A nonsmooth maximum principle for a controlled nonconvex sweeping process

Abstract

For an optimal control problem governed by a controlled nonconvex sweeping process, we provide, using an exponential penalization technique, existence of solution and nonsmooth necessary conditions in the form of the Pontryagin maximum principle. Our results generalize known theorems in the literature, including those in [3] and [23], in several directions. Indeed, the main feature in our sweeping process inclusion is the presence of the subdifferential of a function φ, that is C1,1 in the interior of its domain, instead of the usual normal cone (the subdifferential of the indicator function). Moreover, no convexity is assumed on the function φ and its domain or on the set f(t, x, U), and our control mapping f is merely assumed to be Lipschitz in x.