The Experts below are selected from a list of 4473 Experts worldwide ranked by ideXlab platform
Yannick Privat - One of the best experts on this subject based on the ideXlab platform.
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Geometric and probabilistic results for the observability of the wave equation
2019Co-Authors: Emmanuel Humbert, Yannick Privat, Emmanuel TrélatAbstract:Given any Measurable Subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation Subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}. In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic. The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random Subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.
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Observability properties of the homogeneous wave equation on a closed manifold
Communications in Partial Differential Equations, 2019Co-Authors: Emmanuel Humbert, Yannick Privat, Emmanuel TrélatAbstract:We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a Measurable Subset $\omega$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and if the pair $(\omega,T)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $\omega \times (0, T)$. The observability constant $C_T( {\omega})$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a Measurable Subset $\omega$. We also establish that, as $T\rightarrow+\infty$, the ratio $C_T( {\omega})/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions; the second one is of a geometric nature and involves the average time spent in $\omega$ by Riemannian geodesics.
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Non-localization of eigenfunctions for Sturm-Liouville operators and applications
Journal of Differential Equations, 2018Co-Authors: Thibault Liard, Pierre Lissy, Yannick PrivatAbstract:In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators $A_a=-\partial_{xx}+a(\cdot)\operatorname{Id}$ with Dirichlet boundary conditions, where $a(\cdot)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(\cdot)$ on a Measurable Subset $\omega$ of $(0,L)$, where $e(\cdot)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all Subsets $\omega$ having a prescribed measure and all $L^\infty$ potential functions $a(\cdot)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.
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Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Journal of the European Mathematical Society, 2016Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the wave and Schrödinger equations on a bounded open connected Subset $\Omega$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a Measurable Subset $\omega$ of $\Omega$ during a time interval $[0, T]$ with $T>0$. It is well known that, if the pair $(\omega,T)$ satisfies the Geometric Control Condition ($\omega$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in $\omega \times (0, T)$. We address the problem of the optimal location of the observation Subset $\omega$ among all possible Subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the Subset $\omega$ has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions on $\Omega$, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain $\Omega$). We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures. Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
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Complexity and regularity of maximal energy domains for the wave equation with fixed initial data
Discrete and Continuous Dynamical Systems - Series A, 2015Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the homogeneous wave equation on a bounded open connected Subset $\Omega$ of $\R^n$. Some initial data being specified, we consider the problem of determining a Measurable Subset $\omega$ of $\Omega$ maximizing the $L^2$-norm of the restriction of the corresponding solution to $\omega$ over a time interval $[0,T]$, over all possible Subsets of $\Omega$ having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components.
Sandra Saliani - One of the best experts on this subject based on the ideXlab platform.
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$\ell ^2$-Linear independence for the system of integer translates of a square integrable function
Proceedings of the American Mathematical Society, 2012Co-Authors: Sandra SalianiAbstract:We prove that if the system of integer translates of a square integrable function is l^2-linear independent then its periodization function is strictly positivealmost everywhere. Indeed we show that the above inference holds for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) Measurable Subset A of [0, 1], with positive measure,\ud there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded in the uniform norm. This answers a question posed by Guido Weiss
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ell 2 linear independence for the system of integer translates of a square integrable function
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Sandra SalianiAbstract:We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) Measurable Subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.
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$\ell^2$-Linear Independence for the System of Integer Translates of a Square Integrable Function
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Sandra SalianiAbstract:We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) Measurable Subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.
Kerstin Hesse - One of the best experts on this subject based on the ideXlab platform.
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complexity of numerical integration over spherical caps in a sobolev space setting
Journal of Complexity, 2011Co-Authors: Kerstin HesseAbstract:Let r>=2, let S^r be the unit sphere in R^r^+^1, and let C(z;@c):={[email protected]?S^r:[email protected]?z>[email protected]} be the spherical cap with center [email protected]?S^r and radius @[email protected]?(0,@p]. Let H^s(S^r) be the Sobolev (Hilbert) space of order s of functions on the sphere S^r, and let Q"m be a rule for numerical integration over C(z;@c) with m nodes in C(z;@c). Then the worst-case error of the rule Q"m in H^s(S^r), with s>r/2, is bounded below by c"r","s","@cm^-^s^/^r. The worst-case error in H^s(S^r) of any rule Q"m"("n") that has m(n) nodes in C(z;@c), positive weights, and is exact for all spherical polynomials of degree @?n is bounded above by [email protected]?"r","s","@cn^-^s. If positive weight rules Q"m"("n") with m(n) nodes in C(z;@c) and polynomial degree of exactness n have m(n)~n^r nodes, then the worst-case error is bounded above by [email protected]?"r","s","@c(m(n))^-^s^/^r, giving the same order m^-^s^/^r as in the lower bound. Thus the complexity in H^s(S^r) of numerical integration over C(z;@c) with m nodes is of the order m^-^s^/^r. The constants c"r","s","@c and [email protected]?"r","s","@c in the lower and upper bounds do not depend in the same way on the area |C(z;@c)|[email protected]^r of the cap. A possible explanation for this discrepancy in the behavior of the constants is given. We also explain how the lower and upper bounds on the worst-case error in a Sobolev space setting can be extended to numerical integration over a general non-empty closed and connected Measurable Subset @W of S^r that is the closure of an open set.
Władysław Wilczyński - One of the best experts on this subject based on the ideXlab platform.
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Density topologies on the plane between ordinary and strong. II
Tatra Mountains Mathematical Publications, 2015Co-Authors: Elżbieta Wagner-bojakowska, Władysław WilczyńskiAbstract:Abstract Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a Measurable Subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all Measurable Subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.
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A difference between the sets of ordinary and strong density points on the plane
Mathematica Slovaca, 2012Co-Authors: Grażyna Horbaczewska, Władysław WilczyńskiAbstract:It is well known that the difference between the set of all ordinary density points and the set of all strong density points of an arbitrary Measurable Subset of the plane is a null set. It is of interest to check how large can a difference between sections of these sets be. Both measure and category cases are considered.
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Density topologies on the plane between ordinary and strong
Tatra Mountains Mathematical Publications, 2009Co-Authors: Elżbieta Wagner-bojakowska, Władysław WilczyńskiAbstract:Let C0 denote the set of all non-decreasing continuous functions $f : (0, 1] \to (0, 1]$ such that $lim_x \to 0 + f(x) = 0$ and $f(x) \leq x$ for every $x \in (0, 1]$ and let $A$ be a Measurable Subset of the plane. The notions of a density point of $A$ with respect to $f$ and the mapping $D_f$ defined on the family of all Measurable Subsets of the plane were introduced in [3]. This mapping is a lower density, so it allowed us to introduce the topology $\mathcal{T}_f$ , analogously to the density topology. In this note the properties of the topology $\mathcal{T}_f$ and functions approximately continuous with respect to $f$ are considered. We prove that $(\mathbb{R}^2, \mathcal{T}_f)$ is a completely regular topological space and we study conditions under which topologies generated by two functions $f$ and $g$ are equal.
Emmanuel Trélat - One of the best experts on this subject based on the ideXlab platform.
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Geometric and probabilistic results for the observability of the wave equation
2019Co-Authors: Emmanuel Humbert, Yannick Privat, Emmanuel TrélatAbstract:Given any Measurable Subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation Subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}. In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic. The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random Subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.
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Observability properties of the homogeneous wave equation on a closed manifold
Communications in Partial Differential Equations, 2019Co-Authors: Emmanuel Humbert, Yannick Privat, Emmanuel TrélatAbstract:We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a Measurable Subset $\omega$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and if the pair $(\omega,T)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $\omega \times (0, T)$. The observability constant $C_T( {\omega})$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a Measurable Subset $\omega$. We also establish that, as $T\rightarrow+\infty$, the ratio $C_T( {\omega})/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions; the second one is of a geometric nature and involves the average time spent in $\omega$ by Riemannian geodesics.
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Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Journal of the European Mathematical Society, 2016Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the wave and Schrödinger equations on a bounded open connected Subset $\Omega$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a Measurable Subset $\omega$ of $\Omega$ during a time interval $[0, T]$ with $T>0$. It is well known that, if the pair $(\omega,T)$ satisfies the Geometric Control Condition ($\omega$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in $\omega \times (0, T)$. We address the problem of the optimal location of the observation Subset $\omega$ among all possible Subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the Subset $\omega$ has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions on $\Omega$, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain $\Omega$). We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures. Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
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Complexity and regularity of maximal energy domains for the wave equation with fixed initial data
Discrete and Continuous Dynamical Systems - Series A, 2015Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the homogeneous wave equation on a bounded open connected Subset $\Omega$ of $\R^n$. Some initial data being specified, we consider the problem of determining a Measurable Subset $\omega$ of $\Omega$ maximizing the $L^2$-norm of the restriction of the corresponding solution to $\omega$ over a time interval $[0,T]$, over all possible Subsets of $\Omega$ having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components.