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
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
2. Main Result
System (1.1) is said to be controllable on the interval if for every there exists a control such that of (1.1) satisfies
(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)
(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
If is continuously differentiable on , then can be defined as a mild solution to be the Cauchy problem
and the solution of (1.1) is given by
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
Thus (2.6) is well defined and it is called a mild solution of system (1.1).
If hypotheses (H1)–(H6) are satisfied, then the boundary control fractional integrodifferential system (1.1) is controllable on .
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
Since is compact for every , the set is precompact in for every , Furthermore, for we have
By using conditions (H2)–(H6), we get
Thus, system (1.1) is controllable on .
Let be bounded with smooth boundary
Consider the boundary control fractional integropartial differential system
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
We also introduce the nonlinear operator defined by
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 .
- 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
- 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
- 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
- Atmania R, Mazouzi S: Controllability of semilinear integrodifferential equations with nonlocal conditions. Electronic Journal of Differential Equations 2005,2005(75):1-9.MathSciNetGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- Gorenflo R, Mainardi F: Fractional calculus and stable probability distributions. Archives of Mechanics 1998,50(3):377-388.MATHMathSciNetGoogle Scholar
- 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
- 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
- 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
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.