The necessary and sufficient conditions for the exact construction of a Lyapunov function and the asymptotic stability domain

The problem of the necessary and sufficient exact algorithmic conditions for both the construction of a Lyapunov function and the determination of the asymptotic stability domain is resolved in the framework of a large class of time-invariant nonlinear systems with continuous motions. The class of the systems is characterized by the smoothness of their motions, which is precisely defined.