Existence, Uniqueness, and Convergence of optimal control problems associated with Parabolic variational inequalities of the second kind

Let ug the unique solution of a parabolic variational inequality of second kind, with a given g. Using a regularization method, we prove, for all g1 and g2, a monotony property between μug1 + (1 − μ)ug2 and uμg1+(1−μ)g2 for μ ∈ [0, 1]. This allowed us to prove the existence and uniqueness results to a family of optimal control problems over g for each heat transfer coefficient h > 0, associated to the Newton law, and of another optimal control problem associated to a Dirichlet boundary condition. We prove also , when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition.