Abstract and Applied Analysis

Volume 2010 (2010), Article ID 948764, 27 pages

http://dx.doi.org/10.1155/2010/948764

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

^{1}Faculty of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China^{2}Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL, UK

Received 28 March 2010; Accepted 13 May 2010

Academic Editor: Viorel Barbu

Copyright © 2010 Xuewen Xia and Kai 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.

#### 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.

#### 1. Introduction

Consider the linear system on some Banach space
where is the infinitesimal generator of a -semigroup , . A * mild solution* of (1.1) is defined as for any . The null solution of (1.1) is said to be * (exponentially) stable* if, for any initial , the corresponding mild solution, is as . It may be shown that the null solution is stable if and only if there exist positive numbers , such that, for all , . If the null solution of system (1.1) is unstable,then it is important to consider stabilizability problem of its linear control system
where is a bounded linear operator from some Banach space of control parameters into . The mild solution of (1.2) is well defined for every locally integrable control function , , and is given by the form
System (1.2) is said to be * feedback (exponentially) stabilizable* if there exists a bounded linear operator from into such that the system
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 , the stability and stabilizability problem would become complicated. One of the most important situations is to perturb appropriately by a time-delay term so as that a strongly continuous family of bounded linear operators 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 where , generates a -semigroup , , and is the Stieltjes measures given by Here , , and , , are properly defined linear, bounded operators from into (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 where 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 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 , where denotes a -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 by and , , respectively. Also, denotes the set of all nonnegative numbers. For any , the symbols Re and Im denote the real and imaginary parts of complex number , respectively. Let and be complex, separable Banach spaces and , their adjoint spaces with norms and and the dual pairings and , respectively. We use to denote the space consisting of all bounded linear operators from into with domain . When , is denoted by . Each operator norm is simply denoted by when there is no danger of confusion. For any operator , we employ to denote the domain of , and the symbols and will be used to denote the kernel and image of operator , respectively. For any fixed constant , we denote by the space of all -valued equivalence classes of measurable functions which are squarely integrable on . Let denote the Banach space with the norm Let denote the Sobolev space of -valued functions on such that and its distributional derivative belong to .

#### 2. Neutral Control Systems

Consider the following neutral linear functional differential equation on the Banach space : where generates a -semigroup , , , , and , are the Stieltjes measures given by Here , , , , and . Unless otherwise specified, we always use , , to denote the bounded, linear extension of the mapping for any , and the same remark applies to in an obvious way.

We also wish to consider the hereditary neutral controlled system of (2.1) on :
where . A * mild solution * of (2.4) is defined as the unique solution of the following integral equation on the Banach space ,
with , , and .

To ensure the uniqueness and existence of mild solutions, we further assume that, for each , , and , Im such that . Under these conditions, it has been shown in the study by Liu in [9] that there exists a unique mild solution for (2.4) with and .

Note that, for any , the mild solution is continuous for . To see this, it suffices to notice that, for any , , We define a mapping on , , by where for any . It turns out that , , is a strongly continuous semigroup on .

Proposition 2.1. *For any and , the following relation holds:
**
That is,
**
Moreover, is a -semigroup of bounded linear operators on .*

*Proof. *The linearity of is obvious. Strong continuity of on follows from the fact that in as by virtue of (2.5) and (2.6), and on the other hand, it is easy to see that in as . In order to show the semigroup property (2.8), let and
Then from (2.5), it is easy to verify that
On the other hand, we have for that
Thus, by the uniqueness of solutions of (2.5) with , it implies that
Hence, for all , and so in . The semigroup property (2.8) is thus proved and the proof is complete.

Let be the infinitesimal generator of and denote simply by . The next theorem explicitly describes the operator .

Theorem 2.2. *The infinitesimal generator of is described by
**
for any .*

*Proof. *We denote by and the infinitesimal generator of and its domain, respectively. Let and
Since the second coordinate of is the -shift , it follows immediately that
where denotes the right-hand derivative. By redefining on the set of measure zero, we can suppose that is absolutely continuous from to by Theorem , p. 19, of [10]. Since , this implies that and is strongly continuous. Then the functions and are strongly continuous in such that
Applying (2.17) to the first coordinate of (2.15), we obtain that
Hence, exists in ; that is, , and
which shows that
Next we will show the reverse inclusion. Let ; then it is easy to see that for any , from which (2.17) follow. Combining this with we see that
Noting that
for , we obtain by using Hölder inequality that
This implies that exists in and equals . Therefore, we prove that and for , and (2.14) are shown.

For each , define the densely defined, closed linear operator by
The *neutral resolvent set * is defined as the set of all values in for which the operator has a bounded inverse on

Proposition 2.3. *For any , the relation
**
is equivalent to
**
We solve the differential equation (5.7) to obtain
**
Substituting (5.8) into (5.6) and using Fubini's theorem, we have
**
Also, condition (ii) holds if and only if, for any , there exist and such that
**
Assume that (i) holds and let be an arbitrarily given vector. If we put , then by virtue of (5.9) there exist and such that
**
By setting , 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 . If we put
**
then by virtue of (5.10) we have
**
for some and . For such a vector , we define by
**
Then the function satisfies satisfies (5.6) and (5.7), and relation (i) is therefore proved to be valid.*

Proposition 5.3. *For , the following relations are equivalent:*(i)*;*(ii)*;*(iii)*, that is, is dense in *(iv)*, that is, is dense in .*

*Proof. *We first show the equivalence of (i) and (ii). By the very definitions of adjoint operators and operator , we have that, for any and ,
Thus, the condition is equivalent to the condition . Now assume that (i) holds and let . If we set
then, by virtue of Proposition 3.1, we have , so that, by (i), and thus . This proves the implication (i) (ii). To show the converse implication, suppose that (ii) is true and let . Then again by virtue of Proposition 3.1, we have that ,
and ; hence in view of (ii). Then , and thus relation (i) is shown to be true.Now we show the equivalence of (i) and (iii). Define the closed operator by
Here is a complex Banach space equipped with the norm for any . Then by the duality theorem, condition (iii) is equivalent to . By calculating the adjoint operator that involves duality pairings, we can readily verify that the adjoint is given by
It then follows from (5.19) that if and only if
This proves the desired equivalence of (i) and (iii). We note here that the adjoint operator is given by . 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
let and the controller be defined by
For , it is clear that and we can thus denote the basis of the kernel by , where .

Proposition 5.4. *Assume that is given by (5.22). For any , the following conditions are equivalent:*(i)*;*(ii)*.*

*Proof. *First we note that is given by the orthogonal complement
To show the implication (i) (ii), let us suppose contrarily that the rank condition (ii) is not satisfied. Then there exists a nonzero vector such that
If we set , then is nonzero and
Thus . This implies that (i) does not hold. Next we will show the converse implication (ii) (i). Assume that (ii) is valid and let . Suppose that is represented as , , by use of the basis of , and the condition , , are written as
where : . Here means the transpose operation of matrices. So the rank condition (ii) implies that , . Thus , which obviously shows (i).

We can summarize the previous results in the following form.

Theorem 5.5. *Assume that is given by (5.22). For any , the following relations (i)– (v) are equivalent:*(i)*;*(ii)*;*(iii)*;*(iv)*;*(v)*.*

*Proof. *Since is given by (5.22), is finite dimensional. Whereas is finite by , from Theorem by Kato in [12], the operator is Fredholm and hence by Lemma by Kato in [12], the sum is closed. It is also clear that is closed and so is Im for . Then the equivalences (i)–(v) follow from Propositions 5.2, 5.3 and 5.4.

For a finite set , there exists, by assumption, a rectifiable Jordan curve which surrounds and separates and . If we denote by the projection on , then we can decompose the space as where As , each is a pole of and the subspace is finite dimensional by Proposition 4.2. For the mappings and , we define the operators and by Since the operators and are bounded and linear in the finite-dimensional space , the exponential operator is well defined on . Let and . We introduce the following finite-dimensional control system on by In view of the study by Wonham in [14], the dual observed system of (5.30) on the adjoint space is given by where is given by being mirror image of , , is a bounded linear operator on, is given by . And We denote by the exponential operator generated by. It is obvious that .

In finite-dimensional control theory it is well known (cf., Wonham [14]) that the finite-dimensional control system (5.30) is controllable; that is, if and only if the observed system (5.31) is observable; that is, The space is decomposed as (direct sum) so that we have the similar direct sum . The projection is also decomposed as , and hence Therefore, condition (5.33) is equivalent to the statement for each .

Since the eigenvalue of is a pole of with order , we have Hence, forthe equality is equivalent to Recall that, by virtue of Proposition 4.2, we have We denote the basis of by. Then again by Proposition 4.2, the basis of is given by

We set , then holds for each and . Indeed, we have

In order to prove the finite pole assignability theorem we need the following proposition on the rank condition.

Proposition 5.6. *Assume that is given by (5.22). Then the following two statements are equivalent *(i)*The equalities
imply that .*(ii)*The rank condition**
holds.*

*Proof. *Since is given by (5.22), it follows from standard calculations that the adjoint operator is given by
and is given by for . Hence, the equalities (5.43) can be rewritten as
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),
is not satisfied. Then there exists a nonzero vector such that
If we set , then is nonzero and by (5.49)
Since , equalities (5.46) hold owing to (5.50). The relation (cf., (4.6)) yields , and hence for each. This implies equality (5.47). Hence (i) does not hold, which is a contradiction.

Next we show the converse implication (ii) (i). Let . Assume that the rank condition (5.44), or equivalently (5.48), and equalities (5.46) hold. Since andby (4.5), then , so that by (4.6). Then is written as Hence, it follows from (5.41), (5.45), and the last equality in (5.47) that and thus where and denotes the transpose of the vector. Since the rank condition (5.44) is satisfied, (5.53) implies that . That is, . Hence is an element of , so that by (4.6). We can repeat this procedure via (5.44) to obtain Therefore (i) is shown.Recall notation (5.21), and for each , , let be the basis of the null space , where . We may further obtain the following result by virtue of Proposition 5.6.

Theorem 5.7. *Assume that is given by (5.22) in system (5.2). Let a finite set be given. For each , let , , be matrices given by
**
Then the control system (5.2) is pole assignable with respect to for any finite set in if and only if the rank conditions
**
are satisfied.*

*Proof. *Given that , 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 is controllable. Then by Wonham's pole assignment theorem [14], (5.30) on is controllable if and only if there exists a linear operator such that
Define the operator by
It is clear that on and on . Hence on and on , which implies that on and on 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.

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