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

Abstract

Let u g be the unique solution of a parabolic variational inequality of second kind, with a given g . Using a regularization method, we prove, for all g 1 and g 2 , a monotony property between μ u g 1 + ( 1 − μ ) u g 2 and u μ g 1 + ( 1 − μ ) g 2 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.