Boundary Controllability of Nonlinear Fractional Integrodifferential Systems
© Hamdy M. Ahmed. 2010
Received: 21 July 2009
Accepted: 11 January 2010
Published: 4 February 2010
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.
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
2. Main Result
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
and the solution of (1.1) is given by
Thus (2.6) is well defined and it is called a mild solution of system (1.1).
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 .
By using conditions (H2)–(H6), we get
Consider the boundary control fractional integropartial differential system
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
We also introduce the nonlinear operator defined by
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