×
Mao, Xuerong; Lam, James; Xu, Shengyuan; Gao, Huijun

Razumikhin method and exponential stability of hybrid stochastic delay interval systems. (English) Zbl 1127.60072

J. Math. Anal. Appl. 314, No. 1, 45-66 (2006).
Summary: This paper deals with the exponential stability of hybrid stochastic delay interval systems (also known as stochastic delay interval systems with Markovian switching). The known results in this area (see, e.g., X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Autom. Control 47, No. 10, 1604–1612 (2002)] require the time delay to be a constant or a differentiable function and the main reason for such a restriction is due to the analysis of mathematics. The main aim of this paper is to remove this restriction to allow the time delay to be a bounded variable only. The Razumikhin method is developed to cope with the difficulty arisen from the nondifferentiability of the time delay.

Cited in 41 Documents

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations

Keywords:

interval matrix; Razumikhin method; Brownian motion; Markov chain; M-matrix; exponential stability
Cite Review PDF
Full Text: DOI

References:

[1] Basak, G. K.; Bisi, A.; Ghosh, M. K., Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202, 604-622 (1996) · Zbl 0856.93102
[2] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1994), SIAM · Zbl 0815.15016
[3] Han, H. S.; Lee, J. G., Stability analysis of interval matrices by Lyapunov function approach including ARE, Control-Theory and Advanced Technology, 9, 745-757 (1993)
[4] Has’minskii, R. Z., Stochastic Stability of Differential Equations (1981), Sijthoff and Noordhoff · Zbl 0276.60059
[5] Ji, Y.; Chizeck, H. J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control, 35, 777-788 (1990) · Zbl 0714.93060
[6] Kolmanovskii, V. B.; Myshkis, A., Applied Theory of Functional Differential Equations (1992), Kluwer Academic · Zbl 0785.34005
[7] Ladde, G. S.; Lakshmikantham, V., Random Differential Inequalities (1980), Academic Press · Zbl 0466.60002
[8] Liao, X. X.; Mao, X., Exponential stability of stochastic delay interval systems, Systems Control Lett., 40, 171-181 (2000) · Zbl 0949.60068
[9] Mao, X., Stability of Stochastic Differential Equations with Respect to Semimartingales (1991), Longman · Zbl 0724.60059
[10] Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Dekker · Zbl 0851.93074
[11] Mao, X., Stochastic Differential Equations and Their Applications (1997), Horwood: Horwood Chichester · Zbl 0892.60057
[12] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79, 45-67 (1999) · Zbl 0962.60043
[13] Mao, X., Stochastic functional differential equations with Markovian switching, Functional Differential Equations, 6, 375-396 (1999) · Zbl 1034.60063
[14] Mao, X., Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Automat. Control, 47, 1604-1612 (2002) · Zbl 1364.93685
[15] Mao, X.; Matasov, A.; Piunovskiy, A. B., Stochastic differential delay equations with Markovian switching, Bernoulli, 6, 73-90 (2000) · Zbl 0956.60060
[16] Mariton, M., Jump Linear Systems in Automatic Control (1990), Dekker
[17] Shaikhet, L., Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2, 180-184 (1996) · Zbl 0939.60049
[18] Šiljak, D. D., On stochastic stability of competitive equilibrium, Ann. Econ. Soc. Meas., 6, 315-323 (1977)
[19] Sun, Y.; Lee, C.; Hsieh, J., Sufficient conditions for the stability of interval systems with multiple time-varying delays, J. Math. Anal. Appl., 207, 29-44 (1997) · Zbl 0890.93071
[20] Wang, K.; Michel, A. N.; Liu, D., Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices, IEEE Trans. Automat. Control, 39, 1251-1255 (1994) · Zbl 0810.93051
[21] Wang, K.; Michel, A. N.; Liu, D., Necessary and sufficient conditions for the controllability and observability of a class of linear, time-invariant systems with interval plants, IEEE Trans. Automat. Control, 39, 1443-1447 (1994) · Zbl 0800.93165
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.