%0 Journal Article
%T Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay
%A Sanjukta Das
%A Dwijendra N. Pandey
%A N. Sukavanam
%J International Journal of Partial Differential Equations
%D 2014
%R 10.1155/2014/787092
%X 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. 1. Introduction Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth. Neutral differential equations are functional differential equations in which the highest order derivative of the unknown function appears both with and without derivatives. 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. Discrete perturbations are negligible to the total duration of the process which have been studied in [1¨C6]. However noninstantaneous impulses are recently studied by Ahmad [7]. Stimulated by their numerous applications in mechanics, electrical engineering, medicine, ecology, and so forth, noninstantaneous impulsive differential equations are recently investigated. Recently, much attention is paid to partial functional differential equation with state dependent delay. For details see [7¨C12]. As a matter of fact, in these papers their authors assume severe conditions on the operator family generated by , which imply that the underlying space
%U http://www.hindawi.com/journals/ijpde/2014/787092/