The method of normal local stabilization

A problem of nonlinear systems stabilization is studied. Admissible controls are piecewise constant. The notion of normal local stabilizability is proposed. A point $P$ (not necessary equilibrium) is normally locally stabilizable if for any $\tau>0$ there exists such neighborhood $D(P;<\tau)$ of $P$ that any point $x \in D(P;<\tau)$ can be steered, in a time less than $\tau$, to any neighborhood of $P$ and remains there. The constructive method of normal local stabilization of nonlinear autonomous systems is presented. This method involves a special sequence of contracting cylinders containing a trajectory. A domain of attraction of a given point is constructed.