%0 Conference Proceedings
%T On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities
%+ Université Jean Monnet - Saint-Étienne (UJM)
%+ Universidad Austral
%+ Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET)
%A Boukrouche, Mahdi
%A Tarzia, Domingo, A.
%Z Part 2: Control of Distributed Parameter Systems
%< avec comité de lecture
%( IFIP Advances in Information and Communication Technology
%B 25th System Modeling and Optimization (CSMO)
%C Berlin, Germany
%Y Dietmar Hömberg
%Y Fredi Tröltzsch
%I Springer
%3 System Modeling and Optimization
%V AICT-391
%P 76-84
%8 2011-09-12
%D 2011
%R 10.1007/978-3-642-36062-6_8
%K convex combination of solutions
%K regularization method
%K optimal control problems
%K strict convexity of cost functional
%K Parabolic variational inequalities
%Z Computer Science [cs]Conference papers
%X 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 <em>g</em> 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 <em>h</em> > 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 <em>h</em> → + ∞, 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 <em>h</em> > 0, but in this case the convergence when <em>h</em> → + ∞ is still an open problem.
%G English
%Z TC 7
%2 https://inria.hal.science/hal-01347525/document
%2 https://inria.hal.science/hal-01347525/file/978-3-642-36062-6_8_Chapter.pdf
%L hal-01347525
%U https://inria.hal.science/hal-01347525
%~ UNIV-ST-ETIENNE
%~ IFIP
%~ IFIP-AICT
%~ IFIP-TC
%~ IFIP-TC7
%~ IFIP-CSMO
%~ IFIP-AICT-391
%~ UDL