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Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls

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Abstract

We obtain necessary global optimality conditions for classical optimal control problems based on positional controls. These controls are constructed with classical dynamical programming but with respect to upper (weakly monotone) solutions of the Hamilton-Jacobi equation instead of a Bellman function. We put special emphasis on the positional minimum condition in Pontryagin formalism that significantly strengthens the Maximum Principle for a wide class of problems and can be naturally combined with first order sufficient optimality conditions with linear Krotov’s function. We compare the positional minimum condition with the modified nonsmooth Kaśkosz-Lojasiewicz Maximum Principle. All results are illustrated with specific examples.

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References

  1. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., et al., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Fizmatgiz, 1961.

    Google Scholar 

  2. Ioffe, A.D. and Tikhomirov, V.M., An Extension of Variational Problems, Tr. Mosk. Mat. Obshch., 1968, vol. 18, pp. 187–246.

    MATH  Google Scholar 

  3. Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.

    Google Scholar 

  4. Krotov, V.F., Global Methods in Optimal Control Theory, Monographs and Textbooks in Pure and Applied Mathematics, New York: Marcel Dekker, 1996.

    MATH  Google Scholar 

  5. Gurman, V.I., Printsip rasshireniya v zadachakh upravleniya (Extension Principle in Control Problems), Moscow: Nauka, 1997, 2nd ed.

    MATH  Google Scholar 

  6. Krasovskii, N.N., Differential Games. Approximational and Formal Models, Mat. Sb., 1978, vol. 107, no. 4, pp. 541–571.

    MathSciNet  Google Scholar 

  7. Khrustalev, M.M., Necessary and Sufficient Optimality Conditions in the Form of a Bellman Equation, Dokl. Akad. Nauk SSSR, 1978, vol. 242, no. 5, pp. 1023–1026.

    MathSciNet  Google Scholar 

  8. Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi (Control over a Dynamical System), Moscow: Nauka, 1985.

    Google Scholar 

  9. Subbotin, A.I. and Chentsov, A.G., Optimizatsiya garantii v zadachakh upravleniya (Optimizing Guarantees in Control Problems), Moscow: Nauka, 1981.

    MATH  Google Scholar 

  10. Kurzhanskii, A.B., Izbrannye trudy (Selected Works), Dar’in, A.N., Digajlov, I.A., and Rublev, I.V., Eds., Moscow: Mosk. Gos. Univ., 2009.

  11. Subbotin, A.I., Minimaksnye neravenstva i uravneniya Gamil’tona-Yakobi (Minimax Inequalities and Hamilton-Jacobi Equations), Moscow: Nauka, 1991.

    MATH  Google Scholar 

  12. Subbotin, A.I., Obobshchennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii (Generalized Solutions of First Order Partial Differential Equations. Prospect of Dynamical Optimization), Moscow, Izhevsk: Inst. Komp. Issl., 2003.

    Google Scholar 

  13. Clarke, F.H., Ledyaev, Yu.S., and Subbotin, A.I., Universal Positional Control and Proximal Targeting in Control Problems under Disturbances and Differential Games, Tr. Steklov Math. Inst., 1999, vol. 224, pp. 165–186.

    MathSciNet  Google Scholar 

  14. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., et al. Nonsmooth Analysis and Control Theory, New York: Springer-Verlag, 1998.

    MATH  Google Scholar 

  15. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., et al., Qualitative Properties of Trajectories of Control Systems: A Survey, J. Dynam. Control Syst., 1995, vol. 1, no. 1, pp. 1–48.

    Article  MATH  MathSciNet  Google Scholar 

  16. Bardi, M. and Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Boston: Birkhauser, 1997.

    Book  MATH  Google Scholar 

  17. Cannarsa, P. and Sinestrari, C., Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Progress in Nonlinear Differential Equations and Their Appications, Boston: Birkhauser, 2004.

    Google Scholar 

  18. Kaśkosz, B. and Łojasiewicz, S., A Maximum Principle for Generalized Control, Nonlinear Anal.: Theory, Methods Appl., 1985, vol. 9, no. 2, pp. 109–130.

    Article  MATH  Google Scholar 

  19. Kaśkosz, B., Extremality, Controllability, and Abundant Subsets of Generalized Control Systems, J. Optim. Theory Appl., 1999, vol. 101, no. 1, pp. 73–108.

    Article  MATH  MathSciNet  Google Scholar 

  20. Frankowska, H. and Kaśkosz, B., Linearization and Boundary Trajectories of Nonsmooth Control Systems, Can. J. Math., 1988, vol. 11, no. 3, pp. 589–609.

    Article  Google Scholar 

  21. Sussmann, H.J., A Strong Version of the Lojasiewicz Maximum Principle, in Optimal Control of Differential Equations, Lecture Notes in Pure and Applied Mathematics, Pavel, N.H., Ed., New York: M. Dekker, 1994, pp. 1–17.

    Google Scholar 

  22. Loewen, P.D. and Vinter, R.B., Pontryagin-Type Necessary Conditions for Differential Inclusion Problems, Syst. Control Lett., 1997, vol. 9, no. 9, pp. 263–265.

    MathSciNet  Google Scholar 

  23. Artstein, Z., Pontryagin Maximum Principle Revisited with Feedbacks, Eur. J. Control, 2011, vol. 17, no. 1, pp. 46–54.

    Article  MATH  MathSciNet  Google Scholar 

  24. Dykhta, V.A., Hamilton-Jacobi Inequalities in Optimal Control: Smooth Duality and Improvement, Vestn. Tamb. Univ., Ser. Estestv. Tekh. Nauki, 2010, vol. 15, no. 15, pp. 405–425.

    Google Scholar 

  25. Dykhta, V.A., Weakly Monotone and Generating L-Functions in Optimal Control, in Analiticheskaya mekhanika, ustoichivost’ i upravlenie: Vestn. X mezhd. Chetaev. konf. (Analytical Mechanics, Stability, and Control: Proc. X Int. Chetaev Conf.), vol. 3, Section 3, Control, part I, Kazan: Kazan. Gos. Tekh. Univ., 2012, pp. 408–420.

    Google Scholar 

  26. Dykhta, V.A., The Minimum Principle with Counterstrategies and Nonstandard Duality in Optimal Control, Tr. VI mezhd. nauchn. semin. “Obobshchennye postanovki i resheniya zadach upravleniya” (Proc. VI Int. Sci. Seminar “Generalized Settings and Solutions of Control Problems” (GSSCP-2012)), 2012, electronic edition.

    Google Scholar 

  27. Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Fizmatlit, 1974.

    MATH  Google Scholar 

  28. Warga, J., Optimal Control of Differential and Functional Equations, New York: Academic, 1972. Translated under the title Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1979.

    MATH  Google Scholar 

  29. Gabasov, R. and Kirillova, F.M., Printsip maksimuma v teorii optimal’nogo upravleniya (The Maximum Principle in Optimal Control Theory), Moscow: Editorial URSS, 2011.

    Google Scholar 

  30. Srochko, V.A., Multipoint Optimality Conditions for Singular Controls, in Chislennye metody analiza (prikl. mat.) (Numerical Methods of Analysis (Applied Math.)), Irkutsk: Sib. Energ. Inst. Sib. Otd. Akad. Nauk SSSR, 1976, pp. 43–50.

    Google Scholar 

  31. Trunin, D.O. and Buldaev, A.S., The Nonlocal Improvement Method in Polynomial Optimal Control Problems with Terminal Constraints, Upravlen. Bol’shimi Sist., 2008, no. 22, pp. 51–69.

    Google Scholar 

  32. Clarke, F.H., Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988.

    MATH  Google Scholar 

  33. Vinter, R.B., Optimal Control, Boston: Birkhauser, 2000.

    MATH  Google Scholar 

  34. Gel’fand, I.M. and Fomin, S.V., Variatsionnoe ischislenie (Variational Calculus), Moscow: Fizmatlit, 1961.

    Google Scholar 

  35. Arnol’d, V.I., Matematicheskie metody klassicheskoi mekhaniki (Mathematical Methods of Classical Mechanics), Moscow: Nauka, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia

    V. A. Dykhta

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Correspondence to V. A. Dykhta.

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Original Russian Text © V.A. Dykhta, 2014, published in Avtomatika i Telemekhanika, 2014, No. 5, pp. 31–49.

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Dykhta, V.A. Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls. Autom Remote Control 75, 829–844 (2014). https://doi.org/10.1134/S0005117914050038

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