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Journal of Optimization Theory and Applications

, Volume 128, Issue 1, pp 75–102 | Cite as

Zero-Sum Differential Games Involving Hybrid Controls

  • S. Dharmatti
  • M. Ramaswamy
Article

Abstract

We study a zero-sum differential game with hybrid controls in which both players are allowed to use continuous as well as discrete controls. Discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, an autonomous jump set A or a controlled jump set C, where one controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. One player uses all the three types of controls, namely, continuous controls, autonomous jumps, and controlled jumps; the other player uses continuous controls and autonomous jumps. We prove the continuity of the associated lower and upper value functions V and V+. Using the dynamic programming principle satisfied by V and V+, we derive lower and upper quasivariational inequalities satisfied in the viscosity sense. We characterize the lower and upper value functions as the unique viscosity solutions of the corresponding quasivariational inequalities. Lastly, we state an Isaacs like condition for the game to have a value

Keywords

Dynamic programming principle viscosity solutions quasivariational inequalities hybrid control differential games 

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References

  1. 1.
    Branicky, M. S., Borkar, V., Mitter, S. 1998A Unified Framework for Hybrid Control ProblemsIEEE Transactions on Automatic Control433145MathSciNetGoogle Scholar
  2. 2.
    Bensoussan A., Menaldi J.L. (1997). Dynamics of Hybrid Control and Dynamic Programming, Continuous, Discrete, and Impulsive Systems, Vol. 3, pp. 395–442, 1997.Google Scholar
  3. 3.
    Dharmatti, S., and Ramaswamy, M., Hybrid Control Systems and Viscosity Solutions, (to appear). SIAM Journal of Control and Optimization.Google Scholar
  4. 4.
    Bardi, M., Capuzzo Dolcetta, I. 1997Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman EquationsBirkhauserBoston, MassachusettsGoogle Scholar
  5. 5.
    Evans, L. C., Souganidis, P. E. 1984Differential Games and Representation Formulas for Solutions of the Hamilton-Jacobi-Isaac EquationsIndiana University Mathematics Journal33733797MathSciNetGoogle Scholar
  6. 6.
    Yong, J. 1990Differential Games with Switching StrategiesJournal of Mathematical Analysis and Applications145455467CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Yong, J. 1994Zero-Sum Differential Games Involving Impulse ControlsApplied Mathematics and Optimization29243261zbMATHMathSciNetGoogle Scholar
  8. 8.
    Zeidler, E. 1991Applied Functional Analysis, Main Principles, and Their ApplicationsSpringer VerlagNew York, NYApplied Mathematical Sciences, Vol. 109Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. Dharmatti
    • 1
  • M. Ramaswamy
    • 2
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.IISc-TIFR Mathematics ProgramTata Institute of Fundamental Research CenterBangaloreIndia

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