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Moment exponential stability of random delay systems with two-time-scale Markovian switching. (English) Zbl 1263.60062

The authors consider the \(n\)-dimensional system with random delays \[ \dot{x}(t)= f(x(t), x(t- r(t))), \] where \(r(t)\geq 0\) is a continuous-time Markov chain in a finite state space \(\{r_1, r_2, \dotsc, r_m\}\). The system is a switching system along with the \(m\) fixed delay subsystems \[ \dot{y}(t) = f(y(t), y(t- r_i))\text{ for }i_1,2,\dotsc, m. \] The random switching is governed by the Markov chain \(r(t)\). They show that the fast changing part of the Markov switching, yields some average effect with respect to its stationary measure. Using the average system, a stability analysis is then carry out. The authors establish that the system has a weak limit in the sense of weak convergence of probability measures and also the Razumikhin-type theorem on the moment exponential stability.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations
60J28 Applications of continuous-time Markov processes on discrete state spaces
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[1] Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[2] Montestruque, L.; Antsaklis, P., Stability of model-based networked control systems with tiem-varying transmission times, IEEE Transaction on Automatical Control, 49, 1562-1572 (2004) · Zbl 1365.90039
[3] Nilsson, J.; Bernhardsson, B.; Wittenmark, B., Stochastic analysis and control of real-time systems with random time delays, Automatica, 34, 57-64 (1998) · Zbl 0908.93073
[4] Schenato, L., Optimal estimation in networked control systems subject to random delay and packet drop, IEEE Transaction on Automatical Control, 53, 1311-1317 (2008) · Zbl 1367.93633
[5] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press New York · Zbl 0593.34070
[6] Yang, F.; Wang, Z.; Hung, Y. S.; Gani, M., \(H_\infty\) control for networked systems with random communication delays, IEEE Transaction on Automatical Control, 51, 511-518 (2006) · Zbl 1366.93167
[7] Kolmanovskii, V. B.; Maizenberg, T. L.; Richard, J.-P., Mean square stability of difference equations with a stochastic delay, Nonlinear Analysis, 52, 795-804 (2003) · Zbl 1029.39005
[8] Krtolica, R.; Ozguner, U.; Chan, H.; Goktas, H.; Winkelman, J.; Liubakka, M., Stability of linear feedback systems with random communication delays, International Journal of Control, 59, 925-953 (1994) · Zbl 0812.93073
[9] Zhang, L.; Shi, Y.; Chen, T.; Huang, B., A new method for stabilization of networked control systems with random delays, IEEE Transaction on Automatical Control, 50, 1177-1181 (2005) · Zbl 1365.93421
[10] Kolmanovsky, I.; Maizenberg, T. L., Mean-square stability of nonlinear systems with time-varying, random delay, Stochastic Analysis and Applications, 19, 279-293 (2001) · Zbl 0993.93034
[11] Haddock, J. R.; Krisztin, T.; Terjéki, J.; Wu, J. H., An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations, Journal of Differential Equations, 107, 395-417 (1994) · Zbl 0796.34067
[12] Jankovic, M., Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems, IEEE Transactions on Automatic Control, 46, 1048-1060 (2001) · Zbl 1023.93056
[13] Karafyllis, I.; Pepe, P.; Jiang, Z. P., Input-to-output stability for systems described by retarded functional differential equations, European Journal of Control, 14, 539-555 (2008) · Zbl 1293.93668
[14] Teel, A. R., Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Transactions on Automatic Control, 147, 960-964 (1998) · Zbl 0952.93121
[15] Mao, X., Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Processes and their Applications, 65, 233-250 (1996) · Zbl 0889.60062
[16] Mao, X., Razumikhin-type theorems on exponential stability of neutral stochastic functional differential, SIAM Journal on Mathematical Analysis, 28, 389-401 (1997) · Zbl 0876.60047
[17] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050
[18] Mao, X.; Yuan, C., Stochastic Differential Equations with Markovian Switching (2006), Imperial College Press: Imperial College Press London · Zbl 1126.60002
[19] Huang, D.; Nguang, S. K., State feedback control of uncertain networked control systems with random time delays, IEEE Transaction on Automatical Control, 53, 829-834 (2008) · Zbl 1367.93510
[20] Badowski, G.; Yin, G., Stability of hybrid dynamic systems containing singularly perturbed random processes, IEEE Transactions on Automatic Control, 47, 2021-2032 (2002) · Zbl 1364.93841
[21] Khasminskii, R. Z.; Yin, G., On averaging principle: an asymptotic expansion approach, SIAM Journal on Mathematical Analysis, 35, 1534-1560 (2004) · Zbl 1072.34054
[22] Yin, G., Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process, Asymptotic Analysis, 65, 203-222 (2009) · Zbl 1186.60056
[23] Zhu, C.; Yin, G.; Song, Q. S., Stability of random-switching systems of differential equations, Quarterly of Applied Mathematics, 67, 201-220 (2009) · Zbl 1163.93036
[24] Yin, G.; Zhang, Q., Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0896.60039
[25] Kushner, H. J., Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory (1984), MIT Press: MIT Press Cambridge, MA · Zbl 0551.60056
[26] Grossman, S. E., Stability in \(n\)-dimensional differential-delay equations, Journal of Mathematical Analysis and Applications, 40, 541-546 (1972) · Zbl 0211.12302
[27] Grossman, S. E.; Yorke, J. A., Asymptotic behavior and exponential stability criteria for differential delay equations, Journal of Differential Equations, 12, 236-255 (1972) · Zbl 0268.34079
[28] Halanay, A.; Yorke, J. A., Some new results and problems in the theory of differential-delay equations, SIAM Review, 13, 55-80 (1971) · Zbl 0216.11902
[29] Yoneyama, T., On the stability for the delay-differential equation \(\dot{x}(t) = - a(t) f(x(t - r(t)))\), Journal of Mathematical Analysis and Applications, 120, 271-275 (1986) · Zbl 0618.34065
[30] Yoneyama, T., On the 3/2 stability theorem for one- dimensional delay-differential equations, Journal of Mathematical Analysis and Applications, 125, 161-173 (1987) · Zbl 0655.34062
[31] Yoneyama, T.; Sugie, J., On the stability region of differential equations with two delays, Funkcialaj Ekvacioj, 31, 233-240 (1988) · Zbl 0667.34083
[32] Yoneyama, T., Uniform stability for one-dimensional delay-differential equations with dominant delayed term, Tohoku Mathematical Journal, 41, 217-236 (1989) · Zbl 0706.34065
[33] Yoneyama, T., The 3/2 stability theorem for one-dimensional delay-differential equations with unbounded delay, Journal of Mathematical Analysis and Applications, 165, 133-143 (1992) · Zbl 0755.34074
[34] Yorke, J. A., Asymptotic stability for one dimensional differential-delay equations, Journal of Differential Equatons, 7, 189-202 (1970) · Zbl 0184.12401
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