Nonlinear system stability analysis

Question

How to find an appropriate Lyapunov function to show the stability of a system:

$\dfrac{dx}{dt} = -(x-1)(x-2)^2$

around its equilibrium point $x = 1$?

By linearizing this system, it can be shown that the $x=1$ point is locally asymptotically stable.

What is the appropriate Lyapunov function?

Answer 1

$V(x)=(x-1)^2$ is a (strict) Lyapunov function.

Added: Indeed, $V\ge0$ and $V(x)=0$ if and only if $x=1$. Moreover, $$ \dot V(x)=-2(x-1)^2(x-2)^2 $$ and so for $x\in(0,2)\setminus\{1\}$ we have $\dot V(x)<0$. Hence, $V$ is a strict Lyapunov function, which shows that the equilibrium point $x=1$ is asymptotically stable.