EXACT AND APPROXIMATE CONTROLLABILITY OF SEMILINEAR CONTROL SYSTEMS

Abstract

This thesis concerns exact and approximate controllability of the following abstract semilinear delayed control systems as well as the systems described by partial differential equations. 4(0) = x' (t) = Ax(t) + Aix(t — h) + Bu(t) - F f (t, xt,u(t)); 0 _< t < T x0(0) = OM; —h < 0 < 0 (1) where x : [ —h, 7] —> V is the state which takes values in a Banach space V and 7L : [0, 7] U is the control that takes values in a Banach space U. Let C be the Banach space of all continuous functions from an interval [—h, 0] to V with supremum norm. If x is a continuous on [—h, 7] then xt is an element in C de-fined by xt(0) = set + 0); 0 c [—h, 0]. B : U —> V is a bounded linear operator and f : [0, 7] x C x U -- V is a non linear operator. A : D(A) C V -+ V is a closed (not necessarily bounded) linear operator, Al : V --- V is a bounded linear operator and (1) E C. Z = L2[0, T ; V] and Y = L2[0, T; U] are function spaces. A detailed literature survey is given in chapter 1. i 11 Chapter 2 deals with basics and preliminaries which are used in subsequent chap-ters. Chapter 3 contains some results on exact controllability of semilinear thermoelas-tic system and third order semilinear dispersion equation. Exact controllability is carried out, by converting the respective systems into an abstract equation of the form (1), without delay. The results have been proved, by splitting the non-linear parts, using the exact controllability of the associated linear systems. In literature, the exact controllability results for semilinear thermoelastic system and dispersion equation are very few compared to that of linear systems. In the existing results the exact controllability has been proved under the assumption that the associated non-linear function is Lipschitz continuous having sufficiently small Lipschitz constant. In this chapter the exact controllability has been given for an extended class of nonlinear Lipschitz continuous functions. In Chapter 4, the approximate controllability is proved for the semilinear control system of the form (1) in the case when f = f (t, xt) and Al 0. The result is proved for the important case in which operator A need not be a densely defined operator. In controllability literature, it is common to use Fixed-point or Degree theory argu-ment to establish approximate controllability of the semilinear control systems, which makes it necessary to assume certain inequality conditions involving various system constants. In contrast to it, in this chapter, the approximate controllability has been shown using a direct approach. Through this approach, it is possible to prove the 111 approximate controllability of (1) for certain class of nonlinear function f satisfying the Lipschitz condition, without assuming any inequality condition. As a particular case, the approximate controllability has been proved for controlled heat equation and system of infinite ordinary differential equations. Chapter 5 deals with the exact and approximate controllability of abstract delayed control systems governed by functional form (1). Some sufficient conditions for the exact and approximate controllability are given by assuming suitable conditions on the system operators A, A1, B and f. The controllability results are proved, when f is Lipschitz continuous, using Banach contraction principle and Zarantonello surjec-tive theorem. These results relax certain restrictions made by earlier authors. The obtained results are illustrated with the controlled heat and wave equations. Finally in Chapter 6, numerical computation of control for the controlled linear and semilinear heat equation has been carried out. A neural network and genetic al-gorithm based method of computing internal and boundary control is proposed. This method relies upon the function approximation capabilities of feed forward neural network and global optimization capabilities of genetic algorithm. Using this ap-proach, the internal control for linear and semilinear controlled heat equation as well as boundary control for two dimensional linear heat equation have been computed.