Moment exponential stability of random delay systems with two-time-scale Markovian switching.

*(English)*Zbl 1263.60062The 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.

Reviewer: Ruhollah Jahanipur (Kashan)

##### 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 |

##### Keywords:

random delay system; Razumikhin-type theorem; two-time-scale Markov chain; moment exponential stability; stationary distribution
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\textit{F. Wu} et al., Nonlinear Anal., Real World Appl. 13, No. 6, 2476--2490 (2012; Zbl 1263.60062)

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