Controllability and Observability of Nonautonomous Riesz-Spectral Systems

and Applied Analysis 3 Definition 2. For every t ≥ 0, let A(t) be an operator of form (4) on a Hilbert space X. A(t) is called a generalized Rieszspectral operator if A is a Riesz-spectral operator. Definition 2 states that if a is a nonnegative constant function, then A(t) is a Riesz-spectral operator. In the sequel we always assume that, for every t ≥ 0, A(t) is an operator of form (4). The following results are generalization of the results of [21, 22] for autonomous case. Theorem 3. For every t ≥ 0, let A(t) be an operator of (4) where A is a Riesz-spectral operator with simple eigenvalues {λn : n ∈ N} and corresponding eigenvectors {φn : n ∈ N}. If {φn : n ∈ N} are the eigenvectors ofA∗, the adjoint ofA, such that ⟨φn, φm⟩ = δmn, then (a) ρ(A(t)) = {λa(t) : λ ∈ ρ(A)}, σ(A(t)) = {λa(t) : λ ∈ σ(A)}, and for λ ∈ ρ(A(t)), the resolvent operator R(λ, A(t)) is given by R (λ, A (t)) x = ∞ ∑ n=1 1 λ − a (t) λn ⟨x, φn⟩ φn; (5) (b) A(t) has representation A (t) x = a (t) ∞ ∑ n=1 λn ⟨x, φn⟩ φn (6) for x ∈ D = D(A(t)), where D = {x ∈ X : ∞ ∑ n=1 󵄨󵄨󵄨󵄨λn󵄨󵄨󵄨󵄨2 󵄨󵄨󵄨󵄨⟨x, φn⟩󵄨󵄨󵄨󵄨2 < ∞} ; (7) (c) if supn∈NRe(λn) < ∞, then, for every t ≥ 0, A(t) is the infinitesimal generator of a C0-quasi-semigroup R(t, s) given by


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][2][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.
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 wellposedness 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 quasisemigroup 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  0 -quasi-semigroup on .Finally, the advanced properties and some types of stabilities of the C 0 -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  0 -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  0 -semigroup on , then the solution of (1) for  = 0 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 spectrumdetermined growth condition, controllability, observability, stabilizability, and detectability; see, for example, [22,23].
Although the aforementioned researches provide a wellestablished 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 Rieszspectral system.The obtained nonautonomous operator is implemented to study the controllability and observability for the nonautonomous systems.All the studies use the  0quasi-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.

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 L() be the set of all bounded linear operators on Hilbert space .A two-parameter commutative family {(, )} ,≥0 in L() is called a strongly continuous quasi-semigroup, in short  0 -quasi-semigroup, on  if, for each , ,  ≥ 0 and  ∈ , In the sequel, for simplicity we denote the quasisemigroup {(, )} ,≥0 and family {()} ≥0 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 D = D(()) 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 () > 0,  ≥ 0. It is clear that, for every  ≥ 0, () and  have the common domain and eigenvectors.Moreover, if   ,  ∈ N, 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  ≥ 0, let () be an operator of form (4) on a Hilbert space .() is called a generalized Rieszspectral operator if  is a Riesz-spectral operator.
(c) Let  = sup ≥1 Re(  ).Given  ≥ 0 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  = ℎ and  = , we have Theorem 3.7 of [15] implies that () is an infinitesimal generator of a  0 -quasi-semigroup (, ) with We verify that the operators (, ), ,  ≥ 0, given by where () = ∫  0 () and sup ∈N Re   < ∞, are a  0quasi-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  ≥ 0, () is the generalized Rieszspectral generator of a  0 -quasi-semigroup (, ) on a Hilbert space , then for any  0 ∈ D and  ≥ 0 the initial value problem admits a unique solution.
Proof.It follows from Theorem 2.2 of [13] that (23) admits a unique solution.

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  0 -quasi-semigroup (, ) on .In the sequel, we assume that the two requested real numbers  and  always satisfy and 1 <  < ∞, unless specified.
Definition 5. Assume that the state linear system ((), (), −) holds for all initial state  0 ∈  and for all input  ∈   (R + , ).The state is defined to be a mild solution of (1).
Lemma 7. The controllability map in (26) satisfies the following conditions.
Lemma A.5.5 of [21] states that the integral in ( 26) is welldefined.We verify easily that B  is linear.Now, for 0 ≤  ≤  we have According to Lemma VI 2.8 of [24], this is equivalent to the fact that the mapping B   :   →   ([0, ],   ) is injective.The similarity between adjoint and dual operator gives which implies  = 0 almost everywhere.Therefore, if The observability map of the system ((), (), ()) on [0, ] is a bounded linear map C  :  →   ([0, ]; ) defined by for 0 ≤  ≤ .
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.
(b) We can have similar proof to (a) for the matrix on [0, ].

Nonautonomous Sturm-Liouville Systems
In this section we shall discuss nonautonomous Sturm-Liouville systems, the specifically nonautonomous Rieszspectral systems.First let us recall the definition of Sturm-Liouville operators.In the sequel, we set  to be the Hilbert space of  2 [, ].Consider an operator A on its domain where for  ∈ D(A), where , , , and / are real-valued continuous functions on [, ] such that () > 0 and () > 0.
Since  and  are finite, the definition only corresponds to regular Sturm-Liouville problems.We verify that A is a selfadjoint operator with real, countable, and simple eigenvalues   such that 0 <  1 <  2 < ⋅ ⋅ ⋅ (see [25,26]).
We define a nonautonomous Sturm-Liouville operator to be an operator of form (4): where A a Sturm-Liouville operator on its domain D(A) given by (52).

Corollary 15.
For every  ≥ 0, let () be the negative of a nonautonomous Sturm-Liouville operator of the form (54) on its domain D() given by (52).Then Proof.(a) Lemma 1 of [27] gives the fact that () is generalized Riesz-spectral operator.(b) If {  :  ∈ N} is the set of eigenvalues of −A, then sup ∈N Re(  ) < ∞.Hence, Theorem 3 concludes that, for every  ≥ 0, () is the infinitesimal generator of a  0 -quasisemigroup on .
We note that Corollary 15 does not hold when () is a nonautonomous Sturm-Liouville operator.Indeed, () = −()( 2 / 2 ) is a nonautonomous Sturm-Liouville operator, but it does not generate any  0 -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.

Applications
In this section, we consider two examples of applications to confirm the results of the generalized Riesz-spectral operator in the nonautonomous systems.
We are ready to show that the problem has a unique solution.Let  be a Hilbert space of  2 [1, 𝑏].Problem (55) can be written as We verify that operator  is not self-adjoint on D. Furthermore, we obtain the eigenvalues and corresponding eigenvectors of  as for all 1 ≤  ≤ ,  ∈ N, 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, (),  ≥ 0, is a generalized Riesz-spectral operator.
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 [0, ] if and only if Equation (71) demonstrates that the control points   for which sin(  ) = 0 affect the loss of approximate controllability.This is also the case when () = 0, that is, at the zeros of  on interval [0, ].
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.
this verifies that  = 0.(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 [0, ] if the initial state can be uniquely constructed from the knowledge of the output in   ([0, ], );(b) approximately observable on [0, ] if the knowledge of the output in   ([0, ], ) determines the initial state uniquely.