equalities σP A? σP Δ? and dim Ker λI ? A . This shows σd A ? ? σd Δ can be proved similarly. σd A In what follows we omit the symbol I 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 de?ned, respectively, by 1 2πi

?1

Pλ

z?A

γλ

dz,

Qλ

1 2π i

z?λ z?A

γλ

?1

dz,

4.3

Abstract and Applied Analysis

15

where γλ is a small circle with center λ such that its interior and γλ contain no points of σ A . Pλ X be the generalized eigenspace corresponding to the eigenvalue λ of A. It is Let Nλ obvious that Qλ

j

1 2πi

?1

z?λ

γλ

j

z?A

?1

dz,

j

1, 2, . . .·

4.4

Further, if λ is a pole of z ? A

of order kλ , then we have

kλ Qλ

O,

Im Qλ ? Nλ , Nλ ∩ Ker Qλ ,

4.5 4.6

Ker λ ? A

where O is the null operator, and the direct sum decompositions of X Nλ Ker λ ? A

kλ

,

X

Nλ ⊕ Im λ ? A

kλ

4.7

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 Ker λ ? A φ0 , eλ· φ0 : Δ λ φ0 0 . 4.8

Note that λ is a pole and each Ker λ ? A and Nλ are ?nite dimensional if λ ∈ σd Δ . Let Λ ? σd A be a ?nite set of isolated spectrum. Suppose that there exists a recti?able Jordan curve γΛ which surrounds Λ and separates Λ and C1 \ Λ. We de?ne the projection PΛ on Λ by PΛ 1 2πi z?A

γΛ ?1

dz.

4.9

Then the following decomposition of X holds: X where NΛ PΛ X , RΛ I ? PΛ X , 4.11 NΛ ⊕ RΛ , 4.10

and I is the identity operator on X. Now we introduce the bounded operator Fλ : X ? → L2 ?r, 0 ; X ? de?ned by

? Fλ φ0 θ ? θ ? B0 θ ?r

eλ τ ?θ dη? τ

θ ?r

? ? λeλ τ ?θ B0 , τ dτ φ0

a.e. θ ∈ ?r, 0 4.12

16

Abstract and Applied Analysis

for any λ ∈ C1 . Then by applying the adjoint version of Proposition 3.1, we have Ker λ ? A? where Δ? λ

? ? ? , Fλ φ0 : Δ? λ φ0 φ0

0 ,

4.13

Δ λ, A? , η? , ζ? and, if λ ∈ σd A? dim Ker λ ? A?

σd Δ? , then < ∞. 4.14

dim Ker Δ? λ

Let λ ∈ σ A be an isolated point. Then, by Kato 12 , λ ∈ σ A? is also isolated and the Pλ ? X ? P ? X? are well eigenspace Ker λ ? A? and the generalized eigenspace N? λ λ de?ned, and dim Ker λ ? A dim Ker λ ? A? ≤ ∞, dim Nλ dim N? ≤ ∞,

λ

4.15

where P ? is the projection

λ

P?

λ

1 2πi

∨ γλ

z ? A?

?1

dz

4.16

∨ and γλ is the mirror image of γλ in 4.3 . Hence, we have the following result.

Proposition 4.2. For each λ ∈ σd Δ , one has λ ∈ σd Δ? Ker λ ? A? dim Ker λ ? A? dim N?

λ

σd A? and , 4.17

? ? ? , Fλ φ0 ∈ Ker Δ? λ : φ0 φ0

dim Ker Δ? λ dim Nλ < ∞.

< ∞,

5. Pole Assignment

In general, for system 2.1 we have no ideas whether or not the associated generator A in Theorem 2.2 satis?es the spectral determined growth condition sup{Re λ : λ ∈ σ A } lim ln etA . t 5.1

t→∞

Consequently, it is di?cult 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 ?nite pole assignment problem for 2.4 on the product space X. We are concerned about the ?nite pole assignment problem of the control system 2.4 : under what conditions on M can we construct a feedback law such that any ?nite set in σd Δ is shifted to any preassigned set in the complex plane?

Abstract and Applied Analysis

17

To this end, ?rst note that, by means of A, we can reformulate system 2.4 into the space X as a control system without delay: dY t dt AY t Mu t , t ≥ 0, Y 0 φ φ0 , φ1 ∈ X, u ∈ L2 0, ∞ ; U ,

5.2

where M : U → X is de?ned by Mu

Mu, 0 , u ∈ U.

De?nition 5.1. Let Λ0 {λ1 , . . . , λl } ? σd Δ and Λ1 {μ1 , . . . , μl } be ?nite 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 σ A MK σ A \ Λ0 ∪ Λ1 . 5.3

We remark that the operator K has the form

0

Kφ

K0 φ0

?r

K1 θ φ1 θ dθ,

φ

φ0 , φ1 ∈ X,

5.4

where K0 ∈ L X, U and K1 ∈ L2 ?r, 0 ; L X, U . We will show three results which are important in the subsequent ?nite pole assignability. Proposition 5.2. For arbitrary λ ∈ C1 , the following relations are equivalent: i Im λ ? A ii Im Δ λ Im M Im M X; X.

Proof. Relation i holds if and only if, for any ψ ∈ X, there exist φ φ0 φ1 0 ∈ D A and u ∈ U such that λ?A φ This is equivalent, in view of Proposition 2.3, to λφ1 0 ? Aφ1 0 ?

0 ?r

φ0 , φ1 ∈ D A with

Mu

ψ.

5.5

dζ θ φ1 θ ?

0 ?r

dη θ φ1 θ

Mu

ψ0 ,

φ1 0 ∈ D A ,

5.6

λφ1 θ ?

d φ1 θ dθ

ψ1 θ ,

θ ∈ ?r, 0 .

5.7

18 We solve the di?erential equation 5.7 to obtain eλθ φ1 0

0 θ

Abstract and Applied Analysis

φ1 θ

eλ θ?τ ψ1 τ dτ,

θ ∈ ?r, 0 .

5.8

Substituting 5.8 into 5.6 and using Fubini’s theorem, we have Δ λ φ1 0

0 0

Mu

ψ0

0 ?r

?r

dη θ

θ 0

eλ θ?τ ψ1 τ dτ

λ θ ?τ

0 ?r

dζ θ ψ1 θ 5.9

dζ θ

θ

λe

ψ1 τ dτ,

φ1 0 ∈ D A .

Also, condition ii holds if and only if, for any ?0 ∈ X , there exist κ0 ∈ D A and u ∈ U such that Δ λ κ0 Mu ?0 . 5.10

Assume that i holds and let ?0 ∈ X be an arbitrarily given vector. If we put ψ ψ0 , ψ1 φ1 0 , φ1 ∈ D A and u ∈ U such that ?0 , 0 ∈ X, then by virtue of 5.9 there exist φ Δ λ φ1 0 Mu ?0 , φ φ1 0 , eλ· φ1 0 . 5.11

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

0 0

?0

ψ0

?r

dη θ

θ

eλ θ?τ ψ1 τ dτ

0 ?r

0

dζ θ

θ

λeλ θ?τ ψ1 τ dτ

0 ?r

dζ θ ψ1 θ , 5.12

then by virtue of 5.10 we have Δ λ κ0 Mu ?0 5.13

for some κ0 ∈ D A and u ∈ U. For such a vector κ0 , we de?ne φ1 θ by eλθ κ0

0 θ

φ1 θ

eλ θ?τ ψ1 τ dτ,

θ ∈ ?r, 0 .

5.14

κ0 ∈ D A , φ φ1 0 , φ1 · Then the function φ1 θ satis?es φ1 0 and 5.7 , and relation i is therefore proved to be valid.

∈ D A satis?es 5.6

Abstract and Applied Analysis Proposition 5.3. For λ ∈ C1 , the following relations are equivalent: i Ker λ ? A? ∩ Ker M? ii Ker Δ λ ∩ Ker M iii iv Im λ ? A Im Δ λ Im M

? ?

19

{0}; {0}; X, that is, Im λ ? A Im M is dense in X

Im M

X , that is, Im Δ λ

Im M is dense in X .

Proof. We ?rst show the equivalence of i and ii . By the very de?nitions of adjoint operators ? ? φ0 , φ1 ∈ X? and u ∈ U, and operator M, we have that, for any φ?

? u, M? φ0 U,U? ? Mu, φ0 X,X ? , X,X? ? Mu, φ0 X,X ? .

u, M? φ?

U,U?

Mu, φ?

X,X?

? ? Mu, 0 , φ0 , φ1

5.15

? Thus, the condition φ? ∈ Ker M? is equivalent to the condition φ0 ∈ Ker M? . Now assume ? that i holds and let φ0 ∈ Ker Δ? λ ∩ Ker M? . If we set

φ?

? ? φ0 , φ1

? ? , ? B0 · φ0

· ?r

? eλ θ?· dη? θ φ0

· ?r

? ? λeλ θ?· B0 θ dθφ0

∈ X? ,

5.16

then, by virtue of Proposition 3.1, we have φ? ∈ Ker λ ? A? ∩ Ker M? , so that, by i , φ? ? ? ? , φ1 0 and thus φ0 0. This proves the implication i ? ii . To show the converse φ0 ? ? implication, suppose that ii is true and let φ? φ0 , φ1 ∈ Ker λ ? A? ∩ Ker M? . Then ? again by virtue of Proposition 3.1, we have that φ0 ∈ Ker Δ? λ , φ?

? ? φ0 , ? B0 · · ?r ? eλ θ?· dη? θ φ0 · ?r ? ? λeλ θ?· B0 , θ dθφ0

5.17

? ? ∈ Ker M? ; hence φ0 0 in view of ii . Then φ? 0, and thus relation i is shown to and φ0 be true.Now we show the equivalence of i and iii . De?ne the closed operator λ ? A, M : D A × U ? X × U → X by

λ ? A, M φ, u

λ?A φ

Mu,

φ, u ∈ D A × U.

5.18

φ X u U Here X × U is a complex Banach space equipped with the norm φ, u X×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 λ ? A, M

? ?

φ

λI ? A? φ? , M? φ? ,

?

φ ? ∈ X? .

5.19

It then follows from 5.19 that Ker λ ? A, M

{0} if and only if {0}. 5.20

Ker λ ? A? ∩ Ker M?

20

Abstract and Applied Analysis

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

Λ0

{λ1 , . . . , λl } ? σd Δ ,

Λ1

μ1 , . . . , μl ? Cl ,

5.21

let U

CN and the controller M : CN → X be de?ned by

N

Mu

i 1

uk bk ,

u

u1 , . . . , uN ∈ CN , bk ∈ X, k

1, . . . , N.

5.22

σd Δ? and we can thus denote the basis of the For λ ∈ σd Δ , it is clear that λ ∈ σd A? ? dλ ? dim Ker Δ? λ . kernel Ker Δ λ by {?λj }j 1 , where dλ Proposition 5.4. Assume that M is given by 5.22 . For any λ ∈ σd Δ , the following conditions are equivalent: i Ker Δ? λ ∩ Ker M?

X,X ?

{0}; 1, . . . , dλ dλ .

ii Rank bk , ?? λj

:k

1, . . . , N, j

Proof. First we note that Ker M? is given by the orthogonal complement Ker M?

⊥

Im M

{bk : 1 ≤ k ≤ N }⊥ .

5.23

To show the implication i ? ii , let us suppose contrarily that the rank condition ii is not satis?ed. Then there exists a nonzero vector z z1 , . . . , zdλ ∈ Cdλ such that

dλ j 1

zj bk , ?? λj

X,X ?

0,

k

1, . . . , N.

5.24

If we set ??

dλ j 1

zj ?? , then ?? ∈ Ker Δ? λ is nonzero and λj

bk , ??

dλ X,X ?

bk ,

j 1

zj ?? λj

X,X ?

dλ j 1

zj bk , ?? λj

X,X ?

0,

k

1, . . . , N.

5.25

Abstract and Applied Analysis

21

Ker M? . This implies that i does not hold. Next we will show Thus ?? ∈ {bk : 1 ≤ k ≤ N }⊥ the converse implication ii ? i . Assume that ii is valid and let ?? ∈ Ker Δ? λ ∩ {bk : 1 ≤ dλ ? ? 1 k ≤ N }⊥ . Suppose that ?? is represented as ?? j 1 zj ?λj , zj ∈ C , by use of the basis {?λj } of Ker Δ? λ , and the condition bk , ?? ? 0, . . . , 0

T dλ j 1 X,X ?

0, k

1, . . . , N , are written as

dλ

?T zj ?? λj

X,X ?

? b1 , ? ?

zj ?? λj

X,X ?

,...,

bN ,

j 1

? ?T 5.26

dλ j 1

zj b1 , ?? λj

T

dλ X,X

,..., ?

j 1

zj bN , ?? λj

X,X ?

?

Bλ z1 , . . . , zdλ

,

in CN ,

where Bλ bk , ?? : k 1, . . . , N, j 1, . . . , dλ . Here · T means the transpose operation λj X,X ? of matrices. So the rank condition ii implies that zj zj 0, j 1, . . . , dλ . Thus ?? 0, which obviously shows i . We can summarize the previous results in the following form. Theorem 5.5. Assume that M is given by 5.22 . For any λ ∈ σd Δ , the following relations i – v are equivalent: i Im λ ? A ii Im Δ λ

?

Im M Im M

X; X; {0}; {0}; 1, . . . , N, j 1, . . . , dλ dλ .

?

iii Ker λ ? A? ∩ Ker M? iv Ker Δ λ ∩ Ker M v Rank bk , ?? λj

X,X ?

:k

Proof. Since M is given by 5.22 , Im M is ?nite dimensional. Whereas dim Pλ X dim Mλ is ?nite by λ ∈ σd A , from Theorem 5.28 by Kato in 12 , the operator λ ? A is Fredholm and hence by Lemma 1.9 by Kato in 12 , the sum Im λ ? A Im M is closed. It is also clear that Im Δ λ is closed and so is Im Δ λ Im M for λ ∈ σd Δ . Then the equivalences i – v follow from Propositions 5.2, 5.3 and 5.4. For a ?nite set Λ0 {λ1 , . . . , λl } ? σd Δ , there exists, by assumption, a recti?able 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 X where NΛ0 PΛ0 X, RΛ0 I ? PΛ0 X. 5.28 NΛ0 ⊕ RΛ0 , 5.27

22

Abstract and Applied Analysis

As Λ0 ? σd Δ , each λi ∈ Λ0 is a pole of z ? A ?1 and the subspace NΛ0 is ?nite dimensional by Proposition 4.2. For the mappings M and A, we de?ne the operators AΛ0 and MΛ0 by AΛ0 PΛ0 A, MΛ0 PΛ0 M. 5.29

Since the operators AΛ0 and MΛ0 are bounded and linear in the ?nite-dimensional space NΛ0 , the exponential operator etAΛ0 SΛ0 t is well de?ned on NΛ0 . Let φ ∈ X and u ∈ L2 R ; CN . We introduce the following ?nite-dimensional control system on MΛ0 by dY0 t dt AΛ0 Y0 t Y0 0 MΛ0 u t , t ≥ 0,

5.30

PΛ0 φ ∈ 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 dZ0 t ? AΛ Z0 t , t ≥ 0, 0 dt PΛ0 ? φ? ∈ NΛ0 ? N? , Z0 0

Λ0

5.31

Y0 t

?

M? Λ0 Z0 t ,

t ≥ 0, 1 ∨ z ? A? ?1 dz, γΛ being ∨ 0 2πi γΛ0 ? ? P A is a bounded linear operator

Λ0

where PΛ0

P ? , Λ0

Λ0 Λ0 N

{λ1 , . . . , λl } is given by P ? P ? X? , A? Λ0

Λ0

mirror image of γΛ0 , N?

PΛ0 A

?

Λ0

? ? is given by M? M? P ? ψ ? , ψ ? ∈ X? . And We denote by on N? , M? Λ0 ∈ L X , C Λ0 ψ Λ0 Λ0 ? ? ? SΛ0 t the exponential operator generated by AΛ0 . It is obvious that SΛ t S? t P ? . 0 Λ0 In ?nite-dimensional control theory it is well known cf., Wonham 14 that the ?nitedimensional control system 5.30 is controllable; that is,

t t>0 0

SΛ0 t ? s MΛ0 u s ds : u ∈ L2

0, t ; CN

NΛ0

5.32

if and only if the observed system 5.31 is observable; that is,

? ? M? Λ0 SΛ0 t ψ

0,

t ≥ 0, ψ ? ∈ N? implies that ψ ? Λ

0

0.

5.33

The space NΛ0 is decomposed as NΛ0 Nλ1 ⊕· · ·⊕Nλl direct sum so that we have the similar N? ⊕· · ·⊕N? . The projection P ? is also decomposed as P ? P? · · · P? , direct sum N? Λ0 Λ0 Λ0 λ1 λl λ1 λl and hence

? ? M? Λ0 SΛ0 t ψ ? M ? S ? t PΛ ψ?

0

M? S ? t P ? ψ ?

λ1

···

M? S ? t P ? ψ ? .

λl

5.34

Abstract and Applied Analysis Therefore, condition 5.33 is equivalent to the statement M? S ? t ψ ? 0, t ≥ 0, ψ ? ∈ N? implies that ψ ?

λi

23

0

5.35

for each i 1, . . . , l. Since the eigenvalue λi of A? is a pole of z ? A? S? t P ? 1 2πi etλi · etz z ? A?

γλ

i

?1

with order kλi , we have

?1

λi

dz

?1

1 2πi

et z?λi z ? A?

γλ

i

dz 5.36

etλi

kλ i ? 1 j j 0

1 t · j ! 2πi

kλ i ? 1 j j 1

z ? λi

γλ

i

j

z ? A?

?1

dz

? etλi ?P ?

λi

?

j

t Q? λi j!

?.

Hence, for ψ ? ∈ N? the equality

λi

M? S ? t ψ ? is equivalent to M? P ? ψ ?

λi

0,

t ≥ 0,

5.37

0,

M? Q ?

j

λi

ψ?

0,

j

1, . . . , kλi ? 1.

5.38

Recall that, by virtue of Proposition 4.2, we have di dim Ker λi ? A dim Ker λi ? A? dim Ker Δ? λi < ∞. 5.39

We denote the basis of Ker Δ? λi

the basis of Ker λi ? A? is given by

? ? by{?? i1 , . . . , ?idi } ? D A . Then again by Proposition 4.2,

? ? ? ?? i1 , Fλi ?i1 , . . . , ?idi , Fλi ?idi

? X? .

5.40

We set Φ? ij

? ?? ij , Fλi ?idi , then

bk , ?? ij

X,X ?

bk , 0 , Φ? ij

X,X?

5.41

24 holds for each k 1, . . . , N and j bk , 0 , Φ? ij

X,X?

Abstract and Applied Analysis 1, . . . , di . Indeed, we have bk , 0 ,

? ?? ij , Fλi ?idi X,X?

bk , ?? ij

X,X ? .

5.42

In order to prove the ?nite pole assignability theorem we need the following proposition on the rank condition. Proposition 5.6. Assume that M is given by 5.22 . Then the following two statements are equivalent i The equalities M? P ? ψ ?

λi

0,

M? Q ?

j

λi

ψ?

0,

j

1, . . . , kλi , ψ ? ∈ N? ,

λi

5.43

imply that ψ ?

0.

ii The rank condition Rank holds. Proof. Since M is given by 5.22 , it follows from standard calculations that the adjoint ? ∈ L X ? , CN is given by operator M0

? ? M0 ψ0 ? ? b1 , ψ0 , . . . , bN , ψ0 ? ? M0 ψ0 for ψ ?

bk , ?? ij

X,X ?

:k

1, . . . , N, j

1, . . . , di

di

5.44

,

? ψ? 0 ∈X ,

5.45

and M? ∈ L X? , CN is given by M? ψ ? 5.43 can be rewritten as bk , P ? ψ ?

λi

? ? ψ0 , ψ1 ∈ X? . Hence, the equalities

0 X,X ?

bk , 0 , P ? ψ ?

λi j

X,X?

0, k

k

1, . . . , N, 1, . . . , kλi .

5.46 5.47

bk ,

Q?

j

λi

ψ?

0

X,X ?

bk , 0 , Q?

λi

ψ?

X,X?

0,

1, . . . , N, j

We ?rst show the implication i ? ii . Suppose to the contrary that the rank condition 5.44 , or equivalently by the rank condition 5.41 , Rank bk , 0 , Φ? ij

X,X?

:k

1, . . . , N, j

1, . . . , di

di

5.48

is not satis?ed. Then there exists a nonzero vector z

di

z1 , . . . , zdi ∈ CN such that 0, k 1, . . . , N. 5.49

zj

j 1

bk , 0 , Φ? ij

di X,X?

bk , 0 ,

j 1

zj Φ? ij

X,X?

Abstract and Applied Analysis If we set Φ?

di j 1 ? ? zj Φ? ? N? is nonzero and by 5.49 ij , then Φ ∈ Ker λi ? A λi

25

bk , 0 , Φ?

X,X?

0,

k

1, . . . , N.

5.50

Since P ? Φ?

λi

Φ? , equalities 5.46 hold owing to 5.50 . The relation Ker λi ? A?

N? ∩

λi

Ker Q? cf., 4.6 yields Q? Φ? 0, and hence Q? j Φ? 0 for each j λi λi λi equality 5.47 . Hence i does not hold, which is a contradiction.

1, 2, . . .. This implies

Next we show the converse implication ii ? i . Let ψ ? ∈ N? . Assume that the rank condition 5.44 , or equivalently 5.48 , and equalities 5.46 hold. Since Q? and Im Q? ? M? by λi λi Then ?? is written as 1 4.5 , then ?? 1 ≡ Q ? kλ i ? 1 ψ ? λi ∈ Ker Q? , λi so that ?? 1 ∈ Ker λi ? A

λi kλ i λi ?

O

by 4.6 .

?? 1

di j 1

zj Φ? ij ,

zj ∈ C1 ,

j

1, . . . , di .

5.51

Hence, it follows from 5.41 , 5.45 , and the last equality in 5.47 that ? M? Q ?

kλ i ? 1 λi

ψ?

M? ? ? ? ?

di j 1

? ? zj Φ? ij

0 di di

? zj Φ? ij ?

X,X?

b1 , 0 ,

j 1 di j 1

zj Φ? ij

X,X?

,...,

bN , 0 ,

j 1 di

? 5.52

? b1 ,

zj ?? ij

X,X ?

,...,

bN ,

j 1

zj ?? ij

X,X ?

?

0, . . . , 0 ∈ CN ,

and thus

T

Bi z1 , . . . , zdi

0, . . . , 0 T ,

5.53

where Bi

bk , ?? ij

X,X ?

and ·T denotes the transpose of the vector. Since the rank condition ··· zdi 0. That is, ?? 1

? 0. Hence ?? 2 ≡ Q kλ i ? 2 λi

5.44 is satis?ed, 5.53 implies that z1

ψ?

26

Abstract and Applied Analysis

? is an element of Ker Q? , so that ?? by 4.6 . We can repeat this procedure via 2 ∈ Ker λi ? A λi 5.44 to obtain

?? 2

0,

Q?

kλ i ? 3

λi

ψ?

0, . . . ,

Q? ψ ?

λi

0,

P? ψ? λ

i

ψ?

0.

5.54

Therefore i is shown.Recall notation 5.21 , and for each λi , i the basis of the null space Ker Δ? λ , where di following result by virtue of Proposition 5.6.

di ? 1, . . . , l, let {?? ij }j 1 ? X be

dim Ker Δ? λ . We may further obtain the {λ1 , . . . , λl } ?

Theorem 5.7. Assume that M is given by 5.22 in system 5.2 . Let a ?nite set Λ0 σd Δ be given. For each λi ∈ Λ0 , let Bi , i 1, . . . , l, be N × di matrices given by Bi bk , ?? ij

X,X ?

:k

1, . . . N, j

1, . . . , di .

5.55

Then the control system 5.2 is pole assignable with respect to Λ0 , Λ1 for any ?nite set Λ1 {μ1 , . . . , μl } in C1 if and only if the rank conditions Rank Bi are satis?ed. {μ1 , . . . , μl } ? C1 , by Theorem 5.5 and Proposition 5.6 we have the Proof. Given that Λ1 equivalences of 5.32 , 5.33 , 5.35 , 5.43 , 5.44 and 5.56 . Condition 5.32 means that the ?nite-dimensional control system 5.30 on NΛ0 is controllable. Then by Wonham’s pole assignment theorem 14 , 5.30 on NΛ0 is controllable if and only if there exists a linear operator K0 ∈ L NΛ0 , CN such that σ AΛ0 De?ne the operator K ∈ L X, CN by ? ?K0 φ, ?0, φ ∈ NΛ0 , φ ∈ RΛ0 . BΛ0 K0 Λ1 , on NΛ0 . 5.57 di , for each i 1, . . . , l, 5.56

Kφ

5.58

It is clear that MK O on RΛ0 and MK MΛ0 K0 on NΛ0 . Hence A MK A on RΛ0 and σ A \ Λ0 on RΛ0 and A MK AΛ0 MΛ0 K0 on MΛ0 , which implies that σ A MK 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.

Acknowledgment

The authors like to express their sincere thanks for Professor S. Nakagiri’s constructive comments and suggestions which greatly improved the original manuscript.

Abstract and Applied Analysis

27

References

1 Z. Luo, B. Guo, and O. Morgul, Stability and Stabilization of In?nite Dimensional Systems with Applications, Springer, New York, NY, USA, 1999. 2 J. Wu, Theory and Applications of Partial Functional-Di?erential Equations, vol. 119 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996. 3 M. Yamamoto, “On the stabilization of evolution equations by feedback with time-delay: an operatortheoretical approach,” Journal of the Faculty of Science. University of Tokyo. Section IA, vol. 34, no. 1, pp. 165–191, 1987. 4 S. Nakagiri and M. Yamamoto, “Feedback stabilization of linear retarded systems in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 262, no. 1, pp. 160–178, 2001. 5 G. Da Prato and A. Lunardi, “Stabilizability of integrodi?erential parabolic equations,” Journal of Integral Equations and Applications, vol. 2, no. 2, pp. 281–304, 1990. 6 J. Jeong, “Stabilizability of retarded functional-di?erential equation in Hilbert space,” Osaka Journal of Mathematics, vol. 28, no. 2, pp. 347–365, 1991. 7 D. Salamon, Control and Observation of Neutral Systems, vol. 91 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1984. 8 J. Burns, T. Herdman, and H. Stech, “Linear functional-di?erential equations as semigroups on product spaces,” SIAM Journal on Mathematical Analysis, vol. 14, no. 1, pp. 98–116, 1983. 9 K. Liu, “The fundamental solution and its role in the optimal control of in?nite dimensional neutral systems,” Applied Mathematics and Optimization, vol. 60, no. 1, pp. 1–38, 2009. 10 V. Barbu, Nonlinear Semigroups and Di?erential Equations in Banach Spaces, Noordho?, Leyden, The Netherlands, 1976. 11 J. Hale, Functional Di?erential Equations, Springer, New York, NY, USA, 1971. 12 T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 2nd edition, 1976. 13 H. Tanabe, Functional Analytic Methods for Partial Di?erential Equations, vol. 204 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1997. 14 W. M. Wonham, Linear Multivariable Control, vol. 10, Springer, New York, NY, USA, 3rd edition, 1985.

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