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

Abstract

Controllability of distributed parameter systems, essentially of dymiamnical systems governe(l by partial differential equations, has evolved into a widely researched topic in less than t11r0e decades. Despite generating a (hstmctive identity and philosophy as a part of the theory of dynamical systems, this research field has played a significant role in the advancement of the extensive theory of partia.l differential equations. In last few decades, control theory has contributed enormously to study of realistic problems of elasticity such as thcrrnoelasticity, acroelasticity, problems depicting interactions between fluids and elastic structures and real world problems of fluid 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 significant to the (lesigners and the practitioners. This poses novel challenges that need to be addressed. lii our present work we focuss on the existence, uniqueness and controllablity of nonlinear functional differential equations. We use theory of sernigroup, cosine family, measure of noncompactness and fixed point theorems to ol)tain the results. The results can be applied to a class of functional differential equations, appearing in the mathematical models of several physical phenomena to which the prototype of partial differential equations modeling the phenomena., belongs. rfll(s layout of the thesis, containing 10 chapters, is as follows. Chapter 1 is introductory in nature. The delay differential equations and their applications are discussed. The objective of work done, current status of the field and layout of the t11e5is is also presented in this chapter. Chapter 2 illustrates some basic properties of semigroup theory, cosine family, measure of noncompactness, controllability, fractional and stochastic differential equations. In chapter 3 we study a functional differential equation with deviating argument and finite delay to establish that it is approximately controllable. The results of this chapter are published as 'Approximate Controllability of a Funct. ioimal l)ilferential Equation with Deviated Argument' in Nonlinear Dynamics and Systems Theory, Imifor Math, volume 14, no. 3, (2014), 265-277. In chapter 4 existence of mild solution of a second order partial neutral (hffcreutial equation with state dependent delay and non-instantaneous impulses is investigated. We use Ilausdoril measure of nonconipactness and Darbo Sadovskii fixed point theorem to prove the existence. The results of this chapter are published as 'Existence of Solution for a Second-Order Neutral Differential Equation with State Dependent Delay and Non-instantaneous Impulses' in International JournaI of Nonlinear Science, World Scientific, volume 18, no.2, (2014). 145-155. Chapter 5 consists of two parts. The first part deals with the existence of mild solution of an instantaneous impulsive second order differential equation with state dependent delay. In second part non-instantaneous impulsive conditions are studied. We introduce new non-instantaneous impulses with fixed delays. The results of this chapter are in revision as 'Existence of Solution of Impulsive Second-Order Neutral Integro-Differential Equation with State Delay' 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 differential equation with state dependent delay. The conditions for approximate controllability are investigated for the distributed second order neutral differential system with respect to the approximate controllability of the corresponding linear system in a Ihilbert space. The results of this chapter are published as 'Approximate Controllability of a Seeond Order Neutral Differential Equation with State Dependent Delay' in Differential Equations and I)ynamical Systems, Springer, DOI 10.1007/.s12591 - 014 - 0218 - 6, (2014). Chapter 7 is divided in two parts. In the first, part we study a second order neutral differential equation with state dependent delay and non-instantaneous impulses. The existence and uniqueness of the mild solution are investigated via Flausdorif measure of norl-cOlnl)actlless and Darbo Sadovskii fixed 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 Hi approximate controllability. The results of this chapter are published as 'Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay' in Internatiorial Journal of Partial Differential Equations, Hindawi, volume 2014 (2014), Article ID 787092, 12 pages. In chapter 8 we study a fractional neutral differential equation with deviating argument to establish the existence and uniqueness of mild solution. The approximate controllability of a class of fractional neutral differential equation with deviating argumdnt is discussed by assuming a simple range condition. The results of this chapter arc published as 'Approximate Controllability of a Fractional Neutral System with Deviated Argument in Banach Space' in Differential Equations and Dynamical Systems, Springer, DOI : 10.1007/812591 - 015 —0237— y, (2015). In chapter 9 the approximate controllability of an impulsive fractional stochastic neutral integro-differential equation with deviating argument and infinite delay is studied. The control parameter is also included inside the nonlinear term. Only Schauder fixed point theoremim and a few fundamental hypotheses are used to prove our result. The results of this chapter are published as 'Approximate controllability of an unpulsive neutral fractional stochastic differential equation with deviated argument and infinite delay' in Nonlinear Studies, volume 22, no. 1, 1-16, (2015), CSP - Cambridge, UK; 1&S - Florida, USA. In chapter 10 the existence, uniqueness and convergence of approximate solutions of a stochastic fractional differential equation with deviating argument is established. Analytic semigroup theory is used along with fixed 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 'Approximations of Solutions of a Fractional Stochastic Differential Equations with Deviated Argument' in Journal of Fractional Calculus and Applications in 2015.