On nonlinear control systems with multiple time scales (Q1426953)

Consider the multiscale control system \[ \begin{aligned} \dot x_0 & =f_0 (x_0,x_1,\dots, x_n,u(t)),\\ \varepsilon_1\dot x_1 & =f_1(x_1,\dots,x_n, u(t)),\\ \dots \dots & \quad \dots \dots\\ \varepsilon_n\cdots\varepsilon_1\dot x_n & =f_n (x_n,u(t))\end{aligned} \] on the bounded time horizon \([0,1]\), where the control function \(u\) is measurable. The author proposes an order reduction procedure based on a refined two-scale averaging permitting a re-iteration. For vanishing control range, the results reduce to the well-known Tikhonov theorem on order reduction for singularly perturbed ordinary differential equations.