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

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.