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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.

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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