Lyapunov stability is applicable to only unforced (no control input) dynamical systems. It is used to study the behaviour of dynamical systems under initial perturbations around equilibrium points.
Lets consider that origin is the equilibrium point (EP) of the system. Consider two spheres of radius ε and δ around origin such that δ < ε. A system is said to be stable in the sense of Lyapunov if
The system is said to be asymptotically stable if as
Lyapunov stability theorems
Lyapunov stability theorems give only sufficient condition.