1
vote
0answers
87 views

null controllability of linear wave equation

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
1
vote
0answers
53 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
1
vote
2answers
249 views

Exponential stability in nonlinear differential equations

I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to ...
3
votes
1answer
158 views

Avalanche Principle for higher dimensional unimodular matrices ?

Hello everyone, I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this ...
0
votes
1answer
383 views

Simple system of ODEs with periodic coefficients

I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs: $f'(t) = P \cos(k t + \Phi_1) g(t)$ $g'(t) = Q \cos(k t + ...
4
votes
3answers
646 views

Homogeneous linear differential equation system with simple periodical coefficient matrix

Hello, I encountered the following system of linear first-order differential equations: $y'(z)=A(z) y(z)$ where $y(z): R \rightarrow R^2$ and $A(z)=\begin{pmatrix} 0 & B Cos(\alpha z + \Phi_b) ...
1
vote
1answer
182 views

sum of Perron-Frobenius operators

My operator is the transfer operator $P$ on $L^1$ functions defined on compact $X$. It is the pre-dual of the operator $U:L^∞ \rightarrow L^∞$ defined by $U(ϕ)=ϕ\circ f$, for a fixed map f on X. I ...
22
votes
3answers
2k views

Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
5
votes
1answer
515 views

Spectrum of a generic integral matrix.

My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms. These are given by integral matrices A with determinant 1 and without eigenvalues on the unit ...
4
votes
1answer
315 views

Is there a name for this differential operator and/or its corresponding spectrum?

Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the ...