Stabilization of nonlinear systems in the plane (Q1117190)

It is shown that every small-time locally controllable system in the plane can (locally) be asymptotically stabilized by employing Hölder continuous feedback laws, as essentially was conjectured by E. Sontag. An explicit algorithm for the construction of such feedback laws is given. Typical is the system \(\dot x=u\), \(\dot y=y-x^ 3\); which cannot be asymptotically stabilized by any \(C^ 1\)-feedback law, but is asymptotically stabilized by every feedback law \(u=-ax+by^{1/3}\) with \(b>a>1\).