A STUDY ON EXISTENCE OF SOLUTION AND CONTROLLABILITY OF DELAY DIFFERENTIAL SYSTEMS

Abstract

Controllability of distributed parameter systems, essentially of dynamical systems governed by partial di erential equations, has evolved into a widely researched topic in less than three decades. Despite generating a distinctive identity and philosophy as a part of the theory of dynamical systems, this research eld has played a signi - cant role in the advancement of the extensive theory of partial di erential equations. In last few decades, control theory has contributed enormously to study of realistic problems of elasticity such as thermoelasticity, aeroelasticity, problems depicting interactions between uids and elastic structures and real world problems of uid dynamics, to name but a few. Such real world problems present new mathematical challenges. For instance, the mathematical foundations of basic theoretical issues have to be enriched, along with the development of conceptual insights signi cant to the designers and the practitioners. This poses novel challenges that need to be addressed. In our present work we focuss on the existence, uniqueness and controllablity of nonlinear functional di erential equations. We use theory of semigroup, cosine family, measure of noncompactness and xed point theorems to obtain the results. The results can be applied to a class of functional di erential equations, appearing in the mathematical models of several physical phenomena to which the prototype of partial di erential equations modeling the phenomena, belongs. The layout of the thesis, containing 10 chapters, is as follows. Chapter 1 is introductory in nature. The delay di erential equations and their applications are discussed. The objective of work done, current status of the eld and layout of the thesis is also presented in this chapter. Chapter 2 illustrates some basic properties of semigroup theory, cosine family, measure of noncompactness, controllability, fractional and stochastic di erential equations. In chapter 3 we study a functional di erential equation with deviating argument and nite delay to establish that it is approximately controllable. The results of this chapter are published as 8Approximate Controllability of a Functional Di erential Equation with Deviated Argument0 in Nonlinear Dynamics and Systems Theory, Infor Math, volume 14, no. 3, (2014), 265-277. ii In chapter 4 existence of mild solution of a second order partial neutral differential equation with state dependent delay and non-instantaneous impulses is investigated. We use Hausdor measure of noncompactness and Darbo Sadovskii xed point theorem to prove the existence. The results of this chapter are published as 8Existence of Solution for a Second-Order Neutral Di erential Equation with State Dependent Delay and Non-instantaneous Impulses0 in International Journal of Nonlinear Science, World Scienti c, volume 18, no.2, (2014), 145-155. Chapter 5 consists of two parts. The rst part deals with the existence of mild solution of an instantaneous impulsive second order di erential equation with state dependent delay. In second part non-instantaneous impulsive conditions are studied. We introduce new non-instantaneous impulses with xed delays. The results of this chapter are in revision as 8Existence of Solution of Impulsive Second-Order Neutral Integro-Di erential Equation with State Delay0 in Journal of Integral Equations and Applications. In chapter 6 we establish the existence and uniqueness of mild solution and the approximate controllability of a second order neutral partial di erential equation with state dependent delay. The conditions for approximate controllability are investigated for the distributed second order neutral di erential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. The results of this chapter are published as 8Approximate Controllability of a Second Order Neutral Di erential Equation with State Dependent Delay0 in Di erential Equations and Dynamical Systems, Springer, DOI 10:1007=s12591 􀀀 014 􀀀 0218 􀀀 6; (2014): Chapter 7 is divided in two parts. In the rst part we study a second order neutral di erential equation with state dependent delay and non-instantaneous impulses. The existence and uniqueness of the mild solution are investigated via Hausdor measure of non-compactness and Darbo Sadovskii xed point theorem. In the second part the conditions for approximate controllability are investigated for the neutral second order system under the assumption that the corresponding linear system is approximately controllable. A simple range condition is used to prove iii approximate controllability. The results of this chapter are published as 8Existence of Solution and Approximate Controllability for Neutral Di erential Equation with State Dependent Delay0 in International Journal of Partial Di erential Equations, Hindawi, volume 2014 (2014); Article ID 787092; 12 pages. In chapter 8 we study a fractional neutral di erential equation with deviating argument to establish the existence and uniqueness of mild solution. The approximate controllability of a class of fractional neutral di erential equation with deviating argument is discussed by assuming a simple range condition. The results of this chapter are published as 8Approximate Controllability of a Fractional Neutral System with Deviated Argument in Banach Space0 in Di erential Equations and Dynamical Systems, Springer, DOI : 10:1007=s12591􀀀015􀀀0237􀀀 y; (2015): In chapter 9 the approximate controllability of an impulsive fractional stochastic neutral integro-di erential equation with deviating argument and in nite delay is studied. The control parameter is also included inside the nonlinear term. Only Schauder xed point theorem and a few fundamental hypotheses are used to prove our result. The results of this chapter are published as 8Approximate controllability of an impulsive neutral fractional stochastic di erential equation with deviated argument and in nite delay0 in Nonlinear Studies, volume 22, no. 1, 1-16, (2015), CSP - Cambridge, UK; I&S - Florida, USA. In chapter 10 the existence, uniqueness and convergence of approximate solutions of a stochastic fractional di erential equation with deviating argument is established. Analytic semigroup theory is used along with xed point approach. Then we investigate Faedo-Galerkin approximation of solution and establish some convergence results. The results of this chapter are accepted for publication as 8Approximations of Solutions of a Fractional Stochastic Di erential Equations with Deviated Argument0 in Journal of Fractional Calculus and Applications in 2015.