So, okay, we've just wrapped up,

what does it mean to be positive definite, positive semi-definite?

Now we want to tie this back to stability, right?

We care about what if I perturb something slightly?

What if I perturb it a lot?

I don't want to have to linearize.

How do I now apply this?

This is going to fall into place very quickly.

This is the Lyapunov direct method basically.

It says now, [LAUGH] the next step is we're talking about, besides definiteness,

what is a Lyapunov function?

And once we've proven we have a Lyapunov function, actually we've proven stability.

You will see the last step is very anticlimactic.

So a Lyapunov function is always a scalar function like kinetic energy, right?

But it's in terms of whatever all your states are, which could be omegus.

If you use kinetic energy of a rigid body,

it's one over two omega transposed i omega, right?

And we just mentioned i is a symmetric

posit definite matrix which makes kinetic energy actually.

It's only going to be zero at zero speeds, and

everywhere else it's nonzero, that's something we'll be using a lot.

So a Lyapunov function is always a scalar function subject to this dynamical system.

And we're going to throw in our equations of motion, and attitude, and rotation and

everything.

And we're talking, if it is continuous this function and

there exists in neighborhood, such that for

any states, that we are arguing local stability here with Lyapunov.

There is a neighborhood that if you're staying within that one,

this v function is positive definite about x r.

So you know what that means.

It's kind of like x squared is the bowl shape, right?

There is a Lyapunov function, this Lyapunov function has continuous partial

derivatives, that's one of the requirements.

It takes two lectures to explain why, but that's important.

And then the last one is V dot is negative semi definite.

So once you define this V, you can take its time derivative.

Like we've done this with kinetic energy, and

then we took the time derivative kinetic energy which gave us the power equation.

And actually you had to derive that in your exam.

We're reusing the same math here.

It just doesn't have to be explicitly kinetic energy,

you can throw any mathematical function in here that satisfies these properties.

And you will see some things that we do there.

But if it has these properties, then we're good.

Then this function is a Lyapunov function.