Finite-time stability and stabilization of nonlinear stochastic hybrid systems. (English) Zbl 1163.93033

Summary: This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which have impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.


93E03 Stochastic systems in control theory (general)
93E15 Stochastic stability in control theory
60J75 Jump processes (MSC2010)
Full Text: DOI


[1] Kushner, H. I.; Dupuis, P., Numerical Methods for Stochastic Control Problems in Continuous Time (2001), Springer-Verlag: Springer-Verlag New York
[2] Hespanha, J. P., A model for stochastic hybrid systems with application to communication networks, Nonlinear Anal., 62, 1353-1383 (2005) · Zbl 1131.90322
[3] Ji, Y.; Chizeck, H. J., Controllability, stability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control, 35, 777-788 (1990) · Zbl 0714.93060
[4] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79, 45-67 (1999) · Zbl 0962.60043
[5] Hu, L.; Shi, P.; Huang, B., Stochastic stability and robust control for sampled-data systems with Markovian jump parameters, J. Math. Anal. Appl., 313, 504-517 (2006) · Zbl 1211.93131
[6] Dragan, V.; Morozan, T., Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise, Stochastic Process. Appl., 20, 33-92 (2002) · Zbl 1136.60335
[7] Ye, H.; Michel, A. N.; Hou, L., Stability analysis of systems with impulse effect, IEEE Trans. Automat. Control, 43, 1719-1723 (1998) · Zbl 0957.34051
[8] Xie, G.; Wang, L., Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Trans. Automat. Control, 49, 960-966 (2004) · Zbl 1365.93049
[9] Guan, Z. H.; Hill, D. J.; Shen, X., On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Control, 50, 1058-1062 (2005) · Zbl 1365.93347
[10] Li, Z. G.; Wen, C. Y.; Soh, Y. C., Analysis and design of impulsive control systems, IEEE Trans. Automat. Control, 46, 894-897 (2001) · Zbl 1001.93068
[11] Dorato, P., Short time stability in linear time-varying systems, (Proceedings of IRE Int. Convention Record Part 4 (1961)), 83-87
[12] Weiss, L.; Infante, E. F., Finite time stability under perturbing forces and on product spaces, IEEE Trans. Automat. Control, 12, 54-59 (1967) · Zbl 0168.33903
[13] Angelo, H. D., Linear Time-Varying Systems: Analysis and Synthesis (1970), Allyn and Bacon: Allyn and Bacon Boston
[14] Amato, F.; Ariola, M.; Dorato, P., Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 1459-1463 (2001) · Zbl 0983.93060
[15] Amato, F.; Ariola, M., Finite-time control of discrete-time linear systems, IEEE Trans. Automat. Control, 50, 724-729 (2005) · Zbl 1365.93182
[16] Hong, Y. G.; Wang, J. K., Nonsmooth finite-time stabilization of a class of nonlinear systems, Sci. China Ser. E, 35, 663-672 (2005)
[17] Huang, X. Q.; Lin, W.; Yang, B., Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica, 41, 881-888 (2005) · Zbl 1098.93032
[18] Kushner, H. J., Finite-time stochastic stability and the analysis of tracking systems, IEEE Trans. Automat. Control, 11, 219-227 (1966)
[19] Mastellone, S.; Dorato, P.; Abdallah, C. T., Finite-time stability of discrete-time nonlinear systems: Analysis and design, (Proceedings of the 43rd IEEE Conference on Decision and Control (2004)), 2572-2577 · Zbl 1161.65339
[20] Van Mellaert, L. J.; Dorato, P., Numerical solution of an optimal control problem with a probability criterion, IEEE Trans. Automat. Control, 17, 543-546 (1972) · Zbl 0263.93068
[21] Rami, M. A.; Chen, X.; Moore, J. B.; Zhou, X. Y., Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls, IEEE Trans. Automat. Control, 46, 428-440 (2001) · Zbl 0992.93097
[22] Skorohod, A. V., Symptotic Methods in the Theory of Stochastic Differential Equations (2004), American Mathematical Society: American Mathematical Society Providence, RI
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