APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL SEMILINEAR CONTROL SYSTEMS

Abstract

The present research work deals with the existence of solutions and approximate controllability of deterministic semilinear integer order systems with control delays and fractional order systems without delay. To derive the existence and controllability results, various techniques have been applied along with the semigroup, cosine and sine families, fractional calculus, fractional cosine family, fractional resolvent, xed point theory. Some examples are provided for the illustration of the obtained results. Some introductory matter along with literature survey on controllability of nonlinear and linear control systems of fractional and integer orders are given in Chapter 1. Basic concepts and de nitions of control theory, semigroup theory, cosine family, fractional calculus, fractional cosine family and nonlinear functional analysis which are utilized in forthcoming chapters, are given in Chapter 2. In Chapter 3, the existence of mild solutions of rst-order retarded semilinear system with control delay is proved under the locally Lipschitz continuity of nonlinear function and a xed point theorem. Then the approximate controllability of semilinear system is proved provided that the associated linear system without delay is approximately controllable. Controllability results are obtained by using the method of steps and semigroup theory. The results of this chapter are illustrated with controlled heat equation.