Abstract
For the optimal control problem on the coefficients of parabolic equations, we study correctness problems for the setting in the weak topology on the space of controls. We show that in this problem, the objective functional is bounded from below, the set of optimal controls is nonempty, and every minimizing sequence weakly converges to the set of optimal controls. We establish a necessary optimality condition in the form of a generalized Lagrange multiplier rule.
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Lions, J.L., Controle optimal des systèmes gouvernés par des équations aux derivées partielles, Paris: Dunod Gauthier-Villars, 1968. Translated under the title Optimal’noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Moscow: Mir, 1972.
Murat, F., Contre-Exemples pour Divers Problems on le Controle Intervient dans les Coefficients, Ann. Mat. Pura Appl., 1977, vol. 112, pp. 49–68.
Vasil’ev, F.P., Metody optimizatsii (Methods of Optimization), Moscow: Faktorial-Press, 2002.
Iskenderov, A.D. and Tagiev, R.K. Optimization Problems with Controls in the Coefficients of a Parabolic Equation, Differ. Uravn., 1983, vol. 19, no 8, pp. 1324–1334.
Lur’e, K.A., Optimal’noe upravlenie v zadachakh matematicheskoi fiziki (Optimal Control in Problems of Mathematical Physics), Moscow: Nauka, 1975.
Alifanov, O.M., Artyukhin, E.A., and Rumyantsev, S.V., Ekstremal’nye metody resheniya nekorrektnykh zadach (Extremal Methods for Solving Ill-Posed Problems), Moscow: Nauka, 1988.
Egorov, A.I., Optimal’noe upravlenie teplovymi i diffuzionnymi protsessami (Optimal Control for Thermal and Diffuse Processes), Moscow: Nauka, 1978.
Iskenderov, A.D., On Variational Settings of Multidimensional Inverse Problems of Mathematical Physics, Dokl. Akad. Nauk SSSR, 1984, vol. 274, no. 3, pp. 531–533.
Zolezzi, T., Necessary Conditions for Optimal Control of Elliptic or Parabolic Problems, SIAM J. Control, 1972, vol. 4, pp. 594–602.
Sokolowski, J., Remarks on Existence of Solution for Parametric Optimization Problems for Partial Differential Equations of Parabolis Type, Control Cybern., 1978, vol. 7, no. 2, pp. 47–61.
Serovaiskii, S.Ya., Optimal Control Problem in Coefficients for Equations of Parabolic Type, Izv. Vyssh. Uchebn. Zaved., Mat., 1982, no. 12, pp. 44–50.
Novozhenov, M.M. and Plotnikov, V.I., The Generalized Rule of Lagrange Multipliers for Distributed Systems with Phase Constraints, Differ. Uravn., 1982, vol. 18, no. 4, pp. 584–592.
Vasil’eva, V.N., On the Smoothness of Functionals in Optimal Control Problems for Coefficients of Parabolic Equations, in Issledovanita v Mekhanike Sploshnykh Sred (Researches in Mechanics of Continuous Environments), Irkutsk, 1983, pp. 91–99.
Serovaiskii, S.Ya., The Existence of an Optimal Control in Coefficients for Equations of Parabolic Type, in Dinamika Upravlyaemykh Sistem (Dynamics of the Operated Systems), Alma-Ata, 1985, pp. 99–104.
Tagiev, R.K., Optimal Control Problems for Coefficients of a Parabolic Equation, in Prikladnye zadachi funktsional’nogo analiza (Applied Tasks of the Functional Analysis), Baku: Azerbaid. Gos. Univ., 1986, pp. 93–102.
Hew, R.J., Optimal Control of the Convective Velocity Coefficient in a Parabolic Problem, Nonlin. Anal., 2005, vol. 63, pp. 1383–1390.
Tagiyev, R.K., Optimal Control by the Coefficients of a Parabolic Equation, Trans. NAS Azerbaijan, Math. Mech., 2004, vol. 24, no. 4, pp. 247–256.
Tagiyev, R.K., The Problems of Optimal Control by Parabolic Equations Coefficients, Trans. NAS Azerbaijan, Math. Mech., 2007, vol. 27, no. 1, pp. 135–146.
Tagiev, R.K., Optimal Control for Coefficients in Parabolic Systems, Differ. Uravn., 2009, vol. 45, no. 10, pp. 1492–1501.
Tagiev, R.K., Optimal Control for the Coefficients of a Quasilinear Parabolic Equation, Autom. Remote Control, 2009, vol. 70, no. 11, pp. 1814–1826.
Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Equations of Parabolic Type), Moscow: Nauka, 1967.
Zabreiko, P.P., Koshelov, A.I., Krasnosel’skii, M.A., et al., Integral’nye uravneniya (Integral Equations), Moscow: Nauka, 1968.
Vasil’ev, F.P., Metody resheniya ekstremal’nykh zadach (Methods for Solving Extremal Problems), Moscow: Nauka, 1981.
Baranjer, J. and Témam, R., Nonconvex Optimization Problems Depending on a Parameter, SIAM J. Control, 1975, vol. 13, pp. 146–152.
Ekeland, I. and TĂ©mam, R., Convex Analysis and Variational Problems, Amsterdam: North-Holland, 1976. Translated under the title Vypuklyi analiz i variatsionnye problemy, Moscow: Mir, 1979.
Lions, J.J., Control of Distributed Singular Systems, Paris: Gauthier-Villars, 1985. Translated under the title Upravlenie singulyarnymi raspredelennymi sistemami, Moscow: Nauka, 1987.
Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Problems in Mathematical Physics), Moscow: Nauka, 1973.
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Original Russian Text © R.K. Tagiev, S.A. Gashimov, 2015, published in Avtomatika i Telemekhanika, 2015, No. 8, pp. 27–45.
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Tagiev, R.K., Gashimov, S.A. The optimal control problem for the coefficients of a parabolic equation under phase constraints. Autom Remote Control 76, 1347–1360 (2015). https://doi.org/10.1134/S0005117915080020
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DOI: https://doi.org/10.1134/S0005117915080020