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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...

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

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... since the pendulum is asymptotically stable.

enter image description here

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 ?

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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
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    Thanks a lot for your answer. However there are some things I don't understand. I am not a math specialist, so what exactley to you mean by singleton ? – 2017-01-11
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    I find a Invariant set where V_dot =0, I find all the x-Values (x1=...,x2=...) which satify this equation, I plug them into my orignial state equations, and if I the result is the same as at the equilibrium (x_1 ^dot =0, x_2^dot = 0) then my System converges aysmptotically to those Equilibrium points ... is that what one is supposed to do ?? – 2017-01-11
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    Singleton means a set consisting of one element. It is actually not a straightforward procedure to figure out what the maximal invariant set is. Basically, you plug in the conditions of $\dot{V}=0$ into your state equations and try to find out where the trajectories need to start so that they stay there all the time. If only the trivial trajectory $x(t) \equiv 0$ (provided that the equilibrium is the origin) satisfies this condition then all the trajectories have to converge to there eventually. – 2017-01-11
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    okay, I see, and what if the equilibrium is not the origin ? – 2017-01-11
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    Then you simply change the coords :) like this: $ y := x - x_e$ where x_e is your equilibrium. $x_e=0$ is used for simplicity. – 2017-01-11
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    Ah...okay ! Thanks a lot for your help ! :) – 2017-01-11