On the null controllability of a one-dimensional fluid–solid interaction model

We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. The fluid is governed by the Burgers equation and the control is exerted on the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities and fixed point arguments. To cite this article: A. Doubova, E. Fernández-Cara, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Spectral controllability for 2D and 3D linear Schrödinger equations

We consider a quantum particle in an infinite square potential well of Rn, n=2,3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: (Kal) if Ω is the bottom of the well, then for every eigenvalue λ of , the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in Rn. In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time Tmin(Ω)>0 for spectral controllability, i.e., if T>Tmin(Ω), one has spectral controllability in time T if condition (Kal) holds true for (Ω) and, if T<Tmin(Ω) and the dipolar moment is one-directional, then one does not have spectral controllability in time T. We next characterize a necessary and sufficient condition on the dipolar moment insuring that spectral controllability in time T>Tmin(Ω) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory.     

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.     

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).     

A numerical study of the null boundary controllability of a convection diffusion equation

In this paper we study numerically the cost of the null controllability of a linear control parabolic 1-D equation as the diffusion coefficient tends to 0. For this linear control parabolic 1-D equation, we know from a prior work by J.-M. Coron and S. Guerrero (2005), that, when the diffusion coefficient tends to 0, for a small controllability time, the norm of the optimal control tends to infinity and that, if the controllability time is large enough, this norm tends to 0. For controllability times which are not covered by this work, we estimate numerically the norm of the optimal control as the diffusion coefficient tends to 0. To cite this article: A. Salem, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Simultaneous local exact controllability of 1D bilinear Schrödinger equations

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is the N  -tuple of wave functions. The control is the real amplitude of the laser field. For N=1$N=1$, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N?2$N?2$. Still, for N=2$N=2$, we prove that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. This is proved using Coron's return method. We also prove that for N?3$N?3$, local controllability does not hold in small time even up to a global phase. Finally, for N=3$N=3$, we prove that local controllability holds up to a global phase and a global delay.     

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).     

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).     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

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).     

Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system

We analyze the inverse problem of the identification of a rigid body immersed in a fluid governed by the stationary Boussinesq system. First, we establish a uniqueness result. Then, we present a new method for the partial identification of the body. The proofs use local Carleman estimates, differentiation with respect to domains, data assimilation techniques and controllability results for PDEs. To cite this article: A. Doubova et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

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).     

Controllability of the Ginzburg–Landau equation

This Note investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Null-controllability results are derived from a Carleman estimate and an analysis based upon the theory of sectorial operators. To cite this article: L. Rosier, B.-Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A Kalman-type condition for stochastic approximate controllability

We are interested in the approximate controllability property for a linear stochastic differential equation. For deterministic control necessary and sufficient criterion exists and is called Kalman condition. In the stochastic framework criteria are already known either when the control fully acts on the noise coefficient or when there is no control acting on the noise. We propose a generalization of Kalman condition for the general case. To cite this article: D. Goreac, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Carleman estimates and null controllability for boundary-degenerate parabolic operators

Motivated by several examples coming from physics, biology, and economics, we consider a class of parabolic operators that degenerate at the boundary of the space domain. We study null controllability by a locally distributed control. For this purpose, a specific Carleman estimate for the solutions of degenerate adjoint problems is proved. To cite this article: P. Cannarsa et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the controllability for Maxwell's equations in specific media

In this Note we study the problem of exact controllability of the Maxwell's equations in specific media with two different models, on the one hand the so-called Drude–Born–Fedorov model, in the time domain, and on the other hand a simplified bilinear medium.For the first one we prove the non approximate controllability whereas for the second one we are able to prove the exact controllability under the usual conditions of the wave equation. To cite this article: P. Courilleau, T. Horsin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the Boyd–Kadomtsev system for the three-wave coupling problem

The three-wave coupling system is widely used in plasma physics, specially for the Brillouin instability simulations. We study here a related system obtained with an infinite speed of light. After showing that it is well posed, we propose a numerical method which is based on an implicit time discretization. This method is illustrated on test cases and an extension to the problem with finite speed of light is proposed. To cite this article: R. Sentis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Constructive solution of a bilinear control problem

We present an optimization method of a quantum control problem giving rise to a sequence of controls increasing monotonically the values of a cost functional. We first claim some results about the regularity of this cost functional. Those enable to extend an inequality due to ?ojasiewicz to the infinite dimensional case. Lastly, a sequence of inequalities proving the Cauchy character of the monotonic sequence is obtained, and we can also estimate the rate of convergence. The detailed proof will be given in [L. Baudouin, J. Salomon, Constructive solution of a bilinear quantum control problem, 2005, in preparation. [3]]. To cite this article: L. Baudouin, J. Salomon, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On infinite graphs with a specified number of colorings

In this paper we construct a planar graph of degree four which admits exactly N_u 3-colorings, we prove that such a graph must have degree at least four, and we consider various generalizations. We first allow our graph to have either one or two vertices of infinite degree and/or to admit only finitely many colorings and we note how this effects the degrees of the remaining vertices. We next consider n-colorings for n>3, and we construct graphs which we conjecture (but cannot prove) are of minimal degree. Finally, we consider nondenumerable graphs, and for every 3 <n<ω and every infinite cardinal k we construct a graph of cardinality k which admits exactly kn-colorings. We also show that the number of n-colorings of a denumerable graph can never be strictly between N_u and 2^N_u and that an appropriate generalization holds for at least certain nondenumerable graphs.