ADDING AN INTEGRATOR FOR THE STABILIZATION PROBLEM

We study the relationship between the following two properties: P1: The system x = f(x, y), y = upsilon is locally asymptotically stabilizable; and P2: The system x = f(x, u) is locally asymptotically stabilizable; where x is-an-element-of R(n), y is-an-element-of Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension n = 1. Here, using the so called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles's result; (b) we show that, in general, P1 does not imply P2 for dimensions n larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.