Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind

Abstract

Let ug be 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 with the Newton law, and of another optimal control problem associated with a Dirichlet boundary condition. We prove also, when h → +∞, the strong convergence of the optimal controls and states associated with this family of optimal control problems with the Newton law to that of the optimal control problem associated with a Dirichlet boundary condition.