I am looking for a valid Lyapunov function $V$ that can upper bound the euclidean norm.
For example: $\vert\vert x\vert\vert \leq V$
For this upper bound, the less conservative the better
Better yet, if anyone knows of a way to show that the Euclidean norm is a valid Lyapunov candidate for nonlinear systems that would be great! (I found papers that show it for linear systems.)