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

Let ugug be the unique solution of a parabolic variational inequality of second kind, with a given gg. Using a regularization method, we prove, for all g1g1 and g2g2, a monotony property between μug1+(1−μ)ug2μug1+(1−μ)ug2 and uμg1+(1−μ)g2uμg1+(1−μ)g2 for μ∈[0,1]μ∈[0,1]. This allowed us to prove the existence and uniqueness results to a family of optimal control problems over gg for each heat transfer coefficient h>0h>0, associated with the Newton law, and of another optimal control problem associated with a Dirichlet boundary condition. We prove also, when h→+∞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.