Differential Equations

, Volume 45, Issue 5, pp 731–742 | Cite as

Impulse controls in models of hybrid systems

  • A. B. Kurzhanski
  • P. A. Tochilin
Control Theory


We consider a mathematical model of a hybrid system in which the continuous dynamics induced at each instant of time by a certain system of differential equations from a given finite family is alternated with discrete operations sending commands either for an instantaneous switching from one system to another, or for an instantaneous reset from online coordinates to other coordinates, or for both actions together. We present a description of a hybrid system via systems with impulses. We suggest a new scheme, which permits one to reduce the problem of synthesis of controls for a hybrid system to an impulse control problem. The aim of the present paper is to find a control containing both ordinary components (that are not generalized functions) and higher-order generalized functions.


Hybrid System Inclination Angle Impulse Control Instantaneous Switching Impulse Control Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Branicky, M.S., Borkar, V.S., and Mitter, S.M., A Unified Framework for Hybrid Control: Model and Optimal Control Theory, IEEE Trans. Automat. Control, 1998, vol. 43, no. 1, pp. 31–45.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Van der Schaft, A. and Schumacher, H., An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Inform. Sci., no. 251. Berlin, 2000.Google Scholar
  3. 3.
    Emel’yanov, S.V., Sistemy avtomaticheskogo upravleniya s peremennoi strukturoi (Automated Control Systems with Variable Structure), Moscow, 1967.Google Scholar
  4. 4.
    Barbashin, E.A. and Tabueva, V.A., Dinamicheskie sistemy s tsilindricheskim fazovym prostranstvom (Dynamical Systems with Cylindrical Phase Space), Moscow: Nauka, 1969.Google Scholar
  5. 5.
    Kurzhanski, A.B. and Daryin, A.N., Dynamic Programming for Impulse Controls, Ann. Rev. Control, 2008, vol. 32, pp. 213–227.CrossRefGoogle Scholar
  6. 6.
    Dar’in, A.N. and Kurzhanskii, A.B., Synthesis of Controls in the Class of Generalized Higher-Order Functions, Differ. Uravn., 2007, vol. 43, no. 11, pp. 1443–1453.MathSciNetGoogle Scholar
  7. 7.
    Johansson, M., Piecewise Linear Control Systems, Lecture Notes in Control and Inform. Sci., no. 284. Heidelberg, 2003.Google Scholar
  8. 8.
    Kurzhanskii, A.B. and Varaiya, P., Problems of Dynamics and Control in Hybrid Systems, Tr. mezhdunar. sem. “Teoriya upravleniya i teoriya obobshchennykh reshenii uravnenii Gamil’tona-Yakobi” (Proc. Int. Sem. “Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations”), Yekaterinburg, 2005, pp. 21–37.Google Scholar
  9. 9.
    Kurzhanskii, A.B., On Synthesis of Systems with Impulsive Control, Mekhatronika, Avtomatika, Upravlenie, 2006, no. 4, pp. 2–12.Google Scholar
  10. 10.
    Riesz, F. and Sz-Nagy, B., Leçons d’analyse fonctionnelle, Budapest, 1972.Google Scholar
  11. 11.
    Gelfand, I.M. and Shilov, G.E., Obobshchennye funktsii i deistviya nad nimi (Generalized Functions and Related Operations), Moscow, 1959, vols. 1, 4.Google Scholar
  12. 12.
    Kurzhanskii, A.B. and Osipov, Yu.S., On the Control of a Linear System by Generalized Actions, Differ. Uravn., 1969, vol. 5, no. 8, pp. 1360–1370.zbMATHGoogle Scholar
  13. 13.
    Neustadt, L.W., Optimization, a Moment Problem and Nonlinear Programming, SIAM J. on Control, 1964, vol. 2, no. 1, pp. 33–53.zbMATHMathSciNetGoogle Scholar
  14. 14.
    Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Control of Motion), Moscow: Nauka, 1968.Google Scholar
  15. 15.
    Bensoussan, A. and Lions, J.-L., Contrôle impulsionnel et inéquations quasi-variationnelles, Paris, 1982.Google Scholar
  16. 16.
    Dar’in, A.N., Kurzhanskii, A.B., and Seleznev, A.V., The Dynamic Programming Method in Problems of the Synthesis of Impulsive Controls, Differ. Uravn., 2005, vol. 41, no. 11, pp. 1491–1500.MathSciNetGoogle Scholar
  17. 17.
    Kurzhanskii, A.B. and Tochilin, P.A., Weakly Invariant Sets of Hybrid Systems, Differ. Uravn., 2008, vol. 44, no. 11, pp. 1523–1533.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • A. B. Kurzhanski
    • 1
  • P. A. Tochilin
    • 1
  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations