Controllability for Sobolev type fractional integro-differential systems in a Banach space

In this paper, by using compact semigroups and the Schauder fixed-point theorem, we study the sufficient conditions for controllability of Sobolev type fractional integro-differential systems in a Banach space. An example is provided to illustrate the obtained results. MSC:26A33, 34G20, 93B05.


Introduction
A Sobolev-type equation appears in a variety of physical problems such as flow of fluids through fissured rocks, thermodynamics and propagation of long waves of small amplitude (see [-]). Recently, there has been an increasing interest in studying the problem of controllability of Sobolev type integro-differential systems. Balachandran and Dauer [] studied the controllability of Sobolev type integro-differential systems in Banach spaces. Balachandran and Sakthivel [] studied the controllability of Sobolev type semilinear integro-differential systems in Banach spaces. Balachandran, Anandhi and Dauer [] studied the boundary controllability of Sobolev type abstract nonlinear integro-differential systems.
In this paper, we study the controllability of Sobolev type fractional integro-differential systems in Banach spaces in the following form: where E and A are linear operators with domain contained in a Banach space X and ranges contained in a Banach space Y . The control function u(·) is in L  (J, U), a Banach space of admissible control functions, with U as a Banach space. B is a bounded linear operator from U into Y . The nonlinear operators f ∈ C(J × X, Y ), H ∈ C(J × J × X, X) and g ∈ C(J × J × X × X, Y ) are all uniformly bounded continuous operators. The fractional derivative c D α ,  < α <  is understood in the Caputo sense.

Preliminaries
In this section, we introduce preliminary facts which are used throughout this paper. http://www.advancesindifferenceequations.com/content/2012/1/167 The fractional integral of order α >  with the lower limit zero for a function f can be defined as provided the right-hand side is pointwise defined on [, ∞), where is the gamma function.
The Caputo derivative of order α with the lower limit zero for a function f can be written as If f is an abstract function with values in X, then the integrals appearing in the above definitions are taken in Bochner's sense.

The operators
The hypotheses H  , H  and the closed graph theorem imply the boundedness of the linear operator AE - : Y → Y .

Lemma . [] Let S(t) be a uniformly continuous semigroup. If the resolvent set R(λ; A) of A is compact for every λ ∈ ρ(A), then S(t) is a compact semigroup.
From the above fact, -AE - generates a compact semigroup {T(t), t ≥ } in Y , which means that there exists M >  such that Definition . The system (.) is said to be controllable on the interval J if for every x  , x  ∈ X, there exists a control u ∈ L  (J, U) such that the solution x(·) of (.) satisfies (i) For each t ∈ J, the function f (t, ·) : X → Y is continuous, and for each x ∈ X, the function f (·, x) : J → Y is strongly measurable. (ii) For each positive number k ∈ N , there is a positive function g k (·) : [, a] → R + such that the function s → (ts) -α g k (s) ∈ L  ([, t], R + ), and there exists a β >  such that (H  ) For each (t, s) ∈ J × J, the function H(t, s, ·) : X → X is continuous, and for each x ∈ X, the function H(·, ·, x) : J × J → X is strongly measurable.

Lemma . (see [])
The operators S α (t) and T α (t) have the following properties:

Controllability result
In this section, we present and prove our main result.
Proof Using the assumption (H  ), for an arbitrary function x(·), define the control It shall now be shown that when using this control, the operator Q defined by has a fixed point. This fixed point is then a solution of equation (.).
It can be easily verified that Q maps C into itself continuously. For each positive number k > , let Obviously, B k is clearly a bounded, closed, convex subset in C. We claim that there exists a positive http://www.advancesindifferenceequations.com/content/2012/1/167 number k such that QB k ⊂ B k . If this is not true, then for each positive number k, there exists a function x k ∈ B k with Qx k ∈ B k , that is, Qx k ≥ k, then  ≤  k Qx k , and so However, a contradiction. Hence, QB k ⊂ B k for some positive number k. In fact, the operator Q maps B k into a compact subset of B k . To prove this, we first show that the set V k (t) = {(Qx)(t) : x ∈ B k } is a precompact in X; for every t ∈ J: This is trivial for t = , since V k () = {x  }. Let t,  < t ≤ a; be fixed. For  < < t and arbitrary δ > ; take Since u(s) is bounded and T( α δ), α δ >  is a compact operator, then the set V ,δ is a precompact set in X for every ,  < < t, and for all δ > . Also, for x ∈ B k , using the defined control u(t) yields Therefore, as →  + and δ →  + , there are precompact sets arbitrary close to the set V k (t) and so V k (t) is precompact in X.

Now, T(t) is continuous in the uniform operator topology for t >  since T(t)
is compact, and the right-hand side of the above inequality tends to zero as t → τ . Thus, QB k is both equicontinuous and bounded. By the Arzela-Ascoli theorem, QB k is precompact in C(J, X). Hence, Q is a completely continuous operator on C(J, X). From the Schauder fixed-point theorem, Q has a fixed point in B k . Any fixed point of Q is a mild solution of (.) on J satisfying (Qx)(t) = x(t) ∈ X. Thus, the system (.) is controllable on J.

Example
In this section, we present an example to our abstract results.
We consider the fractional integro-partial differential equation in the form where c ∂ α t is the Caputo fractional partial derivative of order  < α < . Take where z n (x) = √ /π sin nx, n = , , . . . , is the orthonormal set of eigenvectors of A and (z, z n ) is the L  inner product. Moreover, for z ∈ X, we get (A  ): The nonlinear operator μ  : J × J × X → X satisfies the following two conditions: (i) For each (t, s) ∈ J × J, μ  (t, s, z) is continuous.
(A  ): The nonlinear operator μ  : J × J × X × X → Y satisfies the following three conditions: (i) For each (t, s, z) ∈ J × J × X, μ  (t, s, z) is continuous.
Define an operator f : J × X → Y by Then the problem (.) can be formulated abstractly as: