ON THE LYAPUNOV STABlLITY OF MECHANICAL SYSTEMS PERTURBED BY WHITE NOISES

A number of problems in mechanics, physics and applications are described by linear Ito stochastic differential equations of the form Ddx = Axdt + B(e) x dW. (1) Yu. A. MITROPOLSKIJ and D. G. KORENEVSKIJ (1985-1986) have given some effective criteria for the asymptotic stability of solutions of (1) wth small e > O. However, by a simple counterexample we have discovered that a criterion of those authors is only sufficient condition, but not a necessarv one. By using a quite different tool, namely the Lyapunov exponent method, in this talk we derive much more effective sufficient conditions ensuring the asymptotic stability of the trivial solution x = 0 of (1). In some cases our conditions are not only sufficient, but also necessary. The effectiveness of our results is illustrated by concrete numerical examples. A general algorithm for solving the problem is also given and it can be easily used on computers