Abstract and Applied Analysis
Volume 2018 (2018), Article ID 4210135, 10 pages
https://doi.org/10.1155/2018/4210135

Research Article

Controllability and Observability of Nonautonomous Riesz-Spectral Systems

Sutrima Sutrima,1,2 Christiana Rini Indrati,² and Lina Aryati²

¹Mathematics Department, Universitas Sebelas Maret, Surakarta, Indonesia
²Mathematics Department, Universitas Gadjah Mada, Yogyakarta, Indonesia

Correspondence should be addressed to Sutrima Sutrima; zutrima@yahoo.co.id

Received 8 February 2018; Accepted 5 April 2018; Published 15 May 2018

Academic Editor: Khalil Ezzinbi

Copyright © 2018 Sutrima Sutrima et al. 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.

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a -quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.

1. Introduction

In the real problems, many underlying transport-reaction phenomena are described by partial differential equations with the time-varying coefficients. The phenomena arise in processes such as crystal growth, metal casting and annealing, solid-gas reaction systems (see [1–3]), and heat conduction of a material undergoing decay or radioactive damage [4]. The others also arise in solid-fluid mechanics and biological systems. The time-dependencies of the system parameters can be caused by changes in the boundary of domain and variances in the diffusion characteristics. The transport-reaction phenomena encourage the emergence of nonautonomous linear control systems.

Let , , and be complex Hilbert spaces. Suppose that and are bounded operators such that and , where denotes the space of bounded operators from to equipped with strong operator topology. We consider the linear nonautonomous control systems on with state , input , and output :

where is an unknown function from real interval into and is a linear closed operator in with domain , independent of and dense in . We denote the state linear system (1)-(2) by . To avoid clutter, we also use notations and if and , respectively.

There is an extensive amount of literatures which have studied controllability for the system (1). Barcenas and Leiva [5] prove some properties of attainable sets for the systems (1) with time-varying constrained controls and target sets. They also characterize the extremal controls and give necessary and sufficient conditions for the normality of the system. Elharfi et al. [6] study well-posedness of a class of nonautonomous neutral control systems in Banach spaces. The systems are represented by absolutely regular nonautonomous linear systems in the sense of Schnaubelt [7]. These works can be considered as the nonautonomous version of the works of Bounit and Hadd [8]. By employing skew-product semiflow technique, Barcenas et al. [9] give necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite-dimensional nonautonomous control systems (1). Ng et al. [10] characterize the some pertinent aspects regarding the controllability and observability of system (1)-(2) which are modelled by parabolic partial differential equations with time-varying coefficients. By using theory of linear evolution system and Schauder fixed point theorem, Fu and Zhang [11] establish a sufficient result of exact null controllability for a nonautonomous functional evolution system with nonlocal conditions. Using evolution operators and concept of Lebesgue extensions, Hadd [12] proposes a new approach which brings nonautonomous linear systems with state, input, and output delays in line with the standard theory. Leiva and Barcenas [13] have established a quasi-semigroup theory as an alternative approach in solving (1). Even the control theory can be developed by this approach although it is still limited to the time-invariant controls [14]. In this context, is an infinitesimal generator of a -quasi-semigroup on . Finally, the advanced properties and some types of stabilities of the -quasi-semigroups in Banach spaces can be determined by Sutrima et al. [15] and Sutrima et al. [16], respectively. These results are important in analysis and applications of the -quasi-semigroups.

In the autonomous case, that is, , , and , independent of , there are many literatures which have been devoted to study of the controllability and observability for the system of (1)-(2). Dolecki and Russell [17] explore the duality relationships between observation and control in an abstract Banach space setting. Investigation is also given to the problem of optimal reconstruction of system states from observations. Zhao and Weiss [18] establish the well-posedness, regularity, exact (approximate) controllability, and exact (approximate) observability results for the coupled systems consisting of a well-posed and regular subsystem and a finite-dimensional subsystem connected in feedback. For neutral type linear systems in Hilbert spaces, Rabah et al. [19] prove that exact null controllability and complete stabilizability are equivalent. The paper also considers the case when the feedback is not bounded. In particular, if is a Riesz-spectral generator of a -semigroup on , then the solution of (1) for can be expressed as an infinite sum of all its eigenvectors which form a Riesz basis (see [20, 21]), and in this case the system is called a Riesz-spectral system. It gives convenience to analyze some problems in infinite-dimensional systems such as spectrum-determined growth condition, controllability, observability, stabilizability, and detectability; see, for example, [22, 23].

Although the aforementioned researches provide a well-established theoretical basis on the nonautonomous Cauchy problems and the controllability and observability theory, there are a relatively scarce number of the researches using quasi-semigroups. Even, there is no research which investigates the Riesz-spectral systems on Hilbert space for the nonautonomous cases. These are challenges to study and to realize the associated control problems, the controllability, and observability, for the nonautonomous infinite-dimensional systems.

In this paper, we are concerned with investigation of sufficient conditions for to induce a nonautonomous Riesz-spectral system. The obtained nonautonomous operator is implemented to study the controllability and observability for the nonautonomous systems. All the studies use the -quasi-semigroup approach. The organization of this paper is as follows. In Section 2, we provide notion of the generalized Riesz-spectral operator and its sufficiency related to the nonautonomous systems. The concepts of controllability and observability for the nonautonomous systems are considered in the Section 3. In Section 4, we confirm the obtained results by the two examples.

2. Generalized Riesz-Spectral Generator

This section is a part of the main results. We first recall the definition of a strongly continuous quasi-semigroups following [13, 14].

Definition 1. Let be the set of all bounded linear operators on Hilbert space . A two-parameter commutative family in is called a strongly continuous quasi-semigroup, in short -quasi-semigroup, on if, for each and ,

(a) , identity operator on ,(b) ,(c) ,(d) there exists a continuous increasing function such that

In the sequel, for simplicity we denote the quasi-semigroup and family by and , respectively.

In this section we investigate sufficient conditions of such that (1) forms a nonautonomous Riesz-spectral system. It is well known that if is a Riesz-spectral operator, then it can be represented as an infinite sum of all its eigenvectors. However, as declared in Section 1 for nonautonomous system (1), we assume that is independent of . This implies that to be a Riesz-spectral operator, has to have eigenvectors which are independent of . A class that meets this criterion is a family of operators whose representation is as follows:

where is a Riesz-spectral operator on and is a bounded continuous function such that , . It is clear that, for every , and have the common domain and eigenvectors. Moreover, if , , is an eigenvalue of , then are the eigenvalues of of (4). Hence, in general may have the nonsimple eigenvalues. In case is a differential operator, then the operator of [10] satisfying the conditions P1 and P2 verifies (4). These urge the following notion.

Definition 2. For every , let be an operator of form (4) on a Hilbert space . is called a generalized Riesz-spectral operator if is a Riesz-spectral operator.

Definition 2 states that if is a nonnegative constant function, then is a Riesz-spectral operator. In the sequel we always assume that, for every , is an operator of form (4). The following results are generalization of the results of [21, 22] for autonomous case.

Theorem 3. For every , let be an operator of (4) where is a Riesz-spectral operator with simple eigenvalues and corresponding eigenvectors . If are the eigenvectors of , the adjoint of , such that , then

(a) , , and for , the resolvent operator is given by

(b) has representation

 for , where

(c) if , then, for every , is the infinitesimal generator of a -quasi-semigroup given by

 where ;(d) the growth bound of the quasi-semigroup at is given by

Proof. Proofs of (a) and (b) follow the proofs of Theorem of [21] replacing and with

respectively, for every . In this context .
(c) Let . Given fixed, for such that Re, from (a)

and by iteration we have

So by the condition b of Lemma 2.3.2 of [21] for and , we have

Theorem 3.7 of [15] implies that is an infinitesimal generator of a -quasi-semigroup with

We verify that the operators , , given by

where and , are a -quasi-semigroup on satisfying (15) with the infinitesimal generator

on domain

(d) By (10) we have

On the other hand, taking in (16) we get

It implies

Therefore

Corollary 4. If, for every , is the generalized Riesz-spectral generator of a -quasi-semigroup on a Hilbert space , then for any and the initial value problem

admits a unique solution.

Proof. It follows from Theorem 2.2 of [13] that (23) admits a unique solution.

3. Nonautonomous Riesz-Spectral Systems

In this section we shall apply the generalized Riesz-spectral operator in the linear nonautonomous control system of (1)-(2), where is the generalized Riesz-spectral operator generating a -quasi-semigroup on . In the sequel, we assume that the two requested real numbers and always satisfy

and , unless specified.

Definition 5. Assume that the state linear system holds for all initial state and for all input . The state

is defined to be a mild solution of (1).

We verify that and the output defined by (2) always belongs to . The definitions of the controllability and observability in this paper follow the definitions for the autonomous case; see, for example, [21].

Definition 6. The linear system is said to be

(a)exactly controllable on if for each there exists a control such that the mild solution of (1) corresponding to satisfies ;(b)approximately controllable on if for each and any there exists a control such that the mild solution of (1) corresponding to satisfies .

A controllability map of on is a bounded linear map defined by

It is easy to show that the system is exactly controllable on if and only if , where denotes the range of . Also, system is approximately controllable on if and only if .

Lemma 7. The controllability map in (26) satisfies the following conditions.

(a)The operator and for .(b)( on .

Proof. (a) Since is strongly continuous and , then the map is measurable on for every . Moreover,

Lemma A.5.5 of [21] states that the integral in (26) is well-defined. We verify easily that is linear. Now, for we have

This shows that is a bounded mapping from to and , , is a bounded mapping from to
(b) The definition of adjoint operator shows that is bounded. Moreover,

This proves that on

Theorem 8. For , the system is exactly controllable on if and only if any one of the following conditions holds for some and all :

(a) .(b) and is closed.

Proof. (a) We set , so . It is enough to prove that . By similarity of adjoint and dual operator in Hilbert space, Corollary 3.5 of [20] states

if and only if there exists such that

for all . So, by condition (b) of Lemma 7 the assertion is confirmed.
(b) The condition shows that is injective, and so . Next, let be a Cauchy sequence in . Condition (a) shows that is a Cauchy sequence in . However, Lemma 7 (a) forces for some . Thus, has a closed range.

Theorem 9. The linear system is approximately controllable on if and only if any one of the following conditions holds:

(a) , , implies (b) .

Proof. (a) We see that the system is approximately controllable on if and only if . According to Lemma VI 2.8 of [24], this is equivalent to the fact that the mapping is injective. The similarity between adjoint and dual operator gives

which implies almost everywhere. Therefore, if

this verifies that .
(b) Condition (a) and condition (b) of Lemma 7 give the desired result.

Complementary to Definition 6, we define the exact observability and the approximate observability as follows.

Definition 10. The linear system is said to be

(a) exactly observable on if the initial state can be uniquely constructed from the knowledge of the output in ;(b) approximately observable on if the knowledge of the output in determines the initial state uniquely.

The observability map of the system on is a bounded linear map defined by

for .

From Definition 10 and the definition of observability map we verify that the system is exactly observable on if and only if is injective and its inverse is bounded on . Also, is approximately observable on if and only if

Lemma 11. For the linear system one has the following duality:

(a)The linear system is approximately observable on if and only if the dual is approximately controllable on .(b)The linear system is exactly observable on if and only if the dual is exactly controllable on .

Proof. As a consequence of Proposition 1.2 and Theorem 1.6 of [14], if generates a -quasi-semigroup on a Hilbert space, then generates the -quasi-semigroup . Furthermore, we verify that

This implies that the range of equals that of the controllability map for the dual system . If denotes the controllability map of the dual system, then or .
(a) We see that is approximately observable on if and only if . Condition (b) of Theorem 9 implies that if and only if is approximately controllable on . This proves the equivalence.
(b) Suppose that is exactly observable on . There exists an inverse on and a constant such that

The exact controllability of follows from Theorem 8.
Conversely, assume that is exactly controllable on . Theorem 8 (a) gives that is injective and is closed. Since , then is injective and is closed. This states that is exactly observable on .

Theorems 8 and 9 and Lemma 11 yield the following conditions for observability.

Corollary 12. For the linear system , one has the following necessary and sufficient conditions for exact and approximate observability:

(a) is exactly observable on if and only if any one of the following conditions holds for some and all :

(i).(ii) and is closed.

(b) is approximately observable on if and only if any one of the following conditions holds:

(i), , implies .(ii).

In the infinite-dimensional system, it is generally easier to prove the approximate controllability and approximate observability than the exact controllability and exact observability. Next, we shall derive easily verifiable criteria for the approximate controllability and approximate observability of the generalized Riesz-spectral systems with finite-rank inputs and outputs.

Consider system (1)-(2) with finite-rank inputs and outputs

where is the generalized Riesz-spectral operator of (4), , , , , and , . The symbol denotes the transpose of . If we set , , then . In this case we have

Let be the Riesz-spectral operator with simple eigenvalues and corresponding eigenvectors . In addition, if are the eigenvectors of such that and , then according to the condition (c) of Theorem 3 is the infinitesimal generator of a -quasi-semigroup given by

where and is the form of (4). In this context we verify that

By Theorem 9, system (37) is approximately controllable on if and only if

implies that .

Next, we have

In virtue of Corollary 12, system (37)-(38) is approximately observable on if and only if

implies that .

These two facts deal with the following theorem which is a generalization of Theorem of [21] for the autonomous case.

Theorem 13. Consider the linear system of (37)-(38), where is a Riesz-spectral operator with simple eigenvalues such that and corresponding eigenvectors . Let be the eigenvectors of such that . Then

(a) is approximately controllable on if and only if for all

 for all ;(b) is approximately observable on if and only if for all

 for all .

Proof. (a) We consider the matrix :

on . By Lemma 3.14 of [20] and (42), is approximately controllable on if and only if for all

implies . Suppose that is not approximately controllable on , there exists an such that and

This gives for all and , and so rank .
Conversely, suppose that rank for some , then , for all and . So we can find a nonzero such that

Thus, (42) is satisfied for . This is equivalent to the fact that is not approximately controllable on .
(b) We can have similar proof to (a) for the matrix

on .

4. Nonautonomous Sturm-Liouville Systems

In this section we shall discuss nonautonomous Sturm-Liouville systems, the specifically nonautonomous Riesz-spectral systems. First let us recall the definition of Sturm-Liouville operators. In the sequel, we set to be the Hilbert space of . Consider an operator on its domain

where and . Operator is called a Sturm-Liouville operator if

for , where , and are real-valued continuous functions on such that and .

Since and are finite, the definition only corresponds to regular Sturm-Liouville problems. We verify that is a self-adjoint operator with real, countable, and simple eigenvalues such that (see [25, 26]).

We define a nonautonomous Sturm-Liouville operator to be an operator of form (4):

where is a Sturm-Liouville operator on its domain given by (52).

Definition 14. The state linear system of (1)-(2) is called a nonautonomous Sturm-Liouville system if is the negative of a nonautonomous Sturm-Liouville operator of form (54).

Corollary 15. For every , let be the negative of a nonautonomous Sturm-Liouville operator of the form (54) on its domain given by (52). Then

(a) is generalized Riesz-spectral operator;(b) is the infinitesimal generator of a -quasi-semigroup on .

Proof. (a) Lemma 1 of [27] gives the fact that is generalized Riesz-spectral operator.
(b) If is the set of eigenvalues of , then . Hence, Theorem 3 concludes that, for every , is the infinitesimal generator of a -quasi-semigroup on .

We note that Corollary 15 does not hold when is a nonautonomous Sturm-Liouville operator. Indeed, is a nonautonomous Sturm-Liouville operator, but it does not generate any -quasi-semigroup (see Section 3 [14]). Corollary 15 also concludes that any nonautonomous Sturm-Liouville system is the nonautonomous Riesz-spectral system. Therefore, all of the results of the controllability and observability in the previous section are applicable on the nonautonomous Sturm-Liouville systems.

5. Applications

In this section, we consider two examples of applications to confirm the results of the generalized Riesz-spectral operator in the nonautonomous systems.

Example 1. Consider the boundary condition problem of the PDE:

We are ready to show that the problem has a unique solution. Let be a Hilbert space of . Problem (55) can be written as

on , where , , and

on with

We verify that operator is not self-adjoint on . Furthermore, we obtain the eigenvalues and corresponding eigenvectors of as

respectively. It is obvious that every eigenvalue is simple and the set forms Riesz basis of . Moreover, is totally disconnected, that is, for , .
The adjoint of is

on . The eigenvalues and corresponding eigenvectors of are

for all , , and satisfy

Next, since the adjoint of any operator is always closed, then is closed. But in this case we have , so is closed. Thus, is a Riesz-spectral operator. In other words, , , is a generalized Riesz-spectral operator.
Since , condition (c) of Theorem 3 forces that is the infinitesimal generator of a -quasi-semigroup given by

Corollary 4 guarantees that for each problem (56) admits a unique solution

Thus, boundary condition problem (55) has a solution

Example 2. Consider the controlled wave equation

where is bounded uniformly continuous and is a distributed control.
We shall analyze the approximate controllability and approximate observability of the system. Problem (66) can be formulated as a linear system:

on the Hilbert space with the inner product

where and In this context , , , and for with

We verify that has the eigenvalues , and the corresponding Riesz basis of eigenvectors , where . We see that for every . Moreover, Example of [21] shows that is a Riesz-spectral operator on . Hence, is the generalized Riesz-spectral operator generating a -quasi-semigroup .
We assume that the system is controlled around the point . So, we may set

where is an indicator function. Theorem 13 shows that system (67) is approximately controllable on if and only if

Equation (71) demonstrates that the control points for which affect the loss of approximate controllability. This is also the case when , that is, at the zeros of on interval .
Next, consider the observation

where is an output function around the sensing point . Following Example of [21] we can reformulate the observation map as an inner product on :

Condition (b) of Theorem 13 gives the fact that is approximately observable on if and only if for all

We verify that

Therefore, the system is approximately observable on if and only if for all

This shows that the system loses the approximate observability at points for which for some or . Specially, for , where is the sampling frequency, then the system loses the approximate observability at sampling frequencies of , for discrete measurement.

Example 3. Consider the controlled nonautonomous heat equation on interval :

where are bounded uniformly continuous and is a distributed control.
We shall analyze the approximate controllability and approximate observability of the system. Let be a Hilbert space of . System (77) can be formulated as a nonautonomous Sturm-Liouville system

on , where , , and is a Sturm-Liouville operator (53) with , and on its domain :

The eigenvalues and corresponding eigenvectors of are

for , respectively. Moreover, is the infinitesimal generator of a -quasi-semigroup given by

As the previous example, we assume that the system is controlled around the point and

Condition (a) of Theorem 13 shows that system (78) is approximately controllable on if and only if, for all ,

Equation (83) confirms that the zeros of on interval affect the loss of approximate controllability.
Next, we locate the measurement at the system output with the point measurement:

where is a continuous function. Condition (b) of Theorem 13 implies that the system is approximately observable on if and only if, for all ,

for some . The Mean Value Theorem for integral implies that (85) is equivalent to

for some .

Remark 4. In this example we consider the nonautonomous regular Sturm-Liouville problem with the Dirichlet boundary condition. Actually, we can verify that all the results remain valid for which the problem has the Neumann boundary condition. Even, the nonautonomous singular Sturm-Liouville problems can be applied for the results. However, the periodic cases do not hold for the theory due to not simpleness of the related eigenvalues.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors are grateful to Research Institution and Community Service of Universitas Sebelas Maret Surakarta for funding. This research is funded by Doctoral Dissertation Research Grant 2018 of Universitas Sebelas Maret Surakarta.

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