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In this paper, we develop the moment and almost-sure stability with a general decay rate for stochastic functional differential equations with semi-Markovian switching signal (SFDESMS). We first present a new equivalent concept of moment stability with general decay rate and its implication to almost-sure stability with general decay rate. Subsequently, using the present equivalent concept and implication relationship, we formulate some explicit sufficient criteria of the moment and almost sure stability with general decay rate for SFDESMS. In addition, we consider the application of our results to two special classes of SFDESMS and obtain the moment and almost-sure stability with general decay rate for two special classes of SFDESMS. At last, we illustrate the obtained theoretical results by numerical examples.

Asymptotic behaviour analysis of stochastic functional differential equations with semi-Markovian switching signal