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Camille Laurent

Publications and source records attributed to Camille Laurent.

32 records · Page 2Linked to original sources
arXiv2018

Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain $ω$ of a bounded domain $M$. The constant is of the form $e^{\frac{K}{T}}$, where $K$ depends only on the geometry of $M$ and $ω$. Luc Miller (JDE, 2004) conjectured that $K$ is (universally) proportional to the square of the maximal distance from $ω$ to a point of $M$. We show in particular geometries that $K$ may blow up like $|\log(r)|^2$ when $ω$ is a ball of radius $r$, hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation. The proofs rely on the study of the maximal vanishing rate of (sums of) eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs. As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.

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arXiv2017

Tunneling estimates and approximate controllability for hypoelliptic equations

This article is concerned with quantitative unique continuation estimates for equations involving a "sum of squares" operator $\mathcal{L}$ on a compact manifold $\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of $\mathcal{L}$, and $(ii)$ the analyticity of $\mathcal{M}$ and the coefficients of $\mathcal{L}$. The first result is the tunneling estimate $\|φ\|_{L^2(ω)} \geq Ce^{- λ^{\frac{k}{2}}}$ for normalized eigenfunctions $φ$ of $\mathcal{L}$ from a nonempty open set $ω\subset \mathcal{M}$, where $k$ is the hypoellipticity index of $\mathcal{L}$ and $λ$ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation $(\partial_t^2+\mathcal{L})u=0$: for $T>2 \sup_{x \in \mathcal{M}}(dist(x,ω))$ (here, $dist$ is the sub-Riemannian distance), the observation of the solution on $(0,T)\times ω$ determines the data. The constant involved in the estimate is $Ce^{cΛ^k}$ where $Λ$ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation $(\partial_t+\mathcal{L})v=1_ωf$ in any time, with appropriate (exponential) cost, depending on $k$. In case $k=2$ (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary $\partial \mathcal{M}$ can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in arxiv:1506.04254.

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arXiv2016

Uniform observability estimates for linear waves

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, which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric control time (for wave equations with possibly lower order terms). Second, we provide (on manifolds with or without boundary) with an explicit dependence of the observability constant with respect to the addition of a bounded potential to the equation.

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arXiv2015

Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves

In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and Hörmander. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold $\mathcal{M}$ with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary $\partial \mathcal{M}$. This allows us to obtain a global stability estimate from any open set $Γ$ of $\mathcal{M}$ or $\partial \mathcal{M}$, with the optimal time and dependence on the observation. This provides the cost of approximate controllability: for any $T>2 \sup_{x \in \mathcal{M}}(dist(x,Γ))$, we can drive any data of $H^1_0 \times L^2$ in time $T$ to an $\varepsilon$-neighborhood of zero in $L^2 \times H^{-1}$, with a control located in $Γ$, at cost $e^{C/\varepsilon}$. We also obtain similar results for the Schrödinger equation.

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arXiv2014

Control and Stabilization of the Benjamin-Ono equation in $L^2(\mathbb T)$

We prove the control and stabilization of the Benjamin-Ono equation in $L^2(\T)$, the lowest regularity where the initial value problem is well-posed. This problem was already initiated in \cite{LinaresRosierBO} where a stronger stabilization term was used (that makes the equation of parabolic type in the control zone). Here we employ a more natural stabilization term related to the $L^2$ norm. Moreover, by proving a theorem of controllability in $L^2$, we manage to prove the global controllability in large time. Our analysis relies strongly on the bilinear estimates proved in \cite{MolinetPilodBO} and some new extension of these estimates established here.

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arXiv2013

Stabilization for the semilinear wave equation with geometric control condition

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor.

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arXiv2013

A note on the global controllability of the semilinear wave equation

We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-Δv(x,t) + f(x,v(x,t))= \Un_ω u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that $f(x,v)v\geq 0$. In this case, a stabilisation term $u=γ(x)v_t$ makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity $f$ can be more general, so that the solutions of the damped equation may converge to another equilibrium than $0$. To prove global controllability, we study the controllability inside a compact attractor and show that it is possible to travel from one equilibrium point to another by using the heteroclinic orbits.

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arXiv2013

Internal control of the Schrödinger equation

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the new difficulties when we consider the Nonlinear Schrödinger equation.

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arXiv2011

The Laplace-Beltrami operator in almost-Riemannian Geometry

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.

math.SP↗
arXiv2010

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard on $\R^3$, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim on the behavior of concentrating waves on manifolds.

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arXiv2010

Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models (by the first author and with J.M. Coron), in non optimal spaces, in long time and the proof relied on the Nash-Moser implicit function theorem in order to deal with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized, showing there is actually no loss of regularity. Then, the same strategy is applied to nonlinear Schrödinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems.

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arXiv2009

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

This paper aims at completing an earlier work of Russell and Zhang to study internal control problems for the distributed parameter system described by the Korteweg-de Vries equation on a periodic domain T^1. In their article, Russell and Zhang showed that the system is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of T^1. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of propagation of compactness and propagation of regularity in Bourgain spaces for solutions of the associated linear system. Then, using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate.

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arXiv2009

Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3

We prove global internal controllability in large time for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. The result is proved under some geometrical assumptions : geometric control and unique continuation. We give some examples where they are fulfilled on $\Tot$, $S^3$ and $S^2\times S^1$. We prove this by two different methods both inherently interesting. The first one combines stabilization and local controllability near 0. The second one uses successive controls near some trajectories. We also get a regularity result about the control if the data are assumed smoother. If the $H^1$ norm is bounded, it gives a local control in $H^1$ with a smallness assumption only in $L^2$. We use Bourgain spaces.

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