Optimal Control of the Sweeping Process with a Nonsmooth Moving Set
Cristopher Hermosilla, Michele Palladino
Abstract
In this paper we prove a fully nonsmooth Pontryagin maximum principle for optimal control problems driven by a sweeping process with drift $\dot{x}\in f(t,x,u)-\mathcal{N}_{C(t)}(x)$. The setting we study is an optimal control problem of Mayer type in which the optimization procedure is carried out by choosing a control function $u(t)$ from a class of admissible controls $\mathcal{U}$. The choice of $u\in \mathcal{U}$ modifies the drift $f$ and the related solution $x(t)$ to the perturbed sweeping process. Here, for the first time, we are able to prove a Pontryagin maximum principle in the case in which the moving set $C(t)$ is both nonsmooth and nonconvex by using a novel exact penalization technique which is able to exploit the controllability properties of the dynamics.