Existence, uniqueness, and convergence of optimal control problems governed by parabolic variational inequalities of second kind

M. BOUKROUCHE D.A. TARZIA

Berlin

Congreso; 25th IFIP TC 7 Conference on System Modeling and Optimization; 2013

Technical Univ. of Berlin

I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term $g$ through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over $g$ for each parameter $h>0$, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot´s inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter $h$. We also prove, when $h ightarrow +infty$, 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. II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems. III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter $h>0$, but in this case the convergence when $h ightarrow +infty$ is still an open problem.