Numerical Solution for a Controlled Nonconvex Sweeping Process

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.