Set-valued Calculus and Dynamic Programming in Problems of Feedback Control

  • Alexander B. Kurzhanski
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 124)


The present paper is a concise overview of some problems of feedback control under uncertainty and state constraints. It emphasizes the application of set-valued techniques for these problems (see [1], [9], [10]) and indicates the connections with Dynamic Programming (DP) approaches, particularly with the “nonsmooth” versions of the latter, (see [5], [6], [18]). A constructive technique based on ellipsoidal calculus as collected in monograph [13] is then described for linear systems with convex-valued hard bounds on the controls and state space variables. Namely, the respective convex compact set-valued constructions (see [8], [11], [12]) are described in terms of ellipsoidal-valued representations. This “ellipsoidal” move leads to rather effective algorithms with possibility of further computer animation. For the problem of control synthesis it particularly allows to present the solutions in terms of analytical designs rather than algorithms as required in the “exact” theory. 1 The last approach also appears to show connections with control techniques for uncertain systems based on applying Liapunov functions, [14].


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  1. [1]
    Aubin, J.-P.: Viability Theory, Birkhäuser, Boston, 1991.zbMATHGoogle Scholar
  2. [2]
    Basar, T.; Bernhard, P.: H∞ Optimal Control and Related Minimax Design Problems ser. SCFA, 2-nd ed., Birkhäuser, Boston, 1995.CrossRefGoogle Scholar
  3. [3]
    Capuzzo-Dolcetta, I.; Lions, P. L.: Viscosity solutions of Hamilton-Jacobi-Bellman equations and and state constraints. Trans. Amer. Math. Soc, v. 318, 1990, pp. 643–683.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Chernousko, F. L.: State Estimation for Dynamic Systems, CRC Press, 1994.Google Scholar
  5. [5]
    Crandall, M. G.; Lions, P. L.: Viscosity Solutions of Hamilton-Jacobi Equations, Trans. Amer. Math. Soc, 277, 1983, pp. 1–42.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Fleming, W. H.; Soner, H. M.: Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, 1993.Google Scholar
  7. [7]
    Knobloch, H.; Isidori, A.; Flockerzi, D.: Topics in Control Theory, Birkhäuser, DMV-Seminar, Band 22, 1993.zbMATHCrossRefGoogle Scholar
  8. [8]
    Krasovski, N. N.: Game-Theoretic Problems on the Encounter of Motions Nauka, Moscow, 1970, (in Russian), English Translation: Rendezvous Game Problems Nat. Tech. Inf. Serv., Springfield, VA, 1971.Google Scholar
  9. [9]
    Krasovski, N. N.; Subbotin, A. N.: Positional Differential Games, Springer-Verlag, 1988.Google Scholar
  10. [10]
    Kurzhanski, A. B.: Control and Observation Under Uncertainty. Nauka, Moscow, 1977.Google Scholar
  11. [11]
    Kurzhanski, A. B.; Nikonov, O. I.: On the Problem of Synthesizing Control Strategies. Evolution Equations and Set-Valued Integration, Doklady Akad. Nauk SSSR, 311, 1990, pp. 788-793, Sov. Math. Doklady, v. 41, 1990.Google Scholar
  12. [12]
    Kurzhanski, A.; Nikonov, O. I.: Evolution Equations for Tubes of Trajectories of Synthesized Control Systems. Russ. Acad. of Sci. Math. Doklady, v. 48, N3, 1994, pp. 606–611.Google Scholar
  13. [13]
    Kurzhanski, A. B.; Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston, ser. SCFA, 1996.Google Scholar
  14. [14]
    Leitmann, G.: One approach to control of uncertain dynamical systems. Proc. 6-th. Workshop on Dynamics and Control, Vienna, 1993.Google Scholar
  15. [15]
    Lions, P.-L.; Souganidis, P. E.: Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaac’s Equations, SIAM J. Cont. Opt., v. 23, 1985, pp. 566–583.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Pontryagin, L. S.: Linear Differential Games of Pursuit. Mat. Sbornik v. 112, (154):3(7), 1980.Google Scholar
  17. [17]
    Rockafellar, R. T.: Convex Analysis, Princeton University Press, 1970.Google Scholar
  18. [18]
    Subbotin, A. I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective, Ser. SC, Birkhäuser, Boston, 1995.Google Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Alexander B. Kurzhanski
    • 1
  1. 1.Faculty of Comput. Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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