#### Abstract

In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated -quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.

#### 1. Introduction

In this paper we focus on linear nonautonomous control systemwhere is the state, is the control, and and are complex Hilbert spaces of the state and control, respectively; is a densely defined operator in with domain , independent of ; and is a bounded operator such that , where and denote the space of bounded operators from to equipped with strong operator topology and the space of bounded measurable functions from to provided with essential supremum norm, respectively.

In the theory of control systems, controllability and stability are the qualitative control problems that play an important role in the systems. The theory was first introduced by Kalman et al. [1] for the finite dimensional of (autonomous) time-invariant systems. On its development, the theory can be generalized into controllability and stabilizability of the nonautonomous (time-varying) control systems see, e.g., [2–4] and the references therein. Idea of this problem is to find an admissible control such that the corresponding solution of the system has desired properties.

There are many various qualitative control problems that can be implemented to study the stabilizability. One of the most commonly applied qualitative control problems is null controllability. The system (1) is said to be null controllable if there exists an admissible control which steers an arbitrary state of the system into . The associated stabilizability problem is to find a control such that the zero solution of the closed-loop system is asymptotically stable in the Lyapunov sense. In this context, the system is said stabilizable and is called the stabilizing feedback control. The complete stabilizability is one of various types of stability that is often applied to characterize the stability of the control systems. This term was first introduced by Wonham [5] which relates to exponential stability of the systems. Next, based on the Lyapunov function techniques, Phat [3] investigated that the null controllability guaranteed the output feedback stabilization for the non-autonomous systems. While Jerbi [6] deal the problem of stabilizability at the origin of a homogeneous vector field of degree three.

Kalman et al. [1] and Wonham [5] have shown that in the finite-dimensional autonomous control system, if the system is null controllable in finite time then it is stabilizable. However, it does not hold for the converse. Furthermore, if the system is completely stabilizable, then it is null controllable in finite time. Investigations of controllability and stabilizability in the infinite dimensional control theory are more complicated, in particular for nonautonomous systems. For non-autonomous control systems of the finite-dimensional spaces, Ikeda et al. [7] proved that if the system is null controllable, then it is completely stabilizable. As extension of the some results of [7], Phat and Ha [4] characterized the controllability via the stabilizability and Riccati equation for the linear nonautonomous systems.

The results of the stabilizability for the finite-dimensional systems can be generalized into infinite-dimensional systems. For the autonomous systems, Phat and Kiet [8] investigated relationship between stability and exact null controllability extending the Lyapunov equation in Banach spaces. The smart characterization of generator of the perturbation semigroup for Pritchard-Salamon systems was provided by Guo et al. [9]. Rabah et al. [10] prove that exact null controllability implies complete stabilizability for neutral type linear systems in Hilbert spaces. The unbounded feedback is also investigated in the paper. For nonautonomous systems, Hinrichsen and Pritchard [11] introduced a concept of radius stability for the systems under structured nonautonomous perturbations. Indeed, this concept is an advanced investigation of the stabilizable theory. In the linear nonautonomous systems in Hilbert spaces, Niamsup and Phat [12] have proved that exact null controllability implies the complete stabilizability. Fu and zhang [13] had established a sufficient result of exact null controllability for a nonautonomous functional evolution system with nonlocal conditions using theory of linear evolution system and Schauder fixed point theorem.

As described in our recent work [14], a quasisemigroup is an alternative approach that can be implemented to investigate the non-autonomous systems (1). This approach was first introduced by Leiva and Barcenas [15]. By this approach, is an infinitesimal generator of a -quasisemigroup on . Sutrima et al. [16] and Sutrima et al. [17] investigated the advanced properties and some types of stabilities of the -quasisemigroups in Banach spaces, respectively. Even, the quasisemigroup approach can be applied to characterize the controllability of the non-autonomous control systems, although it is still limited to the autonomous controls [18]. However, until now there is no research which investigates the qualitative control problems of the nonautonomous control systems implementing -quasisemigroup theory.

In this paper, we are concern on the exact null controllability, stability, stabilizability, complete stabilizability, detectability, and possible relationship among them. In paper, we identify the stability with the uniform exponential stability of the associated -quasisemigroup. The organization of this paper is as follows. In Section 2, we provide the sufficient and necessary conditons for uniform exponential stability of -quasisemigroup which is an extension of [17]. Relationships among stability, stabilizability, and detectability of the linear nonautonomous control systems are considered in Section 3. In Section 4, we discuss connection between exact null controllability and complete stabizability of the linear nonautonomous control systems.

#### 2. Uniform Exponentially Stability of -Quasisemigroups

This section is a part of the main results. We first recall the definition of a strongly continuous quasisemigroups following [15, 18].

*Definition 1. *Let be the set of all bounded linear operators on a Hilbert space . A two-parameter commutative family in is called a strongly continuous quasisemigroup, in short -quasisemigroup, on if for each and : (a), the identity operator on ,(b),(c),(d)there exists a continuous increasing function such that

Let be the set of all such that the following limits exist: For we define an operator on as The family is called the infinitesimal generator of the -quasisemigroup .

In the sequel, for simplicity we denote the quasisemigroup and family by and , respectively.

In this paper, stability is meant to be uniform exponential stability which was introduced by Megan and Cuc [19] and was elaborated by Sutrima et al. [17].

*Definition 2. *A -quasisemigroup on Banach space is said to be uniformly exponentially stable if there exist constants and such that for all and .

Definition 2 implies that if a -quasisemigroup is uniformly exponentially stable, then the classical solution of abstract non-autonomous Cauchy problems converges to exponentially as .

The following theorem is a generalization of Theorem 2.4 of [15] that plays an important role in characterizing stabilizability of the system (1).

Theorem 3. *Let be an infinitesimal generator of -quasisemigroup on a Banach space . If , then there exists a uniquely -quasisemigroup with its infinitesimal generator such thatfor all with and . Moreover, if , then *

*Proof. *We definefor all with , , and , andfor all . Following the proof of Theorem 2.4 of [15] we obtain the assertions. In addition, we have for all .

Corollary 4. *Let be an infinitesimal generator of -quasisemigroup on a Banach space . If , then the -quasisemigroup with its infinitesimal generator satisfies integral equationfor all with and .*

*Proof. *With defined in (10), it is easy to show thatfor all and , where is defined by for all with , , and . In virtue of (11) and (14) we obtain for all with and .

The following example illustrates the existence of the quasisemigroup .

*Example 5. *Let be the space of all bounded continuous real function on with the supremum norm. The -quasisemigroup defined by for all , is generated by , where on domain . There exists a -quasisemigroup satisfying Theorem 3 for some operator .

We define an operator by for all . We see that and the quasisemigroup defined by is a -quasisemigroup with the infinitesimal generator .

The following theorem is an alternative version for sufficient and necessary conditions of uniform exponential stability that was given by Sutrima* et al*. [17].

Theorem 6. *Let be a -quasisemigroup on a Banach space . The following statements are equivalent. *(a)* is uniformly exponentially stable on .*(b)*The uniform growth bound .*(c)*There exists such that for all .*

*Proof. *. We recall that uniform growth bound is a constant defined by where .

A -quasisemigroup is uniformly exponentially stable on if and only if there exist constants and such that for all and . This gives that for all . This implies that .

. Assume that is uniformly exponentially stable on . There exist constants and such that for all . Set , we have the assertion.

. By condition (3), there exists an increasing continuous function such that for all . The hypothesis gives that . Let . Any can be stated as , , . Therefore, we obtain If we set (since ) and , then for all . This concludes that is uniformly exponentially stable on .

For a -quasisemigroup we can define a semigroup on , , relating with . This semigroup is defined as the Howland semigroup, introduced by Chicone and Laushkin [20], which is defined byfor all .

In the sequel we use to denote the semigroup . It is easy to show that is strongly continuous on . Moreover, if is the infinitesimal generator of with domain , then the infinitesimal generator of is given byon domain

Theorem 7. *A -quasisemigroup is uniformly exponentially stable on a Banach space if and only if the semigroup is exponentially stable on , .*

*Proof. *. Assume that is uniformly exponentially stable on the Banach space . There exist constants and such that for all . For any we obtain Hence, This shows that is exponentially stable on .

. Assume that is exponentially stable on . There exist constants and such thatfor all . We choose such that . For any , we choose such that for some . Let , then By (33), there exists such that , for all . So, in virtue of Theorem 6, is uniformly exponentially stable on a Banach space .

The following results are Datko’s version for the sufficient and necessary conditions for the uniform exponential stability of -quasisemigroups which are derived from Theorem 6. These are also alternative versions of Theorem 3.7 and 3.9 of [17] with the weaker conditions.

Corollary 8. *Let . A -quasisemigroup is uniformly exponentially stable on a Banach space if and only if for every there exists a constant such thatfor all (uniformly in ).*

*Proof. *(). Assume that -quasisemigroup is uniformly exponentially stable on a Banach space . There exist constants and such that for all and . Therefore, for we have This states that for every , a function is bounded on . Hence, the function is integrable on the interval and its value depends on . So, we can choose an such that Using transformation of variable, the last inequality is equivalent to (35).

(). Assume that for every there exists an such that (35) is fulfilled. For every , it is defined a linear operator by . By (35), every operator is closed, so every is bounded. By (35) and Uniform Boundedness Theorem, we obtain . Therefore, for every we have This implies the exponential stability of semigroup on . In fact, This shows that for every there exists an such that Thus, according to Lemma 5.1.2 of [21], the semigroup is exponentially stable on . Finally, Theorem 7 concludes that is uniformly exponentially stable on .

Corollary 9. *A -quasisemigroup is uniformly exponentially stable on a Banach space if and only if for every there exist constants and such thatfor all (uniformly in ).*

*Proof. *(). If a -quasisemigroup is uniformly exponentially stable on a Banach space and constants and satisfy Definition 2, then for all and . This implies that for every , a function is bounded on . Hence, the function is integrable on the interval and for all , , and where .

(). Assume that the -quasisemigroup satisfies condition (42). We set and where is a function satisfying (3). Since is a quasisemigroup, then for we have Consequently, for all , where . If we set , then Therefore, for all . Theorem 6 concludes that is uniformly exponentially stable on .

Next, for a -quasisemigroup , it is defined a convolution operator to be a linear operator on byAccording to the definition of semigroup in (27), the operator can be written as

The following result shows that if is bounded on and is the infinitesimal generator of which is given in (28), then is the inverse of .

Theorem 10. *Let be a -semigroup defined in (27) on with its infinitesimal generator given by (28). If , then the following statements are equivalent. *(a)* dan .*(b)*.*

*Proof. *(a) (b). If condition (a) holds, then by condition (d) of Theorem 2.1.10 of [21], for we obtain If we choose such that , then the definition of gives and This proves statement (b).

(b) (a). Assume that the statement (b) holds. If , then definition of implies that While for we have The both results conclude that On the other hand, since then this proves the statement (a).

Lemma 11. *The operator is bounded on if and only if for every , .*

*Proof. *By Closed Graph Theorem, it is enough to show that the mapping is closed operator.

Theorem 12. *A -quasisemigroup is uniformly exponentially stable on a Banach space if and only if is a bounded operator on , .*

*Proof. *(). In virtue of Theorem 7, if is uniformly exponentially stable on the Banach space , then the -semigroup is exponentially stable on . There exist constants and such that By formula (50), for any we have Therefore, This states that is bounded on .

(). If is a bounded operator on , then by Theorem 10, the operator on is injective and is equivalent to . So, is invertible, where . Consequently, by Theorem 3.10 of Ch.IV of [22] we obtain , where denotes the spectral bound of . By Theorem 1.10 of Ch.V of [22], the -semigroup is exponential stable on . Finally, Theorem 7 gives that is uniformly exponentially stable on .

We recall that a -quasisemigroup on a Banach space is said to be exponentially bounded if there are and a function such that for all . In particular, from Theorems 10 and 12 for is exponentially bounded on the Banach space , we have the following corollary.

Corollary 13. *Let is an exponentially bounded -quasisemigroup on a Banach space and is the infinitesimal generator of -semigroup on . The following statements are equivalent. *(a)* is uniformly exponentially stable on .*(b)* is invertible, with .*(c)*.*

#### 3. Stabilizability and Detectability of Linear Non-Autonomous Control Systems

Let , , and be complex Banach spaces. Assume and are bounded operators such that and . We mainly concern on the linear non-autonomous control systems on with state , input , and output :where is an unknown function from interval into and is the infinitesimal generator of a -quasisemigroup on with domain , independent of and dense in . Here, , , and are called the state space, the control space, and the output space, respectively.

In the sequel, we assume that is a real number such that .

*Definition 14. *Assume that the linear non-autonomous control system (61) holds for all initial state and for all input . The stateis defined to be a mild solution of (61).

Let be a -quasisemigroup with infinitesimal generator . We define an input-output mapping for the non-autonomous system (61)-(62) to be an operator byfor all .

*Definition 15. *The linear non-autonomous control system (61)-(62) is said to be: (a)internally stable if the is the infinitesimal generator of a uniformly exponentially stable -quasisemigroup ;(b)input-output stable if the corresponding input-output mapping of (64) is bounded.

*Definition 16. *The linear non-autonomous control system (61)-(62) is said to be: (a)stabilizable if there exists a feedback operator such that is the infinitesimal generator of a uniformly exponentially stable -quasisemigroup ;(b)detectable if there exists an operator such that is the infinitesimal generator of a uniformly exponentially stable -quassemigroup .

In virtue of Theorem 3, if the quasisemigroup and in Definition 16 exist, then each of them satisfies the integral equationandfor all with and , respectively.

In the sequel, if and are operators satisfying the system (61)-(62) and the operators and in Definition 16 exist, we define the multiply operator , , , and with multiplier , , , and by respectively. It is easy to show that the operators , , , and are linear.

Theorem 17. *Let be an exponentially bounded -quasisemigroup on a Banach space with its infinitesimal generator . The following statements are equivalent. *(a)*The system (61)-(62) is internally stable.*(b)*The system (61)-(62) is stabilizable and is a bounded operator from to .*(c)*The system (61)-(62) is detectable and and is a bounded operator from to .*(d)*The system (61)-(62) is stabilizable, detectable, and input-output stable.*

*Proof. *(a) (b), (c), dan (d). By Theorem 12, the operator is bounded. Since and are bounded operators, then operator is bounded. Hence, the uniform exponential stability of and the boundedness of , , , and , guarantee the existence of -quasisemigroup and satisfying the integral equation (65) and (66), respectively.

(b) (a). By hypothesis, there exists a uniformly exponentially stable -quasisemigroup satisfying (65) for some . In this context, is the infinitesimal generator of . Next, we define an operator , whereAccording to Theorem 12, is a bounded operator on . For , by Fubini Theorem and condition (b) of Theorem 2.1 of [15], we obtain This forces thatThe boundedness of operators , , dan implies that the operator is bounded on . Finally, Theorem 12 gives that -quasisemigroup is uniformly exponentially stable on . This proves the assertion.

(c) (a). Detectability of the system guarantees the existence of a uniformly exponentially stable -quasisemigroup satisfying (66) for some . The quasisemigroup is generated by . Analogously with previous part, we define an operator , whereThe similar reason for operators (68) dan (70), the operator and are bounded on . Therefore, the -quasisemigroup is uniformly exponentially stable on , that is the system (61)-(62) is internally stable.

(d) (a). By detectability of the system, there exists a uniformly exponentially stable -quasisemigroup and a bounded operator . Analogously with calculation (70) we have Since , , and are bounded, then operator is bounded. The boundedness of implies that operator is bounded on . Thus, -quasisemigroup is uniformly exponentially stable on . In other word, the system (61)-(62) is internally stable.

We end this section with a simple example to illustrate some results for stabilizability.

*Example 18. *Let be the space of all bounded continuous real functions on with the supremum norm and be the space of . Consider the the linear non-autonomous control systemon the Banach space , where with domain , , and with is the identity operator.

From the Example 5, is the infinitesimal generator of a -quasisemigroup defined by for all . It is easy to show that there exists such that for all