The main object in the paper is the dynamical system \[ dU^\varepsilon(t)/dt= C(k(t/\varepsilon)) U^\varepsilon(t)\tag{1} \] with velocity matrix \(C(x)= (C_{kr}(x): 1\leq k, r\leq d)\) controlled by a stochastic process \(k(t)\), \(t\geq 0\), whose state space is compact. The associated averaged system is given by the equation \[ dU(t)/dt= CU(t).\tag{2} \] Two cases of control processes are considered: (a) \(k(t)\) is an ergodic Markov process; (b) \(k(t)\) is a stationary and uniformly mixing process.
In case (a) \(C= (C_{kr})\), \(C_{kr}= \int_X\pi(dx) C_{kr}(x)\), \(1\leq k\), \(r\leq d\), \(\pi(dx)\) is the stationary and uniformly mixing process.
In case (b) \(C= E_0C(k(t))\), where \(E_0\) is the conditional expectation with respect to \(\sigma\{k(0)\}\).
Under some conditions, different for cases (a) and (b), the solution \(U^\varepsilon(t)\) of (1) weakly converges to the solution \(U(t)\) of (2) as \(k\to 0\).
In case (a) under some conditions, for small \(\varepsilon> 0\), the stochastic systems \(U^\varepsilon(t)\) are asymptotically stable, \[ P\Biggl\{\lim_{t\to\infty} U^\varepsilon(t)= 0\Biggr\}= 1. \] In case (b), for small \(\varepsilon> 0\), the stochastic systems \(U^\varepsilon(t)\) are exponentially stable in mean square, \[ E_0\|U^\varepsilon(t)\|^2\leq A\|U\|^2\exp\{- bt\},\quad b>0,\quad A>0. \] In the conditions of the propositions, Lyapunov functions and Lyapunov operators are substantially used.
For the entire collection see [Zbl 0942.60001].