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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27 views

Equivalence of Lyapunov equation for continuous and discrete case

I am currently studying the original discrete/continous equivalence proof of the Lyapunov equation by Rice 1967. Continuous case: $A^\star L + L A = -C$ for $C \succcurlyeq 0$ Discrete case: $A^\...
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27 views

Relationship between Lyapunov functions and gradient Systems

given a nonlinear system $f(z) = \dot{z}$ that induces gradient dynamics so that $\nabla V(z) = -f(z)$ where V(z) is the potential function of the system. Is the potential function of a gradient ...
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83 views

What is a suitable Lyapunov function for this system?

I have verified using the eigenvalue method that around $(0,0)$ the system \begin{align}\dot x&=y - 3x - x^3 \\ \dot y &= 6x - 2y \end{align} is stable. However, I have been trying to find ...
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How to derive the Lyapunov Equation?

I have a linear dynamical system: $\dfrac{dx}{dt} = Ax$ with $A \in \mathbb{R}^{n\times n}$ and $x \in \mathbb{R}^{n\times1}$. I consider a quadratic function $V(x) = x^T P x$ where $P \in \mathbb{R}...
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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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105 views

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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33 views

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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21 views

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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82 views

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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63 views

Lyapunov exponents: Why do we know that the changes happen at an exponential rate

Let $E$ be a $\mathbb R$-Banach space, $\Omega\subseteq E$ be open, $f:\Omega\to\Omega$ be continuously Fréchet differentiable, $x_0\in\Omega$ and $\varepsilon>0$ with $B_\varepsilon(x_0)\subseteq\...
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91 views

Find the Lyapunov function

We have the dynamical system $$ \dot{x}=-x-2 y^{6}, \quad \dot{y}=3 x^{3} y-2 y^{9} $$ Can we find an Lyapunov function to prove that $(0,0)$ is an asymptotically stable equilibrium? (I'm really new ...
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Showing if all eigenvalues of $A$ have negative real parts then our system has a strong Lyapunov function of the form $x^TSx$.

Can I please have help solving the problem? I am having a tough time working out the details for $S$ and how to do it without assuming diagonability. Thank you! Show that if all eigenvalues of $A$ ...
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Qualitative behavior of a seemingly simple ODE

In case you're curious about context---there isn't one. I am thinking about this out-of-the-blue because it is mysterious and interesting. Let $ a, p, q, y_0 $ be positive constants.  Consider the ...
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Is the function $V=x^{2}$ a positive semi-definite function?

I recently started studying The Mathod of Lyapunov, and I thought I understood how to distinguish a positive definite from a positive semi-definite functions, or negative from semi-definite functions. ...
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24 views

To find class K infinity bounds on given radially unbounded function

Let $f = x_1^2 + x_2^4$. How to find class $\mathcal{K_{\infty}}$ functions $\alpha_1(||x||)$ and $\alpha_2(||x||)$, such that, $\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$ $\...
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84 views

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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13 views

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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29 views

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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The long term asymptotic solution of the SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -\beta is+\alpha r \\ i' &= \beta i s - \gamma i\\ r' &= \gamma i-\alpha r \end{...
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Please help me prove this about Lyapunov Exponents!

Here, $\lambda(y_0)$ denotes the Lyapunov exponent of the logistic orbit starting at $y_0$. I tried starting like this: We have $F(x_k)=ax_k(1-x_k)$. Hence $|F'(x_k)|=|a-2a x_k|$, but I have no idea ...
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Lyapunov Function for non-autonomous systems?

I wanted to find somewhere (a book or a set of online notes would be ideal) that had the exact theorem about lyapunov functions and how they determine stability but in the case of non-autonomous ...
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Lyapunov Exponent of the Logistic Map

So I was working on this problem: Consider the logistic map $F(x)=ax(1-x)$ on the interval $[0,1]$. Show that for $3<a<1+\sqrt6$, $\lambda(y_0)=\frac12\ln|a^2-2a-4|$ for all $y_0\in(0,1)\...
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24 views

Somewhat easy chaos theory question/summation

Show that for $1 <\mu< 3$ and $\mu\neq2$, the Lyapunov exponent of the logistic map $F_\mu(x)=\mu x(1-x)$ is given by $\lambda(x)=\ln|2-\mu|$. My work: We have $F_\mu(x(k))=\mu x(k)(1-x(k))$. ...
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28 views

What is the definition of positive semi-definite function in $R^3$? [duplicate]

The definition of positive definite function is something like: for all nonzero $x\in R^n$ then the quadratic function $V(x)=x^TPx>0$ where $P \in R^n$x$R^n$ is a positive-definite matrix. for ...
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Proving $S = \int_0^\infty e^{\tau A^T} e^{\tau A} d\tau,$ satisfies $A^TS + SA = -I.$

I am trying to solve the following and was wondering what I am missing, thanks! Show if all eigenvalues of $A$ have negative real parts then the system has a strong Lyapunov function of the form $x^...
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16 views

Show that all solutions to the Lorenz equations enter the region H less than or equal to a and never leave it

I need help with 8g. It is the last part involving the last maximum value that I am having trouble. I started it but now I am stuck. Below are what I have done so far. I am kind of thinking that last ...
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50 views

Lyapunov's Direct Method

The following non linear system is given: \begin{align} \dot{x}_1(t)&=x_2(t)\\ \dot{x}_2(t)&=-\frac{g}{l}\sin(x_1)-\frac{k}{m}x_2 \end{align} Is asked to apply the Lyapunov's direct method in ...
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150 views

Lyapunov Function Stability Function

Hi I have this question Consider the system $$\begin{align} \dot x_1 &= -x_1 + x_2 \\ \dot x_2 &= (x_1+x_2)\sin x_1 -3 x_2 \end{align} $$ Use candidate Lyapunov function $V(x)=\frac{1}...
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1answer
56 views

Nonlinear System Stability using Lyapunov

I am required to find the stability of this nonlinear system and for which values of $k$ is the system stable. $\dot x=x.(x^2-1-k)$ I am trying to use quadratic Lyapunov function, and used the ...
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44 views

Finding Lyapunov function to show origin is stable

Consider the system \begin{cases} \dot{x}=-2y+yz \\ \dot{y}=x-xz \\ \dot{z}=xy \end{cases} I need to find a Lyapunov function of the form $V(x,y,z)=ax^2+bx^2+cz^2$ to show that the origin $(0,0,0)$ is ...
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Lyapunov exponent relationship with chaotic behavior

Many papers used positive Lyapunov exponent as an indicator that a map has chaotic behavior and having sensitive dependence on initial conditions. See for example: Hua, Z., Zhou, Y., Pun, C. M., &...
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497 views

Asymptotic probability distribution of product of Matrices

Suppose that $$L=\prod \limits_{i=1}^n M_i,$$ where $M_i$ are $2\times2$ matrices of the form $$M_i=\begin{bmatrix} \alpha_1\exp(j\phi_{1,i}) & \alpha_2\exp(j\phi_{2,i}) \\ \alpha_2\exp(j\phi_{1,i}...
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36 views

Coupled Differential Lyapunov Equation

I have the following system of coupled Lyapunov equations over the time horizont $0 \leq t \leq T$: $$ \begin{align} \dot{Q} &= AQ + QA^\intercal - BB^\intercal\\ \dot{P} &= AP + PA^\intercal ...
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97 views

How to find the Lyapunov function for the following system?

How to find the Lyapunov function for the following ODE system \begin{align} \dot{x} &= ax - x^3 + y\\ \dot{y} &= x^3 \end{align} where $a$ is just a constant.
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Ljapunov function - Convergence rate

I study Ljapunov convergence and some doubt popped out from it. Suppose there is a candidate to Ljapunov function for the dynamical system $\dot{x} = f(x)$, for $x_e$ an equilibrium point, of the type ...
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1answer
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Does Lyapunov stability imply that the Lyapunov function is monotonically decreasing?

Suppose we have a system $$\dot x = Ax$$ And a Lyapunov function $$V (x(t)) \geq 0, \forall t $$ My question is, does the fact $\dot V(x) = \nabla V(x)^\top \dot x < 0$ imply that $V(x(t))$ is ...
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Lyapunov stability check

It is stated that for a system $\dot{x}=f(x)$ asymptotic stability is true if there is a Lyapunov function $V(x)$ such that $V(x)$ is positive definite and its time derivative is negative definite. My ...
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59 views

Find Lyapunov function of a autonomous system

Let $x_1(x), x_2(x), x_3(x)$ satisfy the system $$\begin{cases} x_1' = -2x_1+x_2x_3 \\ x_2' = x_1 - x_1x_3 \\ x_3' = x_1x_2 \end{cases} $$ Construct a Lyapunov function to show that the origin is ...
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Q: Lyapunov Exponents for Discontinous Systems

I'm attempting to characterize the chaotic nature of orbits for a dynamical system I'm studying. I've implemented some code which calculates the entire spectrum of Lyapunov exponents and works well ...
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26 views

Why this system is unstable in the sense of Lyapunov?

Given the following Lyapunov function $$ V(x_1,x_2) = (x^2_1+x^2_2-1)^2 $$ Its time derivative is $$ \dot{V}(x_1,x_2) = -(x^2_1+x^2_2) $$ Why the origin is unstable?
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Is it necessary to have Lyapunov function to equal to zero at the origin for the stability?

Majority of nonlinear control textbooks state the following theorem: It explicitly states that the Lyapunov function $V(\cdot)$ must be zero at the origin and positive elsewhere, yet I came across ...
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76 views

Finding a strict Lyapunov function for a system of ODE's

Regarding this question, of the following system : $$\begin{cases} \dot x=y-x^2-x \\\ \dot y=3x-x^2-y \\ \end{cases}$$ how can I prove that the equilibrium point $(1,2)$ is asymptotically stable ...
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A Question on Lyapunov Functions, Limits, and Being Negative Definite.

Let's say I have a Lyapunov candidate of the following form: $$V(x,y)=\frac{a_1}{2}x(t)^2+\frac{a_2}{2}y(t)^2$$ Where both $a_1$ and $a_2$ are known, positive, non-zero constants. Let's now assume ...
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Investigate the stability of the zero solution of the system using the method of Lyapunov functions

I know there isnt appropriate method of finding Lyapunov function, but there is chance someone may find it. I have tried most common polynomials, but not result at all. System is $$ \dot{x}=x-y \\ \...
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18 views

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 ...
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51 views

Stability of critical points of a system given by a constant matrix

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 ...
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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....