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Questions tagged [lyapunov-functions]

Lyapunov functions are scalar fields that may be used to prove the stability of an equilibrium point of an ODE.

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Counterexample of Lyapunov' Stability Theorem for equilbrium

I know that thanks to Lyapunov' Stability Theorem one can study the stability of an equilibrium by finding a Lyapunov function associated to that equilibrium. But, is the converse true? Is it true ...
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How can this system not be asymptotically stable?

I am currently studying stability of nonautonomous systems using the book Applied Nonlinear Control by Slotine & Li. On page 125, there is example 4.13: $$ \begin{align} \dot{e} &= -e + \...
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Energy-Casimir method

In the Energy-Casimir method, it is said we look for 'conserved quantities' that are a function of the system. Then subsequently $V = \mathcal{H} + \mathcal{C}$ becomes the candidate Lyapunov function,...
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Proving that a second order system is not exponentially stable

This is a homework exercise that I've been struggling to solve. Any help is appreciated. Consider the system: $\dot{x}_1 = x_2 \\ \dot{x}_2 = -x_1 - g(t)x_2$ Where g(t) is continuously ...
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Using a Lyapunov function to classify stability and sketching a phase portrait

Consider the system $$x' = -x^3-xy^{2k}$$ $$y' = -y^3-x^{2k}y$$ Where $k$ is a given positive integer. a.) Find and classify according to stability the equilibrium solutions. $\it{Hint:...
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Manage to Prove Lyapunov Stability for an Unstable System, Why?

Consider system: $\dot x = ax - {e^{\tanh \left( k \right)}}x$ $\dot k = - \frac{1}{{{{{\mathop{\rm sech}\nolimits} }^2}\left( k \right)}}\frac{1}{{{e^{\tanh \left( k \right)}}}}a{x^2}$ where, $a&...
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Using a Lyapunov function to determine stability of equilibria

Given $$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\ y' &= -3x^2y+y^3\end{aligned}\right.$$ find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever ...
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Lyaponov function in dynamic system (in polar cordonates)

Let the following system in $R^2$ \begin{equation} (S) \left\{ \begin{array}{l c } \overset{.}{\rho}=\rho(1-\rho) \\ \overset{.}{\theta}=\sin^2(\frac{\theta}{2}) \end{array} \right. \end{equation} ...
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Compare solutions in some moment of time for two equations by Lyapunov derivatives?

We have two differential equations $\dot x_1=f_1(x_1)$ and $\dot x_2=f_2(x_2)$ which are too complex to solve but we could show by the same Lyapunov function $V$ (and its derivatives) that equilibrium ...
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How to prove Asymptotic Stability in Lyapunov's Stability theorem

Lyapunov's Stability Theorem: Let $x = 0$ be an equilibrium point for the autonomous system $$\dot{x}(t) = f( x(t) ),$$ and $D \subset \mathbb{R} ^ n$ be a domain containing the equilibrium point, i.e....
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Showing the equilibrium point to be globally exponentially stable using Lyapunov indirect method.

We have the system $\ddot{q} + \dot{q} + g(\dot{q},q) + q = 0, \forall t \geq 0$. $x = \begin{bmatrix} x_{1}\\ x_{2}\end{bmatrix} = \begin{bmatrix} q\\ \dot{q}\end{bmatrix}$ $\dot{x} = Ax + h(x)$ ...
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Synchronization in Coupled Nonlinear Oscillators

I hope with this first question I respect the standards of the forum :) I am currently working on an ANN model for Working Memory (Short Term memory in the Brain) and as a first step I am studying a ...
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Proof of Lyapunov Stability Theorem

I am following the proof presented on page 8 here. I think I follow most of the argument, except for one part. The crux of the proof seems to rest on the fact that when we assume a non-zero $c$, $\dot{...
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Picking a Lyapunov function that is dependent on $\dot{x}$

I have a system $\dot{x}=f(x)$. Is it a good idea if I pick a Lyapunov function V that is dependent on $\dot{x}$. So that $V=V(x,\dot{x})$? One of the conditions for stability is that the origin is ...
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How do I solve this integral? Comes from Lyapunov equation and the derivative being negative definite

I am sure this is a simple solution, but I just cant seem to get this solution In sliding mode, following along with this video, https://www.youtube.com/watch?v=x9WxwM6Ebvo&list=PLv8cjLiRoYbivwv0-...
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Linear algebra based proof that if there exists $P\succ 0$ and $P\succ A^TPA$ then $|\lambda_i(A)|<1$

Is there a proof based on linear algebra that shows the following? If there exist $P \succ 0$ and $P \succ A^TPA$, then $| \lambda_i (A) | < 1$ for all $i$. Here, $|\lambda_i(A)|$ denotes the ...
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Lyapunov Function for Competition Equations

I am trying to show that the function $$Q(x,y)=be(x-\bar{x})^2+2ce(x-\bar{x})(y-\bar{y}) +cf(y-\bar{y})^2$$ is a Lyapunov function for the competition equations: $$\dot{x}=x(a-bx-cy)$$ $$\dot{y}=y(d-...
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Inequality manipulation for a vector norm

How can I manipulate the following inequality to reach from $$\dot{V}\leq -4x_1^2 +4x_1x_2 -2x_2^2 $$ to $$\dot{V}\leq -(3-\sqrt{5}) \|x\|^2 $$ where $x=[x_1 \;x_2]^T$ is a 2D vector and $\|x\|$ is ...
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How is Grönwall's inequality applied here?

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ Now, let $(X_t)_{t\ge0}$ be the unique strong ...
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If the solutions to Lorentz equation diverge exponentially, how can they be confined in a strange attractor? [duplicate]

In the book of Chaosbook, at the beginning of chapter 6, it is given that [...]if the attractor is strange, any two trajectories $x(t) = f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ ...
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Explain Lyapunov Stability

In my lecture notes, we have $$\dfrac{d\vec{x}}{dt}=f(\vec{x}),$$where $f$ is polynomial and $\vec{x}=0\in\mathbb{R}^n$ is an equilibrium. In determining whether all trajectories $\vec{x}(t)\...
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Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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Confusing matrix manipulation in Lyapunov stability proof?

I was looking at the proof of the stability of a problem via the Lyapunov function in a paper. I don't think it is necessary to write the whole problem fully so I will focus just on the particular ...
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Find one domain of attraction for this system

Assume the system: \begin{align} \begin{pmatrix} x \\ y \\ \end{pmatrix}' &= \begin{pmatrix} -(1-y)x \\ -(1-x)y \\ \end{pmatrix}...
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How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis? \begin{equation} \left.\begin{aligned} \dot{x_1} &= x_2 \\ \dot{x_2} &= -\frac{x_1}{1 + x_2^2} ...
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How to pick a Lyapunov function and prove stability?

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for autonomous systems. Say we are given the nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_1(t)x_2(t)...
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radially unbounded functions

Is the following function radially unbounded or not? $$V(x) = \frac{x_{1}^2}{1 + x_{1}^2} + x_{2}^2$$ I know that if $x_{2} \to \infty$ in which case $||x|| \to \infty$ and $V(x) \to \infty$ but if $...
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How to solve an ODE of this form $\dot{P}(t)=A^TP(t)+P(t)A$?

Background If $\dot{x}(t)=A \,x(t)$, then we know the solution is $x(t)=e^{At}x(0)$. Question Now let $\dot{P}(t)=A^TP(t)+P(t)A$, what is $P(t)$? Attempt If we take $P(t)$ as common factor (I'm ...
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A choice of Lyapunov function for this 2D system?

I am thinking of a choice of a suitable Lyapunov function$V(x_{1},x_{2})$ which can make the system stable around the fixed point $x_{1}=1,x_{2}=1$ $\dot{x_{1}} = x_{1}x_{2} - x_{1}^2 $ $\dot{x_{2}}...
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What's an example where Lyapunov fails to find the bounds of stability

In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...
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convex combination of two stable matrices

Let $A_1,A_2$ be real matrices with negative eigenvalues. The the Lyapunov operators $$L_1(P) = A_1^T\cdot P + P\cdot A_1$$ and $$ L_2(P) = A_2^T\cdot P + P\cdot A_2 $$ are nonsingular. Let $$ P_1 = ...
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Are Eigenvalues Invariant?

Question: are the eigenvalues of a dynamical system invariant under a change of variables? More specifically, consider a dynamical system A defined on a manifold $M_A$ by the evolution function $\...
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For real square matrices $A, B$ of compatible dimensions, does the following inequality hold?

For real square matrices $A$ and $B$ of same dimension, does this hold? $x^{T}A^{T}B^{T}BAx\leq\|A\|^2x^{T}B^{T}Bx$ for a non-zero vector $x$? I know that $x^{T}A^{T}B^{T}BAx\leq\|A\|^2\|B\|^2\|x\|^...
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Rate of convergence: Adaptive system

Consider the following dynamical system \begin{align} \dot{x}_1&=-ax_1 + w^T(t)x_2,\quad x_1\in\mathbb{R}^1 \\ \dot{x}_2 &= -w(t)x_1, \quad x_2\in\mathbb{R}^n \end{align} where $a>0$ and $w(...
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Stability of dynamical system via Lyapunov

I have a dynamical system which has the following form: $\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$. My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...
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76 views

Positive definite Hessian implies Lyapunov stability

The Lagrange-Dirichlet Theorem is partially reversed by the following result: if the costraints are holonomous, bilateral, ideal, if $q^*$ is a critical point for the potential energy $U$, then the ...
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68 views

Lyapunov exponents without Jacobian?

I am trying to determine the evolution of a non-linear numerical model through time given certain perturbations in the initial conditions (I have the strong suspicion the system is dissipative, but ...
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Positive definiteness of a multivariate polynomial function

Consider $x \in \Bbb{R}^n$ and suppose $f_m: \Bbb{R}^n \to \Bbb{R}$ which is defined as: $$f_m(x)=x^TAx+\alpha_{1}x_1^3+\alpha_{2}x_1^2x_2+\dots+\alpha_{r_1}x_n^3+\alpha_{r_1+1}x_1^4+\alpha_{r_1+2}...
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On the choice of general Lyapunov functions for discontinuous control

Let us consider the following dynamical system: $$ \dot{X} = A\cdot X + B\cdot u$$ where $X,B \in \mathbb{R}^{n\times 1}$. The considered system is linear, but I think that $A\cdot X$ can be ...
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Strange text about a strange attractor

I'm reading a text about the Lorenz equations, for $r>1$ and all the other parameters positive. At one point the author says My questions are: 1) Why is $L$ a Liapunov function? There are ...
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An alternative Lyapunov-like instability proof

Let $\gamma$ and $\omega$ be two positive real numbers. Consider the following 1-dimensional linear time-varying system $$ \dot{x}(t)=\underbrace{\left(-\frac{1}{2}(1+\gamma)-\cos(\omega t) + \frac{1}{...
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If a system is defined in $n$-dimension then it has $n$ Lyapunov exponents

I wanted to know if there is a formal proof to show that if a system is defined in $n$-dimension then it has $n$ Lyapunov exponents. Any links regarding information about this and Kaplan-Yorke ...
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Show that the solution $u \equiv 0$ of $u''+g(u)=0$ where $g(0)=0,g'(0)>0$ is stable but not asymptotically stable

The original question states Let $g$ be a continuously differentiable function such that $g(0)=0$. Show that the solution $u \equiv 0$ of $u''+g(u)=0$ is not stable when $g'(0)<0$, and stable ...
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Asymptotic stability without Lyapunov

Consider the following system of equations $$\begin{cases} x'=-x^3y^2 \\ y'=-2x^2y^3 \\ \end{cases} $$ Using the Lyapunov function $V(x,y)=x^2+y^2$ we get $$\dot{V}(x,y)=-2x^4y^2-4x^2y^4\leq 0$$ So,...
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Determining stability of equilibrium points for a non linear system

Given the system: $$ \left\{ \begin{array}{} \dot x=-x^3y^2 \\ \dot y = -2x^2y^3 \end{array} \right. $$ I need to find the equilibrium points and to determine whether the system is stable around ...
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79 views

Finding Lyapunov function for a stable equilibruim of a non linear system

Given the following system: $$ \left\{ \begin{array}{c} \dot x=y-x^2-x \\ \dot y=3x-x^2-y \end{array} \right. $$ I need to find the equilibrium points, and if stable, to find a Lyapunov function....
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81 views

Lyapunov function instead of linearization

Consider the following system of equations $$\begin{cases} \dot x=y-x^2-x \\ \dot y=3x-x^2-y \\ \end{cases} $$ Then, the equilibriums are $(0,0)$ and $(1,2)$. Using linearization around $(1,2)$ ...
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1answer
60 views

Checking Lyapunov stability of non linear system

I need to check the stability of the equilibrium point of the following system, $n \in \Bbb N$: $$ \left\{ \begin{array} \dot \dot x_1=x_2 \\ \dot x_2=-x_1^n \end{array} \right. $$ I tried using ...
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240 views

Prove that doesn't exist a Lyapunov function

Given the following ODE in polar coordinates \begin{array}{lcl} \frac{dr}{dt} = r\sin(\frac{1}{r}) \\ \frac{d\theta}{dt} = 1\end{array} 1) Show that the origin $(0,0)$ is Lyapunov stable ...
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Linearization method or Lyapunov function - example

I have to find a proper Lyapunov function or use the linearization method, for the following system: $\left\{ \begin{array}{ll} x'=-x-y+xy \\ y'=x-y+x^2+y^2 \\ \end{array} \right.$ So first of all I ...