# 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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### 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 ...
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 ...
1 vote
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 ...
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/...
1 vote
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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 ...
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1 vote
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 ...
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### 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 ...
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 vote
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 ...
55 views

1 vote
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### 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 ...
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 ...
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+(...
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: $\dot{x_1} = x_2$ $\dot{x_2} = x_1 - x_1^3 - \delta x_2$ ...
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 ...
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### 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 $\Omega$. $4.$ $E$ ...
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 ...
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 $P$ ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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 $0 \in D$, $f_k \in C^1(D)$ ...
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)+(...
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 ...
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### 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 ...
1 vote
### 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 ...