On établit ici, suivant [5], une inégalité de Carleman globale optimale pour les solutions faibles (au sens ) d’équations elliptiques générales avec second membre dans et trace non nulle.
La motivation, qui est expliquée dans l’introduction, réside dans l’obtention d’inégalités de Carleman globale pour l’opérateur de Navier-Stokes linéarisé afin, notamment, d’étudier les questions de contrôlabilité exacte sur les trajectoires pour les équations de Navier-Stokes. Une étape majeure...
We study a non standard unique continuation property for the biharmonic spectral problem in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle , and , a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes...
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,...
We study a non standard unique continuation property for the
biharmonic spectral problem in a 2D
corner with homogeneous Dirichlet boundary conditions and a
supplementary third order boundary condition on one side of the
corner. We prove that if the corner has an angle ,
and , a unique continuation
property holds. Approximate controllability of a 2-D linear
fluid-structure problem follows from this property, with a control
acting on the elastic side of a corner in a domain containing...
Data assimilation refers to any methodology that uses partial observational data and the dynamics of a system for estimating the model state or its parameters. We consider here a non classical approach to data assimilation based in null controllability introduced in [Puel, 335 (2002) 161–166] and [Puel, 48 (2009) 1089–1111] and we apply it to oceanography. More precisely, we are interested in developing this methodology to recover the unknown final state value (state value at the end of the measurement...
Data assimilation refers to any methodology that uses partial
observational data and the dynamics of a system for estimating the
model state or its parameters. We consider here a non classical
approach to data assimilation based in null controllability
introduced in [Puel,
(2002) 161–166] and [Puel,
(2009) 1089–1111] and we apply it to oceanography.
More precisely, we are interested in developing this methodology
to recover the unknown final state value (state...
In this paper, we prove the global null controllability of
the linear heat equation completed with linear Fourier
boundary conditions of the form
.
We consider distributed controls with support in a small set and
nonregular coefficients .
For the proof of null controllability, a crucial tool will be a new
Carleman estimate for the weak solutions of the classical heat
equation with
nonhomogeneous Neumann boundary conditions.
This paper is concerned with the global exact controllability of
the semilinear heat equation (with nonlinear terms involving the state and
the gradient) completed with boundary conditions of the form .
We consider distributed controls, with support in a small set.
The null controllability of similar linear systems has been analyzed
in a previous first part of this work.
In this second part we show that, when the nonlinear terms are
locally Lipschitz-continuous and slightly superlinear, one...
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