# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York  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  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  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  Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press New York · Zbl 0593.34070  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  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  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  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  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  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  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  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  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  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  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  Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050  Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press London · Zbl 1126.60002  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  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  Khasminskii, R.Z.; Yin, G., On averaging principle: an asymptotic expansion approach, SIAM journal on mathematical analysis, 35, 1534-1560, (2004) · Zbl 1072.34054  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  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  Yin, G.; Zhang, Q., Continuous-time Markov chains and applications: A singular perturbation approach, (1998), Springer-Verlag New York · Zbl 0896.60039  Kushner, H.J., Approximation and weak convergence methods for random processes, with applications to stochastic systems theory, (1984), MIT Press Cambridge, MA · Zbl 0551.60056  Grossman, S.E., Stability in $$n$$-dimensional differential-delay equations, Journal of mathematical analysis and applications, 40, 541-546, (1972) · Zbl 0211.12302  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  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  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  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  Yoneyama, T.; Sugie, J., On the stability region of differential equations with two delays, Funkcialaj ekvacioj, 31, 233-240, (1988) · Zbl 0667.34083  Yoneyama, T., Uniform stability for one-dimensional delay-differential equations with dominant delayed term, Tohoku mathematical journal, 41, 217-236, (1989) · Zbl 0706.34065  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  Yorke, J.A., Asymptotic stability for one dimensional differential-delay equations, Journal of differential equatons, 7, 189-202, (1970) · Zbl 0184.12401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.