Stability and boundedness of nonlinear hybrid stochastic differential delay equations

One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness (see e.g. [5,10,11,13–15,17,19,21]). The method of Lyapunov functions is one of the most powerful techniques in the study of stability and boundedness. So far, most of the results in this area do not only require the Lyapunov functions in different modes have the same feature (e.g. polynomials with the same degree) but also that the diffusion operator in different modes be bounded by the same type of functions. These requirements are restrictive and often cannot be met by those hybrid SDDEs that have different nonlinear structures in different modes. To study the stability and boundedness of such hybrid SDDEs, we will in this paper use different types of Lyapunov functions (e.g. polynomials with different degrees) for different modes. Moreover, the condition on the diffusion operator is relaxed significantly.