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For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some positive definite function $P.$

Is there a similar result for nonlinear systems? Namely, for a nonlinear dynamical system in the form $\dot x = f(x)$ such that the origin $x=0$ is globally asymptotically stable, is the existence of a Lyapunov function guaranteed? (Of course, even if the Lyapunov function exists, it might be very tricky to find.) If the existence of a Lyapunov function is not guaranteed, then are there any known counter-examples?

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Yes, it is Theorem 4.17 in Khalil's book:

Let $x=0$ be an asymptotically stable equilibrium of $\dot{x}=f(x)$. Then there is a smooth positive definite function $V(x)$ and a continuous positive definite function $W(x)$ such that: $$ \frac{\partial V}{\partial x}\,f(x)=-W(x) $$

(I have omitted some details)

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