Controllability Analysis of a Class of Differential-Variational Inequalities with Nonlocal Conditions

Abstract. This paper is devoted to the controllability analysis of a class of differential variational inequalities subject to nonlocal initial conditions in Hilbert spaces. Our main contributions are twofold, addressing both exact and approximate controllability. First, we establish sufficient conditions for exact controllability by leveraging a fixed-point principle for multivalued condensing mappings, combined with the theory of measures of noncompactness. Second, recognizing the practical limitations of exact controllability for many infinite-dimensional systems, we develop two distinct theorems for approximate controllability. These results are derived from different methodologies: one employs a fixed-point argument based on the resolvent operator technique, while the other utilizes a perturbation approach. Finally, the applicability of our abstract framework is demonstrated by analyzing the approximate controllability of a coupled elliptic-parabolic partial differential system with mixed boundary conditions. This example illustrates the effectiveness of our results, particularly for systems where exact controllability is known to fail.