Open Access

Boundary Controllability of Nonlinear Fractional Integrodifferential Systems

Advances in Difference Equations20102010:279493

https://doi.org/10.1155/2010/279493

Received: 21 July 2009

Accepted: 11 January 2010

Published: 4 February 2010

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 [16]). 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 [1113]).

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 .

Authors’ Affiliations

(1)
Higher Institute of Engineering, El-Shorouk Academy

References

  1. Balachandran K, Anandhi ER: Controllability of neutral functional integrodifferential infinite delay systems in Banach spaces. Taiwanese Journal of Mathematics 2004,8(4):689-702.MATHMathSciNetGoogle Scholar
  2. Balachandran K, Dauer JP, Balasubramaniam P: Controllability of nonlinear integrodifferential systems in Banach space. Journal of Optimization Theory and Applications 1995,84(1):83-91. 10.1007/BF02191736MATHMathSciNetView ArticleGoogle Scholar
  3. Balachandran K, Park JY: Existence of solutions and controllability of nonlinear integrodifferential systems in Banach spaces. Mathematical Problems in Engineering 2003,2003(1-2):65-79. 10.1155/S1024123X03201022MATHMathSciNetView ArticleGoogle Scholar
  4. Atmania R, Mazouzi S: Controllability of semilinear integrodifferential equations with nonlocal conditions. Electronic Journal of Differential Equations 2005,2005(75):1-9.MathSciNetGoogle Scholar
  5. Balachandran K, Sakthivel R: Controllability of functional semilinear integrodifferential systems in Banach spaces. Journal of Mathematical Analysis and Applications 2001,255(2):447-457. 10.1006/jmaa.2000.7234MATHMathSciNetView ArticleGoogle Scholar
  6. Balachandran K, Sakthivel R: Controllability of integrodifferential systems in Banach spaces. Applied Mathematics and Computation 2001,118(1):63-71. 10.1016/S0096-3003(00)00040-0MATHMathSciNetView ArticleGoogle Scholar
  7. Balachandran K, Anandhi ER: Boundary controllability of integrodifferential systems in Banach spaces. Proceedings. Mathematical Sciences 2001,111(1):127-135. 10.1007/BF02829544MATHMathSciNetView ArticleGoogle Scholar
  8. Balachandran K, Leelamani A: Boundary controllability of abstract integrodifferential systems. Journal of the Korean Society for Industrial and Applied Mathematics 2003,7(1):33-45.Google Scholar
  9. El-Borai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons & Fractals 2002,14(3):433-440. 10.1016/S0960-0779(01)00208-9MATHMathSciNetView ArticleGoogle Scholar
  10. Gorenflo R, Mainardi F: Fractional calculus and stable probability distributions. Archives of Mechanics 1998,50(3):377-388.MATHMathSciNetGoogle Scholar
  11. El-Borai MM, El-Said El-Nadi K, Mostafa OL, Ahmed HM: Semigroup and some fractional stochastic integral equations. The International Journal of Pure and Applied Mathematical Sciences 2006,3(1):47-52.Google Scholar
  12. El-Borai MM, El-Said El-Nadi K, Mostafa OL, Ahmed HM: Volterra equations with fractional stochastic integrals. Mathematical Problems in Engineering 2004,2004(5):453-468. 10.1155/S1024123X04312020MATHView ArticleGoogle Scholar
  13. El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(3):197-211. 10.1155/S1048953304311020MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Hamdy M. Ahmed. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.