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Stabilization of a class of nonlinear systems by a smooth feedback control. (English) Zbl 0569.93056
The local stabilizability problem is considered for systems of the form $$\dot x=f(x)+ub$$ with equilibrium point $$x_ e=0$$, where u is a scalar control and $$b\in R^ n$$ constant. If the uncontrollable part of the linearized system has only eigenvalues with real part $$\leq 0$$ and $$r\geq 1$$ eigenvalues on the imaginary axis the stabilizability question cannot be decided by linear analysis only. The author applies results from center manifold theory to study this problem via an associated system of reduced order r. General stabilizability criteria are not obtained but a sufficient criterion is derived for the special case where $$r=2$$ and the linearized system has a pair of purely imaginary uncontrollable eigenvalues. Several interesting examples are discussed and, in particular, a simple two-dimensional system is presented which is locally and globally controllable but not stabilizable by smooth state feedback.
Reviewer: D.Hinrichsen

##### MSC:
 93D15 Stabilization of systems by feedback 37C75 Stability theory for smooth dynamical systems 93C10 Nonlinear systems in control theory 93D99 Stability of control systems 93B05 Controllability 93C15 Control/observation systems governed by ordinary differential equations
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