Approximate controllability of some nonlinear systems in Banach spaces

Nazim I Mahmudov

doi:10.1186/1687-2770-2013-50

Abstract

In this paper, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixed-point theorem under the compactness assumption of the linear operator involved. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces and heat equations.

Highlights

The problems of controllability of infinite dimensional nonlinear systems were studied widely by many authors; see [ – ] and the references therein
The approximate controllability of nonlinear systems when the semigroup S(t), t >, generated by A is compact has been studied by many authors
In this paper we study the approximate controllability of semilinear abstract systems in Banach spaces

Summary

Introduction

The problems of controllability of infinite dimensional nonlinear (fractional) systems were studied widely by many authors; see [ – ] and the references therein. Sukavanam and Kumar [ ] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder’s fixed-point theorem. To prove the approximate controllability of ( ), for each ε > and h ∈ X, we have to seek for a solution of the following equation:. The approximate controllability of ( ) is derived under the compactness assumption of the linear operator involved. It is known that if the operator L is compact, Im QLB = X, that is, linear system ( ) is not exactly controllable. It is easy to see that by Proposition all the conditions of Theorem are satisfied and ( ) is approximately controllable

Application Consider the partial differential system of the form

Conclusion

Methods

Results