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. But that bowl pinches off to 0 again. It gets infinitely close. And it gets to 0 at infinity, but you never reach infinity. So, This is a function that would not be radially unbounded, so even though the states you have might be good for any initial conditions, because V was not radially unbounded you wouldn't be able to argue global stability. It doesn't prove it's not globally stable, it just means I can't argue it is globally stable. It's that if statement again, a one-way arrow, right? So ideally in our functions we want ones that are radially unbounded. That at least keep on growing monotonically as my states errors get bigger and bigger. That's what your looking for. Is bold behavior not this kind of reversing stuff, okay? So as we do controls, this is something else we'll be looking, is it global or is it local? And that's the condition.