Abstract
Stochastic systems are widely used in many fields, such as scientific research, industry, communication, population model, network control and so on. Some issues of deterministic systems are extended to stochastic systems with the research of stochastic systems. As a basic structural property of the system, stability is a core concept and an important problem in the theory of system control. In this paper, we will provide the criteria of mean square stability and mean square asymptotic stability respectively by defining the Lyapunov function of the system and the corresponding diffusion operator. We also construct the auxiliary delayed differential equation, which proves the conclusion by means of the basic theory of stochastic differential equations and the comparison principle of differential equations. The proposed criteria are more generous and can be generalized to random systems with time delay or stochastic hybrid systems. Two examples, the Hopfield neural network and a nonlinear stochastic system with time-delay, will be analyzed by the proposed criteria. It will be illustrated in the second example that the criteria proposed in this paper are suitable for the situation that time delays are time-varying and not continuously differentiable.
Published Version
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