New Lyapunov conditions of stochastic finite-time stability and instability of nonlinear time-varying SDEs

© 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group. In this paper, we further consider the problem of stochastic finite-time stability and instability of nonlinear stochastic differential equations (SDEs) with time-varying drift and diffusion terms. The contributions of the paper are highlighted as follows. Some new conditions via multiple Lyapunov functions are given for stochastic finite-time stability of nonlinear time-varying SDEs. Compared with the existing results on stochastic finite-time stability, the constraint of the differential operator (Formula presented.) is further relaxed in this paper, which enable us to construct the Lyapunov functions much more easily in applications for nonlinear time-varying SDEs. In addition, a stochastic finite-time instability theorem is proposed, and it shows the necessity of (Formula presented.) in the proposed conditions of stochastic finite-time stability. Some examples are presented to illustrate the new results.