Abstract
This paper investigates Nash games for stochastic singular systems with Markovian jumps. Based on the generalized Itô’s formula, the corresponding linear quadratic optimal control problem is studied for the first time. Then, we establish the existence of Nash strategies by means of generalized coupled Riccati algebraic equations. As an application, the stochastic H2/H∞ control with state, control, and external disturbance-dependent noise is discussed.
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References
Boukas, E.K.: Stabilization of stochastic singular nonlinear hybrid systems. Nonlinear Anal. 64(2), 217–228 (2006)
Dragan, V., Morozan, T., Stoica, A.M.: Mathematical Methods in Robust Control of Linear Stochastic Systems. Mathematical Concepts and Methods in Science and Engineering, vol. 50. Springer, New York (2006)
Engwerda, J.C.: Salmah, feedback Nash equilibrium for linear quadratic descriptor differential games. Automatica: A Journal of lfac the International Federation of Automatic Control 48(4), 625–631 (2010)
Mizukami, K., Xu, H.: A note on the open-loop linear-quadratic Nash games for continuous descriptor systems. Trans. Soc. Instrument Control Engineers 28(7), 896–898 (2009)
Mukaidani, H.: Soft-constrained stochastic Nash games for weakly coupled large-scale systems. Automatica J. IFAC 45(5), 1272–1279 (2009)
Mukaidani, H., Yamamoto, T.: Nash strategy for multiparameter singularly perturbed Markov jump stochastic systems. IET Control Theory Appl. 6(14), 2337–2345 (2012)
Zhou, H.Y., Zhu, H.N., Zhang, C.K.: Linear quadratic Nash differential games of stochastic singular systems. J. Sys. Sci. Inf. 2(6), 553–560 (2014)
Zhu, H.N., Zhang, C.K.: Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise. J. Control Theory Appl. 11(4), 629–633 (2013)
Zhu, H.N., Zhang, C.K., Bin, N.: Stochastic Nash games for Markov jump linear systems with state- and control-dependent noise. J. Oper. Res. Soc. China 2(4), 481–498 (2014)
Zhu, H.N., Zhang, C.K., Bin, N.: Infinite horizon linear quadratic stochastic Nash differential games of Markov jump linear systems with its application. Internat. J. Systems Sci. 45(5), 1196–1201 (2014)
Zhang, C.K., Zhu, H.N., Zhou, H.Y.: Non-cooperative Stochastic Differential Game Theory of Generalized Markov Jump Linear Systems. Studies in Systems, Decision and Control, vol. 67. Springer, Berlin (2017)
Acknowledgements
We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript.
Funding
This work was partially supported by NNSF of China (Grant No. 11571126).
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Liu, B., Wang, X. Linear Quadratic Nash Differential Games of Stochastic Singular Systems with Markovian Jumps. Acta Math Vietnam 45, 651–660 (2020). https://doi.org/10.1007/s40306-018-00318-x
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DOI: https://doi.org/10.1007/s40306-018-00318-x