Necessity of the hypotheses of Lyapunov asymptotic stability theorem

Question

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption.

The statement we saw in class is:

Lyapunov asymptotic stability theorem
We consider the system $\dot{x}(t)=f(x(t))$, $x(t)\in \mathbb{R}^n$ and $f:\mathbb{R}^n\in \mathbb{R}^n$ continuous, $f(0)=0$. Let $V:\mathbb{R}^n\rightarrow \mathbb{R}$ be a positive definite continuously differentiable function.
If $\dot{V}$ is negative definite (i.e., $V$ strictly decreasing), then $0$ is asymptotically stable.

Proof
For the sake of brevity, I link you to a proof similar to the one the professor gave us: proof (page 2)

My question is: can I replace the assumption "$V$ strictly decreasing" with "$V$ decreasing and not definitely constant"? Thus avoiding strictly decreasing.

In the proof, it is used that $V$ is strictly decreasing to get a "negative maximum", but can't I use the hypothesis I propose to get the proof anyway?
Second, if it is possible to weaken assumptions in this way: does it make sense? Does it generalize the statement in any way?

I also tried looking for counterexamples, but couldn't find much: I think my skills in ODE are still too immature…

Answer 1

Decreasing and strictly decreasing are usually taken as synonym (in English). You probably meant nonincreasing. Nonincreasing is not enough for asymptotic stability and one needs extra conditions for proving the asymptotic stability of the equilibrium point.

One of them is through the LaSalle invariance principle, which I am stating below:

If $V(x)$ is positive definite and $\dot{V}(x)\le0$ in a neighborhood $D$ of $0$ and the set $\{\dot{V}(x)=0\}\cap D$ does not contain any trajectories of the system besides the trajectory $x(t)=0$, $t\geq 0$, then the local version of the invariance principle states that the origin is locally asymptotically stable.

An application example is the pendulum with friction which is a bit long to reproduce here but can be found there.

In the case of non-autonomous systems (which includes time-varying ones), Barbalat's lemma should be used instead.

Finally, it is worth mentioning that many converse Lyapunov results exist. One which is due to Kurzweil states that:

Assume that $x=0$ is a globally asymptotically stable equilibrium point for $\dot x=f(x)$ where $f$ is a continuous function, then there exists an ininitely differentiable Lyapunov function.

The main issue with Lyapunov results is that it is often difficult to construct Lyapunov functions that satisfy the strict conditions of the Lyapunov theorem(s). In this regard, the result from LaSalle or the use of Barbalat's lemma enlarge the class of possible functions that can be considered, thereby making them easier to find.

There are other ways to relax the Lyapunov conditions such as conditions using integrals or considering a decreasing sequence of values for the Lyapunov function instead of a continuous one, which both allow for the use of non-monotonically continuously decreasing Lyapunov functions.

For more details about Lyapunov functions, check the book by Khalil, "Nonlinear Systems".