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Mathematical Methods of Operations Research

, Volume 51, Issue 3, pp 399–417 | Cite as

Perturbation of linear quadratic systems with jump parameters and hybrid controls

  • Rachid El Azouzi
  • Mohammed Abbad
  • Eitan Altman

Abstract.

We consider the problem of the perturbation of a class of linear-quadratic differential games with piecewise deterministic dynamics, where the changes from one structure (for the dynamics) to another are governed by a finite-state Markov process. Player 1 controls the continuous dynamics, whereas Player 2 controls the rate of transition for the finite-state Markov process; both have access to the states of both processes. Player 1 wishes to minimize a given quadratic performance index, while player 2 wishes to maximize or minimize the same quantity. The problem above leads to the analysis of some linearly coupled set of quadratic equations (Riccati equations). We obtain a Taylor expansion in the perturbation for the solution of the equation for a fixed stationary policy of the player 2. This allows us to solve the game or team problem as a function of the perturbation.

Key words. Singular perturbation, stochastic game, continuous-time Markov chain, lexicographical optimization, averaging, aggregation 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Rachid El Azouzi
    • 1
  • Mohammed Abbad
    • 2
  • Eitan Altman
    • 3
  1. 1.Département de Math & Info, Faculté des sciences, B.P 1014 Rabat, Maroc (e-mail: relazouz@sophia.inria.fr)FR
  2. 2.Département de Math & Info, Faculté des sciences, B.P 1014 Rabat, Maroc (e-mail: abbad@fsr.ac.ma)FR
  3. 3.INRIA B.P 93, 2204 Route des Lucioles, 06902 Sophia Antipolis Cedex, France (e-mail: altman@sophia.inria.fr)FR

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