Controllability of functional differential systems of Sobolev type in Banach spaces

The problem of controllability of linear and nonlinear systems represented by ordinary differential equations in finite dimensional spaces has been extensively studied. Several authors [5,6,12-14] have extended the concept to infinite dimensional systems in Banach spaces with bounded operators. Triggiani [17] established sufficient conditions for controllability of linear and nonlinear systems in Banach spaces. Exact controllability of abstract semilinear equations has been studied by Lasiecka and Triggiani [10]. Kwun et al [8] investigated the controllability and approximate controllability of delay Volterra systems by using a fixed point theorem. Recently Balachandran et al [1-3] studied the controllabilty and local null controllability of nonlinear integrodifferential systems and functional differential systems in Banach spaces and it was shown that the controllability problem in Banach spaces can be converted into one of a fixed-point problem for a single-valued mapping. The purpose of this paper is to study the controllability of Sobolev type partial functional differential systems in Banach spaces. The equation considered here serves as an abstract formulation of Sobolev type partial functional differential equations which arise in many physical phenomena [4,7,9,11,16]. Consider a nonlinear partial functional differential system of the form