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We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold $\Omega,$ with or without boundary, where the observation (or control) domain is…
We establish sharp regional observability results for solutions of the wave equation in a bounded domain of $\Omega \subset \mathbb{R}^n$, in case where the geometric control condition is not satisfied. Assuming that the waves are observed…
In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos-Lebeau-Rauch,…
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…
This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain $\omega$ of a bounded domain $M$. The constant is of the form $e^{\frac{K}{T}}$, where $K$ depends only on the…
In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the weak…
We study the observability of waves on gas giant manifolds which are a class of Riemannian manifolds whose metrics are singular at the boundary. Such manifolds arise naturally in modeling of acoustic wave propagation in gas giant planets.We…
Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{\"o}dinger equation i$\partial$ t u = F (g)u, where g is the…
This paper presents two observability inequalities for the heat equation over $\Omega\times(0,T)$. In the first one, the observation is from a subset of positive measure in $\Omega\times(0,T)$, while in the second, the observation is from a…
We study the propagation properties of the solutions of the finite-difference space semi-discrete wave equation on an uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along…
This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by…
Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $\omega_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit $\varepsilon\rightarrow 0$,…
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$\Omega = (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $\omega=(a,b) \times…
By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp…
Based on a fundamental identity for stochastic hyperbolic-like operators, we derive in this paper a global Carleman estimate (with singular weight function) for stochastic wave equations. This leads to an observability estimate for…
Energy decay is established for the damped wave equation on compact Riemannian manifolds where the damping coefficient is allowed to depend on time. Using a time dependent observability inequality, it is shown that the energy of solutions…
We establish observability inequalities for various problems involving fractional Schr\"odinger operators $(-\Delta)^{\alpha/2}+V$, $\alpha>0$, on a compact Riemannian manifold. Observability from an open set for the corresponding…
The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition is central in the study of the observability of the wave equation linking this property to high-frequency propagation along geodesics that are therays of geometric…
In the study of the observability of the wave equation (here on $(0,T)\times \mathbb{T}^d$, where $\mathbb{T}^d$ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which…
In this paper, we establish some important results for the impulsive wave equation. We begin by proving the existence of a solution. Then, we study the impulse approximate controllability where the control function acts on a subdomain…