Preprint 15-2009

Optimal control of a class of variational inequalities of the second kind

Author(s) : Juan Carlos De Los Reyes

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 15-2009

MSC 2000

49J40 Variational methods including variational inequalities

49K30 Optimal solutions belonging to restricted classes

Abstract :
Optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated. Applications include the optimal control of simplified friction, visco-plastic fluid flow and thin plate deformation. Based on a Tikhonov regularization of the dual problem, a family of primal-dual regularized control problems is introduced and strong convergence of the regularized solutions towards the solution of the original control problem is verified. For each regularized problem an optimality system is derived and an optimality condition for the original control problem is obtained as limit of the regularized optimality systems. The resulting optimality system includes complementarity relations between the variables involved. Since the regularized optimality systems involve Newton differentiable functions, a semi-smooth Newton algorithm is proposed and its numerical performance investigated.

Keywords : Optimal control, variational inequalities of the second kind, primal-dual regularization, semi-smooth Newton methods.