I am reading the review paper named *review on computational methods for lyapunov functions* (Doi: 10.3934/dcdsb.2015.20.2291) which can be seen here

My question regarding the lower part of page 4 in the paper, which specifies certain subsets of basin of attraction. I quote the sentences here:

- Compact sublevel sets of a strict Lyapunov function,

which are completely contained in U, are subsets of the basin of attraction of

the equilibrium.*

I am not sure what does compact sublevel sets means. I know the sublevel sets of strict lyapunov function is $\{x|v(x)<c\}$ given a level $c$, where $v$ denotes lyapunov function. But what is purpose of being compact here?

## Best Answer

The problem is that $v(x)\leq c$ does not ensure convergence to the origin even when $v$ is positive definite and $\dot{v}$ is negative definite.

This is also the reason why for global stability you additionally need that $v$ is radially unbounded.

The problem is that a level $v(x)\leq c$ set can stretch to infinity itself. You may take a look at this answer by Hans Lundmark to a related question.

You avoid this problem when you require that the level sets are compact which just means bounded and closed (note that closed means $\leq c$ not $< c$).