Finite-time partial stability for stochastic dynamical systems

In this paper, we address finite-time partial stability in probability for nonlinear stochastic dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state, and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability in probability. In addition, we show that finite-time partial stability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for almost sure continuity of the settling-time operator of the nonlinear stochastic dynamical system.