Does Lyapunov function need to be defined at zero?

Question

Consider the function $V(x,y) = x-y-y\ln \left(\frac{x}{y}\right)$, where $x,y>0$ and $y$ is fixed. $V$ is constructed to study the stability of equilibrium point $x=y$ for the system $$ \frac{dx(t)}{dt} = y - x(t) .$$ Then it satisfies the properties of Lyapunov function, but not defined at zero. But in one of the textbook it is defined as Lyapunov function. Can we have Lyapunov function which is positive for all non zero points but not defined at zero?

Answer 1

The Lyapunov function needs only be defined in some neighborhood of the equilibrium point that is to be examined. So from that direction your function is admissible, if the equilibrium is at $(1,1)$, then the function values (and if they exist) on the coordinate axis are only of minor concern.

Note that your claim of positivity is wrong, you can write the function as $V(x,y)=f(x)-f(y)$ where $f(x)=x-1-\ln(x)$. As $f$ has a strict minimum at $x=1$, the values of $V(1,y)$ are negative for $y\ne 1$.