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this is a pretty general question. Let's say you have a dynamic nonlinear system
$$\dot x = f(x) \\ y = h(x)$$ that is (asymptotically) stable and you want to show it by using Lyapunov theory. Moreover, you have a nontrivial solution of $x$. Are there methods to systematically determine a Lyapunov function using the solution of $x$ and if not, are there any tricks or approaches, which utilize the solution of $x$ to narrow down possible Lyapunov function candidates? Any examples or references would be appreciated. Thanks in advance for any answers.

Sozialhilfe
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    I think you might need to add some context. At face value, if you already know the system is stable and you have the solution $x(t)$, why would you even need a Lyapunov candidate? Maybe you can clarify what you mean by solution here because without the signal $u$ you usually cannot solve for $x$ (except in special cases). Is it that you know $x(t)$ for some family of $u(t)$? – Rollen S. D'Souza Jul 21 '24 at 14:40
  • Hi thanks for the answer. I just edited it to more clearly describe what I mean and also removed u from the systems equation. The background of my question is this post https://math.stackexchange.com/questions/2971767/how-to-pick-a-lyapunov-function-and-prove-stability?rq=1 where the accepted answer solved the differential equations and used the solution to build a Lyapunov function. I was wondering, if there is a general approach similar to what was done in the post, where one solves the differential equations and uses the solution to construct a Lyapunov function. – Sozialhilfe Jul 21 '24 at 17:24
  • Right. to expand on your clarification, "non-trivial" here means "not visibly asymptotically stable". Hence the need to find a Lyapunov candidate. – Rollen S. D'Souza Jul 21 '24 at 21:11

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