Optimal control of a rate-independent system constrained to parametrized balanced viscosity solutions

TL;DR

This paper studies an optimal control problem for a complex rate-independent system using parametrized balanced viscosity solutions, establishing solution set compactness to ensure the existence of optimal controls.

Contribution

It introduces a framework for controlling rate-independent systems constrained to BV solutions and proves compactness of solution sets for existence results.

Findings

Established existence of optimal controls via solution set compactness.

Analyzed BV solutions as limits of viscous regularizations.

Provided a theoretical foundation for control of rate-independent systems.

Abstract

We analyze an optimal control problem governed by a rate-independent system in an abstract infinite-dimensional setting. The rate-independent system is characterized by a nonconvex stored energy functional, which depends on time via a time-dependent external loading, and by a convex dissipation potential, which is assumed to be bounded and positively homogeneous of degree one. The optimal control problem uses the external load as control variable and is constrained to normalized parametrized balanced viscosity solutions (BV solutions) of the rate-independent system. Solutions of this type appear as vanishing viscosity limits of viscously regularized versions of the original rate-independent system. Since BV solutions in general are not unique, as a main ingredient for the existence of optimal solutions we prove the compactness of solution sets for BV solutions.