Questions tagged [stability]

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

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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 ...
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 ...
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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 ...
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. ...
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 ...
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 ...
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(\...
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(...
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) ...
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 ...
The dynamics of the $j$th system: \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} \...