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AUTHOR(S): 

Ivan G. Ivanov, Nikolay Netov

 

TITLE

The Nash Equilibrium Point in the LQ Game on Positive Systems with Two Players

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ABSTRACT

We consider the linear quadratic differential games for positive linear systems with the feedback information structure and two players. The Newton method to obtain the stabilizing solution of a corresponding set of Riccati equations is presented in the literature. Here, we modify the Newton method and propose a new faster iterative method. Moreover, the convergence properties of the modification are investigated and the sufficient condition to apply the modification is derived. The performances of the proposed algorithm are illustrated on some numerical examples.

KEYWORDS

feedback Nash equilibrium, generalized Riccati equation, stabilizing solution, nonnegative solution

REFERENCES

[1] J. F. do Amaral, T. P. de Lima and M. S. Silva, Positive Solutions of a Discrete-time System and the Leontief Price-Model, 2006, pp. 65– 72, LNCIS, 341, Springer, in Positive Systems, C.Commault and N. Marchand (editors) Proceedings of the Second Multidisciplinary International Symposium on Positive Systems. [1] J. F. do Amaral, T. P. de Lima and M. S. Silva, Positive Solutions of a Discrete-time System and the Leontief Price-Model, 2006, pp. 65– 72, LNCIS, 341, Springer, in Positive Systems, C.Commault and N. Marchand (editors) Proceedings of the Second Multidisciplinary International Symposium on Positive Systems. 

[2] T. Azevedo-Perdicoulis and G. Jank, Linear Quadratic Nash Games on Positive Linear Systems, European Journal of Control, 11, 2005, 1– 13. 

[3] van den Broek, W. (2001). Uncertainty in Differential Games. PhD-thesis Univ. Tilburg, Netherlands. 

[4] W. van den Broek, J. Engwerda and J. Schumacher, Robust Equilibria in Indefinite Linear Quadratic Differential Games, Journal of Optimization Theory and Applications, 119, 3, 2003, pp.565–595. 

[5] J. Engwerda, Linear Quadratic Differential Games: An Overview, Advances in Dynamic Games and Their Applications, Birkhuser Boston, 2009, pp. 1-34. 

[6] V. Dragan, T. Damm, G. Freiling, Lyapunov Iterations for coupled Riccati Differential Equations arising in connection with Nash Differential Games, Math. Reports, 9(59),1, 2007, pp. 35–46. 

[7] D. Filipovic, S. Tappe and J. Teichmann, Term ´ Structure Models Driven by Wiener Processes and Poisson Measures: Existence and Positivity, SIAM J. Financial Math., 1, 2010, pp. 523–554. 

[8] G. Jank and D. Kremer, Open loop Nash games and positive systems- solvability conditions for nonsymmetric Riccati equations, Proceedings of MTNS, 2004, Katolieke Universiteit, Leuven, Belgium. 

[9] L. Metzler, Stability of multiple markets: the Hicks condition, Econometrica, 13(4), 1945, pp. 277–292. I. G. Ivanov, N. Netov

Cite this paper

Ivan G. Ivanov, Nikolay Netov. (2016) The Nash Equilibrium Point in the LQ Game on Positive Systems with Two Players. Mathematical and Computational Methods, 1, 242-246

 

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