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Reliable solution of parabolic obstacle problems with respect to uncertain data. (English) Zbl 1099.35054
A parabolic initial-value variational inequality is considered in an abstract setting. To prove that this problem has a unique solution \(u\), a penalized equation is introduced and its penalization parameter is let to pass to zero. In the next part of the paper, the input data of the variational inequality are assumed to be uncertain, i.e., the convex set of test and trial functions, the initial state, the operator on the left-hand side, and the right hand-side of the inequality can depend on a parameter \(\omega \) from a compact admissible set \(A\). As a consequence, \(u\) is \(\omega \)-dependent. Moreover, a criterion functional \(\Phi \) dependent on both \(u\) and \(\omega \) is considered. The goal is to maximize \(\Phi (\omega ,u(\omega ))\) over \(A\). Still in an abstract setting, it is shown that the maximization problem has at least one solution. In the final part of the paper, the general abstract scheme is applied to a particular obstacle problem.

MSC:
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
49J40 Variational inequalities
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References:
[1] J. P. Aubin: Un théorème de compacité. C. R. Acad. Sci. Paris 256 (1963), 5042-5044. · Zbl 0195.13002
[2] V. Barbu: Optimal Control of Variational Inequalities. Pitman, Boston, 1984. · Zbl 0574.49005
[3] I. Bock, J. Lovíšek: Optimal control of a viscoelastic plate bending. Math. Nachr. 125 (1968), 135-151. · Zbl 0606.73104
[4] H. Brézis: Problèmes unilatéraux. J. Math. Pures Appl. 51 (1972), 1-168. · Zbl 0221.35028
[5] H. Brézis: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland, Amsterdam, 1973. · Zbl 0252.47055
[6] H. Brézis: Analyse fonctionelle. Masson, Paris, 1982.
[7] H. Gajewski, K. Gröger and K. Zacharias: Nichtlineare Operatorgleichungen und Operator-Differentialgleichungen. Akademie, Berlin, 1974.
[8] I. Hlaváček: Reliable solutions of elliptic boundary value problems with respect to uncertain data. Nonlinear Anal. 30 (1997), 3879-3980. · Zbl 0896.35034
[9] I. Hlaváček: Reliable solutions of linear parabolic problems with respect to uncertain coefficients. Z. Angew. Math. Mech. 79 (1999), 291-301. <a href=”http://dx.doi.org/10.1002/(SICI)1521-4001(199905)79:53.0.CO;2-N” target=”_blank”>DOI 10.1002/(SICI)1521-4001(199905)79:53.0.CO;2-N |
[10] J. L. Lions: Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris, 1968. · Zbl 0179.41801
[11] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969. · Zbl 0189.40603
[12] V. C. Litvinov: Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics. Birkhäuser-Verlag, Berlin, 2000. · Zbl 0947.49001
[13] F. Mignot, J.-P. Puel: Optimal control of some variational inequalities. SIAM J. Control Optim. 22 (1984), 466-478. · Zbl 0561.49007
[14] U. Mosco: Convergence of convex sets and solutions of variational inequalities. Adv. Math. 3 (1969), 510-585. · Zbl 0192.49101
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Masson, Paris, 1967. · Zbl 1225.35003
[16] P. Neittaanmäki, D. Tiba: Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications. Pure and Applied Mathematics. Marcel Dekker inc., New York, 1994. · Zbl 0812.49001
[17] T. I. Seidman, Hong Xing Zhou: Existence and uniqueness of optimal controls for a quasilinear parabolic equation. SIAM J. Control Optim. 20 (1982), 747-762. · Zbl 0508.49005
[18] J. Sokolowski, J. P. Zolesio: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer-Verlag, New York, 1992. · Zbl 0487.49004
[19] J. P. Yvon: Contrôle optimal de systèmes gouvernés par des inéquations variationnelles. Rapport de Recherche 22, INRIA, Paris, 1974.
[20] J. Steinbach: A Variational Inequality Approach to Free Boundary Problems with Applications in Mould Filling. Birkhäuser-Verlag, Basel, 2002. · Zbl 1011.35001
[21] G. M. Troianiello: Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York, 1987. · Zbl 0655.35002
[22] E. Zeidler: Nonlinear Functional Analysis and its Applications II/A, Linear Monotone Operators. Springer-Verlag, New York, 1990. · Zbl 0684.47028
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