Robust stability and controllability of stochastic differential delay equations with Markovian switching. (English) Zbl 1040.93069

The authors investigate the almost surely asymptotic stability for nonlinear stochastic differential delay equations with Markovian switching. Most of the existing results on stochastic differential delay equations with Markovian switching are about the moment stability, while little is known on the almost surely asymptotic stability, which is the main topic of the present publication. The results on such stability are then applied to establish a sufficient condition for the controllability of linear stochastic delay equations with Markovian switching. The robust stability is discussed in the last section by using linear matrix inequalities.


93E15 Stochastic stability in control theory
93D09 Robust stability
60J75 Jump processes (MSC2010)
15A39 Linear inequalities of matrices
93B05 Controllability
93D20 Asymptotic stability in control theory
34K50 Stochastic functional-differential equations
Full Text: DOI


[1] Anderson, W. J., Continuous-time Markov chains (1991), Springer: Springer Berlin · Zbl 0721.60081
[2] Arnold, L., Stochastic differential equations: Theory and applications (1972), Wiley: Wiley New York
[3] Basak, G. K.; Bisi, A.; Ghosh, M. K., Stability of a random diffusion with linear drift, Journal of Mathematical Analysis and Applications, 202, 604-622 (1996) · Zbl 0856.93102
[4] Boukas, E. K.; Liu, Z. K., Robust \(H_∞\) control of discrete-dime Markovian jump linear system with mode-dependent time-delays, IEEE Transactions on Automatic Control, 46, 1924-1981 (2001) · Zbl 1005.93050
[5] Boyd, S.; El Ghaoui, L.; Feron, R.; Balakrishnan, V., Linear matrix inequalities in system and control theory (1998), SIAM: SIAM Philadelphia
[6] Cao, Y. Y., Sun, Y. X., & Lam, J. (1998). Delay-dependant robust \(H_∞\)IEEE Proceeding—Control Theory and Applications 145; Cao, Y. Y., Sun, Y. X., & Lam, J. (1998). Delay-dependant robust \(H_∞\)IEEE Proceeding—Control Theory and Applications 145
[7] Dragan, V.; Morozan, T., Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise, Stochastics Analysis and Applications, 20, 33-92 (2002) · Zbl 1136.60335
[8] Gao, Z. Y.; Ahmed, N. U., Feedback stabilizability of nonlinear stochastic systems with state-dependent noise, International Journal of Control, 45, 729-737 (1987) · Zbl 0618.93068
[9] Ghosh, M. K.; Arapostahis, A.; Marcus, S. I., Optimal control of switching diffusions with applications to flexible manufacturing systems, SIAM Journal of Control and Optimization, 31, 1183-1204 (1993) · Zbl 0785.93092
[10] Has’minskii, R. Z., Stochastic stability of differential equations (1981), Sijthoff and Noordhoff: Sijthoff and Noordhoff Alphen
[11] Jeung, E. T.; Kim, J. H.; Park, H. B., \(H_∞\) output feedback controller design for linear systems with time-varying delayed state, IEEE Transactions on Automatic Control, 43, 971-974 (1998) · Zbl 0952.93032
[12] Ji, Y.; Chizeck, H. J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35, 777-788 (1990) · Zbl 0714.93060
[13] Kushner, H., Stochastic stability and control (1967), Academic Press: Academic Press New York · Zbl 0183.19401
[14] Lin, Y.; Sontag, E. D., A universal formula for stabilization with bounded controls, Systems and Control Letters, 16, 393-397 (1991) · Zbl 0728.93062
[15] Lipster, R. Sh.; Shiryayev, A. N., Theory of martingales (1989), Horwood: Horwood Chichester, UK · Zbl 0728.60048
[16] Mao, X., Stochastic differential equations and applications (1997), Horwood: Horwood England · Zbl 0874.60050
[17] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastics Processes and their Applications, 79, 45-67 (1999) · Zbl 0962.60043
[18] Mao, X., A note on the LaSalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 268, 125-142 (2002) · Zbl 0996.60064
[19] Mao, X.; Matasov, A.; Piunovskiy, A. B., Stochastic differential delay equations with Markovian switching, Bernoulli, 6, 73-90 (2000) · Zbl 0956.60060
[20] Mariton, M., Jump linear systems in automatic control (1990), Marcel Dekker: Marcel Dekker New York
[21] Moerder, D. D.; Halyo, N.; Braussard, J. R.; Caglayan, A. K., Application of precomputed control laws in a reconfigurable aircraft flight control system, Journal of Guidance, Control Dynamics, 12, 325-333 (1989)
[22] Pakshin, P. V., Robust stability and stabilization of family of jumping stochastic systems, Nonlinear Analysis, 30, 2855-2866 (1997) · Zbl 0942.93043
[23] Pan, G.; Bar-Shalom, Y., Stabilization of jump linear Gaussian systems without mode observations, International Journal of Control, 64, 631-661 (1996) · Zbl 0857.93095
[24] Park, P., A delay-dependant stability criterion for systems with uncertain linear systems, IEEE Transactions on Automatic Control, 44, 876-877 (1999) · Zbl 0957.34069
[25] Shaikhet, L., Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2, 180-184 (1996) · Zbl 0939.60049
[26] Skorohod, A. V., Asymptotic methods in the theory of stochastic differential equations (1989), American Mathematical Society: American Mathematical Society Providence, RI
[27] Verriest, E. I., Stability and control of time-delay systems, Lecture notes in control and information sciences, Vol. 228 (1998), Springer: Springer Berlin · Zbl 0901.00019
[28] Wang, Z.; Qian, H.; Burnham, K. J., On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters, IEEE Transactions on Automatic Control, 47, 640-646 (2002) · Zbl 1364.93672
[29] Willems, J. L.; Willems, J. C., Feedback stabilizability for stochastic systems with state and control dependent noise, Automatica, 12, 343-357 (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.