Feedback stabilization of single-input nonlinear systems
The feedback stabilization problem for a scalar input nonlinear system on \({\mathbb{R}}^ n,\dot x=f(x)+g(x)u\) is considered. That is, given \(f(0)=0\), when does there exist a smooth feedback function \(u=\alpha (x)\) such that the closed-loop system \(\dot x=f(x)+g(x)\alpha (x)\) has the origin as a stable equilibrium point. The author uses the idea of partial linearization [see \textit{R. Marino}, \textit{W. M. Boothby} and \textit{D. L. Elliot}, Math. Syst. Theory 18, 97-123 (1985; Zbl 0574.93028)] of the system in order to reduce the feedback stabilization problem to a lower dimensional state space. Then standard center manifold techniques are used to give sufficient conditions for the solvability of this stabilization problem.
scientific article; zbMATH DE number 4041753