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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. ...
Sozialhilfe's user avatar
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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 ...
ebenezer's user avatar
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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 ...
Trb2's user avatar
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Lyapunov stability of a periodic system

Consider a planar system $$ \begin{cases} \dfrac{\mathrm{d}x}{\mathrm{d}t}=-y,\\ \dfrac{\mathrm{d}y}{\mathrm{d}t}=(a+\varepsilon\cos t)x, \end{cases} $$ where $a>0$ and $a\notin\{\dfrac{n^2}{4}\ |\ ...
MakaBaka's user avatar
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Lyapunov stability in a one-sided neighborhood?

Consider a switching system $$ \dot x = { - x, \quad {\rm{if}}\quad x \ge 0} $$ $$ \dot x = {v \left( t \right), {\rm{ if}}\quad x < 0} $$ where $ v(t) $ is bounded but indefinite (can be ...
Tag's user avatar
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Please help with literature for Lyapunov stability for non-linear observer

I'm working with an approximation-based observer design for reaction-diffusion PDE, where I apply the Petrov-Galerkin approximation to a non-linear PDE and get the following ODE: $$\dot{\beta} = A\...
Áron Fehér's user avatar
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particle in motion under the influence of friction

Let's consider a particle of mass = 1 Kg moving according to the law $$ \ddot x(t) = -V'(x(t))-\frac{2}{3}\dot x(t) = -x(t)^3+x(t)-\frac{2}{3} \dot x(t). $$ (The potential energy is $V(x)=\frac{x^4}{4}...
dattiluca's user avatar
2 votes
0 answers
58 views

Relationship discrete algebraic Riccati and discrete Lyapunov equation

Suppose that $K$ is the optimal control for an LQR problem with inputs $(A,B,Q,R)$, i.e.: $$ K = -(B^\top P B + R)^{-1} B^\top P A$$ where $P$ solves the discrete algebraic Riccati equation: $$P = A^\...
sdevlin's user avatar
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how to show global asymptotic stability with $V(x)=f(x)^{T}Pf(x)$ as a lyapunov function.

consider the system $f(x)=\dot{x}$ with $f(0)=0$, $f(x)$ is continuously differentiable. $f(x)$ can be written as $f(x)=\int_{0}^{1}\frac{\partial f}{\partial x}(x\sigma)x\partial\sigma$ (The first ...
TiredMechanicalEng's user avatar
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Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]

I am referring to the ISS definition by Sontag of ${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$ I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
LCG's user avatar
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1 answer
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Divergence of a dynamic system after element sign change

Suppose there is a continuous dynamic system of order $n+1$ given by $$\begin{align} \dot{x}_1 &= Ax_1 + F(t)x_1 + G_1(t)x_2 \\ \dot{x}_2 &= kx_2 + G_2(t)x_1 \end{align}$$ where $x_1\in\...
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Lyapunov Asymptotic Stability

I am still new to Lyapunov stability and I have a question: The system is: $\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$ I used $V(x)=\frac 1 2(x_1^2+x_2^2)$ Then, I get $\dot{V}(x)= -...
Johny's user avatar
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Stability of discrete-time dynamical systems using Lyapunov stability where A is function of optimization variable

Hi I am trying to solve a constrained optimization problem using the Lyapunov stability. In the problem we aim to find $\beta$ such that $$\min_\beta ||\beta^TF-y|| \quad \text{s.t.}\quad A^{T}PA-P&...
geo200's user avatar
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Chetaev theorem for discrete time

In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations Theorem 1 seems to prove a discrete-time analog of Chetaev ...
xyz's user avatar
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2 votes
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Convergence with increasing Lyapunov function

Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures ...
xyz's user avatar
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What does a specific function represent in the 2D space?

Consider the following Lyapunov candidate function: $$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \tag1$$ where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - ...
Teo Protoulis's user avatar
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1 answer
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How can I prove this statement regarding a discrete-time algebraic Riccati inequality?

Consider the square matrix $X \in \mathbb R^{n \times n}$. In a paper that I'm currently reading, the authors state that, if the following two conditions are met: $$ \begin{align} &(1) \quad AXA^T ...
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1 answer
110 views

Non vanishing gradient condition in control barrier funcions.

I am reading about barrier functions in control engineering/dynamical systems. These tools are used to prove that the system is forward invariant with respect to a set $\mathcal{C}$ (i.e., starting in ...
Olayo's user avatar
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1 answer
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Undamped oscillations with Lyapunov stability

Let $V: \mathbb R^n \rightarrow \mathbb R$ be a Lyapunov function for system $\dot{x}=f(x)$. Let $\dot V \leq 0$ hold. Do there exist conditions on th system dynamics such that we can ensure that the ...
Trb2's user avatar
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1 answer
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Lyapunov Stability class K functions

I'm reading the book on Nonlinear System Analysis by M. Vidyasagar. I see they define functions of class K as continuous strictly increasing functions such that $\phi(0)=0$ and from there, they define ...
user1880062's user avatar
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Question on Lasalle's Invariance Principle

Consider system $\dot x = f(x)$ and let $\Omega\subset D$ be a positive invariant set. Let $V: D \rightarrow \mathbb R$ be a radially unbounded, positve definite function. Let the derivative fulfill $$...
Trb2's user avatar
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4 votes
1 answer
146 views

Exponential Stability and Lasalle's Invariance Theorem

It is well known that a system $\dot{x}=f(x)$ with $x \in \mathbb{R}^n$ is exponentially stable if there exists a Lyapunov function $V(x)$ which satisfies \begin{align} k_1\Vert x \Vert \leq V(x) &...
Trb2's user avatar
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0 answers
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LMI robustness to small perturbations

Let $P\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, i.e., $P=P^\top$ and $\langle Px,x\rangle\geq0$ for all $x\in\mathbb{R}^n$. Assume that, for a certain $A\in\mathbb{R}^{n\times n}$ ...
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Khalil's proof of local Lyapunov stability incomplete?

This is regarding the proof of a central theorem (Theorem 4.1) in Hassan Khalil's seminal Nonlinear systems book: Theorem 4.1: Let $x = 0$ be an equilibrium point for $\dot{x} = f(x)$ where $f:D\...
user1814274's user avatar
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2 answers
70 views

Question about Cetaev Theorem

Cetaev Theorem states that: "Considering the ode $X'=F(X)$, with an equilibrium $x_{0}$. If there is a function $V$: $U_{0} \rightarrow $ IR and a region $\omega$ in $U_{0}$ that contains $x_{0}$,...
Albi's user avatar
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1 vote
1 answer
63 views

Construction of Lyapunov function given system of differential equations

Consider the system following system of differential equations \begin{equation} \dot{x} = y - x^{3} \\ \dot{y} = -2x -y^{3}(1-y^{2}) \end{equation} Construct a (strict) Lyapunov function of the form $...
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How do I solve this problem with Lyapunov functions?

Hi this is the system i need to work on: $$ \left\{\begin{array}{rcl} \dot x&=&x(1-x)-axy\\ \dot y&=&y(1-y)-bxy \end{array} \right. $$ I found four stationary points: $(0,0)$;$(1,0)$;$(...
Marzio's user avatar
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0 votes
1 answer
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Find a Lyapunov function for this nonlinear system

This was a problem that I encountered about a year ago: Use the Lyapunov function method to determine the stability of the equilibrium of the origin of this system: $$(x_1)' = -x_1 + x_2 - x_{1}^{3}$$ ...
random's user avatar
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1 vote
1 answer
86 views

Phase Diagram - $\Bbb R^2$ Dynamical System

Given the following dynamical system $$\tag{1}\begin{cases}\label{eq_1} \dot x = x^2 +3y^2-2xy-1 \\ \dot y = 3x^2+y^2-2xy-1 \end{cases}$$ Find a constant of motion and sketch a phase diagram. I know ...
Turquoise Tilt's user avatar
2 votes
1 answer
71 views

Lyapunov methods to show the trajectories that do not converge to the origin [closed]

Consider the map $f(z) = z^2$, where $z$ represents complex numbers. what is the function that $f$ corresponds to the map $p(r, \theta) = (r^2, \theta)$ in polar coordinates. And is it true that all ...
lulu's user avatar
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2 votes
1 answer
134 views

Positive/Negative Definite Functions Confusion

I have the Lyapunov function $V(x_1,x_2) = x_1^2+x_2^2$ for the nonlinear system: $$\dot{x_1} = x_2$$ $$\dot{x_2} = -x_1 - x_2-x_2^3$$ Now obviously $V$ is a positive definite function but in order to ...
Ahsan Yousaf's user avatar
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0 answers
94 views

What should I prove to show the states lie within a compact set?

I'm trying to prove the local stability of a nonlinear system and got the following inequality. $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots $(i) where $c_1, c_2, c_3$ are ...
SpaceTAKA's user avatar
  • 165
0 votes
1 answer
90 views

Lyapunov function candidate and Global asymptotic stability

The following second order system is given: $\dot{x}_1 = -2x_1^3 + 2x_2^2x_1$ $\dot{x}_2 = x_1^2x_2 - 2x_2^3$ I want to determine whether the origin is globally asymptotically stable or not. For this, ...
lord voldemort's user avatar
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0 answers
16 views

Negative semi-definiteness stability

I am working on the stability of a system for the estimation $z_i$ predicted from x using the basic observer design. The relation is as follows: \begin{equation} z_i = \frac{\eta_i }{k_i}(\dot{x} - \...
KASSIM S. O.'s user avatar
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0 answers
47 views

Parameter Estimation of Dynamical System when Model is known

Im working on a nonlinear control based on Lyapunov theory and its working really well. I am able to implement it on a dynamical model of the system in simulink. However I think it has a really big ...
SS1's user avatar
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0 votes
2 answers
57 views

Let $P\geq0$, $P\in\mathbb{R}^{n\times n}$. For every $A\in\mathbb{R}^{n\times n}$ there exists $c\geq0$: $A^\top P+PA\leq cP$.

I want to prove/disprove the following claim. Claim. Let $P\geq0$, $P\in\mathbb{R}^{n\times n}$. For every $A\in\mathbb{R}^{n\times n}$ there exists $c\geq0$: $$A^\top P+PA\leq cP.$$ Draft of the ...
ofir_13's user avatar
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0 votes
0 answers
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Necessary and sufficient condition for Lyapunov stability

Suppose $J $ is a real square matrix and matrix $V = V^\textsf{T}$ is positive-definite. Define $ A = V J $. Can we show that $J ^\textsf{T} + J$ is negative-definite if, and only if, $A^\textsf{T} P +...
PerturbedBiologist's user avatar
0 votes
1 answer
79 views

Does this inequality guarantee the global stability in this paper?

I'm reading a very informative paper. But I met some formulations hard to understand. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^...
SpaceTAKA's user avatar
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0 answers
55 views

How is it possible to have discrete-mixed continuous Lyapunov function in this paper?

I'm reading a very informative paper. But I met some formulations hard to accept. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^2$ ...
SpaceTAKA's user avatar
  • 165
2 votes
1 answer
128 views

Lyapunov Exponents for $n$-Dimensional Matrix $A$

I am wondering whether my solve is correct. I know how to solve the 2, or 3 dimension of the state matrix. But what if the state matrix goes to n-dim? Here is what I tried: To find the Lyapunov ...
lulu's user avatar
  • 97
1 vote
1 answer
42 views

Robotic Control and Parameter Uncertainty: The Significance of Incremental Stability Analysis

Question: In the field of control theory and robotics, incremental stability is a concept that extends our understanding of system behavior. Consider a practical example involving a robotic ...
YAKINDA's user avatar
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2 votes
2 answers
84 views

Stability Preservation in Linear Dynamical Systems Under Nonlinear Perturbations: A Comprehensive Analysis and Conditions for Robust Stability

Barrier of the Process: The statement we are investigating concerns the stability of a point under a perturbation. Specifically, we want to determine whether a point that is Lyapunov stable for the ...
YAKINDA's user avatar
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1 vote
1 answer
62 views

Exploring Sensitivity Dependence in Chaotic Systems

After analyzing the sensitivity dependence of orbits under the map $f(x) = 2x \mod 1$, we have found that every initial point has nearby points that eventually diverge by at least $1/2$ unit after ...
simple1's user avatar
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3 votes
1 answer
155 views

What if the level set of Lyapunov function is disconnected? - when estimating region of attration

Consider $\frac{dx}{dt}=f(x)$, where $x\in\mathbb{R}^n$. Suppose $x=0$ is a stable equilibrium. It is classical way to estimate region of attraction of $0$ by finding a $C^1$ function $V(x)$ such that ...
happyle's user avatar
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1 vote
0 answers
64 views

Lasalle's invariance principle for global stability of synchronization state of Kuramoto model

question My question regarding the argument of the proof of theorem 3.1 in this paper. In the proof the Lasalle's invariance principle is used. From what I learned, radially unboundedness must hold ...
happyle's user avatar
  • 183
-2 votes
1 answer
70 views

Are there any concrete application of the Lyapunov theorem for LTI systems?

Consider a LTI system $\dot x = Ax$. This system is globally asymptotical stable iff given any $Q \succ 0$, there exists a unique $P \succ 0$ such that $A^{T}P+PA+Q=0$ holds. https://en.wikipedia.org/...
Fraïssé's user avatar
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1 vote
1 answer
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On exponential stability of fixed points

I am a bit lost in the concepts of stability theory. Consider the (non-linear) ODE $x' = \varphi(x)$ in some Banach space with a unique stationary point $x_*.$ Then we could say that the fixed point ...
zoli's user avatar
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0 answers
57 views

Candidate for a Lyapunov function

I am working on this epidemic model of this SIR type: Any suggestion of a Lyapunov function candidate that can be used to prove the global stability of the disease-free equilibrium ($\frac{\Pi}{k_1+\...
Zizo's user avatar
  • 1,871
2 votes
2 answers
492 views

Stability of discrete-time dynamical systems using Lyapunov stability

I am studying the use of LMIs as an analysis tool for discrete-time dynamical systems. Consider the autonomous discrete-time system given by $$ x_{_{k+1}} = A x_{_k} \tag{1} \label{sys} $$ where $ x \...
AdamsK's user avatar
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1 vote
0 answers
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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 ...
SpaceTAKA's user avatar
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