ABSTRACT This paper studies a class of continuous-time two person zero-sum stochastic differential games characterized by linear It?’s differential equation with state-dependent noise and Markovian parameter jumps. Under the assumption of stochastic stabilizability, necessary and sufficient condition for the existence of the optimal control strategies is presented by means of a system of coupled algebraic Riccati equations via using the stochastic optimal control theory. Furthermore, the stochastic H∞ control problem for stochastic systems with Markovian jumps is discussed as an immediate application, and meanwhile, an illustrative example is presented.
Cite this paper
H. Zhu, C. Zhang and N. Bin, "Infinite Horizon LQ Zero-Sum Stochastic Differential Games with Markovian Jumps," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1321-1326. doi: 10.4236/am.2012.330188.
 V. A. Ugrinovskii, “Robust H∞ Control in the Presence of Stochastic Uncertainty,” International Journal of Control, Vol. 71, No. 2, 1998, pp. 219-237.
 D. Hinrichsen and A. J. Pritchard, “Stochastic H∞,”SIAM Journal on Control and Optimization, Vol. 36, No. 5, 1998, pp. 1504-1538. doi: 10.1137/S0363012996301336
 B.-S. Chen and W. H. Zhang, “Stochastic H2/H∞ Control with State-Dependent Noise,” IEEE Transactions on Automatic Control, Vol. 49, No. 1, 2004, pp. 45-57.
 Y. L. Huang, W. H. Zhang and G. Feng, “Infinite Horizon H2/H∞ Control for Stochastic Systems with Markovian Jumps,” Automatica, Vol. 44, No. 3, 2008, pp. 857-863.
 T. Basar and G. J. Olsder, “Dynamic Noncooperative Game Theory,” 2nd Edition, SIAM, Philadelphia, 1999.
 E. J. Dockner, S. Jorgensen, N. Van Long and G. Sorger, “Differential Games in Economics and Management Science,” Cambridge University Press, Cambridge, 2000. doi:10.1017/CBO9780511805127
 F. Avner, “Differential Games,” Dover Publications, New York, 2006.
 S. Jorgensen and G. Zaccour, “Differential Games in Marketing,” International Series in Quantitative Marketing, Kluwer Academic Publishers, London, 2004.
 T. L. Friesz, “Dynamic Optimization and Differential Games,” Springer, New York, 2009.
 E. K. Boukas, Q. Zhang and G. Yin, “Robust Production and Maintenance Planning in Stochastic Manufacturing Systems,” IEEE Transactions on Automatic Control, Vol. 40, No. 6, 1995, pp. 1098-1102. doi:10.1109/9.388692
 T. Bj?rk, “Finite Dimensional Optimal Filters for a Class of It?-Processes with Jumping Parameters,” Stochastics, Vol. 4, No. 2, 1980, pp. 167-183.
 D. D. Sworder, “Feedback Control of a Class of Linear Systems with Jump Parameters,” IEEE Transactions on Automatic Control, Vol. 14, No. 1, 1969, pp. 9-14.
 V. Dragan and T. Morozan, “Stability and Robust Stabilization to Linear Stochastic Systems Described by Differential Equations with Markovian Jumping and Multiplicative White Noise,” Stochastic Analysis and Applications, Vol. 20, No. 1, 2002, pp. 33-92.
 V. Dragan and T. Morozan, “The Linear Quadratic Optimization Problems for a Class of Linear Stochastic Systems with Multiplicative White Noise and Markovian Jumping,” IEEE Transactions on Automatic Control, Vol. 49, No. 5, 2004, pp. 665-675.
 M. D. Fragoso and N. C. S. Rocha, “Stationary Filter for Continuous-Time Markovian Jump Linear Systems,” SIAM Journal on Control and Optimization, Vol. 44, No. 3, 2006, pp. 801-815.
 X. Li, X. Y. Zhou and M. A. Rami, “Indefinite Stochastic Linear Quadratic Control with Markovian Jumps in Infinite Time Horizon,” Journal of Global Optimization, Vol. 27, No. 2-3, 2003, pp. 149-175.
 X. R. Mao, G. G. Yin and C. G. Yuan, “Stabilization and Destabilization of Hybrid Systems of Stochastic Differential Equations”, Automatica, Vol. 43, No. 2, 2007, pp. 264-273. doi:10.1016/j.automatica.2006.09.006
 A. S. Willsky, “A Survey of Design Methods for Failure Detection in Dynamic Systems,” Automatica, Vol. 12, No. 6, 1976, pp. 601-611. doi:10.1016/0005-1098(76)90041-8
 M. McAsey and L. Mou, “Generalized Riccati Equations Arising in Stochastic Games,” Linear Algebra and Its Applications, Vol. 416, No. 2-3, 2006, pp. 710-723.
 M. A. Rami and X. Y. Zhou, “Linear Matrix Inequalities, Riccati Equations, and Indefinite Stochastic Linear Quadratic Controls,” IEEE Transactions on Automatic Control, Vol. 45, No. 6, 2000, pp. 1131-1143.
 V. Dragan and I. Ivanon, “A Numerical Procedure to Compute the Stabilising Solution of Game Theoretic Riccati Equations of Stochastic Control,” International Journal of Control, Vol. 84, No. 4, 2011, pp. 783-800.
 Z. W. Lin, Y. Lin and W. H. Zhang, “A Unified Design for State and Output Feedback H∞ Control of Nonlinear Stochastic Markovian Jump Systems with State and Disturbance-Dependent Noise,” Automatica, Vol. 45, No. 12, 2009, pp. 2955-2962.
 W. H. Zhang and B.-S. Chen, “State Feedback H∞ Control for a Class of Nonlinear Stochastic Systems,” SIAM Journal on Control and Optimization, Vol. 44, No. 6, 2006, pp. 1973-1991. doi: 10.1137/S0363012903423727