Exact Null Controllability , Stabilizability , and Detectability of Linear Nonautonomous Control Systems : A Quasisemigroup Approach

and Applied Analysis 3 Definition 2 implies that if a C0-quasisemigroup R(t, s) is uniformly exponentially stable, then the classical solution x(t) = R(0, t)x0 of abstract non-autonomous Cauchy problems ?̇? (t) = A (t) x (t) , t ≥ 0, x (0) = x0 ∈ D, (7) converges to 0 exponentially as t 󳨀→ ∞. 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 A(t) be an infinitesimal generator of C0quasisemigroup R(t, s) on a Banach space X. If B(⋅) ∈ L∞(R,L(X)), then there exists a uniquely C0-quasisemigroup RB(t, s) with its infinitesimal generator A(t) + B(t) such that RB (r, t) x = R (r, t) x + ∫t 0 R (r + s, t − s) B (r + s) RB (r, s) x ds, (8) for all t, r, s ≥ 0 with t ≥ s and x ∈ X. Moreover, if ‖R(r, t)‖ ≤ M(t), then 󵄩󵄩󵄩󵄩RB (r, t)󵄩󵄩󵄩󵄩 ≤ M (t) e‖B‖M(t)t. (9) Proof. We define R0 (r, t) x = R (r, t) x, Rn (r, t) x = ∫ t 0 R (r + s, t − s) B (r + s) Rn−1 (r, s) x ds, (10) for all t, r, s ≥ 0 with t ≥ s, x ∈ X, and n ∈ N, and RB (r, t) = ∞ ∑ n=0 Rn (r, t) , (11) for all r, t ≥ 0. Following the proof of Theorem 2.4 of [15] we obtain the assertions. In addition, we have lim s󳨀→0 RB (t, s) x − x s = lim s󳨀→0 [ R (t, s) x − x s + s ∫ s 0 R (t + w, s − w)B (t + w) RB (t, w) x dw] = A (t) x + R (t, 0) B (t) RB (t, 0) x = (A (t) + B (t)) ⋅ x, (12) for all x ∈ X. Corollary 4. Let A(t) be an infinitesimal generator of C0quasisemigroup R(t, s) on a Banach space X. If B(⋅) ∈ L∞(R,L(X)), then the C0-quasisemigroup RB(t, s) with its infinitesimal generator A(t) + B(t) satisfies integral equation RB (r, t) x = R (r, t) x + ∫t 0 RB (r + s, t − s) B (r + s) R (r, s) x ds, (13) for all t, r, s ≥ 0 with t ≥ s and x ∈ X. Proof. With Rn(r, t) defined in (10), it is easy to show that Rn (r, t) = ?̃?n (r, t) , (14) for all t, r ≥ 0 and n ∈ N, where ?̃?n(r, t) is defined by ?̃?0 (r, t) = R (r, t) ?̃?n (r, t) x = ∫ t 0 ?̃?n−1 (r + s, t − s) B (r + s) R (r, s) x ds, (15) for all t, r, s ≥ 0 with t ≥ s, x ∈ X, and n ∈ N. In virtue of (11) and (14) we obtain RB (r, t) x = R (r, t) x + ∞ ∑ n=1 ?̃?n (r, t) x = R (r, t) x


Introduction
In this paper we focus on linear nonautonomous control system ẋ () =  ()  () +  ()  () , (0) =  0 ,  ≥ 0 (1) where () ∈  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 D(()) = D, independent of ; and () :  →  is a bounded operator such that (⋅) ∈  ∞ (R + , L  (, )), where L  (, ) and  ∞ (Ω, ) denote the space of bounded operators from  to  equipped with strong operator topology and the space of bounded measurable functions from Ω 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][3][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  0 of the system into 0. The associated stabilizability problem is to find a control () = ()() such that the zero solution of the closed-loop system ẋ () = [ () +  ()  ()]  () ,  ≥ 0, 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 nonautonomous 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  0 -quasisemigroup on .Sutrima et al. [16] and Sutrima et al. [17] investigated the advanced properties and some types of stabilities of the  0 -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  0 -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  0 -quasisemigroup.The organization of this paper is as follows.In Section 2, we provide the sufficient and necessary conditons for uniform exponential stability of  0 -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.

Uniform Exponentially Stability of 𝐶 0 -Quasisemigroups
This section is a part of the main results.We first recall the definition of a strongly continuous quasisemigroups following [15,18].
Let D be the set of all  ∈  such that the following limits exist: For  ≥ 0 we define an operator () on D as The family {()} ≥0 is called the infinitesimal generator of the  0 -quasisemigroup {(, )} ,≥0 .
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].
Proof.We define for all , ,  ≥ 0 with  ≥ ,  ∈ , and  ∈ N, and for all ,  ≥ 0. Following the proof of Theorem 2.4 of [15] we obtain the assertions.In addition, we have lim for all  ∈ .
The following example illustrates the existence of the quasisemigroup   (, ).
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  0 -quasisemigroup on a Banach space .e following statements are equivalent.(c) ere exists  0 > 0 such that ‖(,  0 )‖ < 1 for all  ≥ 0.
In the sequel we use () to denote the semigroup {()} ≥0 .It is easy to show that () is strongly continuous on   (R + , ).Moreover, if () is the infinitesimal generator of (, ) with domain D, then the infinitesimal generator of () is given by on domain for all  ∈ .For any  ∈   (R + , ) we obtain Hence, This shows that () is exponentially stable on   (R + , ).
(⇐).Assume that () is exponentially stable on   (R + , ).There exist constants  > 0 and  > 0 such that for all  ≥ 0. We choose  ∈  such that ‖‖ = 1.For any  ≥ 0, we choose  ∈   (R + , ) such that ( 0 ) =  for some The following results are Datko's version for the sufficient and necessary conditions for the uniform exponential stability of  0 -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 1 ≤  < ∞.A  0 -quasisemigroup (, ) is uniformly exponentially stable on a Banach space  if and only if for every  ∈  there exists a constant   such that for all  ≥ 0 (uniformly in ).

Corollary 9.
A  0 -quasisemigroup (, ) is uniformly exponentially stable on a Banach space  if and only if for every  ∈  there exist constants  > 0 and  ≥ 1 such that for all  ≥ 0 (uniformly in ).
Next, for a  0 -quasisemigroup (, ), it is defined a convolution operator G to be a linear operator on   (R + , ) by According to the definition of semigroup () in ( 27), the operator G can be written as The following result shows that if G is bounded on   (R + , ) and Γ is the infinitesimal generator of () which is given in (28), then −G is the inverse of Γ. Theorem 10.Let () be a  0 -semigroup defined in ( ) on   (R + , ) with its infinitesimal generator Γ given by ( ).If ,  ∈   (R + , ), then the following statements are equivalent.
The both results conclude that On the other hand, since then this proves the statement (a).
Proof.By Closed Graph Theorem, it is enough to show that the mapping   → G is closed operator.
Theorem 12.A  0 -quasisemigroup (, ) is uniformly exponentially stable on a Banach space  if and only if G is a bounded operator on   (R + , ), 1 ≤  < ∞.
We recall that a  0 -quasisemigroup (, ) on a Banach space  is said to be exponentially bounded if there are  ∈ R and a function   : R + → [1, ∞) such that for all ,  ≥ 0. 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  0quasisemigroup on a Banach space  and Γ is the infinitesimal generator of  0 -semigroup () on   (R + , ). e following statements are equivalent.
We mainly concern on the linear non-autonomous control systems on  with state , input , and output : ẋ () =  ()  () +  ()  () ,  ≥ 0, (0) =  0 , (61) where  is an unknown function from interval [0, ∞) into  and () is the infinitesimal generator of a  0quasisemigroup on  with domain D(()) = D, 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 1 ≤  < ∞.
Definition .Assume that the linear non-autonomous control system (61) holds for all initial state  0 ∈  and for all input  ∈   (R + , ).The state is defined to be a mild solution of (61).
Let (, ) be a  0 -quasisemigroup with infinitesimal generator ().We define an input-output mapping for the non-autonomous system (61)-( 62 Definition .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  0quasisemigroup (, ); (b) input-output stable if the corresponding input-output mapping L of (64) is bounded.
(d) ⇒ (a).By detectability of the system, there exists a uniformly exponentially stable  0 -quasisemigroup   (, ) and a bounded operator G  .Analogously with calculation (70) we have Since L = CGB, K, and G  are bounded, then operator GB is bounded.The boundedness of B implies that operator G is bounded on   (R + , ).Thus,  0 -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.

Conclusions
We have established the sophistication of  0 -quasisemigroups to characterize some qualitative control problems of linear nonautonomous control systems in Banach spaces including the stability, stabilizability, detectability, exact null controllability, and complete stabilizability.There are equivalences of internal stability, stabizability, detectability, and input-output stability.Also, in linear nonautonomous control systems, the exact null controllability implies complete stabilizability.Some of the obtained results are extensions of existing results in the references to infinite-dimensional and autonomous control systems.