Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 787092, 12 pages
http://dx.doi.org/10.1155/2014/787092
Research Article

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

Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand 247667, India

Received 12 June 2014; Revised 10 September 2014; Accepted 15 September 2014; Published 2 October 2014

Academic Editor: Athanasios N. Yannacopoulos

Copyright © 2014 Sanjukta Das et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.