Exact both construction of Lyapunov function and asymptotic stability domain determination

The problem of an accurate one-shot construction of a Lyapunov function and the problem of an exact one-shot determination of the asymptotic stability domain are solved for a nonlinear dynamical system with continuous motions. The solutions are based on O-uniquely bounded sets, which are explained. A choice of an O-uniquely bounded set is arbitrary in a given class to generate the Lyapunov function.