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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References:

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