  In this paper, the problem of assessing the Finite-Time Stability (FTS)
property for general nonlinear systems is considered. First, some necessary and
sufficient conditions that guarantee the FTS of general nonlinear systems are
provided; such conditions are expressed in terms of the existence of a suitable
Lyapunov-like function. Connections of the main theoretical result of given in
this article with the typical conditions based on Linear Matrix Inequalities
(LMI) that are used for Linear Time-Varying (LTV) systems are discussed. An
extension to the case of discrete time systems is also provided. Then, we
propose a method to verify the obtained conditions for a very broad class of
nonlinear systems. The proposed technique leverages the capability of neural
networks to serve as universal function approximators to obtain the
Lyapunov-like function. The network training data are generated by enforcing
the conditions defining such function in a (large) set of collocation points,
as in the case of Physics-Informed Neural Networks. To illustrate the
effectiveness of the proposed approach, some numerical examples are proposed
and discussed. The technique proposed in this paper allows to obtain the
required Lyapunov-like function in closed form. This has the twofold advantage
of a) providing a practical way to verify the considered FTS property for a
very general class of systems, with an unprecedented flexibility in the FTS
context, and b) paving the way to control applications based on Lyapunov
methods in the framework of Finite-Time Stability and Control.
