Abstract
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 → + ∞, 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 → + ∞ is still an open problem.
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Amassad, A., Chenais, D., Fabre, C.: Optimal control of an elastic contact problem involving Tresca friction law. Nonlinear Analysis 48, 1107–1135 (2002)
Barbu, V.: Optimal control of variational inequalities. Research Notes in Mathematics, vol. 100. Pitman (Advanced Publishing Program), Boston (1984)
Ben Belgacem, F., El Fekih, H., Metoui, H.: Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN 37, 833–850 (2003)
Boukrouche, M., El Mir, R.: On a non-isothermal, non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions. Nonlinear Analysis: Real World Applications 9(2), 674–692 (2008)
Boukrouche, M., Tarzia, D.A.: On a convex combination of solutions to elliptic variational inequalities. Electro. J. Diff. Equations (31), 1–10 (2007)
Boukrouche, M., Tarzia, D.A.: Convergence of distributed optimal controls for second kind parabolic variational inequalities. Nonlinear Analysis: Real World Applications 12(4), 2211–2224 (2011)
Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–162 (1972)
Chipot, M.: Elements of nonlinear Analysis. Birkhäuser Advanced Texts (2000)
De Los Reyes, J.C.: Optimal control of a class of variational inequalities of the second kind. SIAM J. Control Optim. 49, 1629–1658 (2011)
Duvaut, G., Lions, J.L.: Les inéquations en Mécanique et en Physique. Dunod, Paris (1972)
Gariboldi, C.M., Tarzia, D.A.: Convergence of distributed optimal controls on the internal energy in mixed elliptic problems when the heat transfer coefficient goes to infinity. Appl. Math. Optim. 47(3), 213–230 (2003)
Gariboldi, C.M., Tarzia, D.A.: Convergence of boundary optimal controls problems with restrictions in mixed elliptic Stefan-like problems. Adv. Diff. Eq. and Control Processes 1, 113–132 (2008)
Kesavan, S., Muthukumar, T.: Low-cost control problems on perforated and non-perforated domains. Proc. Indian Acad. Sci. (Math. Sci.) 118(1), 133–157 (2008)
Kesavan, S., Saint Jean Paulin, J.: Optimal control on perforated domains. J. Math. Anal. Appl. 229, 563–586 (1997)
Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Academic Press, New York (1980)
Lions, J.L.: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968)
Menaldi, J.L., Tarzia, D.A.: A distributed parabolic control with mixed boundary conditions. Asymptotic Anal. 52, 227–241 (2007)
Mignot, F.: Contrôle dans les inéquations variationelles elliptiques. J. Functional Anal. 22(2), 130–185 (1976)
Rodrigues, J.F.: Obstacle problems in mathematical physics. North-Holland, Amsterdam (1987)
Tabacman, E.D., Tarzia, D.A.: Sufficient and/or necessary condition for the heat transfer coefficient on Γ1 and the heat flux on Γ2 to obtain a steady-state two-phase Stefan problem. J. Diff. Equations 77(1), 16–37 (1989)
Tarzia, D.A.: Una familia de problemas que converge hacia el caso estacionario del problema de Stefan a dos fases. Math. Notae 27, 157–165 (1979)
Tröltzsch, F.: Optimal control of partial differential equations: Theory, methods and applications. American Math. Soc., Providence (2010)
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Boukrouche, M., Tarzia, D.A. (2013). On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities. In: Hömberg, D., Tröltzsch, F. (eds) System Modeling and Optimization. CSMO 2011. IFIP Advances in Information and Communication Technology, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36062-6_8
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