# American Institute of Mathematical Sciences

December  2017, 10(6): 1467-1485. doi: 10.3934/dcdss.2017076

## Optimal control of a rate-independent evolution equation via viscous regularization

 1 University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria 2 Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes - CNR, via Ferrata 1, 27100 Pavia, Italy 3 Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany 4 Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany

* Corresponding author: D. Wachsmuth.

Received  July 2016 Revised  October 2016 Published  June 2017

Fund Project: The second and third author were supported by DFG grants within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged.

We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.

Citation: Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
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