Necessary Conditions of Optimality for a Time-Optimal Bi-level Sweeping Control Problem

Abstract

This article concerns a simple instance of an optimal control problem paradigm combining bi-level optimization with sweeping processes, initially investigated in Khalil and Lobo Pereira (2019). This class of problems arises, for instance, in structured crowd motion control problems in a confined space. We propose a specific class of time-optimal bi-level problem with sweeping process dynamics represented in terms of a truncated normal cone at the lower level. We establish the necessary optimality conditions of the Maximum Principle of Pontryagin type in the Gamkrelidze’s form. Two techniques are at the core of the analysis: a) the smooth approximation of the low level sweeping control system, thereby avoiding the absence of Lipschitzianity inherent to the sweeping process, and, b) the flattening of the bi-level structure to a single level problem by using a exact penalization technique involving the value function of the low level problem to incorporate its inherent constraint in the bi-level structure. Necessary optimality conditions are applied to the resulting approximate flattened problem, and the main result of this article is obtained by passing to the limit.