This paper investigates the exponential stability problem for a class of singularly perturbed impulsive systems in which the flow dynamics is unstable and is affected at discrete time instants by impulses that have both destabilizing and stabilizing effects. More precisely the impulses have stabilizing effects on the slow variables but destabilizing effects on the fast ones. Thus, a first contribution of our work is related to stability analysis of singularly perturbed impulsive systems in the case when neither the flow dynamics nor the impulsive one is stable. In order to take full advantage of the jump matrix structure and its stabilizing effects on the slow dynamics, we introduce a new impulse-dependent vector Lyapunov function. This function allows us to better describe the behavior between two consecutive impulses as well as the jumps at impulse instants. Several numerically tractable criteria for stability of singularly perturbed impulsive systems are established based on vector comparison principle. Additionally, upper bounds on the singular perturbation parameter are derived. Finally, the validity of our results is verified by two numerical examples.

本文研究了一类奇异摄动脉冲系统的指数稳定性问题，该系统的流动动力学不稳定，并且在离散的瞬间受到具有稳定和稳定作用的脉冲的影响。更确切地说，脉冲对慢速变量具有稳定作用，但对快速变量具有不稳定作用。因此，在流动动力学和脉冲动力学都不稳定的情况下，我们的工作的第一贡献与奇异脉冲系统的稳定性分析有关。为了充分利用跳跃矩阵结构及其对慢速动力学的稳定作用，我们引入了一种新的脉冲相关矢量Lyapunov函数。此功能使我们可以更好地描述两个连续脉冲之间的行为以及脉冲瞬间的跳跃。基于矢量比较原理，建立了几个奇摄动脉冲系统稳定性的数值可控准则。另外，推导奇异摄动参数的上限。最后，通过两个数值例子验证了我们的结果的有效性。