Abstract
This paper deals with the necessary conditions satisfied by the optimal control of a variational inequality governed by a semilinear operator of elliptic type and a maximal monotone operatorβ in ℝ × ℝ. A nonclassical smoothing ofβ allows us to formulate a perturbed problem for which the original control is anε-solution. By considering the spike perturbations and applying Ekeland's principle we are able to state approximate optimality conditions in Pontryagin's form. Then passing to the limit we obtain some optimality conditions for the original problem, extending those obtained for semilinear elliptic systems and for variational inequalities.
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References
Agmon S., Douglis A., Nirenberg L. (1959) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math. 12:623–727
Barbu V. (1984) Optimal Control of Variational Inequalities. Lecture Notes in Mathematics, Vol. 100. Pitman, London
Barbu V., Tiba D. (1989) Optimal control of abstract variational inequalities. Proc. Fifth IFAC Symp. Control of Distributed Parameter Systems, June 26–29, Perpignan (A. El Jai, ed.)
Bonnans J. F., Casas E. (1991) Un principe de Pontryagine pour le contrôle des systèmes semilinéaires elliptiques. J. Differential Equations 89, to appear
Bonnans J. F., Casas E. (1989) Maximum principles in the optimal control of semilinear elliptic systems. Proc. Fifth IFAC Symp. Control of Distributed Parameter Systems, June 26–29, Perpignan (A. El Jai, ed.)
Bonnans J. F., Casas E. (1989) Optimal control of state constrained multistate systems. SIAM J. Control Optim. 27:446–455
Brezis H. (1972) Problèmes unilatéraux. J. Math. Pures Appl. 51:1–168
Clarke F. H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York
Ekeland I. (1972) Sur les problèmes variationnels. C. R. Acad. Sci. Paris 275:1057–1059
Ekeland I. (1979) Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS) 1:447–474
Haraux A. (1977) How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29:615–631
Ioffe A. D., Tihomirov V. (1974) Theory of Extremal Problems. Nauka, Moscow (in Russian; English translation: 1979, North Holland, Amsterdam)
Kinderlehrer D., Stampacchia G. (1980) An Introduction to Variational Inequalities and Their Applications. Academic Press, New York
Mignot F. (1976) Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22:130–185
Saguez C. (1980) Contrôle optimal de systèmes à frontière libre. Thèse d'état, Université de Compiègne
Tiba D. (1984) Quelques remarques sur le contrôle de la corde vibrante avec obstacle. C. R. Acad. Sci. Paris, 229:1–13
Tiba D. (1984) Some remarks on the control of the vibrating string with obstacle. Rev. Roumaine Math. Pures Appl. 29:899–906
Tiba D. (1987) Optimal control for second-order semilinear hyperbolic equations. Control Theory Adv. Technol. 3:33–43
Tiba D. (1990) Optimal Control of Nonsmooth Distributed Parameter Systems. Lecture Notes in Mathematics, Vol. 1459. Springer-Verlag, Berlin
Yvon J. P. (1973) Etude de quelques problèmes de contrôle pour des systèmes distribués. Thèse d'état, Université de Paris VI
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Communicated by D. Kinderlehrer
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Bonnans, J.F., Tiba, D. Pontryagin's principle in the control of semilinear elliptic variational inequalities. Appl Math Optim 23, 299–312 (1991). https://doi.org/10.1007/BF01442403
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DOI: https://doi.org/10.1007/BF01442403