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Optimal control of a class of variational inequalities of the second kind. (English) Zbl 1226.49008
Summary: Optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated. Applications include the optimal control of viscoplastic fluid flow and of simplified friction. Based on a Tikhonov regularization of the dual problem, a family of primal-dual regularized control problems is introduced, and the convergence of the regularized solutions towards a solution of the original control problem is verified. For each regularized problem an optimality condition is derived, and an optimality system for the original control problem is obtained as a limit of the regularized ones. Thanks to the structure of the proposed regularization, complementarity relations between the variables involved are derived. Since the regularized optimality systems involve Newton differentiable functions, a semismooth Newton algorithm is proposed and its numerical performance is investigated.

MSC:
49J40Variational methods including variational inequalities
49K20Optimal control problems with PDE (optimality conditions)
49K30Optimal solutions belonging to restricted classes
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
65K10Optimization techniques (numerical methods)
49M15Newton-type methods in calculus of variations
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