Necessary and sufficient conditions of asymptotic stability of nonlinear dynamic systems (Q1914125)

It is well known that, for an autonomous differential system \(x'= f(x)\), with \(x(t)\in \mathbb{R}^n\), the usually given condition for the isolated rest point \(x=0\) to be asymptotically stable is sufficient but not necessary, necessary and sufficient conditions (n.a.s.c.) being much more cumbersome. Here, new n.a.s.c. for asymptotic stability are given, which differ from the conditions given earlier by using a wider class of functions that depend on a parameter. With them, the inversion of the Lyapunov theorem on asymptotic stability takes a very clear geometric meaning, easy to visualize. After defining the family of functions to be used, the main theorem is proved. Motivating examples are also given.