We consider an antiplane contact problem modeling the friction between a nonlinearly elastic body of Hencky type and a rigid foundation. We discuss the well-posedness of the model by considering two friction laws. Firstly, Tresca’s law is used to describe the friction force and leads to a variational inequality. Alternatively, a regularizing power law with a positive exponent r is considered and gives, from the mathematical point of view, a variational equation. In both contexts, we address a boundary optimal control problem by minimizing, on a nonconvex set, a cost functional with two arguments. We show the existence of at least one optimal pair for each problem. Finally, we deliver some convergence results proving that the optimal solution of the regular problem tends, when r goes to zero, to an optimal solution of the first one.
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