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.
418
questions
-1
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0
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39
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Lyapunov function for adimensional Lotka-Volterra model [duplicate]
Consider the following adimensional Lotka-Volterra model:
$$
\begin{cases}
\dfrac{dh}{d\tau} = \gamma h(1-p)\\
\dfrac{dp}{d\tau} = -\dfrac{1}{\gamma}p(1-h)
\end{cases}
$$
I'm asked to consider a ...
- 111
2
votes
1
answer
77
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 ...
- 139
1
vote
0
answers
34
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 ...
- 139
-1
votes
1
answer
55
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/...
- 11.1k
1
vote
1
answer
45
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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 ...
- 732
0
votes
0
answers
36
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+\...
- 1,791
2
votes
2
answers
94
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 \...
- 147
1
vote
0
answers
61
views
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 ...
- 153
-1
votes
1
answer
173
views
Find a Lyapunov function and prove that an almost linear closed-loop system is stable
I would like to find a Lyapunov function and prove the following closed-loop system is stable:
$\dot{x}=(A-BK)x+(z(u)+\epsilon)$,
where a function of control input $z(u)$ satisfies $\|z\| < \rho \|...
- 153
1
vote
0
answers
39
views
The Lyapunov inequality for a given matrix $P$
The famous Lyapunov theory says if a system matrix $A$ is stable, then the Lyapunov inequality $$A^TP+PA<0, \qquad P>0$$ is unique which depends on the negative definite matrix $-Q$, which I ...
- 11
0
votes
0
answers
29
views
Globally exponentially stable point
consider this linear, non-autonomic system:
$x_1 ̇=-x_1-f(t)(x_2-x_3 )$,
$\ x_2 ̇= -x_2+x_1$,
$x_3 ̇=-x_3-x_1$
where $f(t)$ is continuously differentiable and satisfies
$0≤f'(t)≤f(t)≤k$ for all $0≤t ...
0
votes
0
answers
29
views
Why solving Lyapunov’s equation proves the stability?
The equation:
A'P + PA = -Q
where A is an n×n matrix, P is a symmetric positive-definite matrix of the same size, and Q is a symmetric positive-semidefinite matrix.
A is the matrix of the system.
I am ...
0
votes
0
answers
34
views
How do I change this to integration over a trajectory?
I'm working through CHAOS: An Introduction to Dynamical Systems by Alligood et. al. and I'm on Challenge 7 step 5. At this point we have a system of ODEs. (Note: a dot over a variable is its ...
- 1,423
1
vote
0
answers
40
views
Differential inclusion for piecewise finite-time lyapunov function
For each segment of a piecewise Lyapunov function that exhibits asymptotic stability, we can utilize the LaSalle-Yoshizawa theorem and solve it using a differential inclusion. This allows us to merge ...
- 533
2
votes
0
answers
55
views
Instability of equilibrium points of system $x'=y^3$, $y'=\cos(x)\sin(x)$
We take a look at the system
$$
x'=y^3, \quad y'= \cos(x)\sin(x).
$$
It has a Hamiltonian $H(x, y) = \frac{1}{4}y^4 - \frac{\sin(x)^2}{2}$ for $(x, y) \in \mathbb{R}^2$.
It is clear that the points $n\...
- 6,056
0
votes
0
answers
37
views
Regarding the computation on lyapunov function for stability
I’m trying to find a suitable lyapunov function for my paper and come accross this paper about SIR model where they used $V(x,y)= (S-S^1)^2 + (I-I^1)^2$ where $S^1$ and $I^1$ are endemic equilibriums. ...
- 1
0
votes
0
answers
26
views
Disprove existence of homoclinic and etheroclinic chains
I have a two dimensional piecewise smooth ODE and I was wondering if there are some known results for disproving the existence of homoclinic and etheroclinic loops (chain of etheroclinic orbits). I am ...
- 257
1
vote
0
answers
39
views
Invariant sets, La Salles invariance principle
I got some trouble understanding La Salles invariance principle, more specifically this exercise where I want to investigate stability of the origin given system
$$\begin{cases}
x'=-y-x^3\\
y'=x^5
\...
- 551
1
vote
1
answer
51
views
Lyapunov function, weak
I got some trouble with a problem about Lyapunov functions. I want to show that $V(x,y)=x^2+y^2$ is a weak Lyapunov function for the system
$$\begin{cases}
x'=y\\
y'=-x-y^3(1-x^2)^2
\end{cases}$$
So ...
- 551
1
vote
1
answer
43
views
Asymptotical stability vs Asymptotic uniform stability for nonautonomous systems
Slotine:
For autonomous systems:
For nonautonomous systems:
The second definition says there is a ball $0<R_2<R_1$, where trajectories that start in $R_1$ will converge into the smaller ball $...
- 43
0
votes
0
answers
51
views
Proving local exponential stability (check my answer)
Let us consider the $1$D system: $ x'=-\sin x $ .We want to prove that it is locally exponentially stable around $0$ using the Lyapunov function: $V(x)= \frac{1}{2} x(t)^2.$
Attempt:
Step 1:
i) $ V $ ...
- 63
2
votes
1
answer
72
views
Deriving the Lyapunov function of synchronization error of Lorenz systems
I am currenty studying the synchronization of Lorenz systems, in which one system transmits one of its coordinates to the other and this drives the other system to converge towards it exponentially, ...
0
votes
1
answer
63
views
Procedure for randomly creating linear dynamical systems with stable dynamics [closed]
I want a procedure for randomly generating a square matrix A such that the linear system
x_t+1 = A x_t
is globally ...
- 77
1
vote
1
answer
66
views
Finding a Lyapunov function to show stability
Consider the dynamical system
$$x'=y^3-x^5$$
$$y'=-x^3-y^5$$
This has only one equilibrium point, which is $(0,0)$.
I would like to find out whether this is stable, asymptotically stable or unstable.
...
- 725
2
votes
2
answers
182
views
Lyapunov analysis of marginally stable linear systems
Before I start, I want to emphasize I'm dealing with marginally stable linear systems, so many theorems about stable systems simply do not apply.
Let $\dot{x} = Ax$ be a marginally stable. That is, ...
- 2,742
2
votes
0
answers
106
views
How to find a Lyapunov function for this non-linear system?
Consider the system:
$$\dot{x}_1 = x_2 $$
$$\dot{x}_2 = -g(k_1 x_1 + k_2 x_2), k_1>0, k_2>0$$
where the nonlinearity g(·) is such that
$$g(y)y > 0, \forall y \neq 0 $$
$$lim_{|y| \to \infty}...
3
votes
1
answer
86
views
Is there a way to show that this ode system is asymptotically stable?
Suppose we have
$$\dot{x} = -\frac{x}{y+a} $$
$$\dot{y} = -y$$ for $a>0$,
Is the above system asymptotically stable?
Now, I know that we can solve for $y$ as
$$y = y(0) e^{-t}$$
and we can choose ...
- 57
0
votes
0
answers
39
views
$V'(\mathbf{x}) f(\mathbf{x}) \leq 0$ implies $V(\mathbf{x}(t)) \leq V(\mathbf{x}(0))$ for all $t \geq 0$
Consider a nonlinear dynamical system
\begin{equation}
\dot{\mathbf{x}}(t) = f(\mathbf{x}(t)), \quad \mathbf{x}(0) = \mathbf{x}_0, \quad t \geq 0
\end{equation}
and Lyapunov function $V: D \rightarrow ...
- 71
1
vote
1
answer
146
views
Stability of a linear time-varying system?
I am interested in finding out the stability of the system
$\dot{x} = -a \begin{bmatrix} \cos^2(t) &\cos(t)\sin(t) \\\cos(t)\sin(t) &\sin^2(t) \end{bmatrix}x$
with $a > 0$, via Lyapunov ...
- 57
2
votes
1
answer
112
views
Any theorems for Input-output or input-state stability for non-asymptotically stable nonlinear systems?
Update for clarification:
Assume $\dot{x_1}=f(x_1 , x_2)+ au$
where $x_1$ is asymptotically stable for all bounded values of $x_2$. If $x_2$ is kept bounded, will input-output stability theorem apply ...
- 21
2
votes
2
answers
135
views
Exponential stability of a time varying system
I want to show that the system below is exponentially stable and I want to estimate its region of attraction
\begin{align}
\dot x_1 &= -x_1+x_2+(x_1^2+x_2^2)\sin(t) \\
\dot x_2 &= -x_1-x_2+(...
- 55
3
votes
1
answer
149
views
Finding Lyapunov function to control the Duffing Oscillator
I'm trying to do this exercise but I'm struggling to find the Lyapunov function.
We have the state version of the Duffing oscillator below:
$\dot{x_1} = x_2$
$\dot{x_2} = x_1 - x_1^3 - \delta x_2$
...
- 55
0
votes
1
answer
116
views
Global Asymptotic stability of an ODE system?
Suppose we have the differential system of the form,
$\dot{x}_1 = -x_1 x_2$
$\dot{x}_2 = \omega(1-a x_2)$, where $\omega > 0$ and $a > 0$ are positive constants.
How can I prove that the system ...
- 57
0
votes
0
answers
41
views
Question regarding LaSalle's Invariance Principle
Consider LaSalle's Invariance Principle:
$1$. $\Omega\subset D$ a compact set that is positively invariant
$2$. $V(.)$ is continuously differentiable on $D$.
$3$. $\dot{V}\leq 0$ on $\Omega$.
$4.$ $E$ ...
- 55
0
votes
1
answer
77
views
Stability proof of a seemingly simple nonlinear differential equation?
How can I prove that the differential equation $\dot{x} = b(x - a x^2)$, where $a, b > 0$, has a asymptotically stable equilibirum at $x^* = 1/a$? One way to do it is via linearization, but I am ...
- 57
-1
votes
2
answers
60
views
Lyapunov criteria for discrete linear system with noise
Consider constant model matrix $A$ and $B$, the Lyapunov criteria for system $x_{k+1}=Ax_k+Bu_k$ with state feedback input $u_k=Kx_k$ (K is designed matrix) is $P-(A+BK)P(A+BK)^\top>0$, where $P$ ...
- 3
1
vote
1
answer
97
views
Convergence of dynamical system to stable equilibrium $(xy-1)^2$ to set $\{xy = 1\}$.
Consider the following dynamical system:
$\dot{x} = -(xy-1)y$ and $\dot{y} = -(xy-1)x.$
I would like to show that $x,y$ converge to set $xy = 1.$
Using Lyapunov function $V = (xy-1)^2,$ we can show ...
- 11
0
votes
1
answer
56
views
Being $V$ is a Lyapunov function, why is $\lim_{t \to \infty} V(x(t))=0 \Rightarrow \lim_{t \to \infty} x(t)=0$ true?
Being $V(x(t))$ a Lyapunov function, why is $\lim_{t \to \infty} V(x(t))=0 \Rightarrow \lim_{t \to \infty} x(t)=0$ ???
I don't know why is true that implication. I don't now from where to start. The ...
- 953
1
vote
0
answers
23
views
A doubt in the definition of the Lyapunov function
I've got the following definition (the definition of the function of lyapunov), but I don't understand it very well:
Let we have $D \subset \mathbb{R}^n$ open where non $0 \in D$, $f_k \in C^1(D)$ ...
- 953
0
votes
0
answers
47
views
Lyapunov Stability?
So I have the following 4 sets of equations where:
\begin{eqnarray}
n_u=s_u+i_u\label{eqnu}\\
n_e=s_e+i_e\label{eqne}
\end{eqnarray}
\begin{eqnarray}
\dot s_u = \frac{d s_u}{dt}=-\beta s_u(i_u + i_e)+(...
- 1
3
votes
1
answer
200
views
Asymptotic stability implies the existence of a strong Lyapunov function
I am having trouble understanding the proof that asymptotic stability implies the existence of a strong Lyapunov function. Taken from the book "Differential Dynamical Systems", chapter 4, by ...
- 147
0
votes
2
answers
66
views
Prove that $V'(x,y)=-(x^2+y^2)+(2Ax+By)F_1(x,y)+(Bx+2Cy)G_1(x,y)$. Liapunov.
We've got the following linear system:
$$\frac{dx}{dt}=a_{11}x+a_{12}y$$
$$\frac{dy}{dt}=a_{21}x+a_{22}y$$
The critical point $(0,0)$ is an assymptotically stable critical point of the system. I have ...
- 953
1
vote
2
answers
77
views
The critical point $(0,0)$ is assymptotically stable. Demostrate $a_{11}+a_{22}<0$ and $a_{11}a_{22}-a_{12}a_{21}>0.$
I've got the following linear system:
$$\frac{dx}{dt}=a_{11}x+a_{12}y$$
$$\frac{dy}{dt}=a_{21}x+a_{22}y$$
The critical point $(0,0)$ is an assymptotically stable critical point of the system.
We have ...
- 953
1
vote
1
answer
90
views
domain of attraction for a Lyapunov function
Let $V$ is a Lyapounov function $V(x,y) = 3x^2 - 2xy + y^2$, $\dot{V}(x,y) = -2(x+1)(x^2+y^2)$. I need to find its domain of attraction.
The minimum of $V$ is $x = -1$, so $\theta (y)$ $ = 3 + 2y + y^...
- 29
0
votes
1
answer
68
views
Show that if $ \lim _{|x| \rightarrow \infty} \int_0^x h(s) d s=+\infty $ then all solutions to this ODE are bounded.
a) Analyze $x^{\prime \prime}+f(x) x^{\prime}+h(x)=0$ where $f(x)>0$ and $x h(x)>0$ for $x \neq 0$ and such that $f, h$ are continuous.
b) Additionally, show that if
$$
\lim _{|x| \rightarrow \...
- 8,368
1
vote
1
answer
101
views
Analytically estimate the region of attraction of a non-linear system
I know this exercise has already been posted, but here I'm reformulating it to ask for help to find an analytical solution. I want to estimate the region of attraction of the following system:
$$
\...
- 31
0
votes
2
answers
87
views
Demonstrating Global Asymptotic Stability of a point
I'm trying to determine whether the origin of the following system is globally asymptotical stable (g.a.s.) or not:
$$
\begin{gathered}
\dot{x}_1 = -x_1-x_2 \\
\dot{x}_2 = -x_1-x_2^3.
\end{gathered}
$...
- 33
2
votes
1
answer
223
views
Estimating the region of attraction of a non-linear system with a Lyapunov Function
I'm trying to estimate the region of attraction of the following system:
$$
\begin{gathered}
\dot{x}_1 = \sin(x_2) \\
\dot{x}_2 = -x_1 - \sin(x_2).
\end{gathered}
$$
From Khalil, I know that if I ...
- 33
2
votes
0
answers
65
views
Is this planer dynamical system asymptotic stable at equilibrium point $(0,0)$?
Recently, I have been confused by the following problem:
Show that$$\frac{dx}{dt}=y^{3}-x^{3}y^{2}$$ $$\frac{dy}{dt}=-x^{3}$$is Lyapunov asymptotic stable at the equilibrium point $(0,0)$.
I can just ...
0
votes
1
answer
89
views
Proof of the Lyapunov Matrix Equation
Assuming that $A^TP+PA = -Q$ holds, I want to prove that $P = e^{A^Tt} P e^{A^Tt} + \int_{0}^{t} e^{A^T\tau} Q e^{A^T\tau}$ is a solution. After doing the substitutions, I end up with:
$A^TP+PA = A^T (...
- 13