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369 results on '"Rousseau, Jérôme"'

1. Measure and continuous vector field at a boundary II: geodesics and support propagation

Author

Burq, Nicolas, Dehman, Belhassen, and Rousseau, Jérôme Le

Subjects
Mathematics - Analysis of PDEs, 34Bxx, 35Q49, 35R05
Abstract

Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral curves is the general rule. It has been known since the works by L. Ambrosio [2] and L. Ambrosio and G. Crippa [3, 4] that such measures can be described as a superposition of $\delta$-measures supported on integral curves. In this article, motivated by some observability questions for the wave equation, we are interested in such transport equations in the case of domains with boundary. Associated with a wave equation with C^1-coefficients are bicharacteristics that are integral curves of a continuous Hamiltonian vector field. We first study in details their behaviour in the presence of a boundary and define their natural generalisation that follows the laws of geometric optics. Then, we introduce a natural class of transport equations with a source term on the boundary, and we prove that any nonnegative measure satisfying such an equation has a union of maximal generalized bicharacteristics for support. This result is a weak form of the superposition principle in the presence of a boundary. With its companion article [7], this study completes the proof of wave observability generalizing the celebrated result of Bardos, Lebeau, and Rauch [5] in a low regularity framework where coefficients of the wave equation (and associated metric) are C^1 and the boundary and the manifold are C^2., Comment: 103 pages, 9 figures

Published
2024

2. Measure and continuous vector field at a boundary I: propagation equations and wave observability

Author

Burq, Nicolas, Dehman, Belhassen, and Rousseau, Jérôme Le

Subjects
Mathematics - Analysis of PDEs, 35L05, 35L20, 35Q49, 35R05, 93B07, 34A99, 35S05
Abstract

The celebrated geometric control condition of Bardos, Lebeau, and Rauch is necessary and sufficient for wave observability [1,7] and exact controllability. It requires that any point in phase-space be transported by the generalized geodesic flow to the region of observation in some finite time. The initial smoothness (\Cinf) required on the coefficients of the metric to prove that exact control and geometric control are essentially equivalent was subsequently relaxed to \Con2-metrics/coefficients and \Con3-domains [2] which is close to the optimal smoothness required to preserve a generalized geodesic flow. In this article, we investigate a natural generalization of the geometric control condition that makes sense for \Con1-metrics and we prove that wave observability holds under this condition. Moreover, we establish that the observability property is stable under rougher (Lipschitz) perturbation of the metric. We also provide a geometric necessary condition for wave observability to hold. Transport equations that describe the propagation of semi-classical measures are at the heart of the arguments. They are natural extensions to geometries with boundaries of usual transport equations. This article is mainly dedicated to the proof of such propagation equations in this very rough context., Comment: 119 pages

Published
2024

3. Semi-classical analysis

Author

Kammerer, Clotilde Fermanian and Rousseau, Jérôme Le

Subjects
Mathematics - Analysis of PDEs
Abstract

We introduce three representative topics in semi-classical analysis. Starting from the correspondence between classical and quantum mechanics, basic semi-classical analysis tools and results are presented. The three topics are investigated in the light of the introduced techniques allowing one to emphasize different aspects of semi-classical analysis., Comment: This article presents some ideas and applications of semi-classical analysis

Published
2024

7. Measure propagation along a $\mathscr{C}^0$-vector field and wave controllability on a rough compact manifold

Author

Burq, Nicolas, Dehman, Belhassen, and Rousseau, Jérôme Le

Subjects
Mathematics - Analysis of PDEs, Mathematics - Optimization and Control, 35L05, 35Q49, 35Q93, 58Jxx, 93B05, 93B07
Abstract

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 optics. This connection is best understood through the propagation properties of microlocal defect measures that appear as solutions to the wave equation concentrate. For a sufficiently smooth metric this propagation occurs along the bicharacteristic flow. If one considers a merely $\mathscr{C}^1$-metric this bicharacteristic flow may however not exist. The Hamiltonian vector field is only continuous; bicharacteristics do exist (as integral curves of this continuous vector field) but uniqueness is lost. Here, on a compact manifold without boundary, we consider this low regularity setting, revisit the geometric control condition, and address the question of support propagation for a measure solution to an ODE with continuous coefficients. This leads to a sufficient condition for the observability and equivalently the exact controllability of the wave equation. Moreover, we investigate the stabililty of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz.

Published
2023

8. Minimal distance between random orbits

Author

Gouëzel, Sébastien, Rousseau, Jérôme, and Stadlbauer, Manuel

Subjects
Mathematics - Dynamical Systems
Abstract

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension (or R{\'e}nyi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

Published
2022

9. Orbits closeness for slowly mixing dynamical systems

Author

Rousseau, Jerome and Todd, Mike

Subjects
Mathematics - Dynamical Systems, 37A50, 37B20, 37D25
Abstract

Given a dynamical system, we prove that the shortest distance between two $n$-orbits scales like $n$ to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit., Comment: To appear in Ergodic Theory and Dynamical Systems

Published
2022

10. Stabilization of the damped plate equation under general boundary conditions

Author

Rousseau, Jérôme Le and Zongo, Emmanuel Wend-Benedo

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis
Abstract

We consider a damped plate equation on an open bounded subset of R^d, or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskii-Sapiro condition. The damping term acts on a region without imposing a geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first though microlocal estimates, then local estimates, and finally a global estimate.

Published
2021

11. R\'enyi entropy and pattern matching for run-length encoded sequences

Author

Rousseau, Jerome

Subjects
Mathematics - Probability, Computer Science - Information Theory, Mathematics - Dynamical Systems
Abstract

In this note, we studied the asymptotic behaviour of the length of the longest common substring for run-length encoded sequences. When the original sequences are generated by an $\alpha$-mixing process with exponential decay (or $\psi$-mixing with polynomial decay), we proved that this length grows logarithmically with a coefficient depending on the R\'enyi entropy of the pushforward measure. For Bernoulli processes and Markov chains, this coefficient is computed explicitly.

Published
2020

12. Shortest distance between multiple orbits and generalized fractal dimensions

Author

Barros, Vanessa and Rousseau, Jerome

Subjects
Mathematics - Dynamical Systems, Computer Science - Information Theory, Mathematics - Probability
Abstract

We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized R\'enyi entropy. Applications to Markov chains, Gibbs states and the stochastic scrabble are given., Comment: arXiv admin note: text overlap with arXiv:1808.00078

Published
2019

13. Elements of Differential Geometry

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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14. Sobolev Spaces and Laplace Problems on a Riemannian Manifold

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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15. Integration and Differential Operators on Manifolds

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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16. Quantified Unique Continuation on a Riemannian Manifold

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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17. Optimal Estimates at the Boundary

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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18. Carleman Estimates with Source Terms of Weaker Regularity

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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19. Elements of Riemannian Geometry

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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20. Spectral Inequality for General Boundary Conditions and Application

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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21. Stabilization of Waves Under General Boundary Damping

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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22. Estimates for General Boundary Conditions

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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23. Stabilization of Waves Under Neumann Boundary Damping

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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24. Sobolev Norms with a Large Parameter on a Manifold

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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25. Selfadjoint Operators Under General Boundary Conditions

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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26. Fredholm Properties of Second-Order Elliptic Operators

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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27. Estimates on Riemannian Manifolds for Dirichlet Boundary Conditions

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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28. Pseudo-Differential Operators on a Half-Space

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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29. Lopatinskiı̆–Šapiro Boundary Conditions

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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30. Introduction to Volume 2

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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31. Invariance Under Change of Variables

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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32. Stabilization of the Wave Equation with an Inner Damping

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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33. A Short Review of Distribution Theory

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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34. Unique Continuation

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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35. Controllability of Parabolic Equations

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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36. Optimality Aspects of Carleman Estimates

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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37. Elliptic Operator with Dirichlet Data and Associated Semigroup

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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38. Some Elements of Functional Analysis

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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39. Carleman Estimate for a Second-Order Elliptic Operator

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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40. Some Elements of Semigroup Theory

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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41. (Pseudo-)Differential Operators with a Large Parameter

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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42. Introduction

Author

Le Rousseau, Jérôme, Lebeau, Gilles, Robbiano, Luc, Brezis, Haïm, Series Editor, Ambrosio, Luigi, Editorial Board Member, Berestycki, Henri, Editorial Board Member, Caffarelli, Luis, Editorial Board Member, Chang, Sun-Yung Alice, Editorial Board Member, Coron, Jean-Michel, Editorial Board Member, del Pino, Manuel, Editorial Board Member, Evans, Lawrence C., Editorial Board Member, Figalli, Alessio, Editorial Board Member, Frank, Rupert, Editorial Board Member, Fusco, Nicola, Editorial Board Member, Klainerman, Sergiu, Editorial Board Member, Kohn, Robert, Editorial Board Member, Lions, Pierre-Louis, Editorial Board Member, Malchiodi, Andrea, Editorial Board Member, Mawhin, Jean, Editorial Board Member, Merle, Frank, Editorial Board Member, Mingione, Giuseppe, Editorial Board Member, Otto, Felix, Editorial Board Member, Rabinowitz, Paul, Editorial Board Member, Toland, John, Editorial Board Member, Vogelius, Michael, Editorial Board Member, Le Rousseau, Jérôme, Lebeau, Gilles, and Robbiano, Luc

Published
2022
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45. Longest common substring for random subshifts of finite type

Author

Rousseau, Jerome

Subjects
Mathematics - Dynamical Systems, Mathematics - Probability
Abstract

In this paper, we study the behaviour of the longest common substring for random subshifts of finite type (for dynamicists) or of the longest common substring for random sequences in random environments (for probabilists). We prove that, under some exponential mixing assumptions, this behaviour is linked to the R\'enyi entropy of the stationary measure. We emphasize that what we establish is a quenched result., Comment: To appear in Ann. Inst. Henri Poincar\'e Probab. Stat

Published
2019

46. Matching strings in encoded sequences

Author

Coutinho, Adriana, Lambert, Rodrigo, and Rousseau, Jérôme

Subjects
Mathematics - Probability, Computer Science - Data Structures and Algorithms, Computer Science - Information Theory, Mathematics - Dynamical Systems
Abstract

We investigate the longest common substring problem for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the R\'enyi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.

Published
2019

47. On the shortest distance between orbits and the longest common substring problem

Author

Barros, Vanessa, Liao, Lingmin, and Rousseau, Jerome

Subjects
Mathematics - Dynamical Systems, Mathematics - Probability
Abstract

In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of Arratia and Waterman on sequence matching to $\alpha$-mixing processes with exponential decay., Comment: Final version. Accepted for publication in Advances in Mathematics

Published
2018

48. Large deviation for return times

Author

Coutinho, Adriana, Rousseau, Jerome, and Saussol, Benoit

Subjects
Mathematics - Dynamical Systems
Abstract

We prove a large deviation result for return times of the orbits of a dynamical system in a $r$-neighbourhood of an initial point $x$. Our result may be seen as a differentiable version of the work by Jain and Bansal who considered the return time of a stationary and ergodic process defined in a space of infinite sequences.

Published
2018
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49. Exponential Law for Random Maps on Compact Manifolds

Author

Haydn, Nicolai, Rousseau, Jerome, and Yang, Fan

Subjects
Mathematics - Dynamical Systems
Abstract

We consider random dynamical systems on manifolds modeled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for the hitting and return times to geometric balls are both exponential for almost every realisation. We then apply this result to random $C^2$ maps of the interval and random parabolic maps on the unit interval.

Published
2017

50. Geometric control condition for the wave equation with a time-dependent observation domain

Author

Rousseau, Jérôme Le, Lebeau, Gilles, Terpolilli, Peppino, and Trélat, Emmanuel

Subjects
Mathematics - Analysis of PDEs
Abstract

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 time-varying. We provide a condition ensuring observability, in terms of propagating bicharacteristics. This condition extends the well-known geometric control condition established for fixed observation domains. As one of the consequences, we prove that it is always possible to find a time-dependent observation domain of arbitrarily small measure for which the observability property holds. From a practical point of view, this means that it is possible to reconstruct the solutions of the wave equation with only few sensors (in the Lebesgue measure sense), at the price of moving the sensors in the domain in an adequate way.We provide several illustrating examples, in which the observationdomain is the rigid displacement in $\Omega$ of a fixed domain, withspeed $v,$ showing that the observability property depends both on $v$and on the wave speed. Despite the apparent simplicity of some of ourexamples, the observability property can depend on nontrivial arithmeticconsiderations.

Published
2016

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