## Abstract

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 ε_{1} and ε_{2}, and the terminology 'approximate saddle-point solution' is used to refer to saddle-point policies that do not depend on ε_{1} and ε_{2}, 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 language | English (US) |
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Title of host publication | Proceedings of the IEEE Conference on Decision and Control |

Publisher | Publ by IEEE |

Pages | 3366-3371 |

Number of pages | 6 |

Volume | 4 |

ISBN (Print) | 0780312988 |

State | Published - 1993 |

Event | Proceedings of the 32nd IEEE Conference on Decision and Control. Part 2 (of 4) - San Antonio, TX, USA Duration: Dec 15 1993 → Dec 17 1993 |

### Other

Other | Proceedings of the 32nd IEEE Conference on Decision and Control. Part 2 (of 4) |
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City | San Antonio, TX, USA |

Period | 12/15/93 → 12/17/93 |

## ASJC Scopus subject areas

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