Advertisement

Cybernetics and Systems Analysis

, Volume 45, Issue 1, pp 123–140 | Cite as

Game approach problems for dynamic processes with impulse controls

  • A. N. Khimich
  • K. A. Chikrii
Systems Analysis

This paper deals with game problems with impulse and geometric constraints on the players' controls. To analyze conflict-controlled processes with discontinuous trajectories, the method of resolving functions is used. This makes it possible to derive sufficient conditions of the game termination in a finite guaranteed time.

Keywords

impulsive system Dirac delta function conflict-controlled process multivalued mapping Minkowski functional Aumann integral Pontryagin condition Cauchy formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. S. Pontryagin, Selected Scientific Works [in Russian], Vol. 2, Nauka, Moscow (1988).Google Scholar
  2. 2.
    N. N. Krasovskii, Motion Meeting Game Problems [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    M. S. Nikol'skii, First Direct Pontryagin Method in Differential Games [in Russian], Izd. MGU, Moscow (1984).Google Scholar
  4. 4.
    B. M. Miller and E. Ya. Rubinovich, Optimization of Dynamic Pulse Control Systems [in Russian], Nauka, Moscow (2005).Google Scholar
  5. 5.
    A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulsive Forcing [in Russian], Vyshcha Shkola, Kyiv (1987).Google Scholar
  6. 6.
    S. T. Zavalishchin and A. N. Sesekin, Pulse Processes. Models and Applications [in Russian], Nauka, Moscow (1991).Google Scholar
  7. 7.
    B. N. Pshenichnyi and V. V. Ostapenko, Differential Games [in Russian], Naukova Dumka, Kyiv (1992).Google Scholar
  8. 8.
    A. A. Chikrii, Conflict Controlled Processes, Kluwer Acad. Publ., Boston-London-Dordrecht (1997).MATHGoogle Scholar
  9. 9.
    A. A. Chirkii, I. S. Rappoport, and K. A. Chikrii, “Multivalued mappings and their selectors in the theory of conflict-controlled processes,” Cybern. Syst. Analysis, 43, No. 5, 719–730 (2007).CrossRefGoogle Scholar
  10. 10.
    N. L. Grigorenko, Mathematical Methods of Control of Several Dynamic Processes [in Russian], Izd MGU, Moscow (1990).Google Scholar
  11. 11.
    V. M. Kuntsevich, Control under Uncertainty Conditions: Guaranteed Results in Control and Identification Problems [in Russian], Naukova Dumka, Kyiv (2006).Google Scholar
  12. 12.
    Yu. G. Krivonos, I. I. Matichin, and A. A. Chikrii, Dynamic Games with Discontinuous Trajectories [in Russian], Naukova Dumka, Kyiv (2005).Google Scholar
  13. 13.
    K. A. Chikrii, Guaranteed Result for Conflict-Controlled Processes [in Russian], Author's Abstract of PhD Thesis, Kyiv (2007).Google Scholar
  14. 14.
    J.-P. Aubin and H. Frankovska, Set-Valued Analysis, Birkhauser, Boston-Basel-Berlin (1990).MATHGoogle Scholar
  15. 15.
    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems [in Russian], Nauka, Moscow (1974).Google Scholar
  16. 16.
    Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach, Basel (1995).MATHGoogle Scholar
  17. 17.
    A. A. Melikyan, Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games, Birkhauser, Boston (1998).MATHGoogle Scholar
  18. 18.
    A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems [in Russian], Nauka, Moscow (1981).MATHGoogle Scholar
  19. 19.
    A. A. Chirkii, I. I. Matichin, and K. A. Chikrii, “Conflict controlled processes with discontinuous trajectories,” Cybern. Syst. Analysis, 40, No. 6, 800–811 (2004).CrossRefGoogle Scholar
  20. 20.
    B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basis Theory, II: Applications, Springer, Berlin-Heidelberg-New York (2006).Google Scholar
  21. 21.
    A. B. Kurzhanskii, Control and Observation under Uncertainty Conditions [in Russian], Nauka, Moscow (1977).Google Scholar
  22. 22.
    V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulse Differential Equations, World Scientific, Singapore (1989).Google Scholar
  23. 23.
    M. Z. Zgurovskii and V. S. Mel'nik, Nonlinear Analysis and Control of Infinite-Dimensional Systems [in Russian], Naukova Dumka, Kyiv (1999).Google Scholar
  24. 24.
    A. Bressan and F. Rampazzo, “Impulsive control systems without commutativity assumptions,” JOTA, 81, No. 3, 435–457 (1994).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. Bensoussan and J.-L. Lions, “Impulse Control and Quasivariational Inequalities,” Gauthier-Villars, Montrouge (1984).Google Scholar
  26. 26.
    A. A. Chikrii, “Minkowski functionals in pursuit theory,” DAN Rossii, 329, No. 3, 281–284 (1993).Google Scholar
  27. 27.
    V. A. Dykhta and O. N. Samsonyuk, Optimal Pulse Control with Applications [in Russian], Fizmatlit, Moscow (2000).Google Scholar
  28. 28.
    A. G. Chentsov, Finite-Additive Measures and Relaxation of Extreme Problems [in Russian], Nauka, Ekaterinburg (1993).Google Scholar
  29. 29.
    A. Halanay and D. Wexler, Qualitative Theory of Impulsive Systems, Acad. Romania, Bucuresti (1968).Google Scholar
  30. 30.
    Yu. V. Orlov, Theory of Optimal Systems with Generalized Control [in Russian], Nauka, Moscow (1988).Google Scholar
  31. 31.
    S. I. Lyashko, Generalized Control of Linear Systems [in Russian], Naukova Dumka, Kyiv (1998).Google Scholar
  32. 32.
    A. A. Chikrii, “Optimization of game interaction of fractional-order controlled systems,” Optimization Methods and Software, 23, No. 1, 39–73 (2008).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations