Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations

Article History: Received 26 July 2018 Accepted 16 May 2019 Available 30 July 2019 The aim of this article is to investigate the existence of mild solutions as well as approximate controllability of non-autonomous Sobolev type differential equations with the nonlocal condition. To prove our results, we will take the help of Krasnoselskii fixed point technique, evolution system and controllability of the corresponding linear system.


Introduction
In this article, we discuss the approximate controllability of nonlocal Sobolev type nonautonomous evolution equations in a separable Hilbert space X: (1) where A(t), E are X-valued linear operators with domains are subsets of X, and F is X-valued function defined over J × X, G is D(E)-valued function defined over C(J, X), J = [0, b]. The control function u ∈ L 2 (J, U), U is a Hilbert space and B is X-valued linear and bounded operator defined over U.
The Sobolev type differential equations appears in several fields such as thermodynamics [1], fluid flow via fissured rocks [2], and mechanics of soil [3]. Brill [4] first established the existence of solution for a semilinear Sobolev differential equation in a Banach space. Lightbourne et al. [5] studied a partial differential equation of Sobolev type.
Generalization of classical initial condition which is known as nonlocal condition is more effective to obtain better results. Nonlocal Cauchy problem was first considered by Byszewski [6].
Controllability is an important issue in engineering and mathematical control theory. The problem of exact controllability is to show that there exists a control function, that steers the solution of the system from its initial state to the given final state. However in approximate controllability, it is possible to steer the solution of the system from its initial state to arbitrary small neighbourhood of the the final state. Mostly the problem of controllability for various kinds of differential, integro-differential equations and impulsive differential equations are studied for autonomous systems. For more details, we refer to [7] - [13].
The existence of mild solutions for a nonautonomous nonlocal integro-differential equation *Corresponding Author is investigated by Yan [14] via Banach contraction principle, Schauder's fixed point theorem and the theory of evolution families. Haloi et al. [15] generalized the above results for nonautonomous differential equations with deviated arguments by the use of theory of analytic semigroup and Banach fixed point theorem. Alka et al. [16] generalized the results of [15] for instantaneous impulsive non-autonomous differential equations with iterated deviating arguments. Hamdy [17] studied sufficient conditions for controllability of autonomous Sobolev type fractional integro-differential equations with the help of Schauder's fixed point theorem and the theory of compact semigroup. Mahmudov [18] discussed the approximate controllability of autonomous fractional Sobolev type differential system in Banach space with the help of Schauder's fixed point theorem. Recently, Haloi [19] established sufficient conditions for approximate controllability of non-autonomous nonlocal delay differential systems with deviating arguments by using theory of compact semigroup and Krasnoselskii fixed point theorem.
To the best of our knowledge, no work yet available on approximate controllability of nonautonomous Sobolev type differential systems, inspired by this, we consider the system (1) to find the sufficient conditions for the approximate controllability. The remaining part of the article is organized as following. Section 2 is concerned with some basic notations and definitions, also we will introduce the expression for mild solutions of the system (1). In section 3, we will study our main results. In section 4, we will present an example to illustrate our results. In last section 5, we will discuss the conclusions.

Preliminaries
This section is concerned with some basic assumptions, definitions and theorems required to prove our objectives. For more details, we refer [7], [20] and [21]. Let us denote C(J, X) for the complete norm space of all continuous maps from J to X, for a finite constant r > 0, let Ω r = {x ∈ C(J, X) : x(t) r, t ∈ J}. L p (J, X)(1 p < ∞) is the Banach space of all Bochner integrable functions from J to X with norm x L p (J,X) = ( b 0 x(t) p dt) 1 p . Now, we impose the following restrictions (see [4], [20], [21]).
(A1) The operator A(t) is closed, domain of A(t) is dense in X and independent of t.
(A2) For Re(ϑ) 0, t ∈ J, the resolvent operator of A(t) exists and satisfies R(ϑ; t) ς |ϑ|+1 , for some positive constant ς. (A3) For each fixed τ 3 ∈ J, there are constants K|τ 1 − τ 2 | ρ for any τ 1 , τ 2 ∈ J. (S1) E is closed, bijective operator, and The assumptions (A1), (A2) imply that −A(t) generates an analytic semigroup in B(X), where the symbol B(X) stands for Banach space of all bounded linear operators on X. The closed graph theorem with the above assumptions imply that the linear operator −A(t)E −1 : X → X is bounded, and so for each t ∈ J, −A(t)E −1 generates a semigroup of bounded linear operators and hence a unique evolution system {S(t 1 , t 2 ) : 0 t 2 t 1 b} on X, which satisfies (see [14], [20], [21]): exists in strong operator topology, is strongly continuous in t 1 . Moreover, 4,20]) Let F is a uniformly Hölder continuous function on J with exponent β ∈ (0, 1], and the assumptions (A1)-(A3), (S1)-(S2) hold, then the unique solution for the linear Cauchy problem is given by Definition 1. A mild solution of (1) is a function x ∈ C(J, X) satisfying the following integral equation For the control u and initial data x 0 , use (1) is called approximately controllable on J.
Consider the linear control system: Corresponding to (4), the controllability operator is given as where V(t, s) := E −1 S(t, s), * denotes the adjoint of the operator. Notice that Γ b 0 is a bounded linear operator.
The necessary and sufficient conditions for the linear system (4) to be approximately controllable on J is that, δR(δ, Theorem 3. ( [22]) Let S is a convex bounded closed subset of a Banach space X. Suppose that F 1 , F 2 be be two X-valued operators defined on S such that such that F 1 x + F 2 y ∈ S whenever x, y ∈ S, F 1 is continuous and compact, and F 2 is contraction map. Then F 1 +F 2 has a fixed point in S.

Main results
In this section, we prove the existence of mild solutions and approximate controllability of (1). For x ∈ C(J, X), consider the control function for the system (1) as following : For any λ > 0, define F λ on C(J, X) as following: where Now, we state the assumptions that are useful to prove our objective.
(H1) S(t, s), is a compact evolution system whenever t − s > 0 (0 s < t b). (H2) The function F(·, x) from J to X is Lebesgue measurable for every fixed x ∈ X, and the function F(t, ·) from X to X is continuous for every fixed t ∈ J, and for all ̺ ∈ J, η 1 , η 2 , ∈ X , we have for some constant L 1 > 0. (H3) The function G from C(J, X) to D(E) is continuous and there is a constant L 2 > 0 such that For convenience, we use the following notations: Lemma 1. If the assumption (H2) holds, then for x ∈ Ω r and ̺ ∈ J we have ̺ 0 F(η, x(η)) dη K 1 .
Proof. By assumption (H2), we get Theorem 4. Let the assumptions (H1)-(H4) hold and the functions E(G(0)) is bounded, then a mild solution to the system (1) exists, provided that Proof. The proof is divided into the following steps : Step I: For λ > 0, we have a constant R (depends on λ), satisfying F λ (Ω R ) ⊂ Ω R . For any positive constant r and x ∈ Ω r , if t ∈ J, then by using (6), (H3) and Lemma (1), we have and from (8), (11), we obtain This implies, for large enough r > 0, F λ (Ω r ) ⊂ Ω r holds.
Step III: Ψ λ is continuous in Ω R . Consider {x n } be a sequence in Ω R with lim n→∞ x n = x in Ω R . From continuity of nonlinear term F with respect to state variable, we have lim n→∞ F(η, x n (η)) = F(η, x(η)), for each η ∈ J.
Now, we are ready to discuss the approximate controllability of the system (1). In order to prove it, the following hypotheses are also required: (H5) δR(δ, Γ b 0 ) → 0 whenever δ → 0 + in strong operator topology. (H6) There exist constants L 3 > 0 and L 4 > 0, such that Proof. Theorem 4 guaranteed that F λ has a fixed point in Ω R . Let x λ is a mild solution of (1) under the control u λ (t, x λ ) given by (6) and satisfies where According to the compactness of E −1 , S(t, s), and the uniform boundedness of EG, we see that there exists a subsequence of {V(b, 0)EG(x λ ) : λ > 0}, still denoted by it, converges to some x g ∈ X as λ → 0. Since F is uniformly bounded, we get b 0 F(η, x λ (η)) 2 dη L 2 4 b.
The operator E can be written as following (see [5]) (1 + n 2 ) < x, e n > e n , x ∈ D(E). (22) Furthermore for x ∈ X, we have which is compact. So, the operator −A(t)E −1 generates a compact evolution system of bounded linear operators that is given as Hence assumptions (H1), (H4) hold. By putting x(t) = x(t, ·) which means x(t)(z) = x(t, z), t ∈ [0, 1], z ∈ [0, π] and u(t) = µ(t, ·) is continuous. Let the bounded linear operator B : U → X is defined as Bu(t)(z) = µ(t, z). Further F(t, x(t))(z) = sin x(t, z), So, the system (19) can be formulated into the abstract form of (1). Note that EG(x) = 2e t c(1+e t ) cos x. Observe that the functions F, G satisfies the assumptions (H2), (H3), and also F, EG are uniformly bounded. Now it is needed to check the approximately controllability of the associated linear system, for this we show that where V(t, s) = E −1 S(t, s). Notice that S and E −1 are self adjoint. Indeed, This implies that the condition (25) holds, and hence the assumption (H5). Thus by Theorem 5, the system (19) is approximately controllable on J.

Conclusion
In this work, we have obtained that the mild solutions for non-autonomous Sobolev differential equations with nonlocal condition exist mainly by the help of evolution system of bounded linear operators and Krasnoselskii fixed point technique. Also we have determined the sufficient conditions for approximate controllability by using the controllability of corresponding linear system. The results developed in this article can be extended to the study of existence of mild solutions and approximate controllability for neutral and impulsive differential systems. Moreover the obtained results also can be generalized for fractional Sobolev, neutral and impulsive differential systems.