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Stabilization of nonlinear systems in the plane. (English) Zbl 0666.93103

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 \(b>a>1\).
Reviewer: M.Kawski

MSC:

93D15 Stabilization of systems by feedback
93B05 Controllability
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
93D20 Asymptotic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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