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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Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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The lyapunov function of gradient system

Given a dynamical system $$\frac{dx}{dt}=-\nabla f(x)$$ which $x=0$ is the only equilibrium point, i.e. $-\nabla f(x)|_{x=0}=0$. I am reading this tutorial, and it states: $f(x)$ is a lyapunov ...
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What is a weak Lyapunov function and how can I use one to tell if a system is asymptotically stable?

I have the Lyapunov function $V(x_1,x_2) = x_1^2 + x_2^2$ and the following system. $$ \begin{aligned} \dot{x_1} &= -x_1 + x_2^2\\ \dot{x_2} &= -x_1 x_2 - x_1^2 \end{aligned} $$ In this case, $...
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A question about Comparison Principle in Nonlinear Systems?

A question about Comparison Principle For a general system, we have $$ V=x^{2}+y^{2} $$ where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are two independent states, and $V$ is a Lyapunov function. ...
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2 answers
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construct different ode systems but with the same lyapunov function

I am thinking of whether there are some ode systems that are different with each other, suppose all of them have zero as an equilibrium point. Moreover, they have a common lyapunov function that can ...
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32 views

Asymptotic stability of a unrelated equation

I am studying stability of dynamical systems and Lyapunov theory, and I am trying to solving the following exercise: Provide sufficient conditions for the asymptotic stability of the dynamical systems ...
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1 answer
88 views

Gradient of a (Lyapunov) function

For $x\in\mathbb{R}^n$, define $\hat{r}(x) = \left\{ \begin{array}{ll} \vec{0} ,& x = 0\\ \frac{x}{||x||} ,& x \ne0\\ \end{array} \right. $ and for some $r>0$, $x_0\in\...
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Lyapunov function to prove globally asymptotically stable

I have the system $x'=-x^3+2y^3$ and $y'=-2xy^2$. I need to prove that the point $(0,0)$ is asymptotically globally stable. Here's what I did: if we have a Lyapunov function $v(x,y)=ax^2+bxy+cy^2$, ...
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3 votes
1 answer
79 views

Finding Lyapunov function for particular system

I am trying to find Lyapunov function for $$\begin{cases}\dot{t} = y\\\dot{y} = t^2-t\end{cases}$$ I tried common examples but, maybe I was wrong in my computations, couldn't derive anything. Could ...
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Lyapunov function for a second order system involving trigonometric functions

I am studying the stability of the following system: \begin{aligned} \dot{x}_{1} &= -x_{1}^{2} - \sin x_{2}\\ \dot{x}_{2} &= x_{1} - \frac{\cos x_{2}}{x_{1}}\\ \end{aligned} The system ...
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144 views

Is the solution to $\theta''+0.021\,\text{sgn}(\theta')\sqrt{|\theta'|}+0.02\sin(\theta)=0,\,\theta_0=\pi/2,\,\theta'_0=0$ of finite duration?

Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration? I ...
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How to pick a Lyapunov function and estimate PID gains? [closed]

I am currently trying to estimate the range of PID gains by developing a Lyapunov function for a nonlinear 6-Dof quadrotor system. The system is of the following form: $$M(q)\ddot{q}+C(q,\dot q)\dot q+...
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Any idea or direction on how to go about estimating tight region of attraction for this nonlinear system?

I am working with the following discrete nonlinear system: $\begin{gathered} {\delta _{s + 1}} &= &{\theta _s}{\delta _s} \hfill \\ {\theta _{s + 1}} &=& {\theta _s} + c\left( {1 - ...
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A doubt about Lyaponov exponents

I'm studying the spectral theory and I read the following assertion: " since the spectral measure is purely absolutely continuous, the Lyapunov exponents are positive". I taught about it ...
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Analysis and Stability Non Linear System

I have this system and i want study stability: $$ \left\{ \begin{array}{c} \dot x_1 = x_3 \\ \dot x_2=x_4\\ \dot x_3 =\frac{1}{I}[u-bx_3-k(x_1-x_2)]\\ \dot x_4 =\frac{1}{mL^2}[k(x_1-x_2)-mgL\sin(x_2)]...
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1 answer
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Lyapunov function for system of differential equations

I am searching for a proper Lyapunov function, supporting the following system of differential equations. $$ \begin{array} {} \dot{x_1}=a\;x_1-b\;x_1^3 \\ \dot{x_2}=a\;x_2-b\;x_2^3 \end{array} $$ $a,b\...
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Lyapunov function for an arbitrary equilibrium point

Typically, Lyapunov function assumes $0$ as an equilibrium and require $V(0)=0$. If we wanted to analyze the stability of a nonzero equilibrium point $x_0$, most references asks to do a state ...
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1 answer
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Lyapunov Stability Analysis

so my analysis is quite lacking so I wanted to check with everyone here. I need a sanity check and see if my analysis is grounded. I am looking at stability of 1-D function to the quadratic-type ...
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Stability and Boundedness of Solutions of LTV system

I have the differential equation: $$\dot{x} = \Phi(x,t)[Ax+f(t)] $$ where: $$A \in R^{n\times n}- \ Hurwitz $$ $$ f(t) \in R^n \ and \ \Vert{f(t)\Vert < \infty}; $$ and $$ \Phi(x,t) = diag\{\frac{(...
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2 answers
64 views

Bound the norm of a matrix function related to discrete algebraic Riccati equation

I was going through the following paper on perturbation analysis of the discrete Riccati equation. https://dml.cz/bitstream/handle/10338.dmlcz/124552/Kybernetika_29-1993-1_2.pdf. The perturbation ...
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1 answer
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A function where an infinitesimal difference in intial conditions grows into a finitesimal difference in final conditions, within finite time?

The typical functions I see with finite Lyapunov times are merely exponential; they only generate (e times larger) infinitesimal differences in final conditions from infinitesimal differences in ...
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Study instability of a system with Lyapunov functions in terms of a parameter

I am given the following system of odes \begin{gather} \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} - b(...
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3 votes
1 answer
130 views

Determine the stability of equilibrium point with Lyapunov function

I want to determine the stability of $(0,0)$ (stable, asymptotically stable or unstable) in the nonlinear system: $$ \begin{aligned} \dot{x} &= y + xy \\ \dot{y} &= -y + \sin^2(x)...
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Question about LaSalle's invariance principle

I have a question about LaSalle's invariance principle in the following version, which is the one I have to work with: Let $D⊂R^n$ be open and let $f:D->R^n$ with $f(0) = 0$ be locally Lipschitz-...
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1 answer
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Does Lyapunov function need to be defined at zero?

Consider the function $V(x,y) = x-y-y\ln \left(\frac{x}{y}\right)$, where $x,y>0$ and $y$ is fixed. $V$ is constructed to study the stability of equilibrium point $x=y$ for the system $$ \frac{dx(t)...
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How to show that $\Omega_{c}=\left\{x \in \mathbb R^{n}: V(x) \leq c\right\}$ is compact?

Consider an ODE system $$\dot{x}=f(x),$$ having a candidate Lypunov function, which satisfies $V(x)\geq0$, $V(0)=0$, and $\dot{V}(x)\leq0$. How to show that the set $$\Omega_{c}=\left\{x \in \mathbb R^...
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4 votes
1 answer
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Trapping Region for ODE System.

I am working on the following problem, given the system of two differential equations $x′=2x+y−2x^3−3xy^2,$ $y′=−2x+4y−4y^3−2x^2y,$ So far, I have tackled similar problems by trying to find a trapping ...
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2 votes
2 answers
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Behaviour of the solutions of a system of differential equations at infinity.

Consider the following system of differential equations: $$ \begin{cases} u'=-u+uv\\ v'=-2v-u^2 \end{cases} $$ I'm able to prove that solutions must tend to $0$ if $t\to 0$ by the use of Lyapunov ...
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3 votes
1 answer
141 views

How does this expression follow algebraically from the last one? (continued)

Continuing from here: How does this expression follow algebraically from the last one? The "new" system: \begin{align*} \dot S &= \Lambda - (\beta_1 S I_2 +\beta_2 S J+ \beta_3 S A )-\mu ...
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6 votes
1 answer
195 views

How does this expression follow algebraically from the last one?

I was reading this paper: Global stability for an HIV/AIDS epidemic model with different latent stages and treatment Everything is understood apart from on page 7 of the pdf (page 1486 in the document)...
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1 vote
0 answers
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The definition of Lyapunov order number in Hartman ode book.

In the Hartmann ode book: following are the definition of "Lyapunov order number" consider a function $y(t)$ for $t\geq 0$.(Hartman use $y(t)$ as vector function in $\mathbb{R^d}$) A number $...
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2 votes
1 answer
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local stability of a planar autonomous system

I have been trying to prove that origin $(\theta,k) = 0$ of the non-autonomous system $$ \begin{aligned} \dot{\theta}(t) &= -b \theta(t) + k(t) u \\ \dot{k}(t) &= \epsilon \text{ } h(\theta(t),...
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2 votes
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Extremum Seeking Control vs Lyapunov Stability

Consider a dynamical system $$\dot{x}=f(x)+g(x)u$$ and suppose the goal is to stabilize the state $x$ to the origin using a control input $u$. The Lyapunov approach is to formulate a Control Lyapunov ...
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2 answers
68 views

Derivative of a time-varying Lyapunov function

When using Lyapunov functions to prove stability and the Lyapunov function is dependent on time, eg: $$V(x,t) = x_1\sin(t)$$ and I want to show that the derivative is negative $\dot{V}(x,t) < 0$, ...
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0 answers
27 views

Is there a stability theorem akin to Lyapunov's direct or indirect method for differential algebraic equations (DAE)?

Is there a stability theorem akin to Lyapunov's direct or indirect method for differential algebraic equations (DAE)? Specifically, is there a stability theorem that does not require transforming the ...
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1 vote
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Lyapunov functions : why we don’t just use Newtonian mechanics?

I am studying nonlinear control theory, and I wanted to ask a question about Lyapunov functions please. So there is the Lyapunov theorem for stability of $\dot{x}=f(x)$ using Lyapunov functions, but ...
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3 votes
1 answer
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Understanding Lyapunov functions and notation

I'm trying to wrap my mind around Lyapunov functions, but I'm having an hard time and I need some help. The reference I'm using is Khalil's "Nonlinear systems" book. Let's consider the ...
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Lyapunov dimension

I have a nonlinear differential equation system composed of 4 equations. I calculated Lyapunov's dimension of each of the states to be a little bit over 3 (say 3.11, 3.1, 3.13, 3.14). How can I ...
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1 answer
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Find a Lyapunov function of the form $ax^2+by^2+cz^2$

Find a Lyapunov function of the form $ax^2+by^2+cz^2$ to determine whether the equilibrium points of the differential equation: $\dot{x}=2y(z-1)$ $\dot{y}=-x(z-1)$ $\dot{z}=-z^3$ are stable, ...
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3 votes
1 answer
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Stability analysis of non-linear system outside of equilibrium point

I have a non-linear system that comes from the transformation matrix for angular rotational rate from Euler -> Body frame: $$ \dot{r} = A + sin(r)tan(p)B + cos(r)tan(p)C \\ \dot{p} = cos(r) B - sin(...
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0 answers
28 views

Picking a Lyapunov function that is dependent on time

I have the following system: $$ \begin{cases} \dot{x_{1}}(t) = x_{2} - x_{1}^{3}\\ \dot{x_{2}}(t) = -0.5 x_{1} - x_{2} + d\sin(t) \end{cases} $$ where $d$ is some constant. I want to show $(0,0)$ is ...
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1 answer
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What is the meaning of entire solution in LaSalle's invariance principle and how to apply this principle?

This is from Hirsch/Smale/Devaney's textbook "Differential Equations and Introduction to Chaos" Page 199 Theorem. (Lasalle’s Invariance Principle) Let $X^∗$ be an equilibrium point for $X^\...
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Asymptotically stable fixed point that cannot be certified by norm?

Let $F$ be a map on $\mathbb{R}^n$ and consider the dynamical system induced by $F$, i.e., all the orbits $x^k=F^k(x_0)$ with $x_0 \in \mathbb{R}^n$. Suppose that $x^\star$ is an asymptotically stable ...
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1 vote
1 answer
118 views

Inverse dynamics control: Proof of asymptotic stability of error system

The inverse dynamics control in robotic applications yields the error system \begin{equation} \ddot{\mathbf{e}} + \mathbf{K}_1 \dot{\mathbf{e}} + \mathbf{K}_0 {\mathbf{e}} = \mathbf{0} \end{equation} ...
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1 vote
1 answer
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H2 norm and trace

To show: $tr(CPC^{T}) = tr (B^{T}QB)$ with the help of Lyapunov equations $0 = AP + PA^{T} + BB^{T}$ and $0 = A^{T}Q + QA + C^{T}C$. As shown in the text below; $||H||_2$ relates Lyapunov equation ...
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49 views

How to prove that the system is locally stable using Lyapunov function?

3 equilibrium points are obtained in this system $$ \begin{align} \dot{x}&=y+1.6xy-2.7x\\ \dot{y}&=0.625x-xy+1.2y^2-0.7y \end{align} $$ as in this streamplot graph (0,0) in blue (0.3,0.5) is ...
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1 answer
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Doubt about Lyapunov's theorem proof

Given the autonomous system $\dot x=f(x)$ and an equilibrium point $\bar x$, we know that it is stable if $\exists\phi:U_0\to \mathbb R$, $\phi\in\mathcal C^1(U_0;\mathbb R)$, with $ U_0$ open nbh of $...
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1 vote
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Lyapunov function of a transformed system

I have a system, which has the form $\frac{da_i}{dt}=\nu(\eta_i)(\sigma(\eta_i)-a_i)$, where $\eta=WA+V$ and $A,V,\eta\in\mathbb{R}^N$ are vectors and $W\in\mathbb{R}^{N\times N}$ is some square ...
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4 votes
1 answer
154 views

Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?

For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some ...
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2 votes
1 answer
33 views

LMI-Solution Invariant to the Initial Conditions

It is known from R. Bellman that the value of the functional $J = \int_{0}^{\infty}xWx \ dt, \ W>0$ along the solution of the linear time-invariant system $\dot x = Ax, \ x(0)=x_0$, with Schur ...
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