Practical exponential stability and closeness of solutions for singularly perturbed systems via averaging

This paper studies the behavior of singularly perturbed nonlinear differential equations with boundary-layer solutions that do not necessarily converge to an equilibrium. Using the average for the derivative of the slow state variables and assuming the boundary-layer solutions converge exponentially fast to a bounded set, which is possibly parameterized by the slow variable, results on the closeness of solutions of the singularly perturbed system to the solutions of the reduced average and boundary-layer systems over a finite time interval are presented. The closeness of solution error is shown to be of order O(ε) where ε is the perturbation parameter. Moreover, under the additional assumption of exponential stability of the reduced average system, practical exponential stability of the solutions of the singularly perturbed system is provided.