The asymptotic stochastic stability in large of the composite stochastic systems (Q1087858)

The author studies the asymptotic stochastic stability of the stochastic system described by \[ dy_ i/dt=f_ i(y_ i,t)+\sum^{m}_{k=1}B^ k_ i(y_ i,t)\eta_ k+g_ i(y_ 1,...,y_ N,t)+\sum^{m}_{k=1}h^ k_ i(y_ 1,...,y_ N,t)\eta_ k, \] i\(=1,...,N\); \(y_ i\in R^{n_ i}\), in which \(\eta_ k\) are random processes and others are deterministic. The feature of this paper is that the ideas of \textit{R. Z. Khas'minskij} [Stochastic stability of differential equations (1980; Zbl 0441.60060)] on asymptotic stochastic stability are widely used. Several criteria or sufficient conditions for asymptotic stochastic stability in the large of the systems are given, in which the integrable random process \(| \eta (t,\omega)|\) obeys the law of large numbers and \(\sup_{t>0}E| \eta (t,\omega)|\) is bounded by a certain value. An example of a very low order system \((i=1,2)\) is given to illustrate the application of these criteria.