When Lyapunov's Stability Theorem is not enough...

There are cases where Lyapunov's stability theory is not enough in the sense that it can prove merely stability for equilibrium points that are actually asymptotically stable (after days and nights struggling to find a Lyapunov function for our system). The pendulum with linear friction term is the prime example of such a case: The total energy function of the system is used as a Lyapunov function. However, this absolutely natural choice fails to provide the expected asymptotic stability result. At that point La Salle's invariance principle chimes in to accommodate the aforementioned theoretical shortage.

Lyapunov on a stamp

Note on La Salle's invariance principle: 

Download the PDF file from http://users.ntua.gr/chvng/ac2/LaSalle_Invariance_Principle.pdf 
Read also: The Realm of State Space Systems