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Pontryagin's maximum principle in optimal control theory

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References

  1. E. R. Avakov, “Maximum principle for abnormal optimal control problems,”Dokl. Akad. Nauk SSSR,298, No. 6, 1289–1292 (1988).

    MATH  MathSciNet  Google Scholar 

  2. E. R. Avakov, “Necessary conditions of maximum for nonregular problems in Banach spaces. Maximum principle for abnormal optimal control problems,”Tr. Mat. Inst. Akad. Nauk SSSR,68, 3–29 (1989).

    MathSciNet  Google Scholar 

  3. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin,Optimal Control [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  4. A. V. Arutyunov, “Perturbation of extremal problems with constraits and necessary conditions of optimality,” In:Itogi Nauki i Tekhn. Mat. Analiz, Vol. 27, All-Union Institute for Scientific and Technical Information, Moscow (1989), pp. 147–235.

    Google Scholar 

  5. A. V. Arutyunov, “On the necessary conditions of optimality in problems with phase constraints,”Dokl. Akad. Nauk SSSR,280, No. 5, 1033–1037 (1985).

    MATH  MathSciNet  Google Scholar 

  6. A. V. Arutyunov, “On the theory of maximum principle in optimal control problems with phase constraints,”Dokl. Akad. Nauk SSSR,304, No. 1, 11–14 (1989).

    MATH  MathSciNet  Google Scholar 

  7. A. V. Arutyunov, “First-order necessary conditions in optimal control problems with phase constraints,”Tr. Inst. Prikl. Mat. Tbilissk. Univ.,27, 46–59 (1986).

    MathSciNet  Google Scholar 

  8. A. V. Arutyunov, “Maximum principle and first-order necessary conditions in delayed optimal control problems,”Soobshch. Akad. Nauk GSSR,122, No. 2, 265–268 (1986).

    MATH  MathSciNet  Google Scholar 

  9. A. V. Arutyunov and S. M. Aseev, “Maximum principle in optimal control problems with phase constraints. Nondegeneracy and stability,”Dokl. Rossiisk. Akad. Nauk.,334, No. 2, 134–137 (1994).

    MathSciNet  Google Scholar 

  10. A. V. Arutyunov, S. M. Aseev, and V. I. Blagodatskikh, “First-order necessary conditions in optimal control by differential inclusion under phase constraints,”Mat. Sb.,184, No. 6, 3–32 (1993).

    Google Scholar 

  11. A. V. Arutyunov and V. I. Blagodatskikh, “Maximum principle for differential inclusions with phase constraints,”Tr. Mat. Inst. Akad. Nauk SSSR,200, 4–26 (1991).

    Google Scholar 

  12. A. V. Arutynov and N. T. Tynyanskii, “On the maximum principle in problems with phase constraints,”Izv. Akad. Nauk SSSR. Ser. Tekh. Kibern.,4, 60–68 (1984).

    Google Scholar 

  13. A. R. Afanas'ev, V. V. Dikusar, A. A. Milyutin, and S. A. Chukanov,Necessary Condition in Optimal Control Problem [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  14. G. A. Bliss,Lectures on the Calculus of Variations [Russian translation], Inostr. Lit., Moscow (1950).

    Google Scholar 

  15. V. G. Boltyanskii,Mathematical Methods of Optimal Control [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  16. D. Varga,Optimal Control by Differential and Functional Equations [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  17. F. P. Vasil'ev,Numerical Methods of Solving Extremal Problems [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  18. R. Gabasov and F. M. Kirillova, “Methods of Optimal Control,” In:Itogi Nauki i Tekhn. Sovremennye Problemy Matematiki, Vol. 6, All-Union Institute for Scientific and Technical Information, Moscow (1976), pp. 133–261.

    Google Scholar 

  19. R. V. Gamkrelidze, “High-speed optimal processes under bounded phase coordinates,”Dokl. Akad. Nauk SSSR,125, No. 3, 475–478 (1959).

    MATH  MathSciNet  Google Scholar 

  20. R. V. Gamkrelidze,Principles of Optimal Control [in Russian], Izd. Tbilissk. Univ., Tbilisi (1977).

    Google Scholar 

  21. I. V. Girsanov,Lectures on Mathematical Theory of Extremal Problems [in Russian], Izd. Mosk. Univ., Moscow (1970).

    Google Scholar 

  22. V. V. Dikusar and A. A. Milyutin,Qualitative and Numerical Methods in the Maximum Principle [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  23. A. V. Dmitruk, “Maximum principle for the general optimal control problem with phase and regular mixed constraints,” In:Optimality of Dynamic Control Systems [in Russian], Vol. 14, Moscow (1990), pp. 26–42.

  24. A. Ya. Dubovitskii,Integral Maximum Principle in the General Optimal Control Problem [in Russian], Deposited at the All-Union Institute for Scientific and Technical Information. No. 2639-74, Moscow (1974).

  25. A. Ya. Dubovitskii and V. A. Dubovitskii,Maximum Principle for Trajectories with Bounds on the Phase Boundary [in Russian], Preprint, Chernogolovka, Russia (1988).

    Google Scholar 

  26. A. Ya. Dubovitskii and V. A. Dubovitskii, “Necessary conditions of strong minimum in optimal control problems with degenerate endpoints and phase constraints,”Usp. Mat. Nauk,40, No. 2, 175–176. (1985).

    MathSciNet  Google Scholar 

  27. A. Ya. Dubovitskii and V. A. Dubovitskii, “Pointwise nontriviality condition for the maximum principle in a regular optimal control problem,”Tr. Inst. Mat. Tbilissk. Univ.,27, 60–87 (1988).

    MathSciNet  Google Scholar 

  28. A. Ya. Dubovitskii and A. A. Milyutin, “Extremal value problem under constraints,”Dokl. Akad. Nauk SSSR,149, No. 4, 759–762 (1963);Zh. Vychisl. Mat. Mat. Fiz.,5, No. 3, 395–453 (1965).

    Google Scholar 

  29. A. Ya. Dubovitskii and A. A. Milyutin, “Theory of the maximum principle,” In:Methods of Extremal-Value Problems in Economics [in Russian], Nauka, Moscow (1981), pp. 138–177.

    Google Scholar 

  30. A. D. Ioffe and V. M. Tikhomirov,Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  31. F. Clark,Optimization and Nonsmooth Analysis [Russian translation], Nauka, Moscow (1988).

    Google Scholar 

  32. A. N. Kolmogorov and S. V. Fomin,Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  33. A. B. Kurzhanskii and Yu. S. Osipov, “On the problem of control with bounded phase coordinates,”Prikl. Mat. Mekh.,32, No. 2, 194–202 (1968).

    MathSciNet  Google Scholar 

  34. E. B. Lie and L. Marcus,Elements of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).

    Google Scholar 

  35. A. S. Matveev, “On the necessary conditions of extrenum in optimal control problems with phase constraints,”Differents. Uravn.,23, No. 4, 629–639 (1987).

    MATH  Google Scholar 

  36. I. P. Natanson,Theory of Functions of a Real Variable [Russian translation], Nauka, Moscow (1974).

    Google Scholar 

  37. L. S. Pontryagin,The Maximum Principle in Optimal Control [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  38. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  39. R. Rockafellar,Convex Analysis [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  40. A. M. Ter-Krikorov,Optimal Control in Mathematical Economics [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  41. A. M. Filippov, “Certain questions in optimal regulation,”Vestn. Mosk. Univ.,1, No. 2, 25–38 (1959).

    Google Scholar 

  42. G. L. Kharatishvili, “Maximum principle in the theory of optimal processes with delay,”Dokl. Akad. Nauk SSSR,136, No. 1, 39–42 (1961).

    Google Scholar 

  43. A. V. Arutyunov and S. M. Aseev, “Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems wih state constraints,”SIAM J. Contr. Opt. (in press).

  44. V. G. Boltyanskii, “The maximum principle: how it came to be,” Report No. 526, Mathematisches Institut, Technische Universitat, Munchen (1994), pp. 1–30.

    Google Scholar 

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  1. A. V. Arutyunov
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 42, Dinamicheskie Sistemy-6, 1997.

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Arutyunov, A.V. Pontryagin's maximum principle in optimal control theory. J Math Sci 94, 1311–1365 (1999). https://doi.org/10.1007/BF02365017

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