Classical converse theorems in Lyapunov's second method

Lyapunov's second or direct method is one of the most widely used techniques forinvestigating stability properties of dynamical systems. This technique makes use of anauxiliary function, called a Lyapunov function, to ascertain stability properties for a specificsystem without the need to generate system solutions. An important question is theconverse or reversability of Lyapunov's second method; i.e., given a specific stability propertydoes there exist an appropriate Lyapunov function? We survey some of the available answers to thisquestion.