Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions (Q807555)

The author considers single input nonlinear systems \(\dot x=F(x)+uG(x),\quad x\in R^ n,\quad u\in R,\) and provides sufficient conditions for the existence of control Lyapunov functions. The author assumes that the dynamics F and G have the form \(F=(f_ 1',f_ 2')\), \(G=(0,g)'\), where \('\) stands for transpose, the mappings \(f_ 1: R^ n\to R^{n-1}\), \(f_ 2: R^ n\to R\), g: \(R^ n\to R\) are Lipschitz continuous and the origin \(0\in R^ n\) is an equilibrium for the uncontrolled term F, i.e., \(F(0)=0\). Special emphasis is given for the planar case \((n=2)\), where in addition zero is also an equilibrium for the controlled term G.