According to the proof of Lyapnuov's theorem given in [1] the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not.
I wonder if it is possible to relax the continuity of the partial derivatives and proove the theorem as well. The continuity, in fact, is only needed to use the Extreme Value Theorem, and stating that the derivatives have a maximum along the system trajectories. I was wondering what happens if we consider a supremum and not a maximum.
Thank you for your attention!
[1]. Khalil, Hassan K. Nonlinear control. Vol. 406. New York: Pearson, 2015.
First of all, it is important to mention that there are several converse Lyapunov results which prove the existence a continuously differentiable decreasing Lyapunov function under some assumptions on the system. So, in many scenarios, the Lyapunov conditions are necessary and sufficient. The difficulty only lies at the level of the construction of a Lyapunov function that satisfies those conditions.
This was the reason behind the introduction of LaSalle's invariance principle or the use of Barbalat's lemma in order to deal with Lyapunov function which are not decreasing.
When it comes to decreasing Lyapunov sequences for proving stability this has been considered using looped-functionals and/or hybrid stability analysis methods (assuming completeness of the solutions and some extra growth-assumptions). This allows us to consider non-monotonically decreasing Lyapunov functions. Other results for non-monotonic Lyapunov functions have also been considered by considering higher-order derivatives of the Lyapunov function.