Optimization of a Perturbed Implicit Sweeping Process by Constrained Discontinuous Controls

Abstract

This paper studies optimal control problems governed by an implicit sweeping process with velocity-dependent constraint geometry, where the dynamics are described by a differential inclusion involving the normal cone to a moving convex polyhedral set. Such systems form a challenging class of nonsmooth differential inclusions due to the implicit coupling between the state, velocity, and constraint reaction. To address these difficulties, we use a method of discrete approximations together with tools of variational analysis and generalized differentiation. This approach allows us to derive necessary optimality conditions for local minimizer in discrete-time problems and subsequently pass to the limit to obtain optimality conditions for the original continuous-time system. The obtained results extend existing optimal control theory for sweeping processes to a class of implicit dynamical systems with state-and velocity-dependent constraints.