Pendulum with **friction** : La Sall's Invariance Theorem

Question

I am trying to understand LaSalle's Invariance Theorem, which is used to prove that a system is asymptotically stable.

Can someone help me to understand it, by making an example of how to use it. For instance, a pendulum with friction might be a perfect candidate...

$$\begin{align} \dot x_1 &= x_2 \\ \dot x_2 &=-g\sin(x_1)-\dfrac{k}{m}x_2 \end{align}$$

... since the pendulum is asymptotically stable.

A Lyapunov candidate would be:

$$V = E_{cin}+E_{pot} =\dfrac{1}{2}mx_2^2 + mg(1-\cos(x_1))$$

One can find this exampled worked out here:

https://en.wikipedia.org/wiki/LaSalle's_invariance_principle#Example:_the_pendulum_with_friction

But I don't understand the last part, when LaSalle's Invariance Theorem is used to proof the aysmptotic stability.

How exactly (step by step) does one have to procced to show it ?

Answer 1

The algorithm is this:

1. You find a set where $\dot{V}$ vanishes
2. Then, determine which subset of it is invariant. That is, which subset has the property that every trajectory starting there, stays there
3. If the largest invariant subset happens to be the singleton containing the equilibrium, then you conclude (by simple contradiction) that the trajectory has to converge to it eventually