Ingham Inequality Research Articles

Indirect stability of the wave equation with a dynamic boundary control

AbstractIn this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in of the energy. In a second step, under appropriated conditions on the boundary, called the multiplier control conditions, we establish a polynomial decay in of the energy. Later, we show in a particular case that such a polynomial decay is available even if the previous conditions are not satisfied. For this aim, we consider our system on the unit square of the plane. Using a method based on a Fourier analysis and a specific analysis of the obtained 1‐d problems combining Ingham's inequality and an interpolation method, we establish a polynomial decay in of the energy for sufficiently smooth initial data. Finally, in the case of the unit disk, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained decay is optimal in the domain of the operator.

Prony’s Method Under an Almost Sharp Multivariate Ingham Inequality

The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.

Exact controllability for quasilinear perturbations of KdV

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by H\"ormander.

MUSIC for single-snapshot spectral estimation: Stability and super-resolution

This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set.In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix.The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.

Attachment observability of a rotating body-beam

Abstract In this article we analyze the admissibility and the exact observability of a body-beam system when the output is taken in a point of attachment from the beam to the body. The single output case chosen here is the practical measurement of the strength, its velocity or its moment. We prove the exact observability for the moment and the admissibility for the other cases. These results are obtained by the spectral properties of rotating body beam system operator and Ingham's inequalities.

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuni- form meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work (Castro and Micu, Numer. Math. 102 (2006) 413-462) to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.

Discrete Ingham Inequalities and Applications

In this paper we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer, the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical issues. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh-size in suitable classes of numerical solutions in which the high frequency components have been filtered. We also discuss the optimality of these results in connection with the dispersion diagrams of the numerical schemes.

A uniformly controllable and implicit scheme for the 1-D wave equation

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt . We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h 2 and Δt 2 . Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2 (0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L 2 -norm control. The results are illustrated with several numerical experiments.

Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to h 2 , where h denotes the discretization step in space. Using a discrete version of Ingham's inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Discrete Ingham inequalities and applications

In this Note we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical applications. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh size in suitable classes of numerical solutions in which the high frequency components have been filtered. To cite this article: M. Negreanu, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Controllability of star-shaped networks of strings

In this note we consider a star-shaped network of vibrating strings. The problem of controllability when one control acts on the junction point is considered. A simple proof is given that, in particular, does not use Ingham inequalities, of the fact that the set of reachable data is dense, whenever the lengths of the strings are mutually irrational. The proof is based on an observability inequality with suitable weights on the Fourier coefficients that is easily obtained as a consequence of the time-periodicity of the solutions. Those weights can be estimated under certain algebraicity conditions imposed on the lengths of the strings. This allows to proof exact controllability results in Sobolev spaces of appropriate order. Further results are also presented concerning the control from a free end and diffusion processes.

Control of networks of Euler-Bernoulli beams

We consider the exact controllability problem by boundary action of hyperbolic systems of networks of Euler-Bernoulli beams. Using the multiplier method and Ingham's inequality, we give sufficient conditions insuring the exact controllability for all time. These conditions are related to the spectral behaviour of the associated operator and are sufficiently concrete in order to be able to check them on particular networks as illustrated on simple examples.

Analytic Controllability of a Linear Hybrid System

In this paper, we consider a boundary control problem for a model of a fluid-structure hybrid system. This model has been introduced by Micu and Zuazua in connection with the works of Banks et al. They have given explicit values for the spectral data in [S. Micu and E. Zuazua, SIAM J. Math. Anal., 29 (1998), pp. 967--1001] and results for the control problem in [S. Micu and E. Zuazua, SIAM J. Control. Optim., 35 (1997), pp. 1614--1637]. In the latter paper, they use variable separation and Ingham inequalities to prove an observation estimate that implies, through the Hilbert uniqueness method, that initial data can be controlled within finite time. This paper improves these results by using a stronger form of Ingham inequality for low frequencies. Indeed, we prove that any analytic function is controlled in a finite time whose dependence on the analyticity can be sharply estimated.

Singular Internal Stabilization of the Wave Equation

We consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve. In the two dimensional case our main result asserts that generically (i.e., for almost all interior curves) the solutions decay to zero in the energy space. When the domain is strictly convex we show that, whatever the interior curve is, the decay is not uniform. We generalize in this way results known in one space dimension. Our main improvement of existing one-dimensional results consists in giving sharp decay rates, provided that the initial data are regular and the damping term is linear. A crucial intermediate step is the proof of a generalization of Ingham's inequality on nonharmonic Fourier series.