On asymptotic convergence and boundedness of stochastic systems with time-delay

Analyzing the long-term behavior of a nonlinear dynamical system is an important but challenging task, especially in the presence of noise and delays. This paper investigates asymptotic convergence and boundedness properties of stochastic systems with time-delay. We are particularly interested in asymptotic convergence that may not be exponential, as observed in many practical applications. General criteria for checking both moment and almost sure asymptotic convergence and boundedness are established. Such criteria are mainly motivated by stochastic systems with time-varying coefficients and multiple delays and/or different orders of nonlinearities. As shown by several examples, the existing results cannot be applied to analyze such systems. Thus, we generalize the existing theory by allowing the diffusion operator associated with a Lyapunov function to satisfy a weaker assumption, which involves time-varying coefficients and several auxiliary functions to cope with possible multiple delays and/or different orders of nonlinearities. The results presented in this paper provide an effective tool for checking asymptotic convergence and boundedness properties of general stochastic nonlinear systems with time-delay.