Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions

In this paper, we study the approximate controllability of Sobolev-type fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results.


1.
Introduction. It is well known that controllability problem is one of the most fundamental issues in mathematical control theory. Since the beginning of sixties in the last century, controllability problems for dynamical systems have been reported in many literature and monographs.
Fractional calculus (FC for short) has been introduced since the end of the nineteenth century by famous mathematicians Liouville and Riemann, but the concept of non-integer calculus, as a generalization of the traditional integer order calculus was mentioned already in 1695 by Leibnitz and L'Hospital. The subject of FC has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences, engineering to biological sciences and economics.
FC has recently been proved to be a vital tool in the modeling of many physical phenomena, and fractional differential equations (FDE for short) have attracted great attention of several researchers. For more details on this topic, see the literature on fractional calculus and its application [2], [7], [10], [16], [17], [22] and [26], and monographs on FDE [3], [6], [13], [24], [32] and the survey [29]. There are presented fundamental results on FC and on the existence, uniqueness and stability of solutions of FDE with applications [14,30,31]. Moreover, discrete fractional equations are also investigated.
Controllability is one of the important fundamental issues in mathematical control theory. Recently, controllability problems of fractional evolution systems in infinite dimensional spaces have been attracted by many researchers. There are some interesting and important controllability results concerning semilinear differential systems involving the Caputo fractional derivative. For example, Debbouche and Baleanu [4], Fečkan et al. [9] and [28] initiated to study complete controllability of two classes of Sobolev-type fractional functional evolution equations by constructing two new characteristic solution operators via the well-known Schauder Fixed Point Theorem. Meanwhile, Sakthivel and Ren [25], Debbouche and Torres [5], Kerboua et al. [11] and [12], Mahmudov [19], Mahmudov and Zorlu [20] and [21] pay attention to studying approximate controllability for different types of fractional evolution systems. The above controllability results are derived with the help of semigroup theory and fixed point technique.
Note that the concept of approximate controllability enables to steer the system to arbitrary small neighborhood of final state. Thus, approximate controllable problems are more prevalent in fields of control engineering. There is no work yet reported on approximate controllability of Sobolev-type fractional evolution equations with non-local conditions. Inspired by the above recent contributions, we propose to discuss approximate controllability of Sobolev-type fractional evolution systems with classical non-local conditions in Hilbert spaces. To handle our task, we first impose an approximate controllability of the corresponding linear system. Then we develop the approach in [18] and rewrite our control problem as a fixed point equation for an appropriate nonlinear operator. Using the Schauder Fixed Point Theorem on this equation, sufficient conditions of approximate controllability of the desired problem are presented.
To end this section, we would like to claim the original ideas of this paper as follows. We establish a uniform framework to investigate approximate controllability for non-local problems for Sobolev-type fractional evolution systems in Hilbert spaces by introducing a suitable Gramian controllability operator and adopting a fixed point theorem, which extend the classical method for Cauchy problems for integer order systems.

2.
Preliminaries. Let X be a Hilbert space with a scalar product ·, · and the corresponding norm · . We consider the following Sobolev-type fractional evolution system: where C 0 D q t is the generalized Caputo fractional derivative of order 0 < q < 1 with the lower limit zero (see Kilbas et al. [13]), E and A are two linear operators with domains contained in X and ranges still contained in X, the pre-fixed points t k satisfies 0 = t 0 < t 1 < t 2 < · · · < t m < t m+1 = a and a k are real numbers.
In order to guarantee that −AE −1 : X → X generates a semigroup {W (t), t ≥ 0}, we consider that the operators A and E satisfy the following conditions: is continuous. Now we note that (S 4 ) implies that E is closed; (S 3 ) implies (S 4 ). It follows from (S 1 ), (S 2 ), (S 4 ) and the closed graph theorem that −AE −1 : X → X is bounded, which generates a uniformly continuous semigroup {W (t), t ≥ 0} of bounded linear operators from X to itself.
Denote by ρ(−AE −1 ) the resolvent set of −AE −1 . If we assume that the resolvent The state x(t) takes values in X and the control function u(·) is given in U , the Banach space of admissible control functions, where U := L p (J, U ), for q ∈ ] 1 p , 1[ with 1 < p < ∞ and U is a Hilbert space. Moreover, B : U → X is a bounded linear operator and f : J × X → X will be specified later.
Consider a probability density function ξ q (see [8]) defined on ]0, ∞[: Obviously, Then, we define the following two operators: Similar to the proof in Fečkan et al. [9] and Zhou and Jiao [33], one has the following results.
Next, we define Using Pazy [23, Lemma 10.1, p. 38], we have: Assume that there exists a continuous linear operator Θ on X given by where I is the identity operator.
Remark 1. One can give a sufficient condition to guarantee the existence of Θ. For example, set M 1 m k=1 |a k | < 1. Indeed, applying Neumann lemma, we get

Now we introduce a Green function:
Hence, we set Inspired by the concept of mild solutions in [27], [28] and [33], we introduce the following definition.
Definition 2.4. For each u ∈ U , by a mild solution of the system (1) we mean that there exists a function x ∈ C(J, X) satisfying Remark 2. To explain the above formula, like in [9, Lemma 3.1], one can integrate the first equation of the system (1) via Laplace transform to derive which implies that Now using the non-local initial condition in the system (1) one can solve which leads to the desired formula of mild solution.
In what follows, we turn to consider the following linear system: Using the mild solution of (3), we get Define a linear operator P : U → X by By using Lemmas 2.1, 2.2 and Hölder inequality, we obtain Thus, P is bounded. Furthermore, (3) is approximately controllable if and only if cl(P (U )) = X. This is equivalent to ker Hence, we derive if we want to compose P and P * . This is satisfied, when p ≤ p * = p p−1 so 1 < p ≤ 2. Recall q ∈] 1 p , 1[ which gives a restriction 1 2 < q < 1. Define Gramian controllability operator Noting Lemma 2.3, it is straightforward that Γ a 0 is a linear bounded operator. In fact, it follows (4) that Γ a 0 ≤ P P * ≤ M 2 P . Now we recall the following result. We apply Theorem 2.5 with Γ a 0 . Then for any x * ∈ X, we have Note P * : X → U * = L p * (J, U ) ⊂ L p (J, U ) = U ⊂ L 2 (J, U ), since 1 < p ≤ 2. So the above last integral is well defined. We also get that x * , Γ a 0 x * > 0 iff P * x * = 0, i.e., x * / ∈ ker P * . Consequently, Γ a 0 is positive iff ker P * = {0}, i.e., Γ a 0 is positive if and only if the linear system (3) is approximately controllable on J. Setting R( , Γ a 0 ) := ( I + Γ a 0 ) −1 : X → X, > 0, by Theorem 2.5, we arrive at the following result (see also [18]). Theorem 2.6. Let 1 2 < q < 1. (3) is approximately controllable on J iff R( , Γ a 0 ) → 0 in the strong topology as ↓ 0.
3. Main results. In this section, we study the approximate controllability of the system (1) by imposing that the corresponding linear system is approximately controllable and using the Schauder Fixed Point Theorem.
Definition 3.1. Let x(a; x(0), u) be the state value of the system (1) at terminal time a corresponding to the control u ∈ U and non-local initial condition x(0). The system (1) is said to be approximately controllable on the interval J iff the closure cl(R(a, x(0))) = X.
(H 2 ): f : J × X → X is continuous such that (3) is approximately controllable on J.
Recalling the condition (H 3 ) and Theorem 2.6, for any x ∈ C(J, X) and h ∈ X, we define the following control formula: with For each k > 0, define Of course, B k is a bounded, closed, convex subset in C(J, X). Using the above control u in (5), we consider an operator P : B k → C(J, X) given by Now we can prove the following important result.
Proof. We divide the proof into several steps.

Claim 1. For an arbitrary
If it is not true, then for each k > 0, there would exist x ∈ B k andt k ∈ J such that P x)(t k ) > k. Using the following fact for c i > 0 and then we obtain Dividing both sides by k and taking the lower limit as k → ∞, we derive a contradiction 1 ≤ 0. Then f (·, x m ) − f (·, x) ∞ → 0 as m → ∞. Next, following the above estimations, we first obtain and then we get for any t ∈ J. This yields that P is continuous.
Next, by Claim 1 for x ∈ B k , we derive Let x ∈ B k and t , t ∈ J be such that t < t . Note that Then, and by the Hölder inequality, so the terms K 1 and K 2 tend to zero as t ↓ t . It is elementary to see that the term K 3 tends to zero as t ↓ t , as well. As a result, P is compact due to the Arzela-Ascoli Theorem. From above, P is completely continuous operator for all > 0. By using the Schauder Fixed Point Theorem, P has at least one fixed point which rises at least one mild solution of the system (1).
In the sequel, we need the following compactness result. Proof. We can write the linear operator Q as Then following computations for (4), we derive So Q 0 : L r (J, X) → X is continuous. Then (S 3 ) and (7) imply that Q is compact. The proof is finished. Now we are ready to present the main result in this paper. Proof. By Theorem 3.2, there is a fixed point x of P in B k( ) , which is a mild solution of the system (1) under the control u (t, x ) in (5) and satisfies for From the reflexivity of L r (J, X), there is a subsequence {f (t, x i (t))} ∞ i=1 , i → 0 as i → ∞ that converges weakly to say f ∈ L r (J, X). Define by Lemma 3.3 we find that the right-hand side of (9) tends to zero as i → ∞. Thus, it follows from Theorem 2.6, (8) and (9) that This proves the approximate controllability of the system (1). 4. An example. Take X = U = L 2 [0, π] and p = 2. Consider the following fractional partial differential equation with control x, x y are absolutely continuous, x yy ∈ X, x(0) = x(π) = 0}. Then Ax = − ∞ n=1 n 2 x, x n , x ∈ D(A) and Ex = ∞ n=1 (1 + n 2 ) x, x n x n , x ∈ D(E), respectively (see [15]), where x n (y) = 2 π sin ny, n = 1, 2, · · · is the orthonormal set of eigenfunctions of A. Moreover, (1 + n 2 ) −1 x, x n x n , x ∈ D(E) is compact since lim n→∞ 1 1+n 2 = 0. Thus, E −1 is compact, and bounded. Hence, (H 1 ) is satisfied.
Next, the bounded operator −AE −1 generates a strongly continuous semigroup {W (t), t ≥ 0} written as x, x n x n , Define an operator f : J × X → X by f (t, x)(y) = µ cos(2πt) sin x(y). It is easy to verify (H 2 ). Next, B : U → X is defined by B = I. Now, the system (10) can be abstracted as = E −1 −

5.
Conclusions. Approximate controllability of Sobolev-type fractional evolution systems, with classical non-local conditions in Hilbert spaces, are investigated. By imposing an approximate controllability of linear system and rewriting the control problem as a fixed point equation for an appropriate nonlinear operator, sufficient conditions of approximate controllability are presented via the Schauder Fixed Point Theorem.