# 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 the domain of stability of a dynamic system

Suppose I have the system: $\dot{x} = -x^3 - y^2$ $\dot{y} = xy - y^3$ ... and am asked to find the domain of stability of the system. Is my attempt and reasoning below deemed a correct approach? ...
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### Lyapunov Candidate function to derive parameter estimation law

I have a system and a reference model represented in state space in the following form: \begin{gather} \dot{x} = Ax+Bu \\\ u = -Kx+k_rr \\\ K,k_r : constants - controller \ gains \\ A_m = A-BK \\\ \...
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### Problem understanding Center Manifold theory

I am studying stability for non linear control systems, and I am focusing on the Center manifold theory . In particular, I am trying to understand an example which is also in the Hassan K.Khalil book ...
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### Is it necessary for a Lyapunov Candidate to be Differentiable at an Equilibrium Point?

For example, given a general nonlinear system where we want to show that the error system is stable $e=x-x_d$ is it necessary for the Lyapunov candidate to be continuously differentiable at the ...
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### Conditions to get bounded solutions in the future

Let's take the Liénard equation: $$x''+h(x)x'+g(x)=0$$ where $x(t)\in\mathbb{R}$ and $h,g\in\mathcal{C}^1(\mathbb{R})$ are lipstchitz locally. That equation is equivalent to the following plane ...
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### Problem understanding Chetaev theorem

I am studying control theory, and I am focusing on the Lyapunov stability. In particular, I am looking the Chetaev theorem, but I have some problems understanding it well. I know that the Cheatev ...
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### Looking for a way to upper bound the euclidean norm with a valid Lyapunov function

I am looking for a valid Lyapunov function $V$ that can upper bound the euclidean norm. For example: $\vert\vert x\vert\vert \leq V$ For this upper bound, the less conservative the better Better ...
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### Why does the Lyapunov criterion only gives sufficient conditions for stability?

I am studying stability for control systems, and I have written in the notes of my professor that the Lyapunov Criterion only gives sufficient conditions for stability, and not necessary and ...
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### Why for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin?

I am studying the concept of stability for linear and for nonlinear systems. While studying the stability for a linear system I found this definition from the notes of my professor: for a linear ...
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### what is the geometric representation of Lyapunov stability

Take a look at the theorems below, in a dynamical systems linear or nonlinear, we construct a function usually the energy-function of a physical system, compute its derivative and apply the theorem to ...
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### Lyapunov function for specific $2$D autonomous system with no potential

Consider the following two-dimensional autonomous system $$\left\{ \begin{array}{cc} \dot{u} = -2v+v^2 \\ \dot{v} = -3u^2 +6u \end{array} \right.$$ Determine whether the critical point $(0,0)$ ...
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### Confirming Lyapunov function

I have \begin{align} x' &= y\\ y' &= -f(x)y -xg(y)\\ \end{align} $f,g$ are functions, especially $f \ge 0$. I could confirm that $(0,0)$ is equilibrium point easily. I want to confirm that ...
We are given the homogeneous system $$y'=\begin{bmatrix}5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4\end{bmatrix}y$$ with the initial value $y(0)=(1,1,0)$ and are supposed to ...
### What can I claim about $f(y(x))$ when $\frac{d (e^{-\gamma x}f(y(x)))}{dx}=-e^{-\gamma x}(y^2(x)+q(a))$, $\gamma >0$ and $q(a)$ is positive definite.
I have an inequality from control theory, which goes like this, \begin{equation} \frac{d(e^{-\gamma x}f(y(x)))}{dx}=-e^{-\gamma x}[y^2(x)+q(a)] \leq 0 \end{equation} where $q(a)$ is positive definite....