75 research outputs found
Boundary stabilization and control of wave equations by means of a general multiplier method
We describe a general multiplier method to obtain boundary stabilization of
the wave equation by means of a (linear or quasi-linear) Neumann feedback. This
also enables us to get Dirichlet boundary control of the wave equation. This
method leads to new geometrical cases concerning the "active" part of the
boundary where the feedback (or control) is applied. Due to mixed boundary
conditions, the Neumann feedback case generate singularities. Under a simple
geometrical condition concerning the orientation of the boundary, we obtain a
stabilization result in linear or quasi-linear cases
Variants of global Carleman weights in one-measurement inverse problems and fluid-structure controllability problems
We review some recent results on variants of global Carleman weights and Carleman inequalities applied to singular controllability and inverse problems partially developed in collaboration with the authors in a series of papers. First of all, we explain how we can modify weights to study one measurement inverse problems for the heat and wave equations with discontinuous coefficients in the principal part, in a case of locally supported boundary observations for recovering coefficients in the wave equation and we mention also some recent results for the Sch¨odinger equation. As another important application, we show how time-variable global Carleman weights are applied to study the null- controllability for a Navier-Stokes-rigid solid problem in moving domains
Exact Controllability of the Time Discrete Wave Equation: A Multiplier Approach
In this paper we summarize our recent results on the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. It is shown that the projection of the solution in an appropriate space in which the high frequencies have been filtered is exactly controllable with uniformly bounded controls (with respect to the time-step). By classical duality arguments, the problem is reduced to a boundary observability inequality for a time-discrete wave equation. Using multiplier techniques the uniform observability property is proved in a class of filtered initial data. The optimality of the filtering parameter is also analyzed
Energy decay for solutions of the wave equation with general memory boundary conditions
We consider the wave equation in a smooth domain subject to Dirichlet
boundary conditions on one part of the boundary and dissipative boundary
conditions of memory-delay type on the remainder part of the boundary, where a
general borelian measure is involved. Under quite weak assumptions on this
measure, using the multiplier method and a standard integral inequality we show
the exponential stability of the system.
Some examples of measures satisfying our hypotheses are given, recovering and
extending some of the results from the literature.Comment: 14 pages, submitted to Diff. Int. Eq
Approximate controllability for a linear model of fluid structure interaction
Abstract. We consider a linear model of interaction between a viscous incompressible fluid and a thin elastic structure located on a part of the fluid domain boundary, the other part being rigid. After having given an existence and uniqueness result for the direct problem, we study the question of approximate controllability for this system when the control acts as a normal force applied to the structure. The case of an analytic boundary has been studied by Lions and Zuazua in [9] where, in particular, a counterexample is given when the fluid domain is a ball. We prove a result of approximate controllability in the 2d-case when the rigid and the elastic parts of the boundary make a rectangular corner and if the control acts on the whole elastic structure. Resume. Nous considerons un modele lineaire d’interaction entre un fluide visqueux incompressible et une structure elastique mince situee sur une partie de la frontiere du domaine fluide, l’autre partie de la frontiere etant rigide. Apres avoir donne un resultat d’existence et d’unicite pour le probleme direct, nous etudions la question de la contrôlabilite approchee pour ce systeme lorsque le contrôle agit comme une force normale appliquee a la structure. Le cas d’une frontiere analytique a ete etudie par Lions et Zuazua dans [9] ou, en particulier, un contre exemple est donne lorsque le domaine fluide est une boule. Nous montrons un resultat de contrôlabilite approchee dans le cas 2d quand les parties rigide et elastique de la frontiere forment un angle droit et si le contrôle agit sur toute la structur
Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers
The rotated multipliers method is performed in the case of the boundary
stabilization by means of a(linear or non-linear) Neumann feedback. this method
leads to new geometrical cases concerning the "active" part of the boundary
where the feedback is apllied. Due to mixed boundary conditions, these cases
generate singularities. Under a simple geometrical conditon concerning the
orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure
Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation
We establish geometrical conditions for the inverse problem of determining a stationary potential in the wave equation with Dirichlet data from a Neumann measurement on a suitable part of the boundary. We present the stability results when we measure on a part of the boundary satisfying a rotated exit condition. The proofs rely on global Carleman estimates with angle type dependence in the weight functions
On Carleman and observability estimates for wave equations on time-dependent domains
We establish new Carleman estimates for the wave equation, which we then
apply to derive novel observability inequalities for a general class of linear
wave equations. The main features of these inequalities are that (a) they apply
to a fully general class of time-dependent domains, with timelike moving
boundaries, (b) they apply to linear wave equations in any spatial dimension
and with general time-dependent lower-order coefficients, and (c) they allow
for significantly smaller time-dependent regions of observations than allowed
from existing Carleman estimate methods. As a standard application, we
establish exact controllability for general linear waves, again in the setting
of time-dependent domains and regions of control.Comment: 62 pages. Accepted versio
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