# American Institute of Mathematical Sciences

September  2015, 5(3): 697-719. doi: 10.3934/mcrf.2015.5.697

## Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074 2 Department of Mathematics, Wayne State University, Detroit, Michigan 48202 3 Department of Electrical and Computer Engineering, Wayne State University, MI 48202, United States

Received  March 2014 Revised  June 2014 Published  July 2015

This work develops moment exponential stability of functional differential equations (FDEs) with Markovian switching, in which a two-time-scale (real time $t$ and fast time $t/\epsilon$ with a small parameter $\epsilon>0$) continuous-time and finite-state Markov chain is used to represent the switching process. The essence is that there is a time-scale separation, which is motivated by the consideration of networked control systems and manufacturing systems. Under suitable conditions, we establish a Razumikhin-type theorem on the $p$th moment exponential $\epsilon$-stability for the small parameter $\epsilon$. By virtue of the Razumikhin-type theorem, we further deduce mean-square exponential stability results for delay differential equations (DDEs) and ordinary differential equations (ODEs) with two-time-scale Markovian switching. These stability results show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable. It is noted that in the presence of the Markovian switching, the stationary distribution of the fast changing part of the Markov chain plays an important role. Explicit conditions for the mean-square exponential stability of linear equations are derived and illustrative examples are provided to demonstrate our results.
Citation: Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
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