  This technical note studies Lyapunov-like conditions to ensure a class of
dynamical systems to exhibit predefined-time stability. The origin of a
dynamical system is predefined-time stable if it is fixed-time stable and an
upper bound of the settling-time function can be arbitrarily chosen a priori
through a suitable selection of the system parameters. We show that the studied
Lyapunov-like conditions allow to demonstrate equivalence between previous
Lyapunov theorems for predefined-time stability for autonomous systems.
Moreover, the obtained Lyapunov-like theorem is extended for analyzing the
property of predefined-time ultimate boundedness with predefined bound, which
is useful when analyzing uncertain dynamical systems. Therefore, the proposed
results constitute a general framework for analyzing predefined-time stability,
and they also unify a broad class of systems which present the predefined-time
stability property. On the other hand, the proposed framework is used to design
robust controllers for affine control systems, which induce predefined-time
stability (predefined-time ultimate boundedness of the solutions) w.r.t. to
some desired manifold. A simulation example is presented to show the behavior
of a developed controller, especially regarding the settling time estimation.
