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Linear Quadratic Pareto Game of the Stochastic Systems in Infinite Horizon

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Abstract

This paper investigates the necessary/sufficient conditions for Pareto optimality in the infinite horizon linear quadratic stochastic differential game. Based on the necessary and sufficient characterization of the Pareto optimality, the problem is transformed into a set of constrained stochastic optimal control problems with a special structure. Under the assumption about the Lagrange multipliers, utilizing the stochastic Pontryagin maximum principle, the necessary conditions for the existence of the Pareto efficient strategies are presented. Furthermore, a condition is introduced to guarantee that the element zero does not belong to the Lagrange multiplier set. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional and a transversality condition provide the sufficient conditions for a control to be Pareto efficient. The characterization of Pareto efficient strategies and Pareto solutions is also studied. If the system is stabilizable, then the solvability of the related generalized algebraic Riccati equation provides a sufficient condition under which all Pareto efficient strategies can be obtained by the weighted sum optimality method and all Pareto solutions can be derived based on the solutions of an introduced algebraic Lyapunov equation.

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References

  1. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Math. Mex. 5(2), 102–119 (1960)

    MathSciNet  Google Scholar 

  2. Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. Academic Press, New York (1999)

    MATH  Google Scholar 

  3. Neumann, J.V., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)

    MATH  Google Scholar 

  4. Acemoglu, D.: Introduction to Modern Economic Growth. Princeton University Press, New Jersey (2008)

    MATH  Google Scholar 

  5. Dockner, E.J., Jørgensen, S., Long, N.V., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  6. Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38(152), 543–559 (1928)

    Article  Google Scholar 

  7. Engwerda, J.C.: The regular convex cooperative linear quadratic control problem. Automatica 44(9), 2453–2457 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Engwerda, J.C.: Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games. SIAM J. Control Optim. 48(6), 3859–3881 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Reddy, P.V., Engwerda, J.C.: Pareto optimality in infinite horizon linear quadratic differential games. Automatica 49(6), 1705–1714 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Reddy, P.V., Engwerda, J.C.: Necessary and sufficient conditions for Pareto optimality in infinite horizon cooperative differential games. IEEE Trans. Autom. Control 59(9), 2536–2542 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, B.S., Lee, H.C., Wu, C.F.: Pareto optimal filter design for nonlinear stochastic fuzzy systems via multiobjective \(H_2/H_\infty \) optimization. IEEE Trans. Fuzzy Syst. 23(2), 387–399 (2015)

    Article  Google Scholar 

  12. Chen, B.S., Ho, S.J.: Multiobjective tracking control design of T-S fuzzy systems: Fuzzy Pareto optimal approach. Fuzzy Sets Syst. 290, 39–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mukaidani, H., Xu, H.: Pareto optimal strategy for stochastic weakly coupled large scale systems with state dependent system noise. IEEE Trans. Autom. Control 54(9), 2244–2250 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mukaidani, H.: Dynamic games for stochastic systems with delay. Asian J. Control 15(5), 1251–1260 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, W.H., Lin, Y.N., Xue, L.R.: Linear quadratic Pareto optimal control problem of stochastic singular systems. J. Frankl. Inst. 354(2), 1220–1238 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, Y.N., Zhang, T.L., Zhang, W.H.: Infinite horizon linear quadratic Pareto game of the stochastic singular systems. J. Frankl. Inst. 355(10), 4436–4452 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, Y.N., Jiang, X.S., Zhang, W.H.: Necessary and sufficient conditions for Pareto optimality of the stochastic systems in finite horizon. Automatica 94, 341–348 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, Y.N., Zhang, T.L., Zhang, W.H.: Pareto-based guaranteed cost control of the uncertain mean-field stochastic systems in infinite horizon. Automatica 92, 197–209 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, Y.N., Zhang, W.H.: Necessary/sufficient conditions for Pareto optimum in cooperative difference game. Optim. Control Appl. Methods 39(2), 1043–1060 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leitmann, G.: Cooperative and Noncooperative Many Player Differential Games. Springer, New York (1974)

    Google Scholar 

  21. Reddy, P.V.: Essays on Dynamic Games. Center for Economic Research, Tilburg (2011)

    Google Scholar 

  22. Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1968)

    Google Scholar 

  23. Peng, S.G.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28(4), 966–979 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Y.Z., Peng, S.G.: Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems. Automatica 38, 1417–1423 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ait Rami, M., Zhou, X.Y.: Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Trans. Autom. Control 45(6), 1131–1143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Engwerda, J.C.: LQ Dynamic Optimization and Differential Games. Wiley, Chichester (2005)

    Google Scholar 

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Acknowledgements

This work was supported by Shandong Provincial Natural Science Foundation (No. ZR2019MF008).

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  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, China

    Yaning Lin

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  1. Yaning Lin
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Correspondence to Yaning Lin.

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George Leitmann.

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Lin, Y. Linear Quadratic Pareto Game of the Stochastic Systems in Infinite Horizon. J Optim Theory Appl 183, 671–687 (2019). https://doi.org/10.1007/s10957-019-01553-4

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