Lyapunov Measure for Almost Everywhere Stability

This paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system. Based upon a stochastic representation of deterministic dynamics, a Lyapunov measure is proposed for these purposes. This measure is shown to be a stochastic counterpart of stability (transience) just as an invariant measure is a counterpart of the attractor (recurrence). It is a dual of the Lyapunov function and is useful for the study of more general (weaker and set-wise) notions of stability. In addition to the theoretical framework, constructive methods for computing approximations to the Lyapunov measures are presented. These methods are based upon set-oriented numerical approaches. Several equivalent descriptions, including a series formula and a system of linear inequalities, are provided for computational purposes. These descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets. Finally, in certain cases, the exact relationship between Lyapunov functions and Lyapunov measures is also given.