Numerical Solution for a Controlled Nonconvex Sweeping Process

Abstract

A numerical method and the theory leading to its success are developed in this letter to solve nonstandard optimal control problems involving sweeping processes, in which the sweeping set C is non-convex and coincides with the zero-sublevel set of a smooth function having a Lipschitz gradient, and the fixed initial state is allowed to be any point of C. This numerical method was introduced by Pinho et al. (2020) for a special form of our problem in which the function whose zero-sublevel set defines C is restricted to be twice differentiable and convex, and the initial state is confined in the interior of their convex set C. The remarkable feature of this method is manifested in approximating the sweeping process by a sequence of standard control systems invoking an innovative exponential penalty term in lieu of the normal cone, whose presence in the sweeping process renders most standard methods inapplicable. For a general setting, we prove that the optimal solution of the approximating standard optimal control problem converges uniformly to an optimal solution of the original problem (see Remark 3). This numerical method is shown to be efficient through an example in which C is not convex and the initial state is on its boundary.