Journal of Optimization Theory and Applications

, Volume 18, Issue 1, pp 141–151 | Cite as

On saddle-point optimality in differential games

  • H. L. Stalford
  • G. Leitmann
Contributed Papers


A family of two-person, zero-sum differential games in which the admissible strategies are Borel measurable is defined, and two types of saddle-point conditions are introduced as optimality criteria. In one, saddle-point candidates are compared at each point of the state space with all playable pairs at that point; and, in the other, they are compared only with strategy pairs playable on the entire state space. As a theorem, these two types of optimality are shown to be equivalent for the defined family of games. Also, it is shown that a certain closure property is sufficient for this equivalence. A game having admissible strategies everywhere constant, in which the two types of saddle-point candidates are not equivalent, is discussed.

Key Words

Differential games playability saddle-point optimality 


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  1. 1.
    Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1965 (p. 38).Google Scholar
  2. 2.
    Berkovitz, L. D.,Łectures on Differential Games, Necessary Conditions for Optimal Strategies in a Class of Differential Games and Control Problems, SIAM Journal on Control, Vol. 5, No. 1, 1967.Google Scholar
  3. 3.
    Berkovitz, L. D.,Differential Games and Related Topics, Edited by H. W. Kuhn and G. P. Szego, North-Holland Publishing Company, Amsterdam, Holland, 1971 (p. 3).Google Scholar
  4. 4.
    Stalford, H.,Sufficiency Conditions in Optimal Control and Differential Games, University of California, Berkeley, California, Operations Research Center, Report No. ORC-70-13, 1970.Google Scholar
  5. 5.
    Stalford, H., andLeitmann, G.,Sufficient Conditions for Optimality in Two-Person Zero-Sum Differential Games with State and Strategy Constraints, Journal of Mathematical Analysis and Applications, Vol. 33, No. 3, 1971.Google Scholar
  6. 6.
    Varaiya, P., andLin, J.,Existence of Saddle Points in Differential Games, SIAM Journal on Control, Vol. 7, No. 1, 1969.Google Scholar
  7. 7.
    Stalford, H., andLeitmann, G.,On the Equivalence of Two Criteria for Optimal Closed-Loop Control, Ricerche di Automatica, Vol. 4, Nos. 2–3, 1973.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • H. L. Stalford
    • 1
  • G. Leitmann
    • 2
  1. 1.Radar DivisionNaval Research LaboratoryWashington, DC
  2. 2.University of CaliforniaBerkeley

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