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

Boundary Sentinels with Given Sensitivity

The notion of sentinels with given sensitivity was introduced by J.-L. Lions in [10] in order to identify parameters in a problem of pollution ruled by a semilinear parabolic equation. He proves that the existence of such sentinels is reduced to the solution of exact controllability problem with constraints on the state. Reconsidering this notion of sentinels in a more general framework, we prove the existence of the new sentinels by solving a boundary null-controllability problem with constraint on the control. Our results use a Carleman inequality which is adapted to the constraint.The notion of sentinels with given sensitivity was introduced by J.-L. Lions in [10] in order to identify parameters in a problem of pollution ruled by a semilinear parabolic equation. He proves that the existence of such sentinels is reduced to the solution of exact controllability problem with constraints on the state. Reconsidering this notion of sentinels in a more general framework, we prove the existence of the new sentinels by solving a boundary null-controllability problem with constraint on the control. Our results use a Carleman inequality which is adapted to the constraint

Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficientes

The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y| log3/2(1 + |y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.Ministerio de Educación y CienciaFondo Nacional de Desarrollo Científico y Tecnológico (Chile

Null controllability of the heat equation with boundary Fourier conditions: the linear case

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n + β y = 0. We consider distributed controls with support in a small set and nonregular coefficients β = β(x, t). 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.Ministerio de Educación y Cienci

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

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 ∂y ∂n + f(y) = 0. 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 has exact controllability to the trajectories.Ministerio de Educación y Cienci

Some controllability results for the N-Dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls

In this paper we deal with some controllability problems for systems of the Navier-Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control N-dimensional systems (N + 1 scalar unknowns in the case of the Navier–Stokes equations) with N − 1 scalar control functions. In a first step, we present some global Carleman estimates for suitable adjoint problems of linearized Navier–Stokes and Boussinesq systems. In this way, we obtain null controllability properties for these systems. Then, we deduce results concerning the local exact controllability to the trajectories. We also present (global) null controllability results for some (truncated) approximations of the Navier–Stokes equations.Ministerio de Educación y Cienci