Lyapunov function for non-autonomous non-linear differential equations

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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.

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Updated on July 17, 2020

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?