This is a slide I added yesterday.

This isn't in your notes.

I'm probably just going to upload this PDF again as a revised one, so it's in there.

But this is we always talk about stability and

then the definition there's local stability and global stability.

And then in the definition, it just meant, if is true for any of the states,

this was good With the optimal theory, any of the states isn't quite good enough.

It's a necessary but not sufficient requirement.

So it'd be sort of system our total energy function that is positive definite for

any x and any x dot, right?

No matter what non-zero x x dot you put in, you're positive.

That curve never comes back down and hits zero again or does something.

So that's good, but, and same thing with v.

v. equals to minus c times x.

squared is negative semi definite, regardless of what x and x.

is. So for any of those states that was true.

But those two arguments by itself don't prove it's globally asymptotically stable.

What you also have to prove is that v itself is what's called the radially

unbounded function.

And what that means essentially is if you're looking at your norm of your

states, if you make your states grow to infinity along any direction.

So you can't just be specific, I've to let X grow but

I wont touch Y because that causes issues.

You know along any of those directions then that's a radially unbounded function

and in the homework you are actually asked to evaluate this for some functions and

say hey is it posi-definite is it posi-semidefinite is it indefinite.

If it is is it a local result or

is it also you know is it a radially unbounded answer as well.

So this would be an example as one that is radially unbounded

as x's go to infinity my v's go to infinity.

And this keep on growing, basically.

But this is an example of one that's not.

x is 0 here, for any finite x, this function is positive.