Lyapunov’s Convexity Theorem, zonoids, and bang-bang

  • S. S. KutateladzeEmail author


This is a short overview of the connections of the Lyapunov Convexity Theorem with the modern sections of analysis, geometry, and optimal control.


Lyapunov Convexity Theorem 


  1. 1.
    A. A. Lyapunov, “On Completely Additive Set Functions. I,” Izv. Akad. Nauk SSSR Ser. Mat. 4, 465–478 (1940).Google Scholar
  2. 2.
    A. A. Lyapunov, “On Completely Additive Set Functions. II,” Izv. Akad. Nauk SSSR Ser. Mat. 10(3), 277–279 (1946).zbMATHGoogle Scholar
  3. 3.
    K. I. Chuĭkina, “On Additive Vector-Functions,” Dokl. Akad. Nauk SSSR 76, 801–804 (1951).Google Scholar
  4. 4.
    E. V. Glivenko, “On the Ranges of Additive Vector-Functions,” Mat. Sb. 34(76), 407–416 (1954).MathSciNetGoogle Scholar
  5. 5.
    Yu. G. Reshentnyak and V. A. Zalgaller, “On Rectifiable Curves, Additive Vector-Functions, and Mixing of Straight Line Segments,” Vestnik Leningrad. Gos. Univ. 2, 45–65 (1954).Google Scholar
  6. 6.
    Z. Artstein, “Yet Another Proof of the Lyapunov Convexity Theorem,” Proc. Amer. Math. Soc. 108(1), 89–91 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    E. Bolker, “A Class of Convex Bodies,” Trans. Amer. Math. Soc. 145, 323–345 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. Elton and Th. Hill, “A Generalization of Lyapunov Convexity Theorem to Measures with Atoms,” Proc. Amer. Math. Soc. 99(2), 97–304 (1987).MathSciNetGoogle Scholar
  9. 9.
    P. Goodey and W. Weil, “Zonoids and Generalizations,” in Handbook of Convex Geometry, Vol. B (North-Holland, Amsterdam, 1993), pp. 1296–1326.Google Scholar
  10. 10.
    H. Halkin, “A Generalization of La Salle’s Bang-Bang Principle,” SIAM J. Control and Optimization 2, 199–202 (1965).MathSciNetGoogle Scholar
  11. 11.
    H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control (Academic Press, New York, 1969).zbMATHGoogle Scholar
  12. 12.
    J. P. LaSalle, “The Time Optimal Control Problem,” in Contributions to the Theory of Non-Linear Oscillations, Vol. 5 (Princeton Univ. Press, Princeton, 1960), pp. 1–24.Google Scholar
  13. 13.
    N. Levinson, “Minimax, Liapunov, and ‘Bang-Bang,” J. Differential Equations 2, 218–241 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    J. Lindenstrauss, “A Short Proof of Liapounoff’s Convexity Theorem,” J. Math. Mech. 15, 971–972 (1966).MathSciNetzbMATHGoogle Scholar
  15. 15.
    L. W. Neustadt, “The Existence of Optimal Control in the Absence of Convexity,” J. Math. Anal. Appl. 7, 110–117 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    R. J. Nunke and L. J. Savage, “On the Set of Values of a Nonatomic, Finitely Additive, Finite Measure,” Proc. Amer. Math. Soc. 3(2), 217–218 (1952).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    C. Olech, “Extremal Solutions of a Control System,” J. Differential Equations 2, 74–101 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Handbook of Measure Theory, Vols. 1 and 2, Ed. by E. Pap (North Holland, Amsterdam, 2002).Google Scholar
  19. 19.
    A. Pietsch, History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007).zbMATHGoogle Scholar
  20. 20.
    D. Ross, “An Elementary Proof of Lyapunov’s Theorem,” Amer. Math. Monthly 112(7), 651–653 (2005).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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