Does Lyapunov function need to be defined at zero?Consider the function V(x,y)=x−y−yln⁡(xy), where x,y&gt;0 and y is fixed. V is constructed to study the stability of equilibrium point x=y for the systemdx(t)dt=y−x(t).

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Does Lyapunov function need to be defined at zero?Consider the function V(x,y)=x−y−yln⁡(xy), where x,y&amp;gt;0 and y is fixed. V is constructed to study the stability of equilibrium point x=y for the systemdx(t)dt=y−x(t).

star04iks7 · Accepted Answer

Step 1The 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≠1.

Does Lyapunov function need to be defined at

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