Feedback stabilization of nonlinear driftless systems with applications to homogeneous-type systems

Abstract

Issues of asymptotic stabilization of nonlinear driftless systems as given by x = g(x)u with applications to homogeneous-type driftless systems are presented. Conditions of the existence of a smooth time-invariant stabilizer for general nonlinear driftless systems are obtained by the construction of quadratic-type Lyapunov functions. The proposed conditions do not contradict Brockett's (1983) result for the existence of a smooth time-invariant stabilizer. These results are then employed to study the stabilization problem of homogeneous-type systems. Sufficient conditions are obtained for the stabilization of planar type homogeneous driftless systems with positive order. It is shown that the single input control driftless systems cannot be asymptotically stabilizable by any continuous control if g(x) is a homogeneous function of even order. Moreover, equivalent conditions for the stabilizability of linear driftless systems and the explicit design of stabilizing control laws for bilinear driftless systems are also presented.