Advertisement

State-Constrained Control Under Higher Impulses

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

Abstract

In these sections, we deal with additional constraints on the solutions to equations of Sects.  7.1 and  7.4, related to control under generalized (higher) impulses . These restrictions are an analogy of state constraints for systems controlled by ordinary impulses of Chap.  5 (see also [1, 7]). Discussing the problem of optimal control under higher impulses and state constraints we describe it first in terms of the theory of distributions [12, 13]. indicating conditions for its solvability. Then, in order to formulate conditions of optimality, we use a reduction of the system to the first-order form under vector measures, as shown in Sect.  7.3.

References

  1. 1.
    Aubin, J.-P.: Viability Theory. SCFA Birkhauser, Boston (1991)zbMATHGoogle Scholar
  2. 2.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory. Wiley-Interscience, New York (1958)zbMATHGoogle Scholar
  3. 3.
    Ekeland, I., Temam, R.: Analyse Convexe et Problemes Variationelles. Dunot, Paris (1973)zbMATHGoogle Scholar
  4. 4.
    Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. Nort-Holland, Amsterdam (1979)Google Scholar
  5. 5.
    Krein, M., Smulian, V.: On regular convex sets in the conjugate to a Banach space. Ann. Math. 41, 556–583 (1940)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kurzhanski, A.B.: The principle of optimality in measurement feedback control for linear systems. In: Rantzer, A., Byrnes, C. (eds.) Directions in Mathematical Systems Theory and Optimization, pp. 193–202. Springer, Berlin (2003)CrossRefGoogle Scholar
  7. 7.
    Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes: a mathematical formalism for uncertain dynamics, viability and control. In: Advances in Nonlinear Dynamics and Control. Progress in Systems and Control Theory, vol. 41, pp. 122–188 (1993)Google Scholar
  8. 8.
    Kurzhanski, A.B., Osipov, Yu.S.: On controlling linear systems through generalized controls. Differ. Uravn. 5(8), 1360–1370 (1969)Google Scholar
  9. 9.
    Leitman, G.: Optimality and reachability with feedback controls. Dynamical Systems and Microphysics: Control Theory and Mechanics. Academic, Orlando (1982)Google Scholar
  10. 10.
    Liapounoff, A.A.: Sur les fonctions-vecteurs completement additives. Bulletin de l’académie des sciences de l’URSS. Série mathématique 4, 465–478 (1940)Google Scholar
  11. 11.
    Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, Princeton, NJ (1999)zbMATHGoogle Scholar
  12. 12.
    Schwartz, L.: Théorie des distributions. Hermann, Paris (1950)zbMATHGoogle Scholar
  13. 13.
    Schwartz, L.: Méthodes mathématiques pour les sciences physiques. Hermann, Paris (1961)zbMATHGoogle Scholar
  14. 14.
    Zavalischin, S.T.: On the question of the general form of a linear equation, I, II. Differ. Equ. 7(5), 791–797; 7(6), 981–989Google 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