Multi-time scale zero-sum differential games with perfect state measurements

Zigang Pan, M Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We obtain necessary and sufficient conditions for the existence of 'approximate' saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters ε1 and ε2, and the terminology 'approximate saddle-point solution' is used to refer to saddle-point policies that do not depend on ε1 and ε2, while providing cost levels within O(ε(1)) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in the H-optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.

Original languageEnglish (US)
Title of host publicationProceedings of the IEEE Conference on Decision and Control
PublisherPubl by IEEE
Number of pages6
ISBN (Print)0780312988
StatePublished - 1993
EventProceedings of the 32nd IEEE Conference on Decision and Control. Part 2 (of 4) - San Antonio, TX, USA
Duration: Dec 15 1993Dec 17 1993


OtherProceedings of the 32nd IEEE Conference on Decision and Control. Part 2 (of 4)
CitySan Antonio, TX, USA

ASJC Scopus subject areas

  • Chemical Health and Safety
  • Control and Systems Engineering
  • Safety, Risk, Reliability and Quality


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