Lyapunov Stability via Sperner's Lemma

We go on whistle stop tour of one of the most fundamental tools from control theory: the Lyapunov function. But with a twist from combinatorics and topology.

For more on Sperner's Lemma, including a simple derivation, please see the following wonderful video, which was my main source of inspiration for covering this topic:

   • NYT: Sperner's lemma defeats the rent...  

Towards the end the video things become really quite advanced. In fact the thought processes and ideas really represent cutting edge research on Lyapunov functions and convex optimisation. So if you are able to grasp some ideas, well done, you are well on your way to becoming a research level control theorist! Hidden in amongst the ideas we cover today are:

- The Bolzano–Weierstrass theorem
- A proof of the Perron-Frobenius Theorem

See if you can spot them! If not, don't worry I will slowly start expanding on these ideas on my blog (though there is nothing there on this yet...):

richardpates.com

I hope to also go into more details onto how to formalise the ideas at the end into an algorithm for producing Lyapunov functions.

Many thanks to Grant Sanderson for making his animation tool Manim available, and for inspiring thousands of people, including myself, to produce mathematical content.

#SoME1