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

The Frobenius Theorem

The Frobenius theorem is of exceptional importance both from the point of view of differential geometry and differential topology but also from the Automatic Control and Nonlinear Systems analysis and design point of view. This text aims to be a self-contained report on the Frobenius theorem. First, some necessary definitions are given and basic facts about distributions are stated. The dual objects of distributions - the codistributions are introduced. We try to describe these notions and engage them to involutiveness and complete integrability, to state and prove the Frobenius Theorem which is of great importance in nonlinear control theory.

Ferdinand Georg Frobenius