Asymptotic stabilization of a class of bilinear systems by a variable structure feedback

We consider the problem of stabilization of a homogeneous bilinear system at zero. We assume that the system can be reduced to a form that admits feedback linearization at all points of the phase space outside a set N of measure zero. For such systems, we construct a variable structure feedback solving the stabilization problem under the condition that N is not an invariant set of the closed system.