Abstract
The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Starr, A. W., andHo, Y. C.,Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, pp. 184–206, 1969.
Rhodes, I. B. On Nonzero-Sum Differential Games with Quadratic Cost Functionals, Proceedings of the First International Conference on the Theory and Applications of Differential Games, Amherst, Massachusetts, 1969.
Lukes, D. L.,Equilibrium Feedback Control in Linear Games with Quadratic Costs, SIAM Journal on Control, Vol. 9, pp. 234–252, 1971.
Krikelis, N. J., andRekasius, Z. V.,On the Solution of the Optimal Linear Control Problems under Conflict of Interest, IEEE Transactions on Automatic Control, AC-16, pp. 140–147, 1971.
Foley, M. H., andSchmitendorf, W. E.,On A Class of Nonzero-Sum, Linear Quadratic Differential Games, Journal of Optimization Theory and Applications, Vol. 7, pp. 357–377, 1971.
Basar, T.,A Counterexample in Linear-Quadratic Games: Existence of Non-Linear Nash Solutions, Journal of Optimization Theory and Applications, Vol. 14, pp. 425–430, 1974.
Vidyasagar, M.,A New Approach to N-Person, Nonzero-Sum, Linear Differential Games, Journal of Optimization Theory and Applications, Vol. 18, pp. 171–175, 1976.
Varaiya, P.,N-Person Nonzero-Sum Differential Games with Linear Dynamics, SIAM Journal on Control, Vol. 8, pp. 441–449, 1970.
Mageirou, E. F.,Values and Strategies for Infinite-Time Linear-Quadratic Games, IEEE Transactions on Automatic Control, Vol. AC-21, pp. 547–550, 1976.
Jacobson, D. H.,On Values and Strategies for Infinite-Time Linear-Quadratic Games, IEEE Transactions on Automatic Control, Vol. AC-22, pp. 490–491, 1977.
Willems, J. C.,Least Squares Stationary Optimal Control and the Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. AC-16, pp. 621–634, 1971.
Kalman, R. E.,Contributions to the Theory of Optimal Control, Boletin de la Sociedad Matematica Mexicana, Vol. 5, pp. 102–119, 1960.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Brockett, R. W.,Finite Dimensional Linear Systems, John Wiley and Sons, New York, New York, 1970.
Rosen, J. B.,Existence and Uniqueness of Equilibrium Points for Concave N-Person Games, Econometrica, Vol. 33, pp. 520–534, 1965.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAG-29-78-C-0016, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division, under Contract No. US-ERDA-EX-76-C-01-2088.
Rights and permissions
About this article
Cite this article
Papavassilopoulos, G.P., Medanic, J.V. & Cruz, J.B. On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games. J Optim Theory Appl 28, 49–76 (1979). https://doi.org/10.1007/BF00933600
Issue Date:
DOI: https://doi.org/10.1007/BF00933600