Infinite-time observability of the wave equation with time-varying observation domains under a geodesic recurrence condition

Our goal is to relate the observation (or control) of the wave equation on observation domains which evolve in time with some dynamical properties of the geodesic flow. In comparison to the case of static domains of observation, we show that the observability of the wave equation in any dimension of space can be improved by allowing the domain of observation to move. We first prove that, for any domain Ω satisfying a geodesic recurrence condition (GRC), it is possible to observe the wave equation in infinite time on a ball of radius ε moving in Ω at finite speed v, where ε > 0 and v > 0 can be taken arbitrarily small, whereas the wave equation in Ω may not be observable on any static ball of radius ε. We comment on the recurrence condition: we give examples of Riemannian manifolds (Ω, g) for which (GRC) is satisfied, and, using a construction inspired by the Birkhoff-Smale homoclinic theorem, we show that there exist Riemannian manifolds (Ω, g) for which (GRC) is not satisfied. Then we prove that on the 2-dimensional torus and on Zoll manifolds, it is possible to observe the wave equation in finite time with moving balls. Finally, we establish a result of spectral observability (or of concentration of eigenfunctions) on time-dependent domains

From internal to pointwise control for the 1D heat equation and minimal control time

International audienceOur goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_ε = (x_0 − ε, x_0 + ε)$, in the limit $ε → 0$, where $x_0 ∈ (0, 1)$. It is known that depending on arithmetic properties of $x_0$ , there may exist a minimal time $T_0$ of pointwise control at x_0 of the heat equation. Besides, for any ε fixed, the heat equation is controllable with control set $ω_ε$ in any time $T > 0$. We relate these two phenomena. We show that the observability constant on $ω_ε$ does not converge to $0$ as $ε → 0$ at the same speed when $T > T_0$ (in which case it is comparable to $ε 1/2$) or $T < T_0$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^2$ null-controls on $ω_ε$ in the limit $ε → 0$

Subelliptic wave equations are never observable

It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$ such that $\text{Lie}(X_1,\ldots,X_m)=TM$ but $\text{Span}(X_1,\ldots,X_m)\subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ has nonempty interior, the wave equation with subelliptic Laplacian $\Delta$ is not observable on $\omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on spiraling geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash \omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space

Exact observability properties of subelliptic wave and Schr{\"o}dinger equations

In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{\"o}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis

Enquête auprès des usagers et des non-usagers de la Bibliothèque. Comparaison des résultats 2007-2014

La bibliothèque de l\u27Université de Paris 8 a conduit, fin 2014, une enquête auprès de ses usagers et non-usagers, afin d’évaluer leur connaissance des services, leur satisfaction et leurs attentes. Cette enquête reprenant en grande partie un questionnaire de 2006, permet une comparaison des résultats des deux enquêtes à 8 ans d\u27intervalle

Quantum limits of sub-Laplacians via joint spectral calculus

We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub-Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we classify all QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear.Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possibly high degeneracy of the spectrum, the spectral theory of sub-Laplacians can be very rich

Nodal sets of eigenfunctions of sub-Laplacians

Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studiedextensively over the past decades. In this note, we initiate the study of nodal sets ofeigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians.A standard example is the sum of squares of bracket-generating vector fields on compactquotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropicway which can be analyzed with standard tools from sub-Riemannian geometry such assub-Riemannian dilations, nilpotent approximation and desingularization at singular points.Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by $C\sqrt{\lambda}$, which is thebound conjectured by Yau for Laplace-Beltrami operators on smooth manifolds

Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

We obtain a general sublinear upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces. In particular, this yields progress on a longstanding conjecture by Colin de Verdi{\`e}re [Colin de Verdi{\`e}re, 1986]. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in [Jiang-Tidor-Yao-Zhang-Zhao, 2021] for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the ''approximate multiplicity'' of eigenvalues, i.e., the number of eigenvalues in windows of size 1 / log^{\kappa}(g), \kappa > 0