# Controllability and Observability of Nonautonomous Riesz-Spectral Systems

Authors: Sutrima Sutrima; Christiana Rini Indrati; Lina Aryati
Journal: Abstract and Applied Analysis
 (2018)

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## 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 aC0-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.


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## Body

## 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.LetX, U, and Y be complex Hilbert spaces. Suppose that B(t):U→X and C(t):X→Y are bounded operators such that B(·)∈L∞(R+,Ls(U,X)) and C(·)∈L∞(R+,Ls(X,Y)), where Ls(V,W) denotes the space of bounded operators from V to W equipped with strong operator topology. We consider the linear nonautonomous control systems on X with state x, input u, and output y:(1)x˙t=Atxt+Btut,t≥0,x0=x0,(2)yt=Ctxt,where x is an unknown function from real interval [0,∞) into X and A(t) is a linear closed operator in X with domain D(A(t))=D, independent of t and dense in X. We denote the state linear system (1)-(2) by (A(t),B(t),C(t)). To avoid clutter, we also use notations (A(t),B(t),-) and (A(t),-,C(t)) if C(t)=0 and B(t)=0, respectively.There is an extensive amount of literatures which have studied controllability for the system(A(t),B(t),-) (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, A(t) is an infinitesimal generator of a C0-quasi-semigroup on X. Finally, the advanced properties and some types of stabilities of the C0-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 C0-quasi-semigroups.In the autonomous case, that is,A(t)=A, B(t)=B, and C(t)=C, independent of t, there are many literatures which have been devoted to study of the controllability and observability for the system (A,B,C) 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 A is a Riesz-spectral generator of a C0-semigroup on X, then the solution of (1) for B=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 (A,B,-) 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 forA(t) 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 C0-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.
LetL(X) be the set of all bounded linear operators on Hilbert space X. A two-parameter commutative family {R(t,s)}s,t≥0 in L(X) is called a strongly continuous quasi-semigroup, in short C0-quasi-semigroup, on X if, for each r,s,t≥0 and x∈X, (a)
R(t,0)=I, identity operator on X,(b)
R(t,s+r)=R(t+r,s)R(t,r),(c)
lims→0+⁡R(t,s)x-x=0,(d)
there exists a continuous increasing functionM:[0,∞)→[0,∞) such that (3)Rt,s≤Ms.In the sequel, for simplicity we denote the quasi-semigroup{R(t,s)}s,t≥0 and family {A(t)}t≥0 by R(t,s) and A(t), respectively.In this section we investigate sufficient conditions ofA(t) such that (1) forms a nonautonomous Riesz-spectral system. It is well known that if A 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(A(t)) is independent of t. This implies that to be a Riesz-spectral operator, A(t) has to have eigenvectors which are independent of t. A class that meets this criterion is a family of operators whose representation is as follows:(4)At=atA,where A is a Riesz-spectral operator on X and a is a bounded continuous function such that a(t)>0, t≥0. It is clear that, for every t≥0, A(t) and A have the common domain and eigenvectors. Moreover, if λn, n∈N, is an eigenvalue of A, then a(t)λn are the eigenvalues of A(t) of (4). Hence, in general A(t) may have the nonsimple eigenvalues. In case A(t) is a differential operator, then the operator A(t,ξ) of [10] satisfying the conditions P1 and P2 verifies (4). These urge the following notion.Definition 2.
For everyt≥0, let A(t) be an operator of form (4) on a Hilbert space X. A(t) is called a generalized Riesz-spectral operator if A is a Riesz-spectral operator.Definition2 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 everyt≥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 of A∗, the adjoint of A, 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(5)Rλ,Atx=∑n=1∞1λ-atλnx,φnϕn;(b)
A(t) has representation(6)Atx=at∑n=1∞λnx,φnϕnforx∈D=D(A(t)), where (7)D=x∈X:∑n=1∞λn2x,φn2<∞;(c)
ifsupn∈NRe(λn)<∞, then, for every t≥0, A(t) is the infinitesimal generator of a C0-quasi-semigroup R(t,s) given by (8)Rt,sx=∑n=1∞eλngt+s-gtx,φnϕn,whereg(t)=∫0ta(ξ)dξ;(d)
the growth bound of the quasi-semigroup att is given by (9)ω0t=infs>01slog⁡Rt,s=atsupn∈NReλn.Proof.
Proofs of (a) and (b) follow the proofs of Theorem2.3.5 of [21] replacing (λI-A)-1 and yN with(10)λI-At-1≔∑n=1∞1λ-atλnx,φnϕn,(11)yNt≔∑n=1N1λ-atλnx,φnϕn,respectively, for every t≥0. In this context R(λ,A(t))=(λI-A(t))-1.
(c) Letω=supn≥1Re(λn). Given t≥0 fixed, for λ such that Re(λ)>a(t)ω, from (a) (12)λI-At-1x=∑n=1∞1λ-atλnx,φnϕnand by iteration we have (13)λI-At-rx=∑n=1∞1λ-atλnrx,φnϕn.So by the conditionb of Lemma 2.3.2 of [21] for m=h and M=H, we have (14)λI-At-rx2≤Hat∑n=1∞1λ-atλn2rx,φn2≤Hhx2Reλ-atω2r.Theorem 3.7 of [15] implies that A(t) is an infinitesimal generator of a C0-quasi-semigroup R(t,s) with(15)Rt,s≤Hheatωs.We verify that the operators R(t,s), t,s≥0, given by(16)Rt,sx=∑n=1∞eλngt+s-gtx,φnϕn∀x∈X,where g(t)=∫0ta(ξ)dξ and supn∈NReλn<∞, are a C0-quasi-semigroup on X satisfying (15) with the infinitesimal generator (17)Atx=at∑n=1∞λnx,φnϕnon domain (18)D=x∈X:∑n=1∞λnx,φn2<∞.
(d) By (10) we have (19)ω0t=infs>01slog⁡Rt,s≤atsupn≥1⁡Reλn.On the other hand, taking x=ϕn in (16) we get (20)1slog⁡Rt,s=1sgt+s-gtReλn∀n∈N.It implies(21)ω0t=infs>01slog⁡Rt,s≥atsupn≥1⁡Reλn.Therefore (22)ω0t=infs>01slog⁡Rt,s=atsupn∈N⁡Reλn.Corollary 4.
If, for everyt≥0, A(t) is the generalized Riesz-spectral generator of a C0-quasi-semigroup R(t,s) on a Hilbert space X, then for any x0∈D and r≥0 the initial value problem(23)x˙t=Ar+txt,x0=x0admits 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(A(t),B(t),C(t)) of (1)-(2), where A(t) is the generalized Riesz-spectral operator generating a C0-quasi-semigroup R(t,s) on X. In the sequel, we assume that the two requested real numbers p and q always satisfy (24)1p+1q=1and 1<p<∞, unless specified.Definition 5.
Assume that the state linear system(A(t),B(t),-) holds for all initial state x0∈X and for all input u∈Lp(R+,U). The state(25)xt=R0,tx0+∫0tRs,t-sBsusds,0≤t<∞,is defined to be a mild solution of (1).We verify thatx∈C(R+,X) and the output y defined by (2) always belongs to Lq(R+,Y). 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(A(t),B(t),-) is said to be (a)
exactly controllable on[0,τ] if for each x0,x1∈X there exists a control u∈Lp([0,τ],U) such that the mild solution x(·) of (1) corresponding to u(·) satisfies x(τ)=x1;(b)
approximately controllable on[0,τ] if for each x0,x1∈X and any ϵ>0 there exists a control u∈Lp([0,τ],U) such that the mild solution x(·) of (1) corresponding to u(·) satisfies x(τ)-x1<ϵ.A controllability map of(A(t),B(t),-) on [0,τ] is a bounded linear map Bτ:Lp([0,τ];U)→X defined by(26)Bτu=∫0τRs,τ-sBsusds.It is easy to show that the system(A(t),B(t),-) is exactly controllable on [0,τ] if and only if ranBτ=X, where ranT denotes the range of T. Also, system (A(t),B(t),-) is approximately controllable on [0,τ] if and only if ranBτ¯=X.Lemma 7.
The controllability map in (26) satisfies the following conditions. (a)
The operatorBτ∈L(Lp([0,τ],U),X) and Bt∈L(Lp([0,τ],U),Lp([0,τ],X)) for 0≤t≤τ.(b)
(Bτ∗x)(s)=B∗(s)R∗(s,τ-s)x on [0,τ].Proof.
(a) SinceR(t,s) is strongly continuous and u∈Lp([0,τ],U), then the map s↦〈x,Rs,τ-sBsus〉 is measurable on [0,τ] for every x∈X. Moreover, (27)∫0τRs,τ-sBsusXds≤∫0τMτ-sB·LU,XuLpds≤τMτB·LU,XuLp<∞.Lemma A.5.5 of [21] states that the integral in (26) is well-defined. We verify easily that Bτ is linear. Now, for 0≤t≤τ we have (28)Btu≤∫0tRs,τ-sBsusXds≤∫0tMt-sB·LU,XuLpds≤tMtB·LU,XuLp.This shows that Bτ is a bounded mapping from Lp([0,τ],U) to X and u↦Btu, 0≤t≤τ, is a bounded mapping from Lp([0,τ],U) to Lp([0,τ],X).
(b) The definition of adjoint operator shows thatBτ∗ is bounded. Moreover, (29)u,Bτ∗xLp=Bτu,xX=∫0τRs,τ-sBsusds,xX=∫0τRs,τ-sBsus,xXds=∫0τus,B∗sR∗s,τ-sxUds=u,B∗·R∗·,τ-·xLp.This proves that Bτ∗x=B∗(s)R∗(s,τ-s)x on [0,τ].Theorem 8.
Foru∈Lp([0,τ],U), the system (A(t),B(t),-) is exactly controllable on [0,τ] if and only if any one of the following conditions holds for some γ>0 and all x∈X: (a)
γB∗(·)R∗(·,τ-·)xLp≥x.(b)
kerBτ∗={0} and ranBτ∗ is closed.Proof.
(a) We setV=Lp([0,τ],U), so Bτ∈L(V,X). It is enough to prove that ranBτ=X. By similarity of adjoint and dual operator in Hilbert space, Corollary 3.5 of [20] states (30)ranBτ=Xif and only if there exists γ>0 such that (31)γBτ∗xLp≥x,for all x∈X. So, by condition (b) of Lemma 7 the assertion is confirmed.
(b) The conditionγBτ∗xLp≥x shows that Bτ∗ is injective, and so kerBτ∗={0}. Next, let Bτ∗xn be a Cauchy sequence in Lp([0,τ],U). Condition (a) shows that (xn) is a Cauchy sequence in X. However, Lemma 7 (a) forces Bτ∗xn→Bτ∗x for some x∈Bτ∗. Thus, Bτ∗ has a closed range.Theorem 9.
The linear system(A(t),B(t),-) is approximately controllable on [0,τ] if and only if any one of the following conditions holds: (a)
B∗(s)R∗(s,τ-s)x=0, 0≤s≤τ, implies x=0.(b)
kerBτ∗={0}.Proof.
(a) We see that the system(A(t),B(t),-) is approximately controllable on [0,τ] if and only if ranBτ¯=X. According to Lemma VI 2.8 of [24], this is equivalent to the fact that the mapping Bτ′:X′→Lq([0,τ],U′) is injective. The similarity between adjoint and dual operator gives (32)B∗·R∗·,τ-·x=0which implies x=0 almost everywhere. Therefore, if (33)B∗sR∗s,τ-sx=0,0≤s≤τ,this verifies that x=0.
(b) Condition (a) and condition (b) of Lemma7 give the desired result.Complementary to Definition6, we define the exact observability and the approximate observability as follows.Definition 10.
The linear system(A(t),B(t),C(t)) is said to be (a)
exactly observable on[0,τ] if the initial state can be uniquely constructed from the knowledge of the output in Lq([0,τ],Y);(b)
approximately observable on[0,τ] if the knowledge of the output in Lq([0,τ],Y) determines the initial state uniquely.The observability map of the system(A(t),B(t),C(t)) on [0,τ] is a bounded linear map Cτ:X→Lq([0,τ];Y) defined by(34)Cτx=CtR0,tx,for 0≤t≤τ.From Definition10 and the definition of observability map we verify that the system (A(t),B(t),C(t)) is exactly observable on [0,τ] if and only if Cτ is injective and its inverse is bounded on ranCτ. Also, (A(t),B(t),C(t)) is approximately observable on [0,τ] if and only if kerCτ={0}.Lemma 11.
For the linear system(A(t),B(t),C(t)) one has the following duality: (a)
The linear system(A(t),-,C(t)) is approximately observable on [0,τ] if and only if the dual (A∗(t),C∗(t),-) is approximately controllable on [0,τ].(b)
The linear system(A(t),-,C(t)) is exactly observable on [0,τ] if and only if the dual (A∗(t),C∗(t),-) is exactly controllable on [0,τ].Proof.
As a consequence of Proposition 1.2 and Theorem 1.6 of [14], if A(t) generates a C0-quasi-semigroup R(t,s) on a Hilbert space, then A∗(t) generates the C0-quasi-semigroup R∗(t,s). Furthermore, we verify that(35)Cτ∗y=∫0τR∗0,sC∗sysds.This implies that the range of Cτ∗ equals that of the controllability map for the dual system (A∗(t),C∗(t),-). If Bτ denotes the controllability map of the dual system, then Cτ∗=Bτ or Cτ=Bτ∗.
(a) We see that(A(t),-,C(t)) is approximately observable on [0,τ] if and only if {0}=kerCτ=kerBτ∗. Condition (b) of Theorem 9 implies that kerBτ∗={0} if and only if (A∗(t),C∗(t),-) is approximately controllable on [0,τ]. This proves the equivalence.
(b) Suppose that(A(t),-,C(t)) is exactly observable on [0,τ]. There exists an inverse Cτ-1 on ranCτ and a constant κ>0 such that (36)xX=Cτ-1Cτx≤κCτx=κBτ∗.The exact controllability of (A∗(t),C∗(t),-) follows from Theorem 8.
Conversely, assume that(A∗(t),C∗(t),-) is exactly controllable on [0,τ]. Theorem 8 (a) gives that Bτ∗ is injective and ranBτ∗ is closed. Since Bτ∗=Cτ, then Cτ is injective and ranCτ is closed. This states that (A(t),-,C(t)) is exactly observable on [0,τ].Theorems8 and 9 and Lemma 11 yield the following conditions for observability.Corollary 12.
For the linear system(A(t),-,C(t)), one has the following necessary and sufficient conditions for exact and approximate observability: (a)
(A(t),-,C(t)) is exactly observable on [0,τ] if and only if any one of the following conditions holds for some γ>0 and all x∈X:
(i)
γC(·)R(0,·)xLq≥x.(ii)
kerCτ={0} and ranCτ is closed.(b)
(A(t),-,C(t)) is approximately observable on [0,τ] if and only if any one of the following conditions holds:
(i)
C(s)R(0,s)x=0, 0≤s≤τ, implies x=0.(ii)
kerCτ={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.Consider system (1)-(2) with finite-rank inputs and outputs(37)x˙t=Atxt+∑i=1mbituit,t≥0,x0=x0,(38)yt=xt,c1t,…,xt,ckttr,where A(t) is the generalized Riesz-spectral operator of (4), bit∈X, i=1,…,m, cit∈X, i=1,…,k, and ui∈Lp(R+), 1<p<∞. The symbol Str denotes the transpose of S. If we set u=(u1,…,um)∈U≔Rm, B(t)u≔∑i=1mbi(t)ui, then u∈Lp(R+,U). In this case we have (39)Ctxt=xt,c1t,…,xt,ckttr.LetA be the Riesz-spectral operator with simple eigenvalues {λn:n∈N} and corresponding eigenvectors {ϕn:n∈N}. In addition, if {φn:n∈N} are the eigenvectors of A∗ such that 〈ϕn,φm〉=δmn and supn∈NRe(λn)<∞, then according to the condition (c) of Theorem 3A(t) is the infinitesimal generator of a C0-quasi-semigroup R(t,s) given by (40)Rt,sx=∑n=1∞eλngt+s-gtx,φnϕn,where g(t)=∫0ta(ξ)dξ and A(t) is the form of (4). In this context we verify that (41)B∗tx=b1t,xX,…,bmt,xX,R∗t,sx=∑n=1∞eλn¯gt+s-gtx,ϕnφn.By Theorem 9, system (37) is approximately controllable on [0,τ] if and only if(42)∑n=1∞eλn¯gτ-gtx,ϕnbit,φn=0,i=1,…,m,0≤t≤τimplies that x=0.Next, we have(43)CtR0,tx=∑n=1∞eλngtx,φnCtϕn.In virtue of Corollary 12, system (37)-(38) is approximately observable on [0,τ] if and only if(44)∑n=1∞eλngtx,φnCtϕn=0implies that x=0.These two facts deal with the following theorem which is a generalization of Theorem4.2.3 of [21] for the autonomous case.Theorem 13.
Consider the linear system(A(t),B(t),C(t)) of (37)-(38), where A is a Riesz-spectral operator with simple eigenvalues {λn:n∈N} such that supn∈NRe(λn)<∞ and corresponding eigenvectors {ϕn:n∈N}. Let {φn:n∈N} be the eigenvectors of A∗ such that 〈ϕn,φm〉=δmn. Then (a)
(A(t),B(t),-) is approximately controllable on [0,τ] if and only if for all n(45)rank⁡b1t,φn,…,bmt,φn=1for allt∈[0,τ];(b)
(A(t),-,C(t)) is approximately observable on [0,τ] if and only if for all n(46)rank⁡ϕn,c1t,…,ϕn,ckt=1for allt∈[0,τ].Proof.
(a) We consider the matrixBn: (47)Bn=b1t,φn,…,bmt,φnon [0,τ]. By Lemma 3.14 of [20] and (42), (A(t),B(t),-) is approximately controllable on [0,τ] if and only if for all n(48)x,ϕnbit,φn=0,i=1,…,m,0≤t≤τimplies x=0. Suppose that (A(t),B(t),-) is not approximately controllable on [0,τ], there exists an n∈N such that 〈x,ϕn〉≠0 and (49)x,ϕnbit,φn=0,i=1,…,m,0≤t≤τ.This gives 〈bi(t),φn〉=0 for all i=1,…,m and t∈[0,τ], and so rank Bn≠1.
Conversely, suppose that rankBn0≠1 for some n0, then 〈bi(t),ϕn0〉=0, for all i=1,…,m and t∈[0,τ]. So we can find a nonzero x∈X such that (50)x,ϕn0bit,φn0=0.Thus, (42) is satisfied for x≠0. This is equivalent to the fact that (A(t),B(t),-) is not approximately controllable on [0,τ].
(b) We can have similar proof to (a) for the matrix(51)Cn=ϕn,c1t,…,ϕn,ckton [0,τ].


## 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 setX to be the Hilbert space of L2[a,b]. Consider an operator A on its domain(52)DA=x∈X:x,dxdξareabsolutelycontinuous,d2xdξ2∈X,a1dxdξa+a2xa=0,b1dxdξb+b2xb=0,where (a1,a2)≠(0,0) and (b1,b2)≠(0,0). Operator A is called a Sturm-Liouville operator if(53)Ax≔1wξ-ddξpξdxdξξ+qξxξ,for x∈D(A), where w,p,q, and dp/dξ are real-valued continuous functions on [a,b] such that p(ξ)>0 and w(ξ)>0.Sincea and b are finite, the definition only corresponds to regular Sturm-Liouville problems. We verify that A is a self-adjoint operator with real, countable, and simple eigenvalues λn such that 0<λ1<λ2<⋯ (see [25, 26]).We define a nonautonomous Sturm-Liouville operator to be an operator of form (4):(54)At=atA,t≥0,where A is a Sturm-Liouville operator on its domain D(A) given by (52).Definition 14.
The state linear system(A(t),B(t),C(t)) of (1)-(2) is called a nonautonomous Sturm-Liouville system if A(t) is the negative of a nonautonomous Sturm-Liouville operator of form (54).Corollary 15.
For everyt≥0, let A(t) be the negative of a nonautonomous Sturm-Liouville operator of the form (54) on its domain D(A) given by (52). Then (a)
A(t) is generalized Riesz-spectral operator;(b)
A(t) is the infinitesimal generator of a C0-quasi-semigroup on X.Proof.
(a) Lemma 1 of [27] gives the fact that A(t) is generalized Riesz-spectral operator.
(b) If{λn:n∈N} is the set of eigenvalues of -A, then supn∈NRe(λn)<∞. Hence, Theorem 3 concludes that, for every t≥0, A(t) is the infinitesimal generator of a C0-quasi-semigroup on X.We note that Corollary15 does not hold when A(t) is a nonautonomous Sturm-Liouville operator. Indeed, A(t)=-a(t)d2/dξ2 is a nonautonomous Sturm-Liouville operator, but it does not generate any C0-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:(55)∂x∂tt,ξ=ξ2t+1∂2x∂ξ2t,ξ+ξt+1∂x∂ξt,ξ,1<ξ<b,t≥0xt,1=xt,b=0,1<b<∞.
We are ready to show that the problem has a unique solution. LetX be a Hilbert space of L2[1,b]. Problem (55) can be written as(56)x˙t=Atxt,t≥0,on X, where A(t)≔a(t)A, a(t)=1/t+1, and (57)Axξ≔ξ2d2xdξ2+ξdxdξon D with (58)D=x∈X:x,dxdξareabsolutelycontinuous,d2xdξ2∈X,x1=xb=0.
We verify that operatorA is not self-adjoint on D. Furthermore, we obtain the eigenvalues and corresponding eigenvectors of A as (59)λn=-n2π2log⁡b,ϕnξ=2sin⁡nπlog⁡ξfor1≤ξ≤b,respectively. It is obvious that every eigenvalue λn is simple and the set {ϕn:n∈N} forms Riesz basis of X. Moreover, {λn,n∈N}¯ is totally disconnected, that is, for c,d∈{λn,n∈N}¯, [c,d]⊈{λn,n∈N}¯.
The adjoint ofA is (60)A∗xξ≔ξ2d2xdξ2+3ξdxdξ+xξon D(A∗)=D(A). The eigenvalues and corresponding eigenvectors of A∗ are (61)μn=-n2π2log⁡b,ψnξ=2ξ-1sin⁡nπlog⁡ξfor all 1≤ξ≤b, n∈N, and satisfy (62)ϕm,ψn=δmn∀m,n∈N.
Next, since the adjoint of any operator is always closed, thenA∗ is closed. But in this case we have A=(A∗)∗, so A is closed. Thus, A is a Riesz-spectral operator. In other words, A(t), t≥0, is a generalized Riesz-spectral operator.
Sincesupn∈NRe(λn)=-π2/log⁡b<∞, condition (c) of Theorem 3 forces that A(t) is the infinitesimal generator of a C0-quasi-semigroup R(t,s) given by (63)Rt,sx=∑n=1∞t+s+1t+1λnx,ψnϕn.Corollary 4 guarantees that for each x0∈D problem (56) admits a unique solution (64)xt=R0,tx0,x0=x0.Thus, boundary condition problem (55) has a solution (65)xt,ξ=∑n=1∞t+1λnx0,ψnϕn.Example 2.
Consider the controlled wave equation(66)∂2x∂t2t,ξ=∂2x∂ξ2t,ξ+btut,ξ,0<ξ<1,t≥0xt,0=xt,1=0,where b:R+→R is bounded uniformly continuous and u(t,·)∈L2[0,1] 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:(67)w˙t=Atwt+Btut,t≥0,on the Hilbert space X=D(A01/2)⊕L2[0,1] with the inner product (68)v,yX=A01/2v1,A01/2y1L20,1+v2,y2L20,1,where v=v1v2 and y=y1y2. In this context A(t)=0I-A00, B(t)=0b(t), w=xdx/dt, and A0h=-d2h/dξ2 for h∈D(A0) with (69)DA0=h∈L20,1:h,dhdξareabsolutelycontinuousd2hdξ2∈L20,1,h0=h1=0.We verify that A(t) has the eigenvalues λn=inπ, n=±1,±2,… and the corresponding Riesz basis of eigenvectors ϕn(ξ)=1/inπsin(nπξ)inπsin⁡(nπξ), where i=-1. We see that ϕn(ξ)=ψn(ξ) for every n. Moreover, Example 2.3.8 of [21] shows that A(t) is a Riesz-spectral operator on X. Hence, A(t) is the generalized Riesz-spectral operator generating a C0-quasi-semigroup R(t,s).
We assume that the system is controlled around the pointξc. So, we may set (70)ut,ξ=ut12ϵχξc-ϵ,ξc+ϵξ,where χ is an indicator function. Theorem 13 shows that system (67) is approximately controllable on [0,τ] if and only if(71)∫01btsin⁡nπξdξ=btnπϵsin⁡nπξcsin⁡nπϵ≠0forn≥1.Equation (71) demonstrates that the control points ξc for which sin⁡(nπξc)=0 affect the loss of approximate controllability. This is also the case when b(t)=0, that is, at the zeros of b on interval [0,τ].
Next, consider the observation(72)yt=∫01ct,ξxt,ξdξ,where c(t,ξ)=1/2νc(t)χ[ξs-ν,ξs+ν](ξ) is an output function around the sensing point ξs. Following Example 4.2.5 of [21] we can reformulate the observation map as an inner product on X: (73)Ctv1v2≔∫01ct,ξv1ξdξ=v1,ct,ξL20,1=v1v2,ctk0X,wherek=A0-112νχξs-ν,ξs+ν.Condition (b) of Theorem 13 gives the fact that (A(t),-,C(t)) is approximately observable on [0,τ] if and only if for all n(74)ϕn,ctk0X≠0.We verify that (75)ϕn,ctk0X=1inπsin⁡nπ·,ct2νχξs-ν,ξs+ν·L20,1.Therefore, the system is approximately observable on [0,τ] if and only if for all n(76)ct∫ξs-νξs+νsin⁡nπξdξ=ctnπνsin⁡nπξssin⁡nπν≠0.This shows that the system loses the approximate observability at points for which sin⁡(nπξs)=0 for some n or c(t)=0. Specially, for c(t)=sin⁡(fs(t)π), where fs(t) is the sampling frequency, then the system loses the approximate observability at sampling frequencies of fs=k, for k=0,1,2,… discrete measurement.Example 3.
Consider the controlled nonautonomous heat equation on interval[1,b]:(77)∂x∂tt,ξ=at∂∂ξξ2∂x∂ξt,ξ+βtut,ξ,t≥0xt,1=xt,b=0,where a,β:R+→R are bounded uniformly continuous and u(t,·)∈L2[1,b] is a distributed control.
We shall analyze the approximate controllability and approximate observability of the system. LetX be a Hilbert space of L2[1,b]. System (77) can be formulated as a nonautonomous Sturm-Liouville system(78)x˙t=Atxt+Btut,t≥0,on X, where A(t)≔-a(t)A, B(t)≔β(t)I, and A is a Sturm-Liouville operator (53) with w(ξ)=1,p(ξ)=ξ2, and q(ξ)=0 on its domain D: (79)D=x∈X:x,dxdξareabsolutelycontinuous,d2xdξ2∈X,x1=xb=0.The eigenvalues and corresponding eigenvectors of -A are (80)λn=-n2π2+14log⁡b,ϕnξ=2ξ-1/2sin⁡nπlog⁡ξ,for 1≤ξ≤b, respectively. Moreover, A(t) is the infinitesimal generator of a C0-quasi-semigroup R(t,s) given by (81)Rt,sx=∑n=1∞t+s+1t+1λnx,ϕnϕn.
As the previous example, we assume that the system is controlled around the pointξc and (82)ut,ξ=ut12ϵχξc-ϵ,ξc+ϵξ.Condition (a) of Theorem 13 shows that system (78) is approximately controllable on [0,τ] if and only if, for all n≥1,(83)βt∫ξc-ϵξc+ϵξ-1/2sin⁡nπlog⁡ξdξ≠0.Equation (83) confirms that the zeros of b on interval [0,τ] affect the loss of approximate controllability.
Next, we locate the measurementy(t) at the system output with the point measurement:(84)yt=ct1ϵ∫b-ϵbxt,ξdξ,where c(t) is a continuous function. Condition (b) of Theorem 13 implies that the system is approximately observable on [0,τ] if and only if, for all n≥1,(85)ct∫b-ϵbξ-1/2sin⁡nπlog⁡ξdξ≠0,for some ϵ>0. The Mean Value Theorem for integral implies that (85) is equivalent to (86)ϵctξn-1/2sin⁡nπlog⁡ξn≠0,for some ξn∈(b-ϵ,b).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.



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