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

Let $u_{g}$ 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 $\mu u_{g_{1}} + (1-\mu)u_{g_{2}}$ and $u_{\mu g_{1} + (1-\mu)g_{2}}$ for $\mu \in [0, 1]$. 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 $h\to +\infty$, 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.