Stability of nonlinear systems with three time scales (Q1092309)

We study the asymptotic stability of a singularly perturbed nonlinear time-invariant system \({\mathcal S}_{\epsilon \nu}\), which has three vastly different time scales. The system \({\mathcal S}_{\epsilon \nu}\) is approximated by three simpler systems over different time intervals. We give a straightforward proof of the fact that the asymptotic stability of \({\mathcal S}_{\epsilon \nu}\) is guaranteed when the equilibrium points of the three simpler systems are exponentially stable and when the parameters \(\epsilon\) and \(\nu\) are sufficiently small.