Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games
Tóm tắt
In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t
f. In this paper, we study the behavior of feedback Nash equilibria for t
f→∞. In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.
Tài liệu tham khảo
Isaacs, R., Differential Games, Parts 1–4, Research Memoranda RM-1391, RM-1399, RM-1411, RM-1468, Rand Corporation, 1956.
Başar, T., and Bernard, P., H ∞-Optimal Control and Related Minimax Design Problems, Birkhäuser, Boston, Massachusetts, 1991.
Stoorvogel, A. A., H ∞-Control Problem: A State-Space Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1992.
Starr, A. W., and Ho, Y. C., Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, pp. 184–206, 1969.
Starr, A. W., and Ho, Y. C., Further Properties of Nonzero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 3, pp. 207–219, 1969.
Başar, T., and Olsder, G. J., Dynamic Noncooperative Game Theory, 2nd Edition, Academic Press, London, England, 1995.
Nash, J., Noncooperative Games, Annals of Mathematics, Vol. 54, pp. 286–295, 1951.
Engwerda, J. C., and Weeren, A. J. T. M., The Open-Loop Nash Equilibrium in LQ-Games Revisited, CentER Discussion Paper 9551, Tilburg University, 1995.
Lukes, D. L., Equilibrium Feedback Control in Linear Games with Quadratic Cost, SIAM Journal on Control and Optimization, Vol. 9, pp. 234–252, 1971.
Papavassilopoulos, G. P., and Cruz, J. B., Jr., On the Uniqueness of Nash Strategies for a Class of Analytic Differential Games, Journal of Optimization Theory and Applications, Vol. 27, pp. 309–314, 1979.
Freiling, G., Jank, G., and Abou-Kandil, H., On Global Existence of Solutions to Coupled Matrix Riccati Equations in Closed-Loop Nash Games, IEEE Transactions on Automatic Control, Vol. 41, pp. 264–269, 1996.
Papavassilopoulos, G. P., and Olsder, G. J., On the Linear-Quadratic, Closed-Loop No-Memory Nash Game, Journal of Optimization Theory and Applications, Vol. 42, pp. 551–560, 1984.
Cruz, J. B., Jr., and Chen, C. I., Series Nash Solution of Two-Person Nonzero-Sum Linear-Quadratic Games, Journal of Optimization Theory and Applications, Vol. 7, pp. 240–257, 1971.
Jodar, L., and Abou-Kandil, H., Kronecker Products and Coupled Matrix Riccati Differential Equations, Linear Algebra and Its Applications, Vol. 121, pp. 39–51, 1989.
Abou-Kandil, H., Freiling, G., and Jank, G., Necessary Conditions for Constant Solutions of Coupled Riccati Equations in Nash Games, Systems and Control Letters, Vol. 21, pp. 295–306, 1993.
Papavassilopoulos, G. P., Medanic, J. V., and Cruz, J. B., Jr., On the Existence of Nash Strategies and Solutions to Coupled Riccati Equations in Linear-Quadratic Games, Journal of Optimization Theory and Applications, Vol. 28, pp. 49–76, 1979.
Li, T. Y., and Gajic, Z., Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games, New Trends in Dynamic Games and Applications, Edited by G. J. Olsder, Birkhäuser, Boston, Massachusetts, pp. 333–351, 1995.
Khalil, H., and Kokotovic, P., Feedback and Well-Posedness of Singularly Perturbed Nash Games, IEEE Transactions on Automatic Control, Vol. 24, pp. 699–708, 1979.
Khalil, H., Multimodel Design of a Nash Strategy, Journal of Optimization Theory and Applications, Vol. 29, pp. 553–564, 1980.
ÖzgÜner, U., and Perkins, W. R., A Series Solution to the Nash Strategy for Large-Scale Interconnected Systems, Automatica, Vol. 13, pp. 313–315, 1977.
Petrovic, B., and Gajic, Z., The Recursive Solution of Linear-Quadratic Nash Games for Weakly Interconnected Systems, Journal of Optimization Theory and Applications, Vol. 56, pp. 463–477, 1988.
Abou-Kandil, H., and Bertrand, P., Analytic Solution for a Class of Linear-Quadratic Open-Loop Nash Games, International Journal of Control, Vol. 43, pp. 997–1002, 1986.
Maskin, E., and Tirole, J., Markov Perfect Equilibrium, 6th International Symposium on Dynamic Games and Applications, Preprint Volume, Edited by M. Breton and G. Zaccour, École des Hautes Études Commerciales, St. Jovite, Québec, Canada, pp. 432–461, 1994.
Başar, T., A Counterexample in Linear-Quadratic Games: Existence of Nonlinear Nash Strategies, Journal of Optimization Theory and Applications, Vol. 14, pp. 425–430, 1974.
Coppel, W. A., A Survey of Quadratic Systems, Journal of Differential Equations, Vol. 2, pp. 293–304, 1966.
Reyn, J. W., Phase Portraits of a Quadratic System of Differential Equations Occurring Frequently in Applications, Nieuw Archief voor Wiskunde, Vol. 5, pp. 107–154, 1987.
Bendixson, I., Sur les Courbes Définies par des Equations Differentielles, Acta Mathematica, Vol. 24, pp. 1–88, 1901.
Perko, L., Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Springer Verlag, New York, New York, Vol. 7, 1991.
PoincarÉ, H., Mémoire sur les Courbes Défines par une Équation Differentielle, Journal de Mathématiques, Vol. 7, pp. 375–422, 1881.
Tabellini, G., Money, Debt, and Deficits in a Dynamic Game, Journal of Economic Dynamics and Control, Vol. 10, pp. 427–442, 1986.
Geerts, A. H. W., and Hautus, M. L. J., The Output-Stabilizable Subspace and Linear Optimal Control, Progress in Systems and Control Theory, Edited by M. A. Kaashoek, J. H. van Shuppen, and A. C. M. Ran, Birkhäuser, Boston, Massachusetts, pp. 113–120, 1990.
Geerts, A. H. W., Structure of Linear-Quadratic Control, PhD Thesis, Eindhoven University of Technology, 1989.