Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

Sanjukta Das,Dwijendra N Pandey,N Sukavanam

doi:10.1155/2014/787092

Sanjukta Das, Dwijendra N Pandey + Show 1 more

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https://doi.org/10.1155/2014/787092

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Abstract

This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.

Highlights

Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth
Second order neutral differential equations model variational problems in calculus of variation and appear in the study of vibrating masses are attached to an electric bar
Impulsive differential equations are known for their utility in simulating processes and phenomena subject to short term perturbations during their evolution

Summary

Introduction

Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth. Much attention is paid to partial functional differential equation with state dependent delay. The literature related to state dependent delay mostly deals with functional differential equations in which the state belongs to a finite dimensional space. The study of partial functional differential equations with state dependent delay is neglected. This is one of the motivations of our paper. The inverse of it does not exist if the state space X is infinite dimensional [17] Another available method in the literature involves the invertibility of operator (αI+Γ0T), where Γ0T is the controllability Gramian and a limit condition which is difficult to check and apply in practical real world problems. B is a bounded linear operator on a Hilbert space U

Preliminaries

Existence and Uniqueness of Mild Solution

Approximate Controllability

Examples