A Maximum Principle for Optimal Control Problems Involving Sweeping Processes with a Nonsmooth Set

Abstract

We generalize a maximum principle for optimal control problems involving sweeping systems previously derived in de Pinho et al. (Optimization 71(11):3363–3381, 2022, https://doi.org/10.1080/02331934.2022.2101111) to cover the case where the moving set may be nonsmooth. Noteworthy, we consider problems with constrained end point. A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in de Pinho et al. (Set Valued Var Anal 27:523–548, 2019, https://doi.org/10.1007/s11228-018-0501-8).

Numerical Solution for a Controlled Nonconvex Sweeping Process

“…For (x 1 , x 2 , x 3 ) ∈ Γ , we have ⟨∇ψ 1 (x 1 , x 2 , x 3 ), ∇ψ 2 (x 1 , x 2 , x 3 )⟩ = −28 < 0, and hence, the maximum principle of[20] cannot be applied to this sweeping set C.…”