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Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise

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Abstract

This paper discusses the infinite time horizon nonzero-sum linear quadratic (LQ) differential games of stochastic systems governed by Itô’s equation with state and control-dependent noise. First, the nonzero-sum LQ differential games are formulated by applying the results of stochastic LQ problems. Second, under the assumption of mean-square stabilizability of stochastic systems, necessary and sufficient conditions for the existence of the Nash strategy are presented by means of four coupled stochastic algebraic Riccati equations. Moreover, in order to demonstrate the usefulness of the obtained results, the stochastic H-two/H-infinity control with state, control and external disturbance-dependent noise is discussed as an immediate application.

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References

  1. V. A. Ugrinovskii. Robust H∞ control in the presence of stochastic uncertainty. International Journal of Control, 1998, 71(2): 219–237.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Hinrichsen, A. J. Pritchard. Stochastic H∞. SIAM Journal on Control and Optimization, 1998, 36(5), 1504–1538.

    Article  MathSciNet  MATH  Google Scholar 

  3. B. S. Chen, W. H. Zhang. Stochastic H2/H∞ control with statedependent noise. IEEE Transactions on Automatic Control, 2004, 49(1): 45–57.

    Article  Google Scholar 

  4. W. H. Zhang, H. S. Zhang, B. S. Chen. Stochastic H2/H∞ control with (x, u, v)-dependent noise: Finite horizon case. Automatica, 2006, 42(11): 1891–1898.

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. C. Ho, A. E. Bryson, S. Baron. Differential games and optimal pursuit-evasion strategies. IEEE Transactions on Automatic Control, 1965, 10(4): 385–389.

    Article  MathSciNet  Google Scholar 

  6. A. W. Starr, Y. C. Ho. Nonzero-sum differential games. Journal of Optimization Theory and Applications, 1968, 3(3): 184–206.

    Article  MathSciNet  Google Scholar 

  7. W. A. van den Broek, J. C. Engwerda, J. M. Schumacher. Robust equilibria in indefinite linear-quadratic differential games. Journal of Optimization Theory and Applications, 2003, 119(3): 565–595.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Dockner, S. Jørgensen, N. Van Long, et al. Differential Games in Economics and Management Science. Cambridge: Cambridge University Press.

  9. J. C. Engwerda. LQ dynamic optimization and differential games. Chichester: John Wiley and Sons, 2005.

    Google Scholar 

  10. T. Basar, G. J. Olsder. Dynamic Noncooperative Game Theory. Philadelphia: SIAM, 2nd edition, 1999.

    MATH  Google Scholar 

  11. J. M. Yong. A leader-follower stochastic linear quadratic differential game. SIAM Journal on Control and Optimization, 2002, 41(4): 1015–1041.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. B. Mou, J. M. Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial and Management Optimization, 2006, 2(1): 93–115.

    MathSciNet  Google Scholar 

  13. M. McAsey, L. B. Mou. Generalized Riccati Equations Arising in Stochastic Games. Linear Algebra and its Applications, 2006, 416(2/3): 710–723.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Sagara, H. Mukaidani, T. Yamamoto. Stochastic Nash Games with State-Dependent Noise. IEEE Conference on Decision and Control. New York: IEEE, 2007: 1635–1638.

    Google Scholar 

  15. T. T. K. An, B. Øksendal. Maximum principle for stochastic differential games with partial information. Journal of Optimization Theory and Applications, 2008, 139(3): 463–483.

    Article  MathSciNet  Google Scholar 

  16. H. Mukaidani. Soft-constrained stochastic Nash games for weakly coupled large-scale systems. Automatica, 2009, 45(5): 1272–1279.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Feng, B. D. O. Anderson. An iterative algorithm to solve stateperturbed stochastic algebraic Riccati equations in LQ zero-sum games. Systems & Control Letters, 2010, 59(1): 50–56.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Øksendal. Stochastic differential equations: An introduction with application[M]. New York: Springer-Verlag, 5th edition, 1998.

    Google Scholar 

  19. L. Qian, Z. Gajic. Variance minimization stochastic power control in CDMA systems. IEEE Transactions on Wireless Communications, 2006, 5(1): 193–202.

    Article  Google Scholar 

  20. M. A. Rami, J. B. Moore, X. Zhou. Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM Journal on Control and Optimization, 2001, 40(4): 1296–1311.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. A. Rami, X. Zhou. Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Transactions on Automatic Control, 2000, 45(6): 1131–1143.

    Article  MathSciNet  MATH  Google Scholar 

  22. W. H. Zhang, B. S. Chen. On stabilizability and exact observability of stochastic systems with their applications. Automatica, 2004, 40(1): 87–94.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Basar. A counterexample in linear-quadratic games: Existence of nonlinear Nash solutions. Journal of Optimization Theory and Applications, 1974, 14(4): 425–430.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. J. N. Limebeer, B. D. O. Anderson, B. Hendel. A Nash game approach to mixed H2/H∞ control. IEEE Transactions on Automatic Control, 1994, 39(1): 69–82.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. H. Zhang, H. S. Zhang, B. S. Chen. Stochastic H2/H∞ control with (x, u, v)-dependent noise. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference. New York: IEEE, 2005: 7352–7357.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. School of Management, Guangdong University of Technology, Guangzhou Guangdong, 510520, China

    Huainian Zhu

  2. School of Economics & Commence, Guangdong University of Technology, Guangzhou Guangdong, 510520, China

    Chengke Zhang

Authors
  1. Huainian Zhu
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  2. Chengke Zhang
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Correspondence to Huainian Zhu.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 71171061) and the Natural Science Foundation of Guangdong Province (No. S2011010004970).

Huainian ZHU was born in 1985. He received his B.S. degree from North China Institute of Science and Technology in 2008, M.S. degree from Guangdong University of Technology in 2010. Currently, he is a Ph.D. candidate at the School of Management, Guangdong University of Technology. His research interests include management system engineering and game theory.

Chengke ZHANG was born in 1964. He is a professor at the School of Economics & Commence, Guangdong University of Technology. His research interests include system engineering, financial engineering, game theory and application.

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Zhu, H., Zhang, C. Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise. J. Control Theory Appl. 11, 629–633 (2013). https://doi.org/10.1007/s11768-013-1182-3

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  • DOI: https://doi.org/10.1007/s11768-013-1182-3

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