Relaxation in a Nonconvex Optimal Control Problem Described by an Integro-differential BV Prox-regular Sweeping Process

Abstract.

We study a nonconvex optimal control problem governed by an integro-differential sweeping process with prox-regular moving sets. The state dynamics involve a Volterra-type integral perturbation and admit solutions of bounded variation, which leads to inherent nonsmoothness and nonexistence of optimal controls for the original problem. We consider a convexification of both the control constraint and the cost integrand and show that the relaxed problem possesses optimal solutions. Moreover, we prove that the optimal value of the relaxed problem coincides with the infimum of the original nonconvex problem. A key part of the analysis establishes approximation of relaxed optimal solutions by admissible solutions of the original system, together with convergence of the associated cost values. Our results provide a relaxation framework for nonsmooth controlled dynamics with unilateral state constraints and memory effects in an infinite-dimensional Hilbert space setting.