Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.Dans cet article nous étudions le lien entre la contrôlabilité à zéro d'un problème parabolique abstrait et la validité d'une inégalité qui est une version quantifiée du test de Fattorini–Hautus. Nous prouvons que cette inégalité permet de caractériser l'existence d'un temps minimal pour le problème de contrôlabilité à zéro et, selon les hypothèses considérées, d'obtenir la valeur de ce temps minimal. Nous prouvons aussi que dans la plupart des exemples connus de problèmes paraboliques ayant un temps minimal de contrôle à zéro, cette inégalité est une condition nécessaire et suffisante de contrôlabilité.Ministerio de Economía y Competitivida

Partial null controllability of parabolic linear systems

International audienceThis paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain Ω in R N (N ∈ N *), we study the effect of m localized controls in a nonempty open subset ω only controlling p components of the solution (p, m n). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2 × 2 parabolic system, we will give positive and negative results on partial null controllability when the coefficients are space dependent

A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems

In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n, with t0 ∈ [0,T], is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A|B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem.Agence Nationale de la rechercheDirección General de Enseñanza Superio

Contrôlabilité de systèmes gouvernés par des équations aux dérivées partielles

Contrôlabilité de systèmes gouvernés par des équations aux dérivées partiellesControllability of systems governed by partial differential equationsBESANCON-Bib. Electronique (250560099) / SudocSudocFranceF

A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems

We present a generalization of the Kalman rank condition to the case of $n\times n$ linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar $2n-$order parabolic equation and deduce from it an observability inequality for our adjoint system

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

International audienceWe consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllabil-ity. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0

Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences

International audienceLet (A,D(A)) a diagonalizable generator of a C0−semigroup of contractions on a com- plex Hilbert space X, B ∈ L(C, Y ), Y being some suitable extrapolation space of X, and u ∈ L2(0,T;C). Under some assumptions on the sequence of eigenvalues Λ = {λk}k≥1 ⊂ C of (A, D(A)), we prove the existence of a minimal time T0 ∈ [0, ∞] depending on Bernstein's con- densation index of Λ and on B such that y′ = Ay+Bu is null-controllable at any time T > T0 and not null-controllable for T < T0. As a consequence, we solve controllability problems of various systems of coupled parabolic equations. In particular, new results on the boundary controllability of one-dimensional parabolic systems are derived. These seem to be difficult to achieve using other classical tools