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Kaïs Ammari

Publications and source records attributed to Kaïs Ammari.

At least 19 recordsLinked to original sources
arXiv2026

Energy decay and weak observability for some evolution systems

In an abstract functional framework, we investigate the equivalence between two fundamental properties: weak observability and energy decay associated with certain classes of dissipative operators. This analysis is motivated by the study of evolution equations and their long-time behavior. Our main result establishes that under appropriate assumptions, these two notions are not only closely related but in fact equivalent. The proof relies on recent advances in the theory of weak observability, particularly those presented in \cite{AT1}, where new techniques have been developed to quantify the weak observability inequalities. These tools allow us to bridge the gap between qualitative decay properties and spectral estimates, offering a unified perspective on stability and control in abstract dynamical systems.

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arXiv2026

Delay is Necessary for a Potential to Achieve Exponential Stabilization of the Wave Equation via Internal Control

In this work, we study the stabilization of the wave equation using an internal delayed potential. Interestingly, the stabilization mechanism is entirely induced by the delay, since exponential stabilization cannot be achieved in its absence. We first prove the well-posedness of the associated initial--boundary value problem. Then, thanks to the parametric analysis of the corresponding quasipolynomial, we design a delayed po tential feedback law which, together with appropriate initial conditions, ensures the exponential decay rate for the resulting closed-loop system. The control of the transverse vibration of a string illustrates the effectiveness of the result.

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arXiv2025

Numerical stabilization method by switching time-delay

In this paper, we propose a new numerical strategy for the stabilization of evolution systems. The method is based on the methodology given by Ammari, Nicaise andPignotti in ''Stabilization by switching time-delay, Asymptot. Anal., 83 (2013), 263--283''. This method is then implemented in 1D by suitable numerical approximation techniques. Numerical experiments complete this study to confirm the theoretical announced results.

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arXiv2025

The Graph Geometric Control Condition

In this paper, we introduce a novel concept called the Graph Geometric Control Condition (GGCC). It turns out to be a simple, geometric rewriting of many of the frameworks in which the controllability of PDEs on graphs has been studied. We prove that (GGCC) is a necessary and sufficient condition for the exact controllability of the wave equation on metric graphs with internal controls and Dirichlet boundary conditions. We then investigate the internal exact controllability of the wave equation with mixed boundary conditions and the one of the Schrödinger equation, as well as the internal null-controllability of the heat equation. We show that (GGCC) provides a sufficient condition for the controllability of these equations and we provide explicit examples proving that (GGCC) is not necessary in these cases.

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arXiv2024

Stability of a degenerate thermoelastic equation

This work is dedicated to the study of a linear model arising in thermoelastic rod of homogeneous material. The system is resulting from a coupling of a heat and a wave equation in the interval $(0,1)$ with Dirichlet boundary conditions at the outer endpoints where the parabolic component is degenerating at the end point $x=0$. Two models are considered the first is with weak degeneracy and the second is with strong degeneracy. We aim to study the well-posedness and asymptotic stability of both systems using techniques from the $C_{0}$-semigroup theory and a use a frequency domain approach based on the well-known result of Prüss in order to prove using some multiplier techniques that the energy of classical solutions decays uniformly as time goes to infinity.

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arXiv2024

Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends

We study the free Schrödinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the $L^1$ to $L^\infty$ time decay rate at least $t^{-1/2}$. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow between the bounded sub-graph and the unbounded ends.

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arXiv2024

Networks of sign-changing metamaterials: existence and spectral properties

We study composite assemblages of dielectrics and metamaterials with respectively positive and negative material parameters. In the continuum case, for a scalar equation, such media may exhibit so-called plasmonic resonances for certain values of the (negative) conductivity in the metamaterial. This work investigates such resonances, and the associated eigenfunctions, in the case of composite conducting networks. Unlike the continuous media, we show a surprising specific dependence on the geometry of the network of the resonant values. We also study how the problem is affected by the choice of boundary conditions on the external nodes of the structure.

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arXiv2024

Well-posedness and stability of abstract thermoelastic delayed systems

In this paper, we consider a stabilization problem of a generalized thermoelastic system (the so called $α-β$ system) with delay in a part of the coupled system. For each case, we prove the well-posedness of the corresponding system using semigroup approach, then under some sufficient conditions we establish some results of exponential and polynomial stability of the system through a frequency-domain approach. The results are applied to concrete examples in thermoelasticity.

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arXiv2024

Stability of the Rao-Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical study

This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $ξ$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.

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arXiv2024

On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

In this paper we perform the analysis of the spectrum of a degenerate operator $A_\var$ corresponding to the stationary heat equation in a $\var$-periodic composite medium having two components with high contrast diffusivity. We prove that although $ A_\var$ is a bounded self-adjoint operator with compact resolvent, the limits of its eigenvalues when the size $\var$ of the medium tends to zero, make up a part of the spectrum of a unbounded operator $ A_0$, namely the eigenvalues of $ A_0$ located on the left of the first eigenvalue of the bi-dimensional Laplacian with homogeneous Dirichlet condition on the boundary of the representative cell. We also show that the homogenized problem does not differ in any way from the one-dimensional problem obtained in the study of the local reduction of dimension induced by the homogenization.

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arXiv2024

Spectral analysis and best decay rate of the wave propagator on the tadpole graph

We consider the damped wave semigroup on the tadpole graph ${\mathcal R}$. We first give a meticulous spectral analysis, followed by a judicious decomposition of the resolvent's kernel. As a consequence, and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of the energy space $\mathcal{H}$, we establish the exponential decay of the corresponding energy, with the optimal decay rate dictated by the spectral abscissa of the relevant operator.

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arXiv2023

Uniform stabilization for the semi-linear wave equation with nonlinear Kelvin-Voigt damping

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood of the boundary according to the Geometric Control Condition. While the second one is a frictional damping and we consider it hurting the geometric condition of control. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability inequality which combines unique continuation properties and the tools of the Microlocal Analysis Theory.

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arXiv2023

Uniform stabilization of an acoustic system

We study the problem of stabilization for the acoustic system with a spatially distributed damping. With imposing hypothesis on the structural properties of the damping term, we identify exponential decay of solutions with growing time.

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arXiv2022

Note on stability of an abstract coupled hyperbolic-parabolic system: singular case

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^αw = 0, \\ w_t + A^αu_t + A^βw = 0,\\ u(0) = u_0, u_t(0) = u_1, w(0) = w_0, \end{array} \right. $$ where $A$ is a self-adjoint, positive definite operator on a complex Hilbert space $H$, and $(α, β) \in [0,1] \times [0,1]$, which is considered in \cite{Amk}, and after, in \cite{liu1}. Our contribution is to identify a fine scale of polynomial stability of the solution in the region $ S_3: = \left\{(α,β) \in [0,1] \times [0,1]; \, β< 2α-1 \right\}$ taking into account the presence of a singularity at zero.

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arXiv2022

Dispersion on certain Cartesian products of graphs

In this short note we prove a sharp dispersive estimate $\|\mathrm{e}^{\mathrm{i} tH} f\|_\infty < t^{-d/3}\|f\|_1$ for any Cartesian product $\mathbb{Z}^d\mathop\square G_F$ of the integer lattice and a finite graph. This includes the infinite ladder, $k$-strips and infinite cylinders, which can be endowed with certain potentials.

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arXiv2022

Positive and negative exact boundary controllability results for the linear Biharmonic Schrödinger equation

In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation $i\partial_ty=-\partial_x^4y+ γ\partial_x^2y$ on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the {left} endpoint, where the parameter $γ<0$. We prove that this system is exactly controllable in time $T>0$, if and only if, the parameter $γ$ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method.

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arXiv2022

Dispersion for Schrödinger operators on regular trees

We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schrödinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schrödinger operators on the real line. We establish a $t^{-3/2}$-decay for both models which is sharp, as we give the first-order asymptotics.

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