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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17 views

Stability and Lyapunov functions

Consider the system $\frac{dx}{dt} = y$, $\frac{dy}{dt} = -x^2y - x^3$ which has $(0,0)$ as a (unique) critical point. I aim to show that $(0,0)$ is stable (at least that's what Wolfram Alpha sketches ...
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43 views

Lyapunov theorem: is there an extension to when $\dot V$ is nd but $V$ is psd?

The well known Lyapunov theorem says given an autonomous system $$\dot x = f(x)$$ with equilibrium $0 = f(\overline x)$, if we can find a function that is positive definite (pd) $V > 0, \forall x ...
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45 views

$A^TPA-P$ is negative definite, does it imply that $APA^T-P$ is negative definite

If $P$ is positive definite, and if $A^TPA-P$ is negative definite, does it imply that $APA^T-P$ is negative definite?
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27 views

Show the asymptotical stability at the origin of a non linear system

Show that at the origin the following system has an asymptotically stable point: $\begin{cases} \dot{x_1}=-\phi_1(x_1)+\phi_2(x_2)\\ \dot{x_2}=\phi_1(x_1)-\phi_2(x_2)\\ \end{cases}$ ,with $x_i^2\...
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50 views

Find the lyapunov function to prove the asymptotic stability

Let's the following non linear system: $\begin{cases} \dot{x_1}=x_2&\\ \dot{x_2}=-x_1^3&\\ \end{cases}$ determine if the origin is asymptotically stable and in this case if it is globally ...
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16 views

Can we always find a dynamic equation so that a Lyapunov function exists and its derivative is specified?

I would like to find out some statements related to the following one. Consider $X \subset \mathbb{R}^n$. For any $h: X \rightarrow \mathbb{R}$, can we find a function $f:X \rightarrow \mathbb{R}^n$ ...
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2answers
107 views

Lyapunov function for a non linear 3 dimensional system

How it is possible to find a Lyapunov function for the following system? \begin{cases} \dot {x_1}=x_2+x_3 & \\ \dot {x_2}=-\sin x_1-x_3 & \\ \dot {x_3}=-\sin x_1+x_2 & \end{...
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41 views

Lyapunov's direct method proof: show that $V(x(t)) \rightarrow 0$ implies $x(t) \rightarrow 0$

In proofs of Lyapunov's direct method (see this lecture, pg 8 for example), the claim is made that because of continuity of $V$, to show $x(t) \rightarrow 0$, it suffices to show that $V(x(t)) \...
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1answer
52 views

How can I find a Lyapunov function for this control system?

I have the non-linear control system: $$ y_1'=y_2^3 $$ $$y_2'=-y_1+u,$$ where $$u \in [-1,1]$$ and I was wondering what's a good Lyapunov function and a good candidate for $u$?
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30 views

Upper bound on trace of solution to discrete lyapunov or discrete matrix ricatti equation

Take a discrete Lyapunov equation $$ P = R + \beta A' P A $$ for $\beta \in (0,1)$, $R$ and $A$ symmetric. Quetsion: Can I put an upper bound on $tr(P)$ given properties of the primitives? In ...
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1answer
38 views

Ergodicity and constant Lyapunov exponents on attractors

The Wikipedia article on Lyapunov exponents says the following: For a [real] dynamical system with evolution equation $\dot{x_i} = f_i(x)$ in an n–dimensional phase space, the spectrum of Lyapunov ...
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27 views

Lyapunov Exponents vs Lyapunov Function

Suppose I have a system of differential equations $\dot {\vec x} = \vec f(\vec x)$, and that there is an equilibrium point $\vec x = 0$. Moreover, suppose I know the Lyapunov exponents of the ...
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114 views

Can we combine this two Lyapunov functions (which imply local stability by separate) to conclude global stability?

Let $x(t)\in\mathbb{R}^n$ constrained to a dynamical system $$ \dot{x}(t) = f(x(t)) $$ for some vector field $f:\mathbb{R}^n\to\mathbb{R}^n$. Moreover, the dynamical system has a unique equilibrium ...
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150 views

Find the Lyapunov function for the nonlinear second order system.

Consider the system $$ \begin{aligned} \dot{x}_1 &= -x_1+x_2,\\ \dot{x}_2 &= -x_2^3. \end{aligned} $$ The origin is obviously globally asymptotically stable. Indeed, $x_2\to 0$ and the ...
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1answer
31 views

Determine Lyapunov stability of an ODE

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t}=-x-3y+2z+yz,\\ \frac{\mathrm{d}y}{\mathrm{d}t}=3x-y-z+xz,\\ \frac{\mathrm{d}z}{\mathrm{d}t}=-2x+y-z+xy.\\ \end{cases} $$ Let Lyapunov function $\...
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1answer
23 views

Classical References on Stability of Autonomous and Non-autonomous Systems and Lyapunov Stability

I ask for classical, but not much advanced books or articles in Stability of Autonomous and Non-autonomous Systems and Lyapunov Stability. I mean: a book where the definitions and examples are in easy ...
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13 views

Monte-Carlo probability of chaotic dynamics - logistic map

Consider the map of the interval $x_{n+1}=4\mu x_{n}(1-x_{n})$, where $\mu\in[0,1]$. Using a Monte-Carlo technique, calculate the probability of finding a chaotic dynamicsin the parameter interval $\...
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21 views

Lyapunov function construction to look for stable equilibrium

I am looking at a system in the form of: $$ \dot{\bf{x}}=\bf{f}(x) $$ and now I wish to find some stability properties of the equilibrium points, i.e. where $\bf f(x)=0$. Particularly, the uniqueness ...
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1answer
42 views

Dynamical System problem

PLS HELP ME. I have a problem with this exercise given by the professor for home. It's about Lyapunov equation and autonomous systems. Here it is: Prove that if the state of equilibrium $x^*=0$ $(x^*\...
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1answer
60 views

Exponential Stability in Lyapunov Razumikhin Theorem

Is there an exponential stability version of the Lyapunov-Razumikhin Theorem for retarded functional differential equations? i.e. analogous to ODEs, given a FDE $\dot{x}(t)=f(x_t)$ does the following ...
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1answer
49 views

Lyapunov Function for system with $f(x,y) = x' = \sin y$

I have the system $x'=\sin y = f(x,y)$ $y'=-2x-3y = g(x,y)$ I am given the Lyapunov function $V(x,y)=15x^2+6xy+3y^2$. Obviously, $V(0,0)=0$. Also, $V(x,y)=6x^2+9x^2+2 \cdot 3x y + y^2 + 2y^2 = 6x^2+2y^...
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142 views

How to find a Lyapunov function

I have the following system: $$\dot{x_1} = x_2(x_3-2) \\ \dot{x_2} = x_1(x_3-2) \\ \dot{x_3}=-x_3^3 $$ and I want to determine its equilibrium points together with their stability. To find the ...
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1answer
26 views

Lyapunov Function showing a unstable critical point of system of ODE to be stable

I am considering the following system of ODE: $$x'=y, y'=x-x^3$$ where (0,0) is a critical point of it. By considering its locally linear system, I found that this critical point is an unstable saddle ...
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1answer
40 views

Lyapunov stability dynamic systems

$\dot{x}=-(x-1)\cdot(x-2)^2$ I want to find the stability with the 2 Lyapunov methods ( linearization and appropriate Lyapunov function). I solved similar exercises with the first method of ...
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3answers
70 views

Dynamic system show that it has only one equilibrium point

I have the following dynamic system $A\cdot\ddot{x}+B\cdot\dot{x}+ C\cdot x=0 $ and A,B,C are positive definite matrices. I have to show that it has only one equilibrium point and that this point is ...
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32 views

Show that $L$ is a strict Lyapunov function w.r.t. $x^e=0$

Consider a system $\dot{x}=F(x)$ with phase flow denoted by $g^t$. Suppose $0$ is an asymptotically stable equilibrium point and for all $t\geq 0$ $$ |g^t(x)|\leq \phi(t)|x| $$ where $\phi\in C([0,\...
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21 views

Consider the stability of a equilibrium point

For $$ \dot{x}=x+f(x,y),\quad \dot{y}=-y+g(x,y) $$ where $f,g\in$ Lip $(\mathbb{R}^2,\mathbb{R})$ and $$ |f(x,y)|\leq c(x^2+y^2),\quad |g(x,y)|\leq c(x^2+y^2) $$ in a neighborhood of the origin, ...
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1answer
36 views

Find a Lyapunov candidate for scalar system

The following scalar system: $$\dot{x} = x^{2} - x^{3} $$ has two equilibrium points, one located at $x = 0$ and the other at $x = 1$. After looking at the phase portrait, setting xdot(2,1) = 0 and ...
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1answer
38 views

Investigate the stability of the equilibrium point (0,0) by constructing a suitable Lyapunov function.

I'm struggling to follow along with any online notes, as I understand there is no general method for constructing a Lyapunov function, any help would be appreciated. Consider the following dynamical ...
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19 views

A couple of questions about Lyapunov functions

Consider the dynamical equation $\dot{x} = f(x)$, where $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+$. Is $V(x) = x^{1+\alpha}$, where $\alpha \in (0,1)$ a Lyapunov function? Since $x(t) \ge 0$ for all $...
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1answer
81 views

Exponential stability of damped wave equation

I like to proof the exponential stability of a damped wave equation with the following form. Let $G \subset \mathbb{R}^n$ be bounded with smooth margin. For $\gamma > 0$ we are given the damped ...
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60 views

Constructing a Lyapunov function for an ODE system that describes epidemic spreading on scale-free networks

I was recently studying an epidemic spreading model, where two competing viruses spread over a scale-free network. $$ \begin{aligned} \frac{dI_{1,k}(t)}{dt} = - I_{1,k}(t) + \psi_1 k (1-I_{1,k} - I_{2,...
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25 views

Cosider the stability of the equilibrium point of the system

Consider the stability of the equilibrium point of the system $$ \dot{x}=\mathrm{e}^{x+y}-\sqrt{1+2x} $$ $$ \dot{y}=\sin(xy)+\ln(1+2x) $$ The equilibrium of the system is $(0,0)$. My problem is to ...
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1answer
64 views

Lyapunov Function for Nonlinear System

Which Lyapunov function should I choose to show the stability (or instability) of equilibrium points? With $k>0$, $K>0$, $\delta >0$. The system is Hurwitz (asymptotically stable) when $k>...
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17 views

attrativity, stability and global stability

I am confusing in understanding the concept of attractivity and stability. What is the difference between attractivity and asymptotically stability? I am searching for an example that is attractive ...
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66 views

How do I prove that the system is asimptotically stable under a control law?

I am consiering the model of a unicycle in polar coordinates: $\dot{r}=-vcos\gamma$ $\dot{\gamma}=v\frac{sin\gamma}{r}-\omega$ $\dot{\delta}=v\frac{sin\gamma}{r}$ where $r$ is the distance from the ...
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29 views

Complementary to Lyapunov Stability Theorem

Lyapunov Stability Theorem states, loosely speaking, that for $x' = Ax + G(x,t)$ where $G(x, t) = o(\|x\|)$, if all eigenvalues of $A$ have negative real parts, then solution $x_0=0$ is asymptotically ...
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1answer
37 views

Prove the system of ODE is asymptotically stable using Lyapunov function.

Given the system of ODE $$x_1'=x_2$$ $$x_2'=-a \sin x_1- b x_2$$ Where $a,b >0$ real numbers. Using the Lyapunov function $$v(x_1,x_2)=\int_0^{x_1} a \sin{s}\,ds+\frac{1}{2} X^TPX$$ where $X=(x_1,...
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1answer
44 views

How to obtain the Lyapunov candidate of this system

I have been trying to find a suitable Lyapunov function for this system $\dot{x_{1}} = \sin(x_{2})$ $\dot{x_{2}} = -x_{1}-x_{2}$ As can be noted it has multiple equilibrium points: Equilibrium Points: ...
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1answer
245 views

Problem designing a control law

I am trying to build a posture regulation control which works with acceleration inputs. Doing it with velocity inputs,I have well understood how it works. Moreover,for now, the controller only ...
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1answer
54 views

Check if $\frac{x'^{2}}{2}-\sin(x)$ is a lyapunov function for $x''=\cos(x)-0.1x'$ near $(x,x')=(\frac{\pi}{2},0)$

Check if $\frac{x'^{2}}{2}-\sin(x)$ is a lyapunov function for $x''=\cos(x)-0.1x'$ near $(x,x')=(\frac{\pi}{2},0)$ I rewrote this into a system: $x'=y$, $y'=\cos(x)-0.1y$, $F(x,y)=\frac{y^2}{2}-\sin(...
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1answer
41 views

Given $dx/dt=\sin(y)+0.1\sin(x),dy/dt=-\sin(x)+0.2\sin(y)$ and $F(x,y)=\cos(x)+\cos(y)$ Determine if $F$ is a lyapunov function near $(0,0)$

Given $dx/dt=\sin(y)+0.1\sin(x),dy/dt=-\sin(x)+0.2\sin(y)$ and $F(x,y)=\cos(x)+\cos(y)$ Determine if $F$ is a lyapunov function near $(0,0)$ I computed the jacobian matrix at $(0,0)$ as $\begin{...
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1answer
63 views

Comparison principle for differential equations

I am trying to solve example 3.8 in the book Nonlinear systems by Hassan Khalil and I have been unable to figure out how they got the answer for $\frac{\mathrm dv(t)}{\mathrm dt}$ as $-2x^2(t)$. I ...
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1answer
80 views

How to check the Lyapunov stability of the following system?

Given the following nonlinear system: $\dot x = {x^3} - 8{x^2} + 17x + 10$ I want to check Lyapunov stability of equilibrium points.
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22 views

Variable Gradient Method Lyapunov Stability

First, I'm sorry if my topic doesn't look familiar to you. But actually this quite make me confuse. I was studying about Variable Gradient Method which is one of methods to generate lyapunov function. ...
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24 views

Reversed version of de Bruijn's identity/Boltzmann H-theorem for Markov processes

Take a Markov process $X_t$ on a Polish state-space $E$. Suppose the Markov process has invariant measure $\mu$ and is prepared with initial distribution $\mu_0\ll \mu$. De Bruijn's identity says that ...
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1answer
61 views

How to pick a Lyapunov Function and make asymptotically stable?

Dynamics I have , $$\dot{x_1} = x_2$$ $$\dot{x_2} = u$$ if I pick Lyapunov function $$V(x) = \frac{1}{2}*x^2_1 + \frac{1}{2}*x^2_2$$ then $$\dot{V(x)} = x_1*\dot{x_2} +x_2*u$$ and $$u = -x_1 -x_2$$ $$ ...
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47 views

Trying to solve the Lyapunov candidate, but having trouble with some of the steps.

I have this state model: $$\dot{x_1}=-x_1+{x_1}^3+2x_2$$ $$\dot{x_2}=-x_2-x_1^3$$ My Lyapunov candidate is $$x^TPx $$ where P is $$\frac{1}{2}\begin{bmatrix}1 & 1\\1 & 3\end{bmatrix} $$ I know ...
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5 views

Countable sequence of Lyapunov-functions integrating each probability measure

Assuming my answer to this post of mine is correct (which I believe it is), for any Borel probability measure $\mu$ on $\mathbb{R}^d$, there exists a non-negative, $C^2$-function $V_{\mu}$ with ...
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1answer
29 views

Existence of regular Lypanuov function for any probability measure

Given a Borel probability measure $\mu$ on $\mathbb{R}^d$, there exists (since any such $\mu$ is tight) a function $f$ with compact sublevel sets such that $\int f d\mu < \infty$. Such a function ...

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