Abstract
We consider differential games of fixed duration with phase coordinate restrictions on the players. Results of Ref. 1 on games with phase restrictions on only one of the players are extended. Using Berkovitz's definition of a game (Ref. 2), we prove the existence and continuity (or Lipschitz continuity) of the value under appropriate assumptions. We also note that the value can be characterized as the viscosity solution of the associated Hamilton-Jacobi-Isaacs equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ghassemi, K. H.,On Differential Games of Fixed Duration with Phase Coordinate Restrictions on One Player, SIAM Journal on Control and Optimization (to appear).
Berkovitz, L. D.,The Existence of Value and Saddle Point in Games of Fixed Duration, SIAM Journal on Control and Optimization, Vol. 23, pp. 172–196, 1985.
Berkovitz, L. D.,The Existence of Value and Saddle Point in Games of Fixed Duration: Erratum and Addendum, SIAM Journal Control and Optimization, Vol. 26, pp. 740–742, 1988.
Friedman, A.,Differential Games, Wiley-Interscience, New York, New York, 1971.
Friedman, A.,Differential Games, CBMS Regional Conference Series in Mathematics, No. 18, American Mathematical Society, Providence, Rhode Island, 1974.
Scalzo, R. C.,Differential Games with Restricted Phase Coordinates, SIAM Journal on Control and Optimization, Vol. 12, pp. 426–434, 1974.
Subbotin, A. I.,Differential Games with Constraints on Phase States, Soviet Mathematics Doklady, Vol. 11, pp. 933–936, 1970.
Zaremba, L. S.,Existence of Value in Differential Games with Fixed Time Duration, Journal of Optimization Theory and Applications, Vol. 38, pp. 581–598, 1982.
Zaremba, L. S.,Existence of Value in Differential Games with Terminal Cost Function, Journal of Optimization Theory and Applications, Vol. 39, pp. 89–104, 1983.
Soner, H. M.,Optimal Control with State Space Constraint, I, SIAM Journal on Control and Optimization, Vol. 24, pp. 552–561, 1986.
Capuzzo-Dolcetta, I., andLions, P. L.,Hamilton-Jacobi Equations and State-Constrained Problems, Institute for Mathematics and Its Applications, Preprint Series No. 342, 1987.
Berkovitz, L. D.,Differential Games without the Isaacs Condition, Recent Advances in Communication and Control Theory, Edited by R. E. Kalman, G. I. Marchuk, A. E. Roberti, and A. J. Viterbi, Optimization Software, New York, New York, 1987.
Crandall, M. G., andLions, P. L.,Viscosity Solutions of Hamilton-Jacobi Equations, Transactions of the American Mathematical Society, Vol. 277, pp. 1–42, 1983.
Crandall, M. G., Evans, L. C., andLions, P. L.,Some Properties of Hamilton-Jacobi Equations, Transactions of the American Mathematical Society, Vol. 282, pp. 487–502, 1984.
Berkovitz, L. D.,Characterization of the Value of Differential Games, Applied Mathematics and Optimization, Vol. 17, pp. 177–183, 1988.
Ishii, H.,Uniqueness of Unbounded Viscosity Solutions of the Hamilton-Jacobi Equations, Indiana University Mathematics Journal, Vol. 33, pp. 721–748, 1984.
Haji-Ghassemi, K.,On Differential Games of Fixed Duration with Phase Coordinate Restrictions, Purdue University, PhD Thesis, 1988.
Capuzzo-Dolcetta, I.,Hamilton-Jacobi Equations with Constraints, Stochastic Differential Systems, Stochastic Control Theory and Applications, Edited by W. Fleming and P. L. Lions, Springer-Verlag, New York, New York, 1988.
Author information
Authors and Affiliations
Additional information
Communicated by L. D. Berkovitz
This work comprises a part of the author's PhD Thesis completed at Purdue University under the direction of Professor L. D. Berkovitz. The author wishes to thank Professor Berkovitz for suggesting the problem and many valuable discussions. During the research for this work, the author was supported by a David Ross Grant from Purdue University as well as by NSF Grant No. DMS-87-00813.
Rights and permissions
About this article
Cite this article
Ghassemi, K.H. Differential games of fixed duration with state constraints. J Optim Theory Appl 68, 513–537 (1991). https://doi.org/10.1007/BF00940068
Issue Date:
DOI: https://doi.org/10.1007/BF00940068