title: Solvability of some partial functional integrodifferential equations with finite delay and optimal controls in Banach spaces.
abstract: In this work, we consider the control system governed by some partial functional integrodifferential equations with finite delay in Banach spaces. We assume that the undelayed part admits a resolvent operator in the sense of Grimmer. Firstly, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of partial functional integrodifferential infinite dimensional control systems. Secondly, it is proved that, under generally mild conditions of cost functional, the associated Lagrange problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies. Our results extend and complement many other important results in the literature. Finally, a concrete example of application is given to illustrate the effectiveness of our main results.
authors.affiliation: Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, B.P. 2390, Marrakech, Morocco.
authors.forename: Khalil
authors.initials: K
authors.lastname: Ezzinbi
authors.affiliation: Department of Mathematics, Faculty of Science, University of Buea, P.O. Box 63, Buea, Cameroon.
authors.forename: Patrice
authors.initials: P
authors.lastname: Ndambomve
keywords: Finite delay; Mild solutions; Optimal controls; Partial functional integrodifferential equations; Resolvent operator; Solvability