Approximate controllability of nonconvex valued semilinear differential inclusion

In this paper, we consider the approximate controllability of semilinear differential inclusion featuring multivalued nonlinearities with nonconvex values. Initially, we explore the approximate controllability result by assuming the underlying state space to be a super-reflexive Banach space. Then, we relax the super-reflexivity condition and prove the approximate controllability in reflexive Banach spaces by employing the relaxation technique. Incorporating multivalued maps with nonconvex values adds complexity to the studied differential inclusions yet enhances their practical applicability, serving as the core motivation for this research endeavor. We conclude by presenting an illustrative example that satisfies the criteria outlined in this paper.