The concepts of stability, attractivity and asymptotic stability
for systems subject to restrictions of the input values are introduced
and analyzed in terms of Lyapunov functions. A comparison
with the well known input-to-state stability property introduced
by Sontag is provided. We use these concepts in order to derive
sufficient conditions for global stabilization for triangular and
feedforward systems by means of saturated bounded feedback controllers
and also recover some recent results due to Teel.