<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Local controllability and non-controllability for a 1D wave equation with bilinear control

We consider a linear wave equation, on the interval (0,1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.

•

For every T>2, this system is locally controllable in H³×H², in time T, with controls in L²((0,T),R).

•

For T=2, this system is locally controllable up to codimension one in H³×H², in time T, with controls in L²((0,T),R): the reachable set is (locally) a non-flat submanifold of H³×H² with codimension one.

•

For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L²((0,T),R), is contained in a non-flat submanifold of H³×H², with infinite codimension.

The proof of these results relies on the inverse mapping theorem and second order expansions.     

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.     

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.     

Uniform controllability properties for space/time-discretized parabolic equations

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ-schemes with \({\theta \in [\frac{1}2,1]}\) . For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions’ Hilbert Uniqueness Method strategy, exploiting the relaxed uniform observability estimate. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove an error bound between the fully discrete and the semi-discrete control functions. This bound is however not uniform with respect to the space discretization. The theoretical results are illustrated through numerical experimentations.     

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).     

Spectral analysis of certain compact factors for Gaussian dynamical systems

For factors of a Gaussian automorphismT determined by compact subgroups of the group of unitary operators acting onL ² of the spectral measure ofT, we prove that the maximal spectral multiplicity is either 1 or infinity. As an application, we show that the maximal multiplicity of those factors an allL ^p, 1<p<+∞, is the same.     

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]). Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L²(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L²-norm control. We illustrate the mathematical results with several numerical experiments. Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations". Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).     

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.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Approximate controllability and homogenization of a semilinear elliptic problem

The L²- and H¹-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L²-approximate controllability or is locally distributed in the case of H¹-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.     

Linearization of analytic vector fields in the transitive case

Given a nonlinear control system, linear in the controls, all of whose terms have a common critical point, Lie algebraic conditions are established for the existence of a real-analytic transformation to coordinates in which the system is bilinear, that is, of type $\text{dx}\text{dt}\text{= ∑ u}{}_{\mathrm{i}}\text{B}{}_{\mathrm{i}}\text{x}$. (¹) The hypotheses used are analyticity, transitivity of the Lie algebra L associated with (¹) (i.e., controllability of (¹)), and isomorphism of L to the Lie algebra of vector fields associated with the original nonlinear system. That the transitivity condition can be replaced by semisimplicity or compactness of L is known from work of Sternberg and Guillemin.     

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).     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

On the observability of time-discrete conservative linear systems

We consider various time discretization schemes of abstract conservative evolution equations of the form , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair (A,B) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443-471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.     

Approximate controllability for a system of Schrödinger equations modeling a single trapped ion

In this article, we analyze the approximate controllability properties for a system of Schrödinger equations modeling a single trapped ion. The control we use has a special form, which takes its origin from practical limitations. Our approach is based on the controllability of an approximate finite dimensional system for which one can design explicitly exact controls. We then justify the approximations which link up the complete and approximate systems. This yields approximate controllability results in the natural space (L²(R))²$(L^2{(\mathbb{R}))}^2$ and also in stronger spaces corresponding to the domains of powers of the harmonic operator.     

Operators associated with soft and hard spectral edges from unitary ensembles

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy-Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy-Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann-Weyl anti-commutation relations and leave invariant the subspaces of L² that are the ranges of projections given by the Tracy-Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.     

Time optimal boundary controls for the heat equation

The fact that the time optimal controls for parabolic equations have the bang–bang property has been recently proved for controls distributed inside the considered domain (interior control). The main result in this article asserts that the boundary controls for the heat equation have the same property, at least in rectangular domains. This result is proved by combining methods from traditionally distinct fields: the Lebeau–Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Müntz spaces and a generalization of Turán?s inequality, on the other side.