Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions

Abstract

We study stability for dynamical systems specified by autonomous stochastic differential equations of the form dX(t)=f(X(t))dt+g(X(t))dW(t), with (X(t))t≥0 an Rd-valued Itô process and (W(t))t≥0 an RQ-valued Wiener process, and the functions f:Rd→Rd and g:Rd→Rd×Q are Lipschitz and vanish at the origin, making it an equilibrium for the system. The concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. The concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we define a probabilistic version of the basin of attraction, the γ-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least γ. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on γ-BOA.