Exponential stability and partial averaging. (Q1404915)

The initial value problem for the system \(\dot{x}(t)=\varepsilon f(\varepsilon t,t, x(t))\), \(x(t_{0})=x_{0}\), is considered, where \(\varepsilon >0\) is a small parameter. The corresponding partially averaged system is \(\dot{z}(t)=\varepsilon f^{0}(\varepsilon t,z(t))\), \(z(t_{0})=x_{0}\), with \(f^{0}(s,x)=\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}f(s,t,x)\,dt\). Under certain uniformity conditions, it is shown that exponential stability of the averaged system is equivalent to exponential stability of the perturbed system for small values of the perturbation parameter \(\varepsilon\). Explicit estimates on both the approximation of single trajectories and the order of the exponential decay are obtained.