Exact Controllability of Hemivariational Inequalities in Banach spaces

Abstract

The paper is concerned with the exact controllability of the problems described by an evolution of hemivariational inequalities within the framework of reflexive state spaces and uniformly convex control spaces, where the controls are drawn from the space \(L^p(I, U)\), \(~1<p<\infty \). We first introduce an appropriate definition for solutions to the hemivariational inequality problem, as this has not been previously established in the literature. Using this solution framework, we demonstrate that the solutions of the associated differential inclusion problem involving the Clarke subdifferential operator also serve as solutions to the original problem. Consequently, we establish the exact controllability of the original problem through the exact controllability of the corresponding differential inclusion problem. This work presents a novel approach by assuming that the control space U is a uniformly convex Banach space, which helps resolve challenges related to convexity in constructing the necessary control–a difficulty that does not arise when U is a separable Hilbert space.