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Let $u_{g}$ the unique solution of a parabolic variational inequality ofsecond 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 $\mu \in [0, 1]$. Thisallowed us to prove the existence and uniqueness results to a family of optimalcontrol problems over $g$ for each heat transfer coefficient $h>0$, associatedto the Newton law, and of another optimal control problem associated to aDirichlet boundary condition. We prove also, when $h\to +\infty$, the strongconvergence of the optimal controls and states associated to this family ofoptimal control problems with the Newton law to that of the optimal controlproblem associated to a Dirichlet boundary condition.

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