On stochastic stability of Markov evolution associated with impulse Markov dynamical systems

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Abstract
This paper deals with the family of Cauchy matrices of a linear differential equation dependent on a step Markov process and an impulse type dynamical system rapidly switched by the above process. Applying the stochastic and deterministic averaging procedures according to the invariant measures of the Markov process one achieves a simpler linear differential equation dependent on simpler dynamical systems such as an ordinary differential equation, a differential equation with the right hand side switched by a merger Markov process or a stochastic Itô differential equation. It is proved that under some hypotheses one may successfully apply these resulting evolution families not only to analyzing the initial family on an arbitrary finite time interval but also to describing a time asymptotic of this family.