# Questions tagged [lyapunov-functions]

This tag is for questions relating to Lyapunov function, which is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. The Lyapunov function method is applied to study the stability of various differential equations and systems.

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### Determining a Lyapunov function based on solution of differential equation [closed]

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. ...
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### Asymptotic stability and Lyapunov functions

I fail to understand a passage in the proof of the following theorem (right after the definition that gives the context of my question): (Definition of Lyapunov function) Let $\Omega$ be a ...
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### Asymptotic stability by comparison with another system

Consider a nonlinear system of the form $$\dot{x} = f(x)x,$$ with $x \in \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$. I know that there exists an asymptotically stable ...
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### How large the error ball of $\|x\|$is, when using the Lyapunov function $x^\top Px$?
I would like to know how large the error ball of $\|x\|$ is when using the Lyapunov function $x^\top Px$: Assumption: I have an almost linear closed-loop system $\dot{x}=(A-BK)x+\epsilon(x)$ with ...