Exponential Stability of Second‐Order Systems With Nonlinear Damping Under Impulsive Effects

In this paper, we examine the issue of global exponential stability pertaining to a class consisting of second‐order systems with state‐dependent nonlinear damping, stiffness, and coupling terms, with a focus on analyzing their dynamic behavior under impulsive effects. Impulsive effects are classified into two categories: impulsive disturbances that may undermine system stability, and impulsive control inputs adopted to achieve stabilization. For the continuous nonlinear dynamics, the paper employs the variable gradient method for constructing a suitable Lyapunov function and establishes explicit sufficient conditions that guarantee its global exponential stability. On this basis, stability criteria built on decay rate, jump gain, and dwell time are given for stable continuous dynamics under impulsive disturbances. For the case where the continuous dynamics may be divergent and thus require impulsive control, an explicit ADT criterion in terms of nonlinear parameter bounds and impulsive gain reveals their inherent link. Unlike Lyapunov methods requiring unknown quadratic forms, this approach provides a direct construction path from physical parameters to stability conditions. Finally, to verify the rationality of the derived results, corresponding numerical examples are presented.