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On the projections of multivalued maps. (English) Zbl 0885.49001
The problem considered in the paper can be described very roughly as follows. There are given an open set \(\Omega\subset{\mathbb{R}}^n,\) an integer \(m,\) a multivalued map \(\Pi:\Omega\rightarrow 2^{{\mathbb{R}}^{m\times n}}\) and the set \(\Pi'\) of square integrable selections of \(\Pi.\) The author gives sufficient conditions for inclusions of the type \(0\in \text{cl } P\Pi'\) to be valid in the case of a piecewise constant map \(\Pi\) and the map \(\Pi\) is given by a Nemitskii operator, respectively, where \(P\) is a certain projection operator. Problems of such a type occur for example in the control theory for partial differential equations, where the main part of the differential operator depends on the control parameter. An application to such a control problem for elliptic systems is given.
Reviewer: M.Goebel (Halle)

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
35J50 Variational methods for elliptic systems
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[1] GAMKRELIDZE, R. V., Principles of Optimal Control, Tbilisi University, Tbilisi, Georgia, 1975 (in Russian).
[2] WARGA, J., Optimal Control of Differential and Functional Equations, Academic Press, New York, New York, 1972. · Zbl 0253.49001
[3] IOFFE, A. D., and TICHOMIROV, V. M., Extensions of Variational Problems, Trudy Moskovskogo Matematicheskogo Obshchestva, Vol. 18, pp. 187–216, 1968 (in Russian).
[4] TARTAR, L., Problèmes de Controle des Coefficients dans les Equations aux Derivées Partielles, Lecture Notes in Economical and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 107, pp. 420–426, 1975.
[5] LURIE, K. A., The Extension of Optimization Problems Containing Controls in Coefficients, Proceedings of the Royal Society of Edinburgh, Vol. 114A, pp. 81–97, 1990. · Zbl 0723.49002
[6] RAITUMS, U. E., Optimal Control Problems for Elliptic Equations, Zinatne, Riga, Latvia, 1989 (in Russian). · Zbl 0693.49003
[7] KAMPOWSKY, W., and RAITUMS, U., Convexification of Control Problems in Evolution Equations, International Series of Numerical Mathematics, Vol. 111, pp. 13–56, 1993. · Zbl 0810.49004
[8] BHATTACHARYA, K., FIROOZYE, N. B., JAMES, R. D., and KOHN, R. V., Restrictions on Microstructure, Proceedings of the Royal Society of Edinburgh, Vol. 124A, pp. 843–878, 1994. · Zbl 0808.73063
[9] BESOV, O. V., ILIN, V. P., and NIKOLSKY, S. M., Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, Russia, 1975 (in Russian).
[10] PRUTKOVSKY, A. S., On the Solution of Stationary Maxwell’s Equations for Cylindrical Waveguides with Anisotropic Filling, Problems in Mathematical Analysis, Integral and Differential Operators, Differential Equations, Edited by V. I. Smirnov, Leningrad University, Leningrad, Russia, Vol. 4, pp. 87–95, 1973 (in Russian).
[11] DACOROGNA, B., Direct Methods in the Calculus of Variations, Springer Verlag, New York, New York, 1989. · Zbl 0703.49001
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