This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Lévy noise (SFDEs-SMS-LN). Based on functional Itô’s formula, multiple degenerate Lyapunov functionals and nonnegative semi-martingale convergence theorem, several new pth moment and almost sure stability criteria with general decay rate for SFDEs-SMS-LN are established. Meanwhile, the diffusion operators are allowed to be controlled by multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients. Furthermore, as applications of the presented stability criteria, new delay-dependent sufficient conditions for general decay stability of the stochastic delayed neural network with semi-Markovian switching and Lévy noise (SDNN-SMS-LN) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are respectively obtained in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

本文研究了具有半马尔可夫切换和Lévy噪声（SFDEs-SMS-LN）的随机泛函微分方程表示的系统的一般衰减稳定性。基于泛函Itô公式，多个简并Lyapunov泛函和非负半-收敛定理，几个新的p建立了SFDEs-SMS-LN的第一个矩和几乎确定的具有一般衰减率的稳定性标准。同时，允许扩散算子由具有时变系数的多个辅助函数控制，这样可以更好地适应具有高阶非线性系数的非自治SFDEs-SMS-LN。此外，作为提出的稳定性标准的应用，具有半马尔可夫切换和Lévy噪声（SDNN-SMS-LN）和标量非自治SFDE-SMS的随机延迟神经网络的一般延迟稳定性的新的依赖于延迟的充分条件根据二元对角矩阵（BDM）和线性矩阵不等式（LMI）分别获得具有非全局Lipschitz条件的-LN。最后，给出了两个数值例子来说明所提出的结果的有效性。