Remarks on exact controllability for Stokes and Navier–Stokes systems

This Note deals with the controllability of Stokes and Navier–Stokes systems with distributed controls with support in possibly small subdomains. We first present a new global Carleman inequality for the solutions to Stokes-like systems that leads to the null controllability at any time T>0. Then, we present a local result concerning exact controllability to trajectories of the Navier–Stokes system. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Remarks on global approximate controllability for the 2-D Navier–Stokes system with Dirichlet boundary conditions

This Note deals with the two-dimensional Navier–Stokes system. In this context, we prove a result concerning its global approximate controllability by means of boundary controls. To cite this article: S. Guerrero et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Application of the exact null controllability of the heat equation to moving sets

We study the lagrangian controllability of the heat equation in several dimensions. In dimension one, we prove that any pairs of intervals are diffeomorphic through the flow of the solution of the heat equation via an adequate control. In higher dimensions we prove a similar controllability result for the flow of the gradient of the solution in a radial case in arbitrary finite time, and for convex domains in a sufficiently large time. To cite this article: T. Horsin Molinaro, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Nulle contrôlabilité régionale d'une équation de type Crocco linéariséeRegional null controllability of a linearized Crocco type equation

We are interested in controllability problems of equations coming from a boundary layer model. This problem is described by a degenerate parabolic equation (a linearized Crocco type equation) where phenomena of diffusion and transport are coupled.First we give a geometric characterization of the influence domain of a locally distributed control. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space–time domain and Carleman type estimates along characteristics. To cite this article: P. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 581–584.     

Contrôlabilité et observabilité unilatérales de systèmes hyperboliques quasi-linéaires

The known theory on the one-side exact boundary controllability and the one-side exact boundary observability for first-order quasilinear hyperbolic systems requires that the unknown variables should be suitably coupled in the boundary conditions at the non-control or non-observation side. In this Note we illustrate, with an inspiring example, that the one-side exact boundary controllability and the one-side exact boundary observability can still be realized by means of a suitable coupling among the unknown variables in the quasilinear hyperbolic system itself. To cite this article: T. Li et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

Sur l'existence des opérateurs d'onde pour l'équation de Wigner dans les espaces L2,pExistence of wave operators for the Wigner equation in L2,p spaces

Basing on the formalism established by Markovich, we show the completeness of wave operators for the Wigner equation in L². In the second part, using estimations proved by Castella and Perthame on the one hand, and the L^p→L^q estimations for the Schrödinger group on the other hand, we prove the existence of the wave operators in L^2,p spaces. To cite this article: H. Emamirad, P. Rogeon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 811–816.     

Controllability of some coupled parabolic systems by one control force

In this Note we present a new approach which allows one to prove new controllability results for some coupled parabolic systems considered in a bounded domain Ω of ${\mathbb{R}}^N$ when one controls by a unique distributed control. We analyze, as a model example, the null controllability of a linear phase field system. First, one controls the system by two controls. Then, one eliminates the introduced fictitious control. Global Carleman estimates and the parabolic regularity are used. To cite this article: M. González-Burgos, R. Pérez-García, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws

In this paper, the one-sided exact boundary null controllability of entropy solutions is studied for a class of general strictly hyperbolic systems of conservation laws, whose negative (or positive) characteristic families are all linearly degenerate. The authors first prove the well-posedness of semi-global solutions constructed as the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions and they establish various properties of both the ε-approximate front tracking solutions and such solutions. By means of essential modifications of the strategy suggested by the first author in [17] originally for the local exact boundary controllability in the framework of classical solutions, the one-sided local exact boundary null controllability of entropy solutions can then be realized via boundary controls acting on one side of the boundary, where the incoming characteristics are all linearly degenerate.     

Exact controllability of a 3D piezoelectric body

In this Note we study the exact controllability of a three-dimensional body made of a material whose constitutive law introduces an elasticity-electricity coupling. We show that, without any geometrical assumption, two controls (the elastic and the electric controls) acting on the whole boundary drive the system to rest in finite time. To cite this article: I. Lasiecka, B. Miara, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Contrôlabilité exacte d'un problème de transmission

By using HUM, we prove the exact controllability for the transmission wave equation in the case a₁<a₂.     

Discrete Ingham inequalities and applications

In this Note we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical applications. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh size in suitable classes of numerical solutions in which the high frequency components have been filtered. To cite this article: M. Negreanu, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Controllability of systems of Stokes equations with one control force: existence of insensitizing controls

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L² norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when the coupling terms concern first and second order derivatives, respectively), we prove this result for some general couplings.     

Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to $h^2$, where h denotes the discretization step in space. Using a discrete version of Ingham's inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications

We prove that the Schrödinger equation is approximately controllable in Sobolev spaces Hs$H^s$, s>0$s>0$, generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere S   in L²$L^2$.     

Polynomial decay and control of a 1−d model for fluid–structure interaction

We consider a linearized and simplified 1?d model for fluid–structure interaction. The domain where the system evolves consists in two bounded intervals in which the wave and heat equations evolve respectively, with transmission conditions at the point of interface. First, we develop a careful spectral asymptotic analysis on high frequencies. Next, according to this spectral analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Finally, we prove the null-controllability of the system when the control acts on the boundary of the interval where the heat equation holds. The proof is based on a new Ingham-type inequality, which follows from the spectral analysis we develop and the null controllability result in Zuazua (in: J.L. Menaldi et al. (Eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198–210) where the control acts on the wave component. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur la densité d'état de l'opérateur de Schrödinger quasi-périodique unidimensionnel

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder-$\frac{1}{2}$ function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Uniform null controllability of the heat equation with rapidly oscillating periodic density

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 those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Contrôlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles

By means of the existence and uniqueness of semi-global C¹ solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues, we present a unified method to establish the exact boundary controllability for 1-D quasilinear wave equations with boundary conditions of different types. To cite this article: T.T. Li, L.X. Yu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).