Advertisement

Open-Loop Impulse Control

  • Alexander B. KurzhanskiEmail author
  • Alexander N. Daryin
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 468)

Abstract

This chapter describes how to find optimal open-loop impulse controls [1, 2, 3, 4, 5, 7, 8, 9]. We begin by defining an impulse control system and proving the existence and uniqueness of its trajectories. Then we set up the basic problem of open-loop impulse control. This is how to transfer the system from a given initial state to a given target state within given time under a control of minimum variation. A key point in solving the open-loop impulse control problem is the construction of reachability sets for the system. Here we indicate how to construct such sets and study their properties. After that we present some simple model examples. The solution to the optimal impulse control problem problem is given by the Maximum Rule for Impulse Controls, an analogue of Pontryagin’s Maximum Principle for ordinary controls [7, 10]. The linearity of the considered system implies that the Maximum Rule indicates not only the necessary, but also some sufficient conditions of optimality. We further describe and prove an important feature of the problem which is that there exists an optimal control as a combination of a finite number of impulses, whose number is not greater than the system dimension [8]. Then we discuss some extensions of the basic impulse control problem. These include the control of a subset of coordinates, problems with set-valued boundary conditions, and the problem of time-optimal impulse control. Finally we treat a problem of the Mayer–Bolza type (a Stiltjes integral-terminal functional), that is further used in solving the problem of closed-loop impulse control.

References

  1. 1.
    Bellman, R.: Introduction to the Mathematical Theory of Controlled Processes, vol. 1/2, Academic Press, New York (1967/1971)Google Scholar
  2. 2.
    Bensoussan, A., Lions, J.L.: Contrôle Impulsionnel et Inéquations Quasi-Variationnelles. Dunod, Paris (1982)zbMATHGoogle Scholar
  3. 3.
    Carter, T.E.: Optimal impulsive space trajectories based on linear equations. J. Optim. Theory Appl. 70(2), 277–297 (1991).  https://doi.org/10.1007/BF00940627MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Daryin, A.N., Kurzhanski, A.B.: Impulse control inputs and the theory of fast controls. In: Proceedings of 17th IFAC World Congress, pp. 4869–4874. IFAC, Seoul (2008)Google Scholar
  5. 5.
    Dykhta, V.A., Samsonuk, O.N.: Optimal Impulsive Control with Applications. Fizmatlit, Moscow (2003). (Russian)Google Scholar
  6. 6.
    Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publicarions, New York (1975)Google Scholar
  7. 7.
    Krasovski, N.N.: The Theory of Control of Motion. Nauka, Moscow (1968)Google Scholar
  8. 8.
    Kurzhanski, A.B., Osipov, YuS: On controlling linear systems through generalized controls. Differ. Uravn (Differential Equations) 5(8), 1360–1370 (1969). (Russian)Google Scholar
  9. 9.
    Neustadt, L.W.: Optimization, a moment problem and nonlinear programming. SIAM J. Control 2(1), 33–53 (1964)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, Princeton (1999)zbMATHGoogle Scholar
  11. 11.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1950)zbMATHGoogle Scholar
  12. 12.
    Schmaedeke, W.W., Russell, D.L.: Time optimal control with amplitude and rate limited controls. SIAM J Control 2(3), 373–395 (1964).  https://doi.org/10.1137/0302030MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Valentine, F.: Convex Sets. MacGrawhill, New York (1954)Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  • Alexander B. Kurzhanski
    • 1
    Email author
  • Alexander N. Daryin
    • 2
  1. 1.Faculty of Computational Mathematics and CyberneticsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Google ResearchZürichSwitzerland

Personalised recommendations