Stability analysis of impulsive systems with impulse-dependent varying delay: A MDI-based looped functional approach

The dynamic behavior of nonlinear systems is often influenced by both time delay and impulsive effects. These systems experience instantaneous state jumps (impulses) at certain moments while also being affected by time delays. In fields such as biology, communications and economics, time delays often depend on impulsive responses. Therefore, conducting stability analysis of such systems provides a scientific basis and technical support for their design and optimization. This paper investigates the stability issue of impulsive systems with impulse-dependent varying delays by applying a monotone-delay-interval-based (MDI-based) looped Lyapunov functional approach. Firstly, two novel integral inequalities are introduced. One is a free-matrix-based integral inequality that exploits two-sided impulsive characteristics, thereby enabling the derived conditions to capture richer information on impulsive intervals. The other is established by integrating a free-matrix formulation with a monotone-delay-interval-based looped functional, which leads to less conservative stability criteria.This combination effectively captures both the two-sided impulsive characteristics and the variation in time delays. Afterward, by leveraging these inequalities and proposed MDI-based looped Lyapunov functional, several stability criteria and two related corollaries are established. Finally, an example of practical application in fishery management and additional numerical examples are provided to verify the applicability and efficiency of the proposed approach.