SOME RESULTS ON THE CONTROLLABILITY OF SEMILINEAR CONTROL SYSTEMS

Abstract

Controllability is an important area in the study of control systems. The present work deals with exact controllability, approximate controllability of semilinear deterministic control systems and S-controllability of non-deterministic semilinear control systems with or without nonlocal initial conditions in the infinite dimensional Banach spaces. In chapter 1, a general introduction about the control theory is given. A brief account of the related work made by various authors in the direction is presented. In chapter 2, basics and preliminaries which are used in subsequent chapters are described. In chapter 3, the exact controllability of semilinear thermoelastic system is proved using the some lemmas and Schauder's fixed point theorem. Exact controllability carried out by converting the respective system into an abstract first order semilinear control system. The result has been proved by splitting the nonlinear part using the exact controllability of the associated linear system. In literature, the exact controllability has been proved under the assumption that the associated nonlinear function is Lipschitz continuous. In this chapter, the exact controllability has been given for an extended class of nonlinear monotone functions. In chapter 4, the exact controllability is proved for mixed volterra-fredholm type integrodifferential third order dispersion equation, using Banach fixed point theorem. In this work, a more general integrodifferential dispersion system is considered which covers the earlier results as particular cases. Chapter 5 deals with the S-controllability of a partially observed abstract semilinear stochastic control system with an additive Gaussian white noise disturbance in which the nonlinear function satisfies monotone condition. The results are obtained by sep-arating the given semilinear stochastic system into two systems namely a semilinear deterministic system and a linear stochastic system. In literature, the above result is 11 proved for linear system but in this work the result is extended for semilinear stochastic system. In chapter 6, the approximate controllability is proved for abstract semilinear first order control system in which operator (generated by Co-semigroup) need not be densely defined operator. In this work, there are two cases. In the first case the nonlinearity is monotone and in the second case the nonlinearity is integral contractor. In this chapter, the approximate controllability has been shown using a direct approach. Through this approach it is possible to prove the controllability without assuming any inequality condition..