# Questions tagged [stability]

Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)

118
questions

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11 views

### Perron - like theorem for LTV systems & BIBO stable

I'm looking for a necessary and sufficient conditions such that a LTV control system
$\dot{x}(t) = A(t) x(t) + u(t), \ x(0)=x_0$, for all $t\geq 0$,
$y(t) = C(t) x(t)$
satisfies the Perron-like result ...

**2**

votes

**1**answer

81 views

### Routh-Hurwitz criterion for matrices

The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half ...

**0**

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**0**answers

57 views

### convergence rate for ergodic Markov chains induced by stable dynamical systems

Consider a deterministic dynamical system on $\mathbb{R}^n$ defined by the recurrence $x_{t+1} = f(x_t)$.
Suppose the dynamical system is stable in the following sense: there exists a $Q : \mathbb{R}^...

**4**

votes

**1**answer

65 views

### Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?

Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...

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143 views

### Subset of the domain of attraction

Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...

**2**

votes

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108 views

### Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...

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88 views

### Reference for matrix Lyapunov function / matrix dynamic system / stability

We usually consider $\dot{x} = f(x)$, where $x$ is a vector.
Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...

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**0**answers

48 views

### Lyapunov theory in coupled nonlinear dynamic system with input

Suppose I have the following nonlinear coupled dynamic system
\begin{align*}
&\dot{x}_1 = f_1(x_1,x_2)\\
&\dot{x}_2 = f_2(x_2) + u
\end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...

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55 views

### Uniform stability of linear operators - reference request

Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result:
Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent:
(...

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28 views

### Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...

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75 views

### Von Neumann analysis on a finite difference hyperbolic scheme

I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations).
The finite difference scheme is:
$$
\mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...

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35 views

### Terminology: Almost stable states

I have a question about fixed points which are almost stable.
I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...

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**1**answer

99 views

### Observability inequality for the 1D transport equation

Let $(a,b) \subset (0,1)$. Consider the following transport equation
$$z_t+z_x=0, \ (t,x)\in(0,T)\times(0,1), \\z(t,0)=0, \ z(0,x)=z_0(x).$$
It is clear that the solution to the above equation is ...

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**1**answer

134 views

### Exact solution to a periodic linear ODE sought

We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...

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34 views

### Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...

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55 views

### Stability of a continuous piecewise linear map

I am studying random perturbation of a system that is continuous and piecewise linear. More precisely: I am given a map $\Phi_1:\mathbb{R}^d\to \mathbb{R}^d$ such that
$$ \Phi_1(x) = \left\{\begin{...

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**0**answers

70 views

### Stability and capacity, error in the book of Adams-Hedberg?

I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...

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24 views

### Lower bounds of kappa class functions

I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of
Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R}
^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any ...

**2**

votes

**1**answer

192 views

### Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...

**2**

votes

**0**answers

42 views

### (1, 2) stability and Hausdorff dimension

Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following:
$$
W_0^{1,2}(E) = W_0^{1,2}(E^0),
$$
where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...

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60 views

### What is a suitable state feedback adaptive law in discrete time?

I am trying to find a discrete adaptive law and prove stability in an analogous way to a well-known case in continuous time. Let me give a rough explanation of the background.
In continuous time, let ...

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77 views

### Is there any equivalent Foster-Lyapunov Theorem where expected Lyapunov drift is negative though not uniformly upper bounded by a negative number

I have a process $\{X_t\}_t$, where $x_t$ in $[0,1]$, and a Lyapunov function $V(x)=x$ such that the drift
$$E[X_{t+1}|x_t] - x_t < -k(1-x_t)$$
for $x_t \in (1-\epsilon,1]$, where $k>0$. Thus ...

**2**

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**1**answer

91 views

### Is there any foster-Lyapunov criterion for time varying Markov processes?

Suppose I have 2 Markov processes with transition kernels Q_1(y|x) and Q_2(y|x). Suppose i also have Lyapunov functions V_1, V_2 for these processes w.r.t. a common set, i.e. there exists a compact ...

**1**

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**1**answer

204 views

### Stability of holomorphic vector bundles

I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling ...

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143 views

### Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it.
Problem definition: Let $f(\xi) \in \...

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34 views

### Attractivity of a system with state-dependent transitions

Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system:
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,
$$
where $\max\{\cdot\}$ acts ...

**4**

votes

**1**answer

99 views

### almost linear ODE

Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$
$$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$
where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...

**4**

votes

**2**answers

366 views

### What is the current status on methods to find limit cycles?

What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...

**1**

vote

**0**answers

19 views

### Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{
& {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...

**2**

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51 views

### Strong stability of the wave equation with time depending potential

It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...

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59 views

### Part 2: When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters

Examine the polynomial
$$
x^{\tau+1}-\left(1+m\right)x^{\tau}+mx^{\tau-1}+\left(1-m\right)\alpha=0\,.
$$
with positive parameters $\tau,\alpha$ and $m<1$, and denote $\left|x_{\max}\left(\tau,\...

**4**

votes

**1**answer

109 views

### When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters

Examine the polynomial
$$
x^{\tau+1}-x^{\tau}+\alpha=0\,
$$
and denote by $\left|x_{\max}\left(\tau,\alpha\right)\right|$ as the
maximal magnitude of a root of this equation. For $\tau>1$, I ...

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**0**answers

75 views

### Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking
...

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**0**answers

69 views

### conditions for asymptotic comparison to hold

I have the following simple dynamical system:
\begin{align}
x_1' &= a - f(x_2)x_1\\
x_2' &= bx_1 - cx_2,
\end{align}
where all parameters and initial conditions are positive. $f(x_2)$ is a ...

**0**

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**0**answers

70 views

### Global stability question for system with a unique locally-asymptotically-stable steady state

I have an ordinary differential system of dimension 3 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is strictly-positively invariant and bounded.
Now, ...

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**1**answer

313 views

### How to study the global stability for this 3D system?

I am studying a biological system (HIV) and arrived at this simplified dynamical system:
\begin{align}
x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\
x_2' &= a_5\...

**0**

votes

**1**answer

43 views

### Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?

What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...

**0**

votes

**1**answer

99 views

### Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...

**1**

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**1**answer

176 views

### Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this ...

**3**

votes

**1**answer

153 views

### Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...

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**0**answers

83 views

### 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 ...

**3**

votes

**1**answer

292 views

### Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space.
...

**0**

votes

**1**answer

414 views

### Von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here
...

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**1**answer

338 views

### Center manifold theorem and case of all zero eigenvalues

Is the center manifold theorem applicable if say for a planar (2D) system of non-linear ODE, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector.
If not, what is ...

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votes

**1**answer

757 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

**1**

vote

**1**answer

138 views

### Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...

**0**

votes

**1**answer

189 views

### Stability of eigenvectors for diagonal perturbations

In a previous question I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) ...

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**0**answers

67 views

### Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...

**2**

votes

**1**answer

122 views

### On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...

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183 views

### Stability when linearization fails

The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...