Louis T. Tebou

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• Ph.D.
• Professor (Full) at Florida International University

We prove new Carleman estimates, (first with boundary observation, then with internal observation), for a transmission system of two nonconservative wave equations in the one-dimensional setting. The wave equations have nonconstant coefficients; this feature requires the introduction of weight functions adapted to the coefficients. Our Carleman est...

We study stability issues for a dynamical system consisting of a wave equation and a quasilinear parabolic equation. The nonlinearity involves the p-Laplacian, and the coupling involves a fractional Laplacian with exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{...

The presentation revolves around stability of regularity issues for semigroups corresponding to Euler-Bernoulli plate with localized structural of Kelvin-Voigt damping. New Gevrey regularity as well as exponential and polynomial stability results are established.

We consider an Euler-Bernoulli plate equation with Kelvin-Voigt damping in a bounded domain. The damping is localized in an appropriate open strict subset ω of the domain Ω. While it is known that the solutions of this model with a full damping ω = Ω generates an analytic semigroup, this property is no longer valid for locally distributed damping....

We consider two wave equations coupled through a singular Kelvin-Voigt damping mechanism in a bounded domain. We are interested in investigating stability issues for this system.We prove the polynomial stability of the semigroup if the damping region is big enough, and logarithmic stability of the semigroup if the damping region is an arbitrarily s...

We consider a structurally damped Euler-Bernoulli plate equation in a bounded domain. The damping is localized in an appropriate open subset of the domain. The damping coefficient is smooth and satisfies some structural conditions. Using the frequency domain approach combined with interpolation inequalities and multipliers technique, we show that t...

We consider two uncoupled wave equations with potentials on an interval; they both have the same Dirichlet boundary control at the left endpoint. First, we discuss the conservative case, and by transmutation, we obtain a simultaneous null controllability for the corresponding uncoupled heat equations. Afterward, in the nonconservative case, we prov...

In this note, first, we consider a one-dimensional fluid–structure model where the parabolic component is weakly degenerate, meaning that the degeneracy exponent lies in the open interval (0, 1). For this model, we show that the underlying semigroup is polynomially stable, and provide a decay rate of O(t-(2-α)1-α), which is conjectured to be optima...

This presentation discusses some semigroup stability and regularity results for some toy models of fluid-structure interaction. The parabolic component is degenerate and the stability and regularity estimates depend in a precise way on the degeneracy parameter.

In this paper, we examine regularity issues for two damped abstract elastic systems; the damping and coupling involve fractional powers μ,θ$$ \mu, \theta $$ of the principal operators, with 0<μ,θ≤1$$ 0<\mu, \theta \le 1 $$. The matrix defining the coupling and damping is nondegenerate. This new work is a sequel to the degenerate case that we discus...

In this note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power θ in [0, 1], The damping matrix is degenerate, which makes the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers...

We consider a damped wave equation in a bounded domain. The damping is nonlinear and is homogeneous with degree p -- 1 with p > 2. First, we show that the energy of the strong solution in the supercritical case decays as a negative power of t; the rate of decay is the same as in the subcritical or critical cases, provided that the space dimension d...

In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power \(\theta \), with \(\theta \) in \([-1,1]\), of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for...

In this paper, we examine regularity issues for two damped abstract elastic systems; the damping and coupling involve fractional powers $\mu, \theta$, with $0 \leq \mu , \theta \leq 1$, of the principal operators. The matrix defining the coupling and damping is nondegenerate. This new work is a sequel to the degenerate case that we discussed recent...

We consider a damped plate model with rotational forces in a bounded domain. The plate is either clamped or hinged. The rotational forces and damping involve the spectral fractional Laplacian with powers θ in [0, 1] and δ in [0, 2], respectively. Using the frequency domain approach, and appropriate interpolation inequalities, we show that the under...

We consider a system of thermoelastic plates with the same conductivity. The coupling in each plate component involves the average temperature of the system. We show that if the coefficients of flexural rigidity are pairwise distinct, then the system underlying semigroup is exponentially stable. This is done first for the Fourier model, and then fo...

In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power $\theta$, with $\theta$ in $[-1,1]$, of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for $\theta$...

In this talk, finite thin elastic plates involving fractional rotational forces are considered. Using resolvent estimates, new regularity and stability results for the underlying semigroups are established. First, I examine a thermoelastic plate with fractional rotational forces, and prove Gevrey regularity, as well as exponential stability of the...

We consider a coupled system consisting of a Kirchhoff thermoelastic plate and an undamped wave equation. It is known that the Kirchhoff thermoelastic plate is exponentially stable. The coupling is weak. First, we show that the coupled system is not exponentially stable. Afterwards, we prove that the coupled system is polynomially stable, and provi...

In this note, we investigate stability issues for a mixture of two elastic solids. The governing equations consist of two coupled systems of elastodynamic equations with a weak damping. The damping involves the difference of velocities, so that the damping operator is degenerate. First, we show that the underlying semigroup is strongly stable under...

We consider a locally damped wave equation in a bounded domain. The damping is nonlinear, involves the Laplace operator, and is localized in a suitable open subset of the domain under consideration. First, we discuss the well-posedness and regularity of the solutions of the system by using a combination of the nonlinear semigroup theory and the Fae...

In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $0\le\ta\le 1$. The model includes both the Euler-Bernoulli ($\ta=0$) and Kirchhoff ($\ta=1$) models for thermoelastic plate as special cases. First, we show that the underlying semigro...

In this presentation, I discuss some recent simultaneous stabilization results for multi-component systems, including elastodynamic systems, plate/wave systems, Timoshenko beam.

In this presentation, I discuss some indirect control problems involving coupled systems of wave equations, the Mindlin-Timoshenko plate, and a weakly coupled system of plate/wave equations.

In this presentation, I discuss some open problems on the simultaneous and indirect control of second order evolution systems.

We investigate the indirect stabilization of a weakly coupled system consisting of a Kirchhoff plate equation involving free boundary conditions and the wave equation with Dirichlet boundary conditions in a bounded domain. The distributed damping is frictional and appears in one of the equations only. First, we consider the case where the damping o...

We consider a linear Euler–Bernoulli beam equation introduced by Russell (Appl Math Optim 46:291–312, 2002) that equation describes the motion of a tape moving axially between two sets of rollers. We discuss an underlying hybrid model accounting for the mass of the roller assembly and show that the system is uniformly stable; this result generalize...

This presentation reviews old and recent results about the stabilization of the wave equation with localized damping.

In this note, we investigate stability issues for a Mindlin-Timoshenko plate with internal dissipation distributed everywhere in the domain under consideration. The damping occurs only in the elasticity equations describing the motion of the angles of rotation of a filament; the vertical deflection equation has no damping, but the effect of the dam...

We consider the wave equation with two types of locally distributed damping mechanisms: a frictional damping and a Kelvin-Voigt type damping. The location of each damping is such that none of them alone is able to exponentially stabilize the system; the main obstacle being that there is a quite big undamped region. Using a combination of the multip...

In this paper, we study the existence at the H¹-level as well as the stability for the damped defocusing Schrödinger equation in Rd. The considered damping coefficient is time-dependent and may vanish at infinity. To prove the existence, we employ the method devised by Özsarı, Kalantarov and Lasiecka [27], which is based on monotone operators theor...

We consider the wave equation with mixed boundary conditions in a bounded domain; on one portion of the boundary, we have dynamic Wentzell boundary conditions, and on the other portion, we have homogeneous Dirichlet boundary conditions. First, using an appropriate geometric partition of the boundary, we prove some Carleman estimates for this system...

We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show th...

The feedback control of interacting waves through the usual transmission conditions are known to be quite challenging especially when the two wave equations involved have different speeds of propagation. In this contribution, we consider a thermoelasticity system composed of a wave equation and a parabolic equation in a bounded domain surrounding a...

We consider the exact controllability problem for some uncoupled semilinear wave equations with proportional, but different principal operators in a bounded domain. The control is locally distributed, and its support satisfies the geometric control condition of Bardos-Lebeau-Rauch. First, we examine the case of a nonlinearity that is asymptotically...

The stabilization of the Timoshenko beam system with localized damping is examined. The damping involves the sum of the bending and shear angle velocities; this work generalizes an earlier result of Haraux, established for a system of ordinary wave equations, to the Timoshenko system. First, we show that strong stability holds if and only if the bo...

We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introd...

In a bounded domain, we consider an Euler-Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero....

We consider a system consisting of a wave equation and a plate equation, where the rotational forces may be accounted for, in a bounded domain. The system is coupled through the dissipation which is localized in an appropriate portion of the domain under consideration. First, we show that this system is strongly stable; the feedback control region...

We consider a system of two coupled nonconservative wave equations. For this system, we prove several observability estimates. Those observability estimates are sharp in the sense that they lead by duality to the controllability (exact or approximate) of the coupled system with a single control acting through one of the equations only while keeping...

First, we discuss the simultaneous controllability of uncoupled wave equations with different speeds of propagation in a bounded domain. The control is locally distributed, and the control region satisfies the geometric control condition of Bardos-Lebeau-Rauch. Thanks to the Hilbert uniqueness method of Lions, the controllability problem is reduced...

We consider the wave equation with Kelvin-Voigt clamping in a bounded domain. The clamping is localized in a suitable open subset of the domain under consideration. The exponential stability result proposed by Liu and Rao for that system assumes that the damping is localized in a neighborhood of the whole boundary, and the damping coefficient is co...

We consider an Euler-Bernoulli equation in a bounded domain with a local dissipation of viscoelastic type involving the p-Laplacian. The dissipation is effective in a suitable nonvoid subset of the domain under consideration. This equation corresponds to the plate equation with a localized structural damping when both the parameter p and the space...

We consider the Euler-Bernoulli equation coupled with a wave equation in a bounded domain. The Euler-Bernoulli has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is distributed everywhere in the domain under consideration acts through one of the equations only; its effect is transmitted to the...

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos-Lebeau-Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations. we prove an observability estimate for that system. Applying Lions Hilbert uniqueness method (...

We consider a parabolic equation with fast oscillating periodic coefficients, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new in a bounded domain, and their proof relies on a judicious smoothing of t...

First, we consider a semilinear hyperbolic equation with partially known initial data in a bounded domain. For this system, we construct a locally distributed control that desensitizes a certain norm of the state. This result is new, and the method for proving it combines a judicious application of the Carleman estimate and a localization technique...

First, we consider a coupled system consisting of the wave equation and the heat equation in a bounded domain. The coupling involves an operator parametrized by a real number $\mu$ lying in the interval [0,1]. We show that for $0\leq\mu<1$, the associated semigroup is not uniformly stable. Then we propose an explicit non-uniform decay rate. For $\m...

We consider an N-dimensional plate equation in a bounded domain with a locally distributed nonlinear dissipation involving the Laplacian. The dissipation is effective in a neighborhood of a suitable portion of the boundary. When the space dimension equals two, the associated linear equation corresponds to the plate equation with a localized viscoel...

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use...

We consider an abstract second order semilinear evolution equation with a bounded dissipation. We establish an equivalence between the stabilization of this system and the observability of the corresponding undamped system. Our technique of proof relies on an appropriate decomposition of the solution, and the energy method. Our result gener-alizes...

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007)...

We consider the wave equation with partially known initial data in a bounded domain. For this system, we construct locally distributed controls that desensitize a certain norm of the state. This result is new in space dimensions greater than one. The method of proof combines a judicious application of the Carleman estimate, and a localization techn...

We consider the dynamic elasticity equations with a locally distributed damping in a bounded domain. The local dissipation of the form a(x)yt involves coefficients a that vanish on a negligible portion of the subset where the damping is effective. Using multiplier techniques, interpolation inequalities, and a judicious application of Hölder inequal...

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of...

We consider the dynamic elasticity equations with a locally distributed damping in a bounded domain. The local dissipation of the form $a(x)y_{t}$ allows coefficients $a$ that lie in some $L^r(\Omega)$, with $(r>2)$. Using multiplier techniques, interpolation inequalities, and a judicious application of the Hölder inequality, we prove sharp energy...

First, we consider a semilinear wave equation with a locally distributed damping in a bounded domain. Using the Carleman estimate, we devise an elementary proof of the exponential decay of the energy of this system. Afterwards we apply the same technique to the stabilization of the same type of equation in the whole space. Our proofs are constructi...

In this paper we study the null-controllability of a beam equa- tion with hinged ends and structural damping, the damping depending on a positive parameter. We prove that this system is exactly null controllable in arbitrarily small time. This result is proven using a combination of Ingham- type inequalities, adapted for complex frequencies, and ex...

We consider the 1-D finite-difference space semi-discretization of the heat equation with locally distributed control. First, using a result of Russell and Fattorini on biorthogonal series and a seemingly new trigonometrical inequality, we prove the uniform (with respect to the step size) null controllability of this system. Then we show that the s...

In this article we study a class of robust control problems in fluid mechanics recently proposed in [T. R. Bewley, R. Temam and M. Ziane, Physica D 138, No. 3–4, 360–392 (2000; Zbl 0981.76026)]. Using a method of [V. P. Shutyaev, Russ. J. Numer. Anal. Math. Model. 14, No. 2, 137–176 (1999; Zbl 0931.65069)], we provide another proof of the existence...

In this article we study the convergence of an adjoint-based iterative method recently proposed in [T. R. Bewley, R. Temam, and M. Ziane, Phys. D, 138 (2000), pp. 360--392] for the numerical solution of a class of nonlinear robust control problems in fluid mechanics. Under weaker assumptions than those of [T. Tachim Medjo, Numer. Funct. Anal. Optim...

We consider the Euler{Bernoulli equation in a bounded domain › with a local dissipation ay0. The localizing coe-cient a is of the form a(x) = fi(x)=(d(x;¡))s, (0

We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to...

We study the limit behavior the solution of a nonlinear problem in an open set perforated periodically. Results of homogenization that we obtain generalize anterior results of Cioranescu - Donato [3]. In addition to the homogenization results, a theorem of existence and uniqueness of the solution is proved.

We study the limit behavior the solution of a nonlinear problem in an open set perforated periodically. Results of homogenization that we obtain generalize anterior results of Cioranescu-Donato [3]. In addition to the homogenization results, a theorem of existence and uniqueness of the solution is proved.

We consider the wave equation with perturbations of order 4p and 4p + 2, pℕ0. First, we solve a problem raised by Lions in the second volume of his monograph on exact controllability. Second, we generalize his results to the case of perturbations of order greater than 4.

We consider the wave equation with perturbations of order 4p and 4p + 2, Ρ ∈ ℕ\{0}. First, we solve a problem raised by Lions in the second volume of his monograph on exact controllability. Second, we generalize his results to the case of perturbations of order greater than 4.

We prove some decay estimates of the energy of the wave equation in a bounded domain. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. The method of proof is direct and is based on the multipliers technique, on some integral inequalities due to Haraux and Komornik, and on a judicious idea of Con...

In a bounded domain Ω, we consider the wave equation with a local dissipation ay′. Tlw function a belongs to with r greater than 2. We prove that the system is well-posed and we also prove the polynomial decay of its energy. The results obtained improve the existing results proved for Key words.wave equation, decay estimates, local dissipation, deg...

In a bounded domain, we consider the wave equation with a local dissipa- tion. We prove the polynomial decay of the energy for a degenerate dissipation and the exponential decay of the energy for a nondegenerate dissipation. The method of proof is direct and is based on multipliers technique, on some integral inequalities due to Haraux and on a jud...

We prove some decay estintate.s of the energy of the ware equation in a bounded domain. The damping is nonlinear and is effective only in a neighbourhood of a suitable subset of the boundary. The method of proof is direct and is based on multipliers technique and on some integral inequalities due to Haraux and Komornik.

We prove some decay estimates of the energy of the wave equation in a bounded domain. The damping is nonlinear and is effective only in a neighbourhood of a suitable subset of the boundary. The method of proof is direct and is based on multipliers technique and on some integral inequalities due to Haraux and Komornik.

We consider the internal controllability of a generalized wave equation in a periodically perforated domain, with a Fourier type boundary condition on the boundary of the holes. First, we establish by the Hilbert uniqueness method, HUM, introduced by J.L. Lions, the existence of an exact control; afterwards, we prove that the sequence of controls w...

We study by the method H.U.M. of J. L. Lions, the exact internal controllability of the vibrations of a body of small thickness. Afterwards, we let the thickness parameter go to zero and we prove that the limit of the sequence of exact controls is she exact control of the 2-dimensional limit system. We also prove strong convergence results for the...

In [SIAM J. Control Optimization 29, No. 1, 197-208 (1991; Zbl 0749.35018)], V. Komornik gives a rapid decay result for the energy of the wave equation. This result is very interesting when the dimension of the space variables is greater than or equal to 3 according to the principle of Russell as well as the exact controllability results of Lions....

Cette thèse est constituée d'un ensemble de résultats sur la contrôlabilité exacte et la stabilisation ; elle comporte cinq chapitres. Dans les deux premiers chapitres on étudie la contrôlabilité exacte interne de l'équation des ondes dans des domaines perturbés en utilisant la méthode H.U.M. de J.L. Lions. Dans le troisième chapitre, on étudie la...