Invited Session Tue.3.MA 313

Tuesday, 15:15 - 16:45 h, Room: MA 313

Cluster 3: Complementarity & variational inequalities [...]

MPECs in function space II


Chair: Christian Meyer and Michael Hinterm├╝ller



Tuesday, 15:15 - 15:40 h, Room: MA 313, Talk 1

Stanislaw Migorski
An optimal control problem for a system of elliptic hemivariational inequalities


In this paper we deal with a system of two hemivariational inequalities which is a variational formulation of a boundary value problem for two coupled elliptic partial differential equations. The boundary conditions in the problem are described by the Clarke subdifferential multivalued and nonmonotone laws. First, we provide the results on existence and uniqueness of a weak solution to the system. Then we consider an optimal control problem for the system, we prove the continuous dependence of a solution on the control variable, and establish the existence of optimal solutions. Finally, we illustrate the applicability of the results in a study of a mathematical model which describes the static frictional contact problem between a piezoelectric body and a foundation.



Tuesday, 15:45 - 16:10 h, Room: MA 313, Talk 2

Juan Carlos De los Reyes
Optimality conditions for control problems of variational inequalities of the second kind


In this talk we discuss optimality conditions for control problems governed by a class of variational inequalities of the second kind. Applications include the optimal control of Bingham viscoplastic materials and simplified friction problems. If the problem is posed in Rn an optimality system has been derived by J. Outrata (2000). When considered in function spaces, however, the problem presents additional difficulties.
We propose an alternative approximation approach based on a Huber type regularization of the governing variational inequality. By using a family of regularized optimization problems and performing an asymptotic analysis, an optimality system for the original optimal control problem (including complementarity relations between the variables involved) is obtained.
We discuss on the gap between the function space optimality system and the finite-dimensional one, and explore sufficient conditions in order to close the gap.



Tuesday, 16:15 - 16:40 h, Room: MA 313, Talk 3

Gerd Wachsmuth
Optimal control of quasistatic plasticity

Coauthors: Roland Herzog, Christian Meyer


An optimal control problem is considered for the variational inequality
representing the stress-based (dual) formulation of quasistatic
elastoplasticity. The linear kinematic hardening model and the von Mises
yield condition are used. By showing that the VI can be written as an
evolutionary variational inequality, we obtain the continuity of the
forward operator. This is the key step to prove the existence of minimizers.
In order to derive necessary optimality conditions, a family of time
discretized and regularized optimal control problems is analyzed. By
passing to the limit in the optimality conditions for the regularized
problems, necessary optimality conditions of weakly stationarity type
are obtained.
We present a solution method which builds upon the optimality system of
the time discrete and regularized problem. Numerical results which
illustrates the possibility of controlling the springback effect.


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