Lyapunov function for non-autonomous non-linear differential equations

Question

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system.

Suppose there is a non-autonomous non-linear differential equations: $$\frac{dx}{dt}=f(x,t)$$

In order to use Lyapunov’s Second Method for the system, the books state that Lyapunov function $W(x,t)$ is needed. I would like to know whether $W(x,t)$ can be constant of $t$. That is, $W(x,t)$ is just a positive definite $V(x)$ which does not involve $t$ certainly.

Actually, I have constructed a $V(x)$ and shown that $\frac{dV(x)}{dt}=\frac{dV(x)}{dx}\cdot f(x,t)<0, \forall x$ and $\forall t>0$. Is this sufficient to show the system is Lyapunov stable and even asymptotically stable?

Answer 1

I'll take the easy way out and point you to the book where a solution to your problem is given along with its proof: Khalil, Nonlinear Systems.

The short answer to your question is yes. Your Lyapunov function does not have to have an explicit $t$ dependence. The general result goes as follows. If you can find a function $V(t,x)$ that is lower and upper bounded by two positive definite functions $W_1(x)$ and $W_2(x)$ and $\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(x,t) \leq -W_3(x)$ for some positive definite function $W_3(x)$, then the origin is uniformly asymptotically stable. In your case $\frac{\partial V}{\partial t} = 0$ and if your $V(x)$ satisfies the conditions I listed above, then you have uniform asymptotic stability.

Answer 2

A function not depending on $t$ is a particular case of a function depending on $t$. In other words, if you need to find a $W(t,x)$ satisfying certain conditions and you manage to find $W_1(x)$ satisfying these conditions, then everything is ok.

As for showing that the zero solution is (asymptotically) stable, just adopt the same proof as for the case of autonomous system. If you don't have the proof, ask in comments, I'll sketch it.