Asymptotic Stabilization of a Class of Smooth Two-Dimensional Systems

Abstract

This paper studies the asymptotic stabilizability of two-dimensional control systems. The class under consideration includes $C^\infty $-systems that satisfy a certain genericity assumption and all real analytic systems. Necessary and sufficient conditions for feedback stabilization using continuous feedback and a sufficient condition for $C^1 $-feedback stabilization are given. This latter condition is given in terms of an inequality involving two indices. If the direction of the inequality is changed, an obstruction to $C^\infty $-feedback stabilizability is obtained. A subclass of polynomial systems is also studied and given complete necessary and sufficient conditions for global asymptotic stabilization using $C^1 $-feedback.