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Numerical analysis of distributed optimal control problems governed by elliptic variational inequalities

A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy $g$. It was proved the existence and uniqueness of the optimal control and its associated state system. The objective of this work is to make the numerical analysis of the above optimal control problem, through the finite element method with Lagrange's triangles of type 1. We discretize the elliptic variational inequality which define the state system and the corresponding cost functional, and we prove that there exists a discrete optimal control and its associated discrete state system for each positive $h$ (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter $h$ goes to zero.