Boundary Controllability of Nonlinear Fractional Integrodifferential Systems

Abstract

Sufficient conditions for boundary controllability of nonlinear fractional integrodifferential systems in Banach space are established. The results are obtained by using fixed point theorems. We also give an application for integropartial differential equations of fractional order.

1. Introduction

Let and be a pair of real Banach spaces with norms and , respectively. Let be a linear closed and densely defined operator with and let be a linear operator with and , a Banach space. In this paper we study the boundary controllability of nonlinear fractional integrodifferential systems in the form

(1.1)

where and is a linear continuous operator, and the control function is given in a Banach space of admissible control functions. The nonlinear operators and are given and

Let be the linear operator defined by

(1.2)

The controllability of integrodifferential systems has been studied by many authors (see [1–6]). This work may be regarded as a direct attempt to generalize the work in [7, 8].

2. Main Result

Definition 2.1.

System (1.1) is said to be controllable on the interval if for every there exists a control such that of (1.1) satisfies

To establish the result, we need the following hypotheses.

• (H1) and the restriction of to is continuous relative to the graph norm of .

• (H2) The operator is the infinitesimal generator of a compact semigroup and there exists a constant such that

• (H3) There exists a linear continuous operator such that , for all Also is continuously differentiable and for all where C is a constant.

• (H4) For all and , . Moreover, there exists a positive constant such that

• (H5) The nonlinear operators and , for satisfy

  

  (2.1)

  where and

• (H6) The linear operator from into defined by

  

  (2.2)

where is a probability density function defined on (see [9, 10]) and induces an invertible operator defined on and there exists a positive constant and such that and . Let be the solution of (1.1). Then we define a function and from our assumption we see that . Hence (1.1) can be written in terms of and as

(2.3)

If is continuously differentiable on , then can be defined as a mild solution to be the Cauchy problem

(2.4)

and the solution of (1.1) is given by

(2.5)

(see [11–13]).

Since the differentiability of the control represents an unrealistic and severe requirement, it is necessary of the solution for the general inputs Integrating (2.5) by parts, we get

(2.6)

Thus (2.6) is well defined and it is called a mild solution of system (1.1).

Theorem 2.2.

If hypotheses (H1)–(H6) are satisfied, then the boundary control fractional integrodifferential system (1.1) is controllable on .

Proof.

Using assumption (H6), for an arbitrary function define the control

(2.7)

We shall now show that, when using this control, the operator defined by

(2.8)

has a fixed point. This fixed point is then a solution of (1.1). Clearly, which means that the control steers the nonlinear fractional integrodifferential system from the initial state to in time , provided we can obtain a fixed point of the nonlinear operator .

Let and where the positive constant is given by

(2.9)

Then is clearly a bounded, closed, and convex subset of . We define a mapping by

(2.10)

Consider

(2.11)

Since and are continuous and it follows that is also continuous and maps into itself. Moreover, maps into precompact subset of . To prove this, we first show that, for every fixed , the set is precompact in . This is clear for , since . Let be fixed and for define

(2.12)

Since is compact for every , the set is precompact in for every , Furthermore, for we have

(2.13)

which implies that is totally bounded, that is, precompact in . We want to show that is an equicontinuous family of functions. For that, let Then we have

(2.14)

By using conditions (H2)–(H6), we get

(2.15)

The compactness of implies that is continuous in the uniform operator topology for Thus, the right hand side of (2.15) tends to zero as So, is an equicontinuous family of functions. Also, is bounded in , and so by the Arzela- Ascoli theorem, is precompact. Hence, from the Schauder fixed point in any fixed point of is a mild solution of (1.1) on satisfying

(2.16)

Thus, system (1.1) is controllable on .

3. Application

Let be bounded with smooth boundary

Consider the boundary control fractional integropartial differential system

(3.1)

The above problem can be formulated as a boundary control problem of the form of (1.1) by suitably taking the spaces and the operators , and as follows.

Let , , , , the identity operator and , The operator is the trace operator such that is well defined and belongs to for each and the operator is given by , where and are usual Sobolev spaces on We define the linear operator by where is the unique solution to the Dirichlet boundary value problem

(3.2)

We also introduce the nonlinear operator defined by

(3.3)

Choose and other constants such that conditions (H1)–(H6) are satisfied. Consequently Theorem 2.2 can be applied for (3.1), so (3.1) is controllable on .