Viscous Approximation of Optimal Control Problems Governed by Rate-Independent Systems with Non-Convex Energies

TL;DR

This paper introduces a viscous regularization approach for solving optimal control problems governed by non-convex, rate-independent systems, demonstrating convergence of solutions from the regularized to the original problem.

Contribution

It develops a viscous approximation method for complex rate-independent control problems with non-convex energies and analyzes its convergence properties.

Findings

Viscous regularization simplifies solving non-smooth ODE constraints.

Sequences of optimal solutions to regularized problems converge to solutions of the original problem.

The method is effective when the original problem admits a continuous optimal state.

Abstract

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert space, which is substantially easier to solve than the original optimal control problem. The convergence properties of the viscous regularization are investigated. It is shown that every sequence of globally optimal solutions of the viscous problems admits a (weakly) converging subsequence whose limit is a globally optimal solution of the original problem, provided that the latter admits at least one optimal solution with an optimal state that is continuous in time.