Exponential stability of singularly perturbed systems with mixed impulses. (May 2021)

• Record Type: Journal Article
• Authors: Yang, Wu; Wang, Yan-Wu; Morǎrescu, Irinel-Constantin; Daafouz, Jamal
• Is Part Of: Nonlinear analysis. Volume 40(2021)
• Publication Date: 2021-05
• Subjects: Singularly perturbed system -- Mixed impulse -- Exponential stability -- Vector Lyapunov function; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nahs.2021.101023
• Languages: English
• ISSNs: 1751-570X

Abstract

Abstract: 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. Highlights: Neither the flow dynamics nor the jump dynamics of the model is stable. The impulse's stabilizing effect is explored by developing vector Lyapunov function. Several numerically tractable criteria for stability of SPSs are established. The upper bound on the singular perturbation parameter is derived.