Proximal normal analysis approach to optimal control problems in infinite-dimensional spaces. (English) Zbl 0794.49017

Summary: We extend the Pontryagin maximum principle and the transversality conditions to a class of optimal control problems for an evolution system of parabolic type through the analysis of proximal normals to the epigraph of suitable value functions. The paper extends previous results of the same authors to nonconvex target situations.


49K20 Optimality conditions for problems involving partial differential equations
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[1] Basile, N., andMininni, M.,An Extension of the Maximum Principle for a Class of Optimal Control Problems in Infinite-Dimensional Spaces, SIAM Journal on Control and Optimization, Vol. 28, pp. 1113-1135, 1990. · Zbl 0717.49017 · doi:10.1137/0328060
[2] Arnautu, V., Barbu, V., andCapasso, V.,Controlling the Spread of a Class of Epidemics, Applied Mathematics and Optimization, Vol. 20, pp. 297-317, 1989. · Zbl 0691.49024 · doi:10.1007/BF01447658
[3] Basile, N., andMininni, M.,A Vector-Valued Optimization Approach to the Study of a Class of Epidemics, Journal of Mathematical Analysis and Applications, Vol. 155, pp. 485-498, 1991. · Zbl 0727.92024 · doi:10.1016/0022-247X(91)90014-Q
[4] Clarke, F. H., andLoewen, P. D.,State Constraints in Optimal Control: A Case Study in Proximal Normal Analysis, SIAM Journal on Control and Optimization, Vol. 25, pp. 1440-1456, 1987. · Zbl 0637.49007 · doi:10.1137/0325080
[5] Fattorini, H. O.,A Unified Theory of Necessary Conditions for Nonlinear Nonconvex Systems, Applied Mathematics and Optimization, Vol. 15, pp. 141-185, 1987. · Zbl 0616.49015 · doi:10.1007/BF01442651
[6] Fattorini, H. O., andFrankowska, H.,Necessary Conditions for Infinite-Dimensional Control Problems, Proceedings of the 8th International Conference on Analysis and Optimization of Systems, Lecture Notes on Control and Information Science, Vol. 111, pp. 381-392, 1988.
[7] Li, X. J., andYao, Y. L.,Maximum Principle of Distributed-Parameter Systems with Time Lags, Proceedings of the Conference on Control Theory of Distributed-Parameter Systems and Applications, Edited by F. Kappel, K. Kunish, and W. Schappacher, Springer-Verlag, New York, New York, pp. 410-427, 1985.
[8] Loewen, P. D.,The Proximal Normal Formula in Hilbert Spaces, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 11, pp. 979-995, 1987. · Zbl 0647.49010 · doi:10.1016/0362-546X(87)90079-4
[9] Borwein, J. M., andStrojwas, H. M.,Proximal Analysis and Boundaries of Closed Sets in Banach Spaces, Part 1: Theory, Canadian Journal of Mathematics, Vol. 38, pp. 431-452, 1986. · Zbl 0577.46011 · doi:10.4153/CJM-1986-022-4
[10] Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983. · Zbl 0582.49001
[11] Tanabe, H.,Equation of Evolution, Pitman, London, England, 1979. · Zbl 0417.35003
[12] Pazy, A.,Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, New York, 1983. · Zbl 0516.47023
[13] Basile, N., andMininni, M.,A Proximal Normal Analysis Approach to Optimal Control Problems in Infinite-Dimensional Spaces, Dipartimento di Matematica, Università di Bari, Report, 1990.
[14] Di Blasio, G.,Linear Parabolic Evolution Equations in L p -Spaces, Annali di Matematica Pura e Applicata, Series IV, Vol. 138, pp. 55-104, 1984. · Zbl 0568.35047 · doi:10.1007/BF01762539
[15] Lions, J. L.,Contrôle Optimal des Systemes Gouvernés par des Equations aux Derivées Partielles, Dunod-Gauthier-Villars, Paris, France, 1968.
[16] Triebel, H.,Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, Holland, 1978. · Zbl 0387.46032
[17] Dieudonne’, J.,Eléments d’Analyse, Part 1: Fondements de l’Analyse Moderne, Gauthier-Villars, Paris, France, 1968.
[18] Lions, J. L.,Quelques Méthodes de Résolution des Problèmes aux Limites, Dunod-Gauthier-Villars, Paris, France, 1969.
[19] Cannarsa, P., andVespri, V.,On Maximal L p -Regularity for the Abstract Cauchy Problem, Bollettino dell’Unione Matematica Italiana, Vol. 5B, pp. 165-175, 1986. · Zbl 0608.35027
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