Robust exponential stability of non-autonomous systems under small parameter

This work investigates the stability of nonlinear dynamic systems featuring a small parameter and operating near an equilibrium. The central contribution is proving that such a perturbed system can exhibit global uniform exponential stability and global practical uniform exponential stability, provided the perturbation magnitude is adequately small. Our analysis leverages Gronwall–Bellman inequalities, offering a more tractable alternative to Lyapunov's direct method, which requires an often elusive Lyapunov function. This makes the stability criteria straightforward to apply in practice. The validity of the theoretical results is illustrated with several numerical examples.