Strong Stationarity Conditions for Optimal Control Problems Governed by a Rate-Independent Evolution Variational Inequality

TL;DR

This paper establishes strong stationarity conditions for optimal control problems governed by rate-independent evolution variational inequalities, providing a primal-dual multiplier system equivalent to Bouligand stationarity.

Contribution

It introduces a novel approach using Hadamard directional differentiability and temporal polyhedricity to derive strong stationarity conditions for these complex control problems.

Findings

Derived primal-dual multiplier system for optimality conditions

Extended classical ideas with temporal polyhedricity

Compared new conditions with existing elliptic obstacle problems

Abstract

We prove strong stationarity conditions for optimal control problems that are governed by a prototypical rate-independent evolution variational inequality, i.e., first-order necessary optimality conditions in the form of a primal-dual multiplier system that are equivalent to the purely primal notion of Bouligand stationarity. Our analysis relies on recent results on the Hadamard directional differentiability of the scalar stop operator and a new concept of temporal polyhedricity that generalizes classical ideas of Mignot. The established strong stationarity system is compared with known optimality conditions for optimal control problems governed by elliptic obstacle-type variational inequalities and stationarity systems obtained by regularization.