Stochastic Barbalat's lemma and its applications

In the deterministic case, a significant improvement on stability analysis of nonlinear systems is caused by introducing Barbalat's lemma into control area after Lyapunov's second method and LaSalle's theorem were established. This note considers the extension of Barbalat's lemma to the stochastic case. To this end, the uniform continuity and the absolute integrability are firstly described in stochastic forms. It is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily It diffusions. When it is applied to It diffusion processes, many classical results on stochastic stability are covered as special cases.