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

Ulisse Stefanelli; Daniel Wachsmuth; Gerd Wachsmuth

doi:10.3934/dcdss.2017076

Article Contents

2017, Volume 10, Issue 6: 1467-1485. Doi: 10.3934/dcdss.2017076

This issue Previous Article Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity Next Article Cohesive zone-type delamination in visco-elasticity

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.
* Corresponding author: D. Wachsmuth.

Received: June 30, 2016

Revised: September 30, 2016

Published: December 2017

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 49K20; Secondary: 35K87.

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