26 problems

In a model of atmospheric dynamics, suppose that the atmospheric temperature field is observed continuously in time. Charney et al.'s observability conjecture. Knowledge of the con…

Russell–Weiss conjecture. The Hautus test is sufficient for exact controllability of the system.

Uniform observability conjecture. For every $T>0$, there exists a constant $C=C(X,S,T)>0$ such that the observability inequality holds. Moreover, the constant can be chosen to depe…

Let $\mathcal{M}$ be an almost-Riemannian manifold equipped with a measure, and consider the heat equation associated with its Laplace--Beltrami operator. Denote the minimal observ…

Let $u0$, let $boldsymbol{Omega}=boldsymbol{Omega}xtimesboldsymbol{Omega}y$ be the domain, and let $boldsymbol{omega}=boldsymbol{omega}xtimesboldsymbol{Omega}y$ be an open observa…

For each $d\geq 1$, let $Y_0$ be the closure space for Schrödinger waves introduced in the paper, and let $\tilde{Y}_0\subset Y_0$ be its time-independent subspace. Time-independen…

For each $d\geq 1$, let $Y$ be the normed function space introduced in the paper for Schrödinger-wave observability, and let $Y_0$ denote the closure of continuous functions in $Y$…

Let $d\geq 1$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $V\in L^\infty_x(\mathbb{T}^d)$. Let $\mathcal{H}_V=-\Delta+V$, and let $f$ be any eigenfunction of…

Let $d\geq 1$, let $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$, and let $V\in L^\infty_x(\mathbb{T}^d)$ be a bounded potential. For $u_0\in L^2_x(\mathbb{T}^d)$, write…

Let $C_S(T,L)$ denote the best constant in the observability inequality for the one-dimensional Schrödinger equation on $(0,L)$, and define … The known bounds are…

Let $m_d(r)$ be the optimal observability constant for Laplacian eigenfunctions on the $d$-dimensional torus observed on a ball of radius $r$. For $d\geq 4$, let $C_d$ and $C'_d$ b…

Asymptotic error-magnification conjecture. As $t\to\infty$, $\kappa_{ji}^t$ approaches a limit, denoted by $\kappa_{ji}$:

Upstream observability conjecture. For almost every choice of entries of $A$, if there is a directed path from node $i$ to node $j$, then the time series of node $i$ can be reconst…

Let $\omega\subset\mathbb{R}^d$ be an observation set, let $B_R(\phi^t(\rho))$ denote the ball of radius $R$ centered at the point $\phi^t(\rho)$ along the relevant classical flow,…

Let $m,l\five in\text{N}$, and let $P_lambda$ be the spectral projector for the Shubin operator $H=(-Delta)^m+|x|^{2l}$. Suppose that $omega\five subset\real^d$ satisfies the decay…

Let $\tau>0$ and let $\function{P}_\tau$ denote the spectral projector associated with $-Delta+|x|^\tau$. Suppose that $omega\five subset\real^d$ is measurable and that, for some…

Let $M$ be the quotient under consideration of a group of Heisenberg type, let $U\subset M$ be open, and let $T_{\rm GCC}(U)$ denote the geometric control time associated with $U$.…

Let $s$ be the number of adversarial sensor attacks, and consider the problem of verifying the $2s$-sparse observability condition for a linear system. NP-hardness conjecture. Veri…

Let $({\mathcal M},g)$ be a Riemannian manifold and let $\omega\subset {\mathcal M}$ satisfy $\overline{\omega}\neq {\mathcal M}$. Define … Let ${\mathfrak K}_{heat}(\omega)$ denot…

Miller's conjecture. The constant ${\mathfrak K}$ is universally proportional to ${\mathcal L}({\mathcal M},\omega)^2$.

Let $S$ be a subset of observing nodes or variables of a dynamical network, and let $X$ be a network node whose trajectory is to be reconstructed. Consider an ergodic trajectory on…

Let a Boolean control network (BCN) be a finite-state dynamical system with Boolean-valued states, inputs, and outputs. The four observability notions considered in the paper are t…

Let $L>0$ and $T>0$, and let $C_{int}(T,L)$ be a constant for which the finite-time weighted observability estimate … holds for the relevant solutions of the one-dimensional heat e…

Ervedoza–Zuazua conjecture. For every $L>0$ and every $K>L^2/4$, there exists some $C(K)>0$ such that, for every $T$ sufficiently small,

Let $T>\mu>0$, let $a<b$ with $(a,b)\subset(0,1)$, and let $\phi_n(x)=\sqrt{2}\sin(n\pi x)$. For each measurable set $\Xi\subset[0,T]$ of measure $\mu$, consider the largest consta…