CONTROLLABILITY FOR A CLASS OF SEMILINEAR FRACTIONAL EVOLUTION SYSTEMS VIA RESOLVENT OPERATORS

. This paper deals with the exact controllability for a class of fractional evolution systems in a Banach space. First, we introduce a new concept of exact controllability and give notion of the mild solutions of the considered evolutional systems via resolvent operators. Second, by utilizing the semigroup theory, the ﬁxed point strategy and Kuratowski’s measure of noncompactness, the exact controllability of the evolutional systems is investigated without Lip- schitz continuity and growth conditions imposed on nonlinear functions. The results are established under the hypothesis that the resolvent operator is dif- ferentiable and analytic, respectively, instead of supposing that the semigroup is compact. An example is provided to illustrate the proposed results.


1.
Introduction. This paper aims to establish the sufficient conditions for exact controllability of the following fractional evolution systems: C D α x(t) = Ax(t) + f (t, x(t)) + Bu(t), t ∈ I = [0, a], x(0) = x 0 ∈ X, (1.1) where C D α is the Caputo derivative of order 0 < α < 1, A : D(A) ⊂ X → X is the infinitesimal generator of a C 0 -semigroup of bounded linear operators {T (t) : t ≥ 0} defined on a Banach space X. The control function u(·) is given in L 2 (I; U ) with U is a Banach space. B is a bounded linear operator from U into X, and f : I ×X → X is a given continuous function. Fractional calculus and fractional differential equations have been investigated extensively due mainly to their demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering such as physics, economics, control theory, aerodynamics and electromagnetics, etc. Actually, fractional models can present a more vivid and accurate description over things than integral ones, and fractional derivatives can provide an excellent tool for the description of memory and hereditary properties of various materials and processes. For details, see [2,4], and the references therein.
Recently, Hernández et al. [11] pointed out that the concept of mild solutions used in some papers such as [7,13] was not suitable for fractional evolution systems at all. Concerning this, we hold that when we investigate the evolutional systems in infinite-dimensional spaces, the introduction of the concept of mild solutions is the most important step. Generally speaking, there are two main types of definition of mild solutions. The first one was constructed in terms of a probability density function given by El-Borai [10] and was then developed by Zhou et al. [30]. Considering the convergent domain of the probability density function, we know that this way is only applicable to the case 0 < α < 1, where α stands for the fractional derivative of the considered fractional evolution system. The second one was defined by means of an α-resolvent family provided by Araya et al. [1]. Noticing that the solution of a fractional evolution system is in fact a Volterra integral equation, we point out that the concept of the resolvent operator plays an important role in it.
On the other hand, control theory is an interdisciplinary branch of engineering and mathematics that deals with influence behavior of dynamical systems. Controllability plays an important role in the analysis and design of control systems. Many practical problems of control theory such as stabilizability and pole assignment may be solved under the controllable hypothesis. Controllability problems for different kinds of dynamical systems and evolutional systems have been investigated by many authors. For detailed description of some recent results, we refer the reader to papers [15,16,17,18,19,20,21,22,26] and the references therein.
We point out that there are two basic concepts of controllability. The first one is exact controllability (complete controllability) and the other one is approximate controllability. In the case of infinite-dimensional spaces, exact controllability enables to steer the system to arbitrary final state while approximate controllability means that the system can be steered to an arbitrary small neighborhood of final state. The major way of dealing with exact controllability is to transform the controllability problem into a fixed point problem with hypothesis that the invertibility of a controllability operator is satisfied, see [14]. With the help of fixed point theorem and semigroup theory under the assumption that the linear part of the associated nonlinear system is approximately controllable, the approximate controllability is also well investigated, see [25]. It is worth mentioning that in case of finite-dimensional space the concepts of exact and approximate controllability coincide. For example, in [29], J. Wang et al. studied the following fractional differential systems: where C D q t is the Caputo derivative of order 0 < q ≤ 1, A is the infinitesimal generator of a strongly continuous semigroup, B is a bounded linear operator and f is given X-valued functions. The authors obtained the exact controllability under a compact condition and the assumption that f was Lipschitz continuous.
In [9], by means of Banach contraction mapping principle and the Lipschitz continuous assumptions imposed on functions f, g, h, Φ and I i , A. Debbouche et al. established a sufficient condition for the exact controllability of the following impulsive systems: dt α + A(t, u(t))u(t) = (Bµ)(t) + Φ(t, f (t, u(β(t))), t 0 g(t, s, u(γ(s)))ds), where −A(t, .) is a closed linear operator, B is a bounded linear operator, control function µ belongs to the spaces L 2 ([0, a]; U ) with U is a Banach space, and f, g, h, Φ, I i are given functions. Using the Banach contraction mapping principle together with the Lipschitz continuity of the functions f and g, V. Vijayakumar et al. in [28] obtained the exact controllability of the following systems: where D α t is the Caputo derivative of order 1 < α < 2, A, G(t) are closed linear operators, and f, g are appropriate functions.
In [27], the authors considered a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the sth power of a positive definite operator. In this problem, the fractional parameter s serves as the control parameter. Using Tikhonov's compactness theorem, the authors overcame the difficulty that with changing s also the domain of L S changed.
Concerning the above results, we note that when we convert the controllability into a fixed point problem, nonlinear function f and nonlocal item are usually assumed to be compact or Lipschitz continuous. But these assumptions are not satisfied in many practical applications. Else, Hernández and O'Regan point out that some papers on controllability of abstract control systems contain a similar technical error when the compactness of semigroup and other assumptions are satisfied. Motivated by all above mentioned works, in the present paper, we will introduce a new and weaker concept of exact controllability which can be regarded as an extension of the existing notion. In addition, we introduce a suitable concept for mild solutions of the the fractional evolution system (1.1) via resolvent operators which are well developed for Volterra integral equations. It should also be stressed that for equations with unbounded operators in infinite-dimensional spaces the concept of the resolvent is more appropriate since it is a direct generalization of C 0 -semigroups. Meanwhile, to get the exact controllability of system (1.1) in the infinite-dimensional spaces, the differentiable and analytic conditions will be put on the resolvent operators respectively instead of supposing that the semigroup {T (t) : t ≥ 0} is compact. Moreover, by using semigroup theory, the Sadovskii's fixed point theorem and the Kuratowski's measure of noncompactness, we investigate the exact controllability of fractional evolution system (1.1) without the Lipschitz continuity and other growth conditions imposed on the nonlinear function f . Actually, the nonlinear function f is only supposed to be continuous. So our results can be regarded as a development of previous conclusions obtained in [9,14,29].
The rest of present paper is organized as follows. Section 2 gives some necessary preliminaries and lemmas. In Section 3, we establish the sufficient conditions for the exact controllability of fractional evolution system (1.1). Finally, an illustrative example is presented to support the new results.
2. Preliminaries and some lemmas. Let (X, · ) be a real Banach space and A : D(A) ⊂ X → X is the infinitesimal generator of a C 0 -semigroup of bounded linear operators {T (t) : t ≥ 0} defined on X. In the sequel we denote by L(X, Y ) the space of bounded linear operators from X into Y endowed with the operator norm denoted by · L(X,Y ) where Y is another Banach space, and we write simply L(X) and · L(X) when X = Y . Let D be the domain of the operator A equipped with the graph norm x D = x + Ax . Obviously, D is a Banach space, and it is continuously and densely embedded into X. We denote by C(I; X) the space of Xvalued continuous functions on I with the sup-norm · C(I;X) . C α (I; X), α ∈ (0, 1), represents the space formed by all the X-valued α-Hölder continuous functions from I into X with the norm x C α (I;X) = x C(I;X) + [|x|] C α (I;X) , where [|x|] C α (I;X) = sup t,s∈I,t =s In this paper, we assume that the integral equation has an associated resolvent operator {S(t)} t≥0 on X. Let us recall the following known definitions and lemmas.

Definition 2.1 ([23]
). The Caputo fractional derivative of order α > 0 of a continuous function u : (0, ∞) → R is given by where n = [α] + 1, [α] denotes the integer part of the real number α, and provided the right side integral is pointwise defined on [0, ∞). Remark 1. (i) Obviously, the Caputo derivative of a constant is equal to zero. (ii) Especially, when 0 < α < 1, we have (iii) The solutions of Caputo-type equations are in general much more abundant than in the classical case. For instance, the Caputo derivative would represent a memory effect if we consider a function depending on time, pointing out that the state of a system at a given time depends on past events. In addition, the solutions of Caputo-type equations can approximate any given smooth function arbitrarily. For details, see [3].
The inequality in the following lemma will be useful in the proof of the main results.
Consider the following nonlinear Volterra integral equation: where h ∈ L 1 (I; X). According to [24], we introduce the concept of mild solution of (2.2) as follows.
The next result compiles different properties related to the mild solution of (2.2).

Lemma 2.6 ([24]). (i) Suppose (2.2) admits a differentiable resolvent operator S(t)
and h ∈ C(I; D). Then the function x(t) defined by is a mild solution of (2.2). (ii) Suppose (2.2) admits a analytic resolvent operator S(t) and h ∈ C α (I; X). Then the function x(t) defined by is a mild solution of (2.2).
To establish our next theorems, we need the following lemmas which are the useful tools to study the equicontinuity of the considered operators. 12]). If w ∈ C(I; D) and W : I → X is the function defined by Lemma 2.8 ( [12]). If w ∈ C(I; X) and W : I → X is the function defined by [|W|] C α (I;X) ≤ 2 αΓ(α) w C(I;X) .
To end this section, we recall some properties of Kuratowski's measures of noncompactness and several lemmas which will be used in the next section. The presentation here can be found in, for example, [6]. Lemma 2.9. Let us denote by X a Banach space.
(1) Let S, T be bounded sets of X and λ ∈ R. Then (i) α(S) = 0 if and only if S is relatively compact; (3) Let D ⊂ C(I; X) be bounded and equicontinuous. Then α(D(t)) is continuous on I, and α(D) = max t∈I α(D(t)).
(4) Let D = {u n } ⊂ C(I; X) be a bounded and countable set. Then α(D(t)) is integrable on I, and α I u n (t)dt : n ∈ N ≤ 2 I α(D(t))dt. The following two fixed-point theorems and the theorem of Ascoli-Arzelà play a key role in our proof of exact controllability. Lemma 2.10 (Sadovskii). Let P be a condensing operator on a Banach space X, i.e. P is continuous and takes bounded sets into bounded sets, and α(P(D)) < α(D) for every bounded set D of X with α(D) > 0. If P(S) ⊂ S for a convex, closed and bounded set S of X, then P has a fixed point in S. Lemma 2.11 (Mönch). Let D be a closed and convex subset of a Banach space E and x 0 ∈ D. Assume that the continuous operator A : D → D has the following property: 3. Exact controllability results. In this section, we establish sufficient conditions for the exact controllability of the fractional evolution system (1.1). As an application, an example is given to illustrate our main results. In the sequel we assume the resolvent operator {S(t)} t≥0 is differentiable and denote by ϕ A the function introduced in Definition 2.3.
We first introduce the mild solution of the fractional evolution system (1.1). Obviously, if x(·) is a solution of (1.1), t ∈ I, then one has (3.1) By Definition 2.5 and the representation (3.1), we give the following concept of mild solution for system (1.1).
Definition 3.1. For each u ∈ L 2 (I; U ), a mild solution of the fractional evolution system (1.1) on J we mean a function x ∈ C(J; X) which satisfies t 0 Definition 3.2 (Exact controllability). The fractional evolution system (1.1) is said to be exactly controllable on the interval I = [0, a], if for every x 0 , x 1 ∈ X, there exists a control function u ∈ L 2 (I, U ) and a constant τ ∈ (0, a] such that a mild solution Remark 3. Compared to the existing concept in [9,14,29] and so on in which τ is equal to a, our definition in which τ ∈ (0, a] is weaker and can be regarded as an extension of the present notion of exact controllability.
For the sake of convenience, we list the main hypotheses and some notations to be used in the paper as follows: (H1) f ∈ C(I × X; D) and takes bounded sets into bounded sets, x 0 ∈ D.
(H2) The linear operator B : has an induced inverse operator W −1 (t) which takes values in L 2 (I; U )/kerW (t) for every t ∈ I and there exist two positive constants M 1 , M 2 > 0 such that Let us take Proof. Denote C = sup{ f (t, x(t)) D : Using hypothesis (H2), for an arbitrary function x(·) ∈ C(I; X), we define the control Put J = [0, τ ]. By considering Lemma 2.6 (i), we shall show that, the operator T : C(J; X) → C(J; X) defined by Obviously, (T x)(τ ) = x 1 which implies that u x steers the system (1.1) from x 0 to x 1 in finite time τ . This means that system (1.1) is exactly controllable on I.
Let Ω = {x ∈ C(J; X) : x(t) ≤ R 0 , t ∈ J}, then Ω is obviously a closed ball in C(J; X). In the following, we divide the proof into several steps.
As a matter of fact, From Lemma 2.7 and it follows that one can obtain

D. ZHAO, Y. LIU AND X. LI
As already done in the estimation of I 1 , we have Note that As already done in the discussion of I 2 , we get Consequently, (T x)(t + h) − (T x)(t) → 0 independently of x ∈ Ω as h → 0, which indicates that the operator T : Ω → Ω is equicontinuous.
Step 3. We prove that T is continuous on Ω.
For this purpose, we assume that y n ∈ Ω with y n → y in Ω. Obviously, for each t ∈ J, It is easy to see that Note that These together with the continuity of f and Lebesgue's domination convergence theorem indicate J 1 → 0, J 2 → 0, as n → +∞. On the other hand, for each t ∈ J, we can easily get it follows from the similar discussion of J 1 and J 2 that Therefore, for each t ∈ J, we have Thus, from Lemma 2.12, it is not difficult to see that {T y n } is relatively compact in C(J; X). We now prove T y n − T y C(J;X) → 0, as n → +∞. Since {T y n } is relatively compact, there is a subsequence of {T y ni } which converges in C(J; X) to some v ∈ C(J; X). No loss of generality, we assume that {T y ni } itself converges to v: In view of (3.4) and (3.7), we have v = T y, and so, (3.7) contradicts (3.6). Then, (3.5) holds, and the continuity of T on Ω is proved. Denote B = coT (Ω)( i.e. the convex closure of T (Ω)). It is easy to check T (B) ⊆ B and B is equicontinuous.
Step 4. We shall prove that T : B → B is a condensing operator (i.e. α(T D) < α(D) for any bounded D ⊂ B which is not relatively compact).
Introduce the decomposition From [5], for an arbitrary small positive constant ε and any bounded set D ⊂ B, there exits a countable sequence Clearly, In view of Lemma 2.9 (4) and hypothesis (H3), we obtain that and (3.11) For brevity, we take This together with Lemma 2.9 (4), hypotheses (H2) and (H3) implies and (3.13) Since T (D 0 ) ⊂ B is bounded and equicontinuous, one gets from the Lemma 2.9 (3) that α (T (D 0 )) = max t∈J α ((T D 0 ) (t)) . (3.14) Then, from (3.8)-(3.14), it follows that By the arbitrariness of ε and (3.2), we have α ((T D)) ≤ 4L 2 ϕ A L 1 (I) + 1 2 (2M 1 M 2 a α + 1) which implies T : B → B is a condensing operator. Therefore, due to Lemma 2.10, T has at least one fixed point x on B. It is easy to see that x is a mild solution of the fractional evolution system (1.1) on J satisfying x(τ ) = x 1 and thus system (1.1) is exactly controllable on I. This completes the proof.
Remark 4. In Theorem 3.3, f is supposed to be continuous instead of compact or Lipschitz continuous, and f has no other growth conditions. We put the differentiable condition on resolvent operator instead of assuming the semigroup {T (t) : t ≥ 0} is compact. The concept of exact controllability here is weaker than the existing notion. In addition, if f is compact or Lipschitz continuous, then condition (H3) is satisfied. Therefore, our results extend some previous conclusions such as in [9,14,29].
In the following, we shall suppress the assumptions that f and x 0 are D-valued and suppose that the resolvent operator {S(t)} t≥0 is analytic. To this end, we need to replace the hypothesis (H1) and (H2) with the following hypotheses: (H1) f ∈ C(I × X; X) and takes bounded sets into bounded sets, x 0 ∈ X.
(H2) The linear operator B : L 2 (I; U ) → L 1 (I; X) is bounded, W (t) defined by Proof. We let C = sup{ f (t, x(t)) : For an arbitrary function x(·) ∈ C(I; X), t ∈ I, by hypothesis (H2) , we can define the following control Put J = [0, τ ]. By Lemma 2.6 (ii), we shall show that the operator T : C(J; X) → C(J; X) defined by has a fixed point.
For any x ∈ Ω, s, t ∈ J with s ≤ t, it is easy to see that
Since S(·)x 0 is continuous, one has I 1 → 0, as h → 0. Using Lemma 2.8, we have As already done in the discussion of I 4 in Theorem 3.3, one can obtain Note that Just by the same way, as h → 0, we also have Then, we assert that (T x)(t+h)−(T x)(t) → 0, as h → 0, for all x ∈ Ω. Therefore, the operator T : Ω → Ω is equicontinuous.
Step 3. The operator T is continuous on Ω.
Assume that y n ∈ Ω with y n → y in Ω. Obviously, for each t ∈ J,
Remark 5. (i) We suppress the hypotheses that f , x 0 are D-valued and assume that f , x 0 are X-valued. The analytic condition is put on the resolvent operator instead of the compactness of semigroup {T (t) : t ≥ 0}. (ii) We can also use Mönch fixed point theorem to demonstrate our results. In fact, one can assume that C ⊂ B is countable, and C ⊂ co({z} T C), where z ∈ B. It is easy to see that α(C(t)) ≤ α((T C)(t)). (3.27)
For ξ ∈ (0, π), we assume that the linear operator W (t) is given by Moreover, it is easy to check that (H3) are satisfied with L = 2. Consequently, from Theorem 3.4, it follows that the fractional evolution system (3.28) is exactly controllable on [0, 1].