APPROXIMATE CONTROLLABILITY OF SEMILINEAR DELAY CONTROL SYSTEMS

Abstract

Controllability is an important area in the study of control systems. The present work deals with the approximate controllability of deterministic and stochastic semilinear delayed first order systems and fractional order systems in Banach spaces. In chapter 1, a general introduction about the control theory is given. A brief account of the related work by various authors in this direction is presented. Chapter 2, contains basic concepts and definitions of control theory and nonlinear functional analysis that will be used in subsequent chapters. In chapter 3, we studied the approximate controllability of semilinear system with state delay. Instead of a CO-semigroup associated with the mild solution of the system, we use the so-called fundamental solution. Controllability results are obtained by using sequential approach and the operator semigroup theory. In chapter 4, we discuss the approximate controllability of retarded semilinear stochastic system with nonlocal conditions. Using the infinite dimensional controllability operator the control function for the system is constructed. By using this control function, Banach fixed point theorem and stochastic analysis, some results for proposed problems in Hubert space are presented. The objective of this chapter is to study the approximate controllability of semilinear fractional stochastic control system with delay. Sufficient conditions are obtained by separating the given fractional semilinear stochastic system into two systems viz, a fractional linear stochastic system and a fractional semilinear deterministic system. To prove our results Schauder fixed point theorem has been applied. 1 11 Chapter 6 contains two sections. In the first section we studied the approximate controllability of fractional order semilinear system of order c E (1, 2] in Hubert spaces. The results of first section are obtained by using Schauder's fixed point theorem. In the second section we studied approximate controllability of fractional order semilinear delay system of order o E (1, 21. The results of second section are obtained by using the theory of strongly continuous a-order cosine family and Gronwall's Inequality. In chapter 7, we studied the approximate controllability of semilinear fractional control system of order a E (1,2] with infinite delay. The results are obtained with the help of strongly continuous a-order cosine family and sequence method. In chapter 8, we studied the approximate controllability of fractional semilinear stochastic system of order (1, 2] in L spaces. A set of sufficient conditions is obtained using the theory of strongly continuous a-order cosine family, Banach fixed point theorem and stochastic analysis techniques.