HOMOGENEOUS LYAPUNOV FUNCTION FOR HOMOGENEOUS CONTINUOUS VECTOR FIELD

The goal of this article is to provide a construction of a homogeneous Lyapunov function VBAR associated with a system of differential equations x = f(x), x is-an-element-of R(n) (n greater-than-or-equal-to 1), under the hypotheses: (1) f is-an-element-of C(R(n), R(n)) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function VBAR(x) tends to infinity with \\x\\, and belongs to C(infinity)R(n)\{0}, R) and C(p)(R(n), R), with p is-an-element-of N* as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.