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On constrained impulsive control problems

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Abstract

This paper considers constrained impulsive control problems for which the authors propose a new mathematical concept of control required for the impulsive framework. These controls can arise in engineering, in particular, in problems of space navigation. We derive necessary extremum conditions in the form of the Pontryagin maximum principle and also study conditions under which the constraint regularity clarifications become weaker. In the proof of the main result, Ekeland’s variational principle is used.

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References

  1. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  2. A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems, Kluwer Academic Publisher (2000).

  3. A. V. Arutyunov, “Some properties of quadratic mappings,” Vestn. MGU, Vychisl. Mat. Kibern., 2, 30–32 (1999).

    Google Scholar 

  4. A. V. Arutyunov and D. Yu. Karamzin, “Necessary conditions of the minimum in an impulse optimal control problem,” In: Nonlinear Dynamics and Control [in Russian], 4, Fizmatlit, Moscow (2004), pp. 205–240.

    Google Scholar 

  5. A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “A nondegenerate maximum principle for the impulse control problem with state constraints,” SIAM J. Control Optim., 43, No. 5, 1812–1843 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “Maximum principle in problems with mixed constraints under weak assumptions of regularity,” In: Theoretical and Applied Problems of Nonlinear Analysis [in Russian], Computational Center of Russian Academy of Sciences (2008), pp. 1–33.

  7. A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “Necessary optimality conditions for problems with equality and inequality constraints: The abnormal case,” J. Optim. Theory Appl., 1 (2009) (in press).

  8. J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York (2000).

    MATH  Google Scholar 

  9. A. Bressan and F. Rampazzo, “Impulsive control systems with commutative vector fields,” J. Optim. Theory Appl., 71, 67–83 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York (1983).

    MATH  Google Scholar 

  11. V. A. Dykhta and O. N. Samsonyuk, Optimal Impulse Control with Applications [in Russian], Fizmatlit, Moscow (2000).

    Google Scholar 

  12. I. Ekeland, “On the variational principle,” J. Math. Anal. Appl., 47, 324–353 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  13. M. W. Hirsch, Differential Topology, Springer, New York (1976).

    MATH  Google Scholar 

  14. A. D. Ioffe and V. M. Tikhomirov, “Several remarks on variational principles,” Mat. Zametki, 61, No. 2, 305–311 (1997).

    MathSciNet  Google Scholar 

  15. A. A. Kirillov and A. D. Gvishiani, Theorems and Problems of Functional Analysis [in Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  16. A. B. Kurzhanskii, Optimal Systems with Impulse Controls, Differential Games and Control Problems [in Russian], Preprint, UNTs, Akad. Nauk SSSR, Sverdlovsk, 131–156 (1975).

  17. B. M. Miller, “Optimality conditions in the problems of generalized optimization, I (necessary conditions)”, Automat. Remote Control, 53, No. 3, 50–58 (1992).

    MathSciNet  Google Scholar 

  18. A. A. Milyutin, Maximum Principle in a General Optimal Control Problem [in Russian], Fizmatlit, Moscow (2001).

    Google Scholar 

  19. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Springer (2005).

  20. B. S. Mordukhovich, “Maximum principle in problems of time optimal control with nonsmooth constraints,” Appl. Math. Mech., 40, 960–969 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  21. M. D. R. de Pinho and R. V. Vinter, “Necessary conditions for optimal control problems involving nonlinear differential algebraic equations,” J. Math. Anal. Appl., 212, 493–516 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  22. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1983).

    MATH  Google Scholar 

  23. R. W. Rishel, “An extended Pontryagin principle for control systems whose control laws contain measures,” J. SIAM. Ser. A. Control, 3, No. 2, 191–205 (1965).

    MATH  MathSciNet  Google Scholar 

  24. R. Rockafellar, Convex Analysis, Princeton University Press (1970).

  25. R. B. Vinter and F. L. Pereira, “A maximum principle for optimal processes with discontinuous trajectories,” SIAM J. Control Optim., 26, 205–229 (1988).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Russian University of People’s Friendship, Moscow, Russia

    A. V. Arutyunov

  2. Computational Center of the Russian Academy of Sciences, Moscow, Russia

    D. Yu. Karamzin

  3. University of Porto, FEUP, rua do Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal

    F. L. Pereira

Authors
  1. A. V. Arutyunov
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  2. D. Yu. Karamzin
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  3. F. L. Pereira
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Corresponding author

Correspondence to A. V. Arutyunov.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 65, Mathematical Physics, Combinatorics, and Optimal Control, 2009.

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Arutyunov, A.V., Karamzin, D.Y. & Pereira, F.L. On constrained impulsive control problems. J Math Sci 165, 654–688 (2010). https://doi.org/10.1007/s10958-010-9834-z

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