Existence of solution and approximate controllability for neutral differential equation with state dependent delay
type: Item
instance of: scholarly article
title: Existence of solution and approximate controllability for neutral differential equation with state dependent delay
zbMATH Open document ID: 1314.35204
DOI: 10.1155/2014/787092
author: Sanjukta Das
author: N. Sukavanam
author: Dwijendra N. Pandey
published in: International Journal of Partial Differential Equations
publication date: 2015-03-27T00:00:00Z
review text: Summary: 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 [\textit{K. Balachandran} et al., Comput. Math. Appl. 46, No. 8--9, 1313--1324 (2003; Zbl 1054.45006)], 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 [\textit{J. P. Dauer} and \textit{N. I. Mahmudov}, J. Math. Anal. Appl. 273, No. 2, 310--327 (2002; Zbl 1017.93019)], which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.
Mathematics Subject Classification ID: 35Q93
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 6420101
zbMATH Keywords: controllability
zbMATH Keywords: neutral partial differential equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/787092
OpenAlex ID: W2045926190
Wikidata QID: Q59051642
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description: scientific article; zbMATH DE number 6420101