Optimization of generalized solutions of nonlinear hybrid (discrete-continuous) systems

  • Boris M. Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1386)


The optimal control problem for hybrid (discrete-continuouis) system is considered in the case when the continuous behavior can be controled and discontinuities arise when the system achives the boundary of some set. We suppose that discontinuities can be considered as a result of some impulsive inputs, which can be represented in feedback form as the intermediated conditions. Meanwhile, variuos types of irregulariries such as: nonextandability of solution or sliding mode can arise. However, if the jumps of solution are described by some shift operator, as for hybrid system satisfying the robustness condition, one can reduce this problem to the standard problem of nonsmooth optimization and the representation of solution by differential equation with a measure and the existence theorem for optimal solution can be obtained.


Optimal Control Problem Hybrid System Auxiliary Problem Admissible Control Feedback Form 
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.
    B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Lecture Notes in Control and Information Sciences, No 220. Springer-Verlag (1996).Google Scholar
  2. 2.
    F. H. Clarke, Optimization and Nonsmooth Analysis, Jonh Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore (1983).Google Scholar
  3. 3.
    M. Jean and J. J. Moreau, Dynamics of elastic or rigid bodies with frictional contact: numerical methods Publications of Laboratory oif Mechanics and Acoustics, Marseille, April, No 124, (1991).Google Scholar
  4. 4.
    E. B. Lee and L. Markus, Foundations of Optimal Control Theory John Wiley and Sons, Inc., New York, London, Sydney (1967).Google Scholar
  5. 5.
    B. M. Miller, “Method of Discontinuous Time Change in Problems of Control for Impulse and Discrete-Continuous Systems,“ Autom. Rem. Control, 54 (1993) 1727–1750.Google Scholar
  6. 6.
    B. M. Miller., “The generalized solutions of nonlinear optimization problems with impulse controls,“ SIAM J. Control Optirn., 34, No. 4, 1420–1440 (1996).Google Scholar
  7. 7.
    B.-M. Miller., “The generalized solutions of ordinary differential equations in the impulse control problems,“ Journal of Mathematical Systems, Estimation, and Control, 6, No 4, 415–435, (1996).Google Scholar
  8. 8.
    B. M. Miller, “Representation of robust and non-robust solutions of nonlinear discrete-continuous systems“ in Proccedings of International Workshop on Hybrid and Real-Time Systems (H ART'97), Grenoble, France. March 26–28, (1997).Google Scholar
  9. 9.
    J. J. Moreau, “Unilateral contacts and dry friction in finite freedom dynamics”, in Nonsmooth-Mechanics and Applications, CIMS Course and Lectures, No 302, Springer-Verlag, Wien. New York, pp; 1–82, (1988).Google Scholar
  10. 10.
    Yu. V. Orlov, Theory of Optimal Systems with Generalized Controls. [in Russian], Nauka, Moscow (1988).Google Scholar
  11. 11.
    A. Ph. Phillipov, Differential Equations with Discontinuous Right-Hand-Side. [in Russian], Nayka, Moscow (1985).Google Scholar
  12. 12.
    L. C. Young, Lectures on Variational Calculus and the Theory of Optimal Control, W. B. Saunders Company, Philadelphia, Londod, Toronto, 1969.Google Scholar
  13. 13.
    S. T. Zavalishchin and A. N. Sesekin, Impulsive Processes. Models and Applications [in Russian], Nauka, Moscow (1991).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Boris M. Miller
    • 1
  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations