Prescribed-time stability of stochastic nonlinear delay systems

This paper investigates the prescribed-time stability and stabilization problem for stochastic nonlinear delay systems. We introduce a new definition of prescribed-time mean-square stability which includes stability in probability and prescribed-time convergence to zero. Utilizing the prescribed-time adjustment function and some stochastic analysis techniques, we establish Lyapunov theorems of prescribed-time mean-square stability for stochastic nonlinear delay systems. An appealing feature of the new theorems is that the solution of prescribed-time stable stochastic nonlinear delay systems can converge to zero at any preset time irrespective of initial data and design parameters. Moreover, under the local Lipschitz condition and the Khasminskiitype condition, we prove that the controlled stochastic nonlinear delay system has a unique solution and achieves prescribed-time mean-square stability. Two simulation examples demonstrate the effectiveness of the theoretical analysis.