Optimal control of non-convex rate-independent systems via vanishing viscosity – The finite dimensional case

Abstract

We investigate an optimal control problem governed by the evolution of a rate-independent system in finite dimensions. The rate-independent system is determined by a (possibly) non-convex energy, which contains the controllable, external load, and a dissipation potential, which is assumed to be positively homogenous of degree one. Under the several di˙erent concepts of solutions for these rate-independent systems, we bear on the so-called normalized parametrized BV solutions and prove the existence of a globally optimal solution of the optimal control problem constrained by this notion of solution. Our main result however concerns the approximation of optimal solutions by means of viscous regularization. The crucial issue in this context is that normalized parametrized BV solutions are in general non-unique and lack regularity, whereas the viscous solutions are unique and time-continuous. With the help of an additional regularity assumption on at least one optimal solution and a tailored penalization of the energy, one can nonetheless show that global minimizers of the viscous optimal control problems converge to an optimal solution of the original problem as the viscosity parameter tends to zero.