Sophie Dubuisson-Quellier, La Consommation engagée Compte-rendu de Julien Pongerard La consommation engagée, vaste ensemble de « formes de mobilisation » qui « ont en commun de faire de la consommation un mode d'expression de choix politiques » (p. 7), faisait en 2009 l'objet d'un ouvrage de Sophie Dubuisson-Quellier devenu depuis un manuel de référence sur le sujet1. À l'heure où la mobilisation des consommateurs semble étendre constamment son domaine -des labels du commerce équitable dans les rayons des supermarchés aux appels au boycott sur les réseaux sociaux, en passant par les circuits courts de distribution -la réédition de cette synthèse didactique paraît opportune. Quoique l'actualisation de l'ouvrage reste limitée à quelques ajouts ponctuels et une réorganisation du dernier chapitre, elle permet à Dubuisson-Quellier de mettre l'accent sur des formes originales de mobilisation politique des consommateurs, par un changement de leurs modes de vie ; et d'insister sur des pistes de réflexion pour une étude critique de la consommation engagée. Bien qu'il ne présente pas de nouveaux résultats de recherche, l'ouvrage constitue toujours un excellent guide pour s'orienter dans le vaste univers des études sur l'engagement politique à travers la consommation, et ouvre à la voie à de futurs travaux.

En soulignant le caractère diffus et disparate des formes de consommation engagée, Dubuisson-Quellier présente au fil de l'ouvrage une grande variété de modes d'action qui en relèvent, et interroge les diverses significations et portées de ces actes. Le premier chapitre est consacré à un retour historique sur différentes formes de consommation engagée qui apparaissent à la fin du XIXe siècle dans plusieurs pays. Les évolutions de la consommation engagée sont mises en parallèle avec celles des structures de la consommation elle-même, notamment avec l'avènement, dans les années 1950 aux États-Unis, d'une « république des consommateurs »2, un nouveau pacte social rendu possible par une économie productiviste et qui s'étend ensuite dans le monde occidental. Dans les chapitres 2 et 3, les modes d'action de la consommation engagée sont décrits au prisme de deux principales causes d'engagement des consommateurs : l'environnement et la justice sociale. On retrouve ici un répertoire d'action classique, et normalisé, avec par exemple les dispositifs de boycotts et buycotts, que l'auteure présente comme visant non seulement à exercer une pression économique sur des entreprises pour qu'elles abandonnent des pratiques socialement ou écologiquement néfastes, mais aussi et surtout à stigmatiser ces entreprises, choisies pour leur notoriété, afin que des pratiques plus vertueuses se diffusent par entraînement. L'auteure se penche enfin, dans les chapitres 4 et 5, sur des formes d'engagement plus « singulières » (p. 10), qui passent par une modification profonde des modes de vie, la construction de systèmes d'échange alternatifs au marché, ou des actions sortant du répertoire normalisé de la consommation engagée. Ces modes d'action invitent à repenser la structure des circuits d'échange, qu'ils soient refondés à l'échelle locale (circuits courts), sur d'autres unités de valeur que la monnaie (systèmes d'échange local), ou en mettant en oeuvre un rejet de la propriété privée et des logiques marchandes néolibérales (habitats collectifs). Plutôt que des tentatives de changer le système marchand de l'intérieur, ces modes d'action constituent des attaques, symboliques ou matérielles, contre les structures de la consommation de masse : l'auteure le montre à travers l'exemple de la désobéissance civile des freegans (qui se nourrissent de nourriture jetée par l'industrie agroalimentaire), et celui des actions collectives « anti-pub ».

L'ouvrage fait la part belle à la description des collectifs plus ou moins formels (des associations de consommateurs à la la nébuleuse des anti-pub) et aux enjeux d'organisation qu'ils suscitent. L'auteure montre ainsi au chapitre 1 que les coopératives sont apparues comme « un moyen de mobiliser les masses pour des projets politiques variés » (p. 15), entre coopératives ouvrières à vocation autogestionnaire et coopératives encadrées par le patronat visant à inculquer aux ouvriers des valeurs bourgeoises. Au-delà de ces questions d'organisation, l'ouvrage interroge les mécanismes collectifs à l'oeuvre derrière la dimension individuelle apparente de la consommation. Lorsque les pratiques de consommation sont influencées par l'objectif plus large d'une transformation des structures économiques et sociales, elles prennent un sens politique et s'inscrivent alors dans des dynamiques collectives. L'auteure le montre en s'appuyant sur la grille d'analyse de Hirschman3, qui étudie le changement social à travers le processus de « défection » et de prise de parole ». Les actes individuels de consommation engagée (boycotts, choix de produits labellisés, adoption de régime végétarien, etc.) ne consistent pas simplement à abandonner certains produits ou pratiques, ils s'accompagnent d'une prise de parole, par exemple à travers des campagnes d'opinion menées par des associations militantes ou des syndicats. Cette prise de parole au nom des consommateurs donne aux actes individuels de consommation engagée un sens collectif, et une portée politique ; elle constitue un enjeu à part entière des dispositifs de mobilisation. Or, l'auteure explique que la représentation des consommateurs a été progressivement institutionnalisée, par des « administrations publiques, des associations de défense de consommateurs qui en sont souvent proches », ou les départements marketing de grandes entreprises. Cela a eu pour conséquence d'« orient[er] cette représentation vers la défense des intérêts des consommateurs plutôt que vers la mise en cause de leur responsabilité collective » (p. 12). Ces institutions font en effet partie intégrante d'un modèle social fondé sur la consommation de masse, et promeuvent des objectifs d'engagement plus susceptibles de porter une critique au sein des systèmes marchands, et vers une modification à la marge des comportements de consommation, que contre eux.

Dubuisson-Quellier mène ainsi une analyse critique de la consommation engagée, notamment lorsqu'elle pointe l'ambigüité entre la contestation qui se situe dans des structures marchandes et celle qui opère hors de ce cadre. Cette question est largement décryptée au fil des chapitres et à travers de nombreux exemples. Dans une section significativement nommée « marchandisation de la justice sociale » (p. 68-75), l'auteure présente notamment les deux dispositifs alternatifs de labellisation pour le commerce « équitable » d'une part, et de construction d'une filière intégrée de commerce « éthique », qui comprend des lieux de distribution spécifiques, de l'autre. Le succès (relatif) du commerce équitable labellisé est celui d'une démarche qui s'inscrit dans les structures marchandes plus qu'elle ne les conteste. En reprenant les analyses de Micheletti4, l'auteure explique que les labels sont un symbole de ces dispositifs d'engagement qui « procèdent d'une responsabilisation des consommateurs très en phase avec les idéologies néolibérales en vigueur » (p. 48). Elle souligne en conclusion que la consommation engagée peut contribuer à « étendre le marché » (p. 141), en introduisant de nouveaux biens et services à destination de consommateurs engagés, plutôt qu'à le remettre en cause. C'est en partie en réaction à cette récupération marchande de la contestation que l'on peut comprendre l'émergence des collectifs présentés dans le dernier chapitre -remanié dans cette seconde édition, et accompagné par l'ajout, p. 44-47, d'une section sur d'autres mouvements : « végétarisme, végétalisme, véganisme ». Ces collectifs promeuvent des formes plus radicales d'engagement et la construction de dispositifs marchands alternatifs ; ils cherchent à impliquer les consommateurs eux-mêmes au coeur de la gouvernance des systèmes économiques (comme dans le cas des AMAP, Associations pour le maintien d'une agriculture paysanne). Ils visent explicitement à remettre en cause les structures de la consommation et sa place dans les sociétés contemporaines, mais ne permettent pas de dépasser une autre limite de la consommation engagée : sa segmentation sociale. Dubuisson-Quellier le souligne dans cette seconde édition, en introduisant notamment une référence aux travaux de Bourdieu5. Il apparaît en effet que la consommation engagée est avant tout le fait de femmes de moins de 35 ans possédant un fort capital culturel ; elle compose le « répertoire culturel d'un groupe social très spécifique » (p. 8), un répertoire mobilisable pour des enjeux variés.

On pourra regretter que des enjeux importants évoqués en conclusion ne fassent pas l'objet d'analyses plus développées, comme la proximité culturelle entre militants de la consommation engagée et journalistes, mais aussi et surtout le rôle des réseaux sociaux numériques -comment la consommation engagée s'en saisit, et ce qu'ils lui font. Cette seconde édition à l'ère du web 2.0 aurait pu être l'occasion de s'y pencher. Cependant, neuf ans après la première édition de l'ouvrage, ses conclusions restent d'actualité. Cette synthèse érudite de travaux internationaux constitue une mine de ressources pour les lecteur-trices, et ouvre des perspectives de recherche en pointant çà et là des manques dans les travaux français. Les pistes de réflexion proposées à propos de l'engagement politique et la défense de projets collectifs à travers la consommation offrent un point de vue original sur les tentatives de bouleversement des structures économiques, à l'heure où celles-ci sont souvent analysées à travers les dispositifs marchands d'une économie s'auto-proclamant « collaborative ».


Debayan Maity

Debanjana Mitra

Mythily Ramaswamy

LACK OF COMPLETE STABILIZABILITY OF SOME COUPLED ODE-PARABOLIC SYSTEMS

Keywords: 2010 Mathematics Subject Classification. 93D15, 93C20, 35Q30, 76A10 Stabilizability, linearized compressible Navier-Stokes system, linear viscoelasticity

In this article, we consider several coupled ode-parabolic systems. These systems are known to be not null controllable at any time by localized interior controls. We show that, these systems are not exponentially stabilizable with arbitrary decay rate. And, consequently we recover the known results that they are not null controllable any time.

Introduction

The purpose of this work is to discuss exponential stabilizability property of some coupled odeparabolic linear systems such as linearized compressible Navier-Stokes, linear viscoelastic flows etc. For the convenience of the reader, let us first describe some basic concepts of controllability and stabilizability in an abstract framework.

Let H and U be two Hilbert spaces. Let us consider the linear control system

z (t) = Az(t) + Bu(t), t 0, z(0) = z 0 ∈ H, (1.1)
where A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup T on H, and B ∈ L(U, H) is a bounded control operator. It is well known that, given initial data z 0 ∈ H and a control u ∈ L 2 loc (0, ∞; U), the system (1.1) admits a unique solution y ∈ C([0, ∞); H) ∩ H 1 loc (0, ∞; D(A * ) ), where D(A * ) is the dual of D(A) with respect to the pivot space H.

Let us now introduce several notations of stabilizability.

Definition 1.1 (Open loop stabilizable). The system (1.1) or the pair (A, B) is said to be open loop stabilizable if for any initial data z 0 ∈ H there exists a control u ∈ L 2 (0, ∞; U) such that z ∈ L 2 (0, ∞; H).

Definition 1.2 (Exponentially stabilizable). The system (1.1) or the pair (A, B) is said to be exponentially stabilizable if for any initial data z 0 ∈ H there exists a control u ∈ L 2 (0, ∞; U) such that z(t) H Ce -νt z 0 H , t 0, for some constant C > 0 and ν > 0.

Definition 1.3 (Feedback stabilizable). The system (1.1) or the pair (A, B) is said to be feedback stabilizable if there exists an operator K ∈ L(H; U) such that the operator A + BK with the domain D(A + BK) = D(A) generates an exponentially stable semigroup T K on H, i.e., there exist constants M 1 and ν > 0, such that

T K t L(H)
M e -νt , t 0.

In particular, feedback stabilizability is a special case of exponential stabilizability. Obviously feedback stabilizability or exponential stabilizability implies open loop stabilizability. The converse is also true, see for instance [START_REF] Bensoussan | Representation and control of infinite dimensional systems, Systems & Control: Foundations & Applications[END_REF]Part V,Theorem 3.1]. For these reasons, these notions will be used in an exchangeable manner. Definition 1.4 (Complete stabilizability). The system (1.1) or the pair (A, B) is said to be feedback stabilizable with a decay rate ω > 0 if the pair (A + ωI, B) is feedback stabilizable. The system (1.1) or the pair (A, B) is said to be completely stabilizable if it is feedback stabilizable with any decay rate ω > 0.

Finally, we introduce the notion of null controllability. Definition 1.5 (Null controllability). The system (1.1) or the pair (A, B) is said to be null controllable in H at time τ > 0 if for any z 0 ∈ H, there exists a control u ∈ L 2 (0, T ; U), such that z(τ ) = 0.

If the system (1.1) is null controllable at time τ > 0 then it is completely stabilizable. The converse is not true in general. However, if A generates a group of operators, then they are equivalent ( [START_REF] Zabczyk | Mathematical control theory: an introduction, Systems & Control: Foundations & Applications[END_REF]Part IV,Theorem 3.4]).

In this article, we discuss complete stabilizability or their lack of, for some coupled odeparabolic linear system. More precisely, we will consider the following linear systems

(1) Compressible Navier-Stokes system linearized around zero velocity (cf. [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF][START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]) (2) One dimensional blood flow models linearized around constant steady state (cf. [START_REF] Maity | Existence and uniqueness of maximal strong solution of a 1d blood flow in a network of vessels[END_REF]) (3) Linear viscoelastic Jeffreys system (cf. [START_REF] Maity | Lack of null controllability of viscoelastic flows[END_REF]) These systems are known to be not null controllable at any time by localized interior controls or by boundary controls. We refer to the articles [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF][START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF][START_REF] Maity | Lack of null controllability of viscoelastic flows[END_REF][START_REF] Guerrero | Remarks on non controllability of the heat equation with memory[END_REF][START_REF] Ahamed | Lack of null controllability of one dimensional linear coupled transport-parabolic system with variable coefficients[END_REF] for detailed discussions regarding lack of null controllability of such systems. In this article, we prove more stronger result. More precisely, Theorem. The systems mentioned above are not exponentially stabilizable with arbitrary decay rate. In other words, they are not completely stabilizable.

As a consequence of the above result, we recover the fact that, these systems are not null controllable at any time. The proof of these results is based on duality arguments. Recently, a dual characterization of exponential stabilizability has been proved in [START_REF] Trélat | Characterization by observability inequalities of controllability and stabilization properties[END_REF]Theorem 1]. More precisely, Theorem 1.6. [15, Theorem 1] The system (1.1) or the pair (A, B) is exponentially stabilizable if and only if there exist β ∈ (0, 1), τ > 0, C 0 such that

T * τ z 0 H C B * T * t z 0 L 2 (0,τ ;U ) + β z 0 H . (1.2)
Note that, if β = 0, the observability inequality (1.2) coincides with the so-called final state observability of the pair (A * , B * ), which is equivalent to null controllability of (A, B) (see for instance [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Section 11.2]). Thus the observability inequality equivalent to exponential stabilizability is weaker than the one which is equivalent to null controllability. In order to prove our results, we shall construct special solutions known as Gaussian beam solutions, such that corresponding observability inequalities do not hold. There are some results available in the literature where it has been shown that similar coupled ode-parabolic systems are not exponentially stabilizable with any decay rate, see for instance [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF]Corollary 7.12], [START_REF] Breiten | Compensator design for the monodomain equations with the FitzHugh-Nagumo model[END_REF]Theorem 2.7]. However, the proofs are based on explicit computation of the eigenvalues and eigenfunctions of the linear operator, and thus restricted to certain boundary conditions and dimension. The method we present here is quite robust, and it does not depend on explicit computation of the eigenvalues and eigenfunctions of the linear operator. Thus this method can be applied to many such similar models. In this article we only consider few of the examples mentioned earlier.

The remaining part of this work is organized as follows. In Section 2, we study lack of complete stabilizability of compressible Navier-Stokes linearized around constant state in any space dimension. In Section 3, we discuss similar results for two other coupled ODE-parabolic system in one dimension. Section 4 is devoted towards proving lack of complete stabilization for viscoelastic flows. Finally, in Section 5, we discuss some open questions.

Compressible Navier-Stokes linearized around zero velocity

Let Ω be smooth bounded domain in R n , n 1. We consider the compressible Navier-Stokes system linearised around (ρ, 0), ρ > 0 (see [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]Eq. 1.10]). More precisely, we consider the following linear control system

             ∂ t ρ + ρdivu = 1 O 1 f 1 (t, x) ∈ (0, ∞) × Ω, ∂ t u - µ ρ ∆u - α + µ ρ ∇(div u) + aγ ργ-2 ∇ρ = 1 O 2 f 2 (t, x) ∈ (0, ∞) × Ω, u = 0 (t, x) ∈ (0, ∞) × ∂Ω, ρ(0) = ρ 0 , u(0) = u 0 x ∈ Ω, (2.1)
where µ > 0 and α + 2µ > 0. In the above system ρ(t, •) : Ω → R and u(t, •) : Ω → R n represent the fluid density and velocity respectively. Let us set

L 2 m (Ω) = f ∈ L 2 (Ω) | Ω f dx = 0 , H = L 2 (Ω) 1+n , H m = L 2 m (Ω) × L 2 (Ω) n .
We consider the operator A : D(A) → H defined by

D(A) = (ρ, u) ∈ L 2 (Ω) × H 1 0 (Ω) n | µ ρ ∆u + α + µ ρ ∇(div u) -aγ ργ-2 ∇ρ ∈ L 2 (Ω) n , A ρ u = -ρdivu µ ρ ∆u + α + µ ρ ∇(div u) -aγ ργ-2 ∇ρ . (2.2)
We introduce the input space U = H and the control operator B ∈ L(U; H) defined by

B f 1 f 2 = 1 O 1 f 1 1 O 2 f 2 . (2.3)
With the above notations, the system (2.1) can be written as

d dt ρ u = A ρ u + B f 1 f 2 , ρ u (0) = ρ 0 u 0 . (2.4)
The fact that the system we consider is well-posed follows from the following result:

Lemma 2.1. The operator (A, D(A)) is the infinitesimal generator of a strongly continuous semigroup T on H. Moreover, H m is invariant under T, and the operator A can be restricted to

H m . The part of A in H m is the restriction of A to the domain D(A) ∩ H m .
Proof. This lemma can be proved by applying Lumer-Phillips theorem. In dimension one, the details of the proof can be found in [7, Lemma 2.1]. For higher dimension, the same argument can be used by noting that the second order operator -µ ρ ∆u -

α + µ ρ ∇(div u) is a uniformly elliptic operator.
Next, we determine the adjoint of the operator A.

Proposition 2.2. The adjoint of (A, D(A)) in H is defined by

D(A * ) = (σ, v) ∈ L 2 (Ω) × H 1 0 (Ω) n | µ ρ ∆v + α + µ ρ ∇(div v) + ρ∇σ ∈ L 2 (Ω) n , A * σ v = aγ ργ-2 divv µ ρ ∆v + α + µ ρ ∇(div v) + ρ∇σ . ( 2.5)
We now state our main results of this section. Our goal is to show that the system (2.1) is not exponentially stabilizable with arbitrary decay rate. More precisely, we shall prove the following result

Theorem 2.3. Let O 1 ⊂ Ω, O 2 ⊆ Ω, ω 0 := aγ ργ 2µ + α . (2.6)
The system (2.1) is not exponentially stabilizable in H with an exponential decay rate ω ω 0 , by interior controls

f 1 ∈ L 2 (0, ∞; L 2 (O 1 )) and f 2 ∈ L 2 (0, ∞; L 2 (O 2 )) n .
In other words, for ω ω 0 , the pair (A + ωI, B) is not exponentially stabilizable on H.

Let us now give special attention to the case where the control is not active in the first component, i.e., f 1 ≡ 0. In this case, it turns out that, H m is the appropriate space for stabilization than H. For instance, let us consider the system (2.1) with f 1 ≡ 0. Integrating (2.1)

1 we have Ω ρ(t, x)dx = Ω ρ 0 (x) dx for all t 0.
And, therefore

Ω |ρ(t, x)| 2 dx 1 |Ω| Ω ρ 0 (x) dx 2 (t > 0).
Thus, if Ω ρ 0 (x) dx = 0, the system can not be stabilized with the control f 2 only. With this observation in mind, when f 1 ≡ 0, Theorem 2.3 can be reformulated as Theorem 2.4. Assume f 1 ≡ 0, and O 2 ⊆ Ω. The system (2.1) is not exponentially stabilizable in H m with an exponential decay rate ω ω 0 , by interior controls

f 2 ∈ L 2 (0, ∞; L 2 (O 2 )) n . Remark 2.5. From [7, Lemma 2.5],
we observe that in one-dimension, ω 0 in Theorem 2.3 coincides with the accumulation point of the spectrum of the operator A.

Let us now mention some related works from the literature. Controllability of the system (2.1) in dimension one and two has been studied in [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF][START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]. In these articles, it was shown that the system is not null controllable at any time by localized interior controls. Regarding exponential stabilizability, in [START_REF] Arfaoui | Boundary stabilizability of the linearized viscous Saint-Venant system[END_REF][START_REF] Chowdhury | Local stabilization of the compressible Navier-Stokes system, around null velocity, in one dimension[END_REF], it was proved that the system (2.1) in dimension one is exponentially stabilizable with a decay ω < ω 0 , where ω 0 is the same as (2.6), either by boundary control or by localized interior control on the velocity component only. Still in dimension one, in [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF]Corollary 7.12], it was established that the system is not exponentially stabilizable with decay rate ω > ω 0 . Recently, in [START_REF] Chowdhury | Boundary stabilizability of the linearized compressible Navier-Stokes system in one dimension by backstepping approach[END_REF] via backstepping method, it was shown that the decay rate ω 0 is achievable by boundary control, provided initial data belongs to suitable regular space and the system is considered with Neumann type boundary conditions. In fact, in [5, Proposition 3.3], the authors pointed out that, with Dirichlet boundary condition at one end and Dirichlet boundary control on the other end, decay of ρ(t) L 2 is not possible with the feedback control constructed in that paper. In Theorem 2.3 and Theorem 2.4, we show that, albeit for interior control, there is no feedback control that stabilizes the system (2.1) with a decay ω ω 0 .

The proof of the above results relies on a suitable observability inequality of the adjoint system, which is equivalent to the exponential stabilizability. More precisely, let ω 0, and we consider the following adjoint system

             ∂ t σ -ωσ -aγ ργ-2 divv = 0 (t, x) ∈ (0, τ ) × Ω, ∂ t v -ωv - µ ρ ∆v - µ + α ρ ∇(divv) -ρ∇σ = 0 (t, x) ∈ (0, τ ) × Ω, v = 0 (t, x) ∈ (0, τ ) × ∂Ω, σ(0) = σ 0 , v(0) = v 0 x ∈ Ω.
(2.7)

According to [15, Theorem 1], the exponential stabilizability of the pair (A + ωI, B) is equivalent to the following observability inequality: Proposition 2.6. Let ω > 0.

(1) The system (2.1) is exponentially stabilizable with a decay rate ω, by interior controls

f 1 ∈ L 2 (0, ∞; L 2 (O 1 )) and f 2 ∈ L 2 (0, ∞; L 2 (O 2 )
) n , if and only if, there exist β ∈ (0, 1), τ > 0 and C 0 such that, for any (σ 0 , v 0 ) ∈ H, the solution of (2.7) satisfies the following observability estimate

σ(τ, •) L 2 (Ω) + v(τ, •) L 2 (Ω) n C σ L 2 (0,τ ;L 2 (O 1 )) + v L 2 (0,τ ;L 2 (O 2 )) n + β σ 0 L 2 (Ω) + v 0 L 2 (Ω) n . (2.8)
(2) Assume further that f 1 ≡ 0. The system (2.1) is exponentially stabilizable with a decay rate ω, by interior controls

f 2 ∈ L 2 (0, ∞; L 2 (O 2 )
) n , if and only if, there exist β ∈ (0, 1), τ > 0 and C 0 such that, for any (σ 0 , v 0 ) ∈ H m , the solution of (2.7) satisfies the following observability estimate

σ(τ, •) L 2 (Ω) + v(τ, •) L 2 (Ω) n C v L 2 (0,τ ;L 2 (O 2 )) n + β σ 0 L 2 (Ω) + v 0 L 2 (Ω) n .
(2.9)

To prove Theorem 2.3, we shall construct special initial data (σ 0 , v 0 ) belonging to H (or H m ), so that the corresponding solution to system (2.7) violates the observability inequality (2.8) (or (2.9)) for ω ω 0 . To this aim, we first construct highly localised solutions known as "Gaussian beam solutions" to the adjoint of (2.1) in (0, T ) × R n . More precisely, let ω > 0, and we consider the following system :

         ∂ t σ -ωσ -aγ ργ-2 divv = 0 (t, x) ∈ (0, τ ) × R n , ∂ t v -ωv - µ ρ ∆v - µ + α ρ ∇(divv) -ρ∇σ = 0 (t, x) ∈ (0, τ ) × R n , σ(0) = σ 0 := div w 0 , v(0) = v 0 x ∈ R n .
(2.10)

We shall construct a special solution to the above system. Note that in (2.10), the initial data σ 0 is taken the form σ 0 := div w 0 to ensure that the initial conditions in (2.7) belongs to H m . This is needed to prove Theorem 2.4. In dimension 1 and 2, similar solutions were constructed in [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]. We follow a similar approach here. We set

σ(t, ξ) := F x σ(t, x), v(t, ξ) := F x v(t, x), (2.11)
σ 0 (ξ) := F x σ 0 (x), w 0 (ξ) := F x w 0 (x), v 0 (ξ) := F x v 0 (x), (2.12)
where F x (f ) denotes the Fourier transformation of f with respect to the spatial variable x.

Applying Fourier transform to (2.10), we obtain

d dt σ v = ( A(ξ) + ωI n ) σ v , σ v (0) = σ 0 v 0 = iξ • w 0 v 0 t ∈ (0, τ ), (2.13)
where

A(ξ) = 0 iaγ ργ-2 ξ iρξ - µ ρ |ξ| 2 I n - µ + α ρ ξ ⊗ ξ , for all ξ ∈ R n , (2.14)
where I n is the n × n identity matrix and ξ ⊗ ξ is the n × n matrix with (ξ

⊗ ξ) ij = ξ i ξ j for all i = 1, • • • , n and j = 1, • • • , n.
We first study eigenvalues and eigenvectors of A(ξ). We have the following result:

Lemma 2.7. The following holds

(1) The characteristic polynomial of A(ξ) is λ + µ ρ |ξ| 2 n-1 λ 2 + α + 2µ ρ |ξ| 2 λ + aγ ργ-1 |ξ| 2 = 0. (2.15) (2) The eigenvalues of A(ξ) are λ 1 (ξ) = • • • = λ n-1 (ξ) = - µ ρ |ξ| 2 , λ n (ξ) = - (2µ + α)|ξ| 2 2ρ 1 + 1 - 4aγ ργ+1 (2µ + α) 2 |ξ| 2 , δ(ξ) = - (2µ + α)|ξ| 2 2ρ 1 -1 - 4aγ ργ+1 (2µ + α) 2 |ξ| 2 .
(3) There exists ξ 0 > 0, such that for all |ξ| ξ 0 , all the eigenvalues are real. They satisfy

lim |ξ|→∞ λ k (ξ) |ξ| 2 = - µ ρ , k = 1, 2, . . . , n -1 lim |ξ|→∞ λ n (ξ) |ξ| 2 = - 2µ + α ρ , lim |ξ|→∞ δ(ξ) = - aγ ργ 2µ + α := -ω 0 .
(4) For |ξ| ξ 0 , we have

-2 aγ ργ 2µ + α < δ(ξ) < - aγ ργ 2µ + α , i.e. -2ω 0 < δ(ξ) < -ω 0 , and
δ(ξ 2 ) > δ(ξ 1 ) for |ξ 2 | > |ξ 1 | > ξ 0 , and ∆ k ξ δ(ξ) C, (k ∈ N), (2.16)
for some positive constant C independent of ξ.

(

) The eigenfunction of A(ξ) corresponding to δ(ξ) is 1, δ(ξ)ξ iaγ ργ-2 |ξ| 2 . 5
Proof. The proof follows easily from the expression of δ(ξ).

We now construct Gaussian beam solutions for the system (2.10).

Theorem 2.8. Let β ∈ (0, 1), τ > 0, and ω ω 0 . Let ξ = ( 1 c 0 , 0, • • • , 0) ∈ R n with c 0 > 1, and x 0 ∈ R n . Let ψ be a smooth function compactly supported in the unit ball and of unit L 2 (R n ) norm. For any ε > 0, we define

ψ ε (ξ) = ε n 4 ψ √ ε ξ - ξ ε e -ix 0 •ξ , and
w 0 (ξ) = εψ ε (ξ)e 1 , σ 0 (ξ) = iξ • w 0 (ξ), v 0 (ξ) = δ(ξ)ξ iaγ ργ-2 |ξ| 2 σ 0 (ξ), (2.17)
where e 1 = (1, 0, . . . , 0) . Then (σ, v) defined by

σ(t, x) = F -1 ξ σ 0 (ξ)e (δ(ξ)+ω)t , v(t, x) = F -1 ξ v 0 (ξ)e (δ(ξ)+ω)t , (2.18)
satisfies the system (2.10), with

w 0 (x) = F -1 ξ ( w 0 (ξ)) , σ 0 (x) = div w 0 = F -1 ξ ( σ 0 (ξ)) , v 0 (x) = F -1 ξ ( v 0 (ξ)) . ( 2

.19)

Moreover, there exists ε 0 > 0 such that, for all 0 < ε < ε 0 , we have (i) There exists a positive constant C depending on ε 0 , but independent of ε such that

w 0 L 2 (R n ) n = ε (2π) n , σ 0 L 2 (R n ) 1 (2π) n ,
(2.20)

β 1/m c 2 0 (2π) n e (m-1)(ω-ω 0 )τ m σ(τ, •) L 2 (R n ) 1 (2π) n e (ω-ω 0 )τ , m ∈ N, (2.21) v(t, •) H s (R n ) n Cε (1-s) , s ∈ [0, 1), t ∈ [0, τ ]. (2.22)
(ii) For any η > 0, there exists a constant C, depending on ω, τ, η and ε 0 , but independent of ε such that

σ(t, •) L 2 (|x-x 0 | η) Cε k-n/4 , k ∈ N, t ∈ [0, τ ],
(2.23)

w 0 H 1 (|x-x 0 | η)) n Cε k-1-n/4 , k ∈ N, (2.24) v H 1 (0,τ ;H 2 (|x-x 0 | η)) n Cε k-1-n/4 , k ∈ N, (2.25)
Proof. Let us recall

σ(t, x) = 1 (2π) n R n σ 0 (ξ)e ix•ξ e (δ(ξ)+ω)t dξ.
(2.26) By Parseval's relation, for all t ∈ [0, τ ] we have

σ(t, •) 2 L 2 (R n ) = 1 (2π) 2n R n | σ 0 (ξ)| 2 e 2(Reδ(ξ)+ω)t dξ = ε 2+ n 2 (2π) 2n R n ξ 2 1 ψ √ ε ξ - ξ ε 2 e 2(Reδ(ξ)+ω)t dξ = 1 (2π) 2n |ζ| 1 |ζ 1 √ ε + ξ1 | 2 |ψ(ζ)| 2 exp 2Reδ ζ √ ε + ξ ε t e 2ωt dζ.
From the above relation, by choosing ε sufficiently small, we get (2.20). Next, we define

κ = (ω -ω 0 ) - ln β τ .
Note that, by the hypothesis of the theorem we have κ > 0. Let m ∈ N. We choose ε sufficiently small so that for any |ζ| 1, ζ √ ε + ξ ε > ξ 0 , and then from Lemma 2.7 we have

Reδ ζ √ ε + ξ ε = δ ζ √ ε + ξ ε ∈ -ω 0 - κ m , -ω 0 , 1 c 4 0 |ζ 1 √ ε + ξ1 | 2 1 for |ζ| 1.
With the above choice of ε and using the above estimates, it is easy to see that 1

c 2 0 (2π) n e (ω-ω 0 -κ/m)τ σ(τ, •) L 2 (R n ) 1 (2π) n e (ω-ω 0 )τ .
Using the definition of κ the above expression can be simplified as

β 1/m c 2 0 (2π) n e (m-1)(ω-ω 0 )τ m σ(τ, •) L 2 (R n ) 1 (2π) n e (ω-ω 0 )τ .
This completes the proof of (2.21). Recalling the expression of σ from (2.26), we have

σ(t, x) = ε 1+ n 4 (2π) n R n iξ 1 ψ √ ε ξ - ξ ε e i(x-x 0 )•ξ e (δ(ξ)+ω)t dξ = ε -n/4 (2π) n |ζ| 1 i(ζ 1 √ ε + ξ1 )ψ(ζ)e i(x-x 0 )•( ζ √ + ξ ε ) e δ( ζ √ ε + ξ ε )t e ωt dζ.
Note that

∆ k ζ e i(x-x 0 )•( ζ √ ε + ξ ε ) = (-1) k |x -x 0 | 2 ε k e i(x-x 0 )•( ζ √ ε + ξ ε ) k ∈ N.
Thus for |x -x 0 | η > 0 we have

σ(t, x) = - iε k-n/4 (2π) n |x -x 0 | 2k |ζ| 1 ∆ ζ e i(x-x 0 )•( ζ √ ε + ξ ) (ζ 1 √ ε + ξ1 )ψ(ζ)e δ( ζ √ ε + ξ ε )t e ωt dζ = - iε k-n/4 (2π) n |x -x 0 | 2k |ζ| 1 e i(x-x 0 )•( ζ √ ε + ξ ε ) ∆ ζ (ζ 1 √ ε + ξ1 )ψ(ζ)e δ( ζ √ ε + ξ ε )t e ωt dζ. (2.27)
Using Lemma 2.7, we infer that, there exists ε 0 > 0 such that, for ε < ε 0

∆ ζ (ζ 1 √ ε + ξ1 )ψ(ζ)e δ( ζ √ + ξ )t C,
for some constant C independent of ε. Thus for 0 < ε < ε 0 and for |x -x 0 | η > 0 we get

|σ(t, x)| Ce ωt ε k-n/4 (2π) n |x -x 0 | 2k .
(2.28)

This proves (2.23). ( 2.24) can be proved in a similar manner. To prove estimate (2.22), we note that, for t ∈ [0, τ ] and s ∈ [0, 1)

v(t, •) 2 H s (R n ) = 1 (2π) 2n R n (1 + |ξ| 2 ) s |ξ| 2 | σ 0 (ξ)| 2 δ(ξ) aγ ργ-2 i 2 e 2(Reδ(ξ)+ω)t dξ Ce ωt ε 2+ n 2 (2π) 2n R n (1 + |ξ| 2 ) s |ξ| 2 ξ 2 1 ψ √ ε ξ - ξ ε 2 dξ = Ce ωt ε 2(1-s) (2π) 2n |ζ| 1 (|ζ √ ε + ξ| 2 + ε 2 ) s |ζ √ ε + ξ| 2 |ζ 1 √ ε + ξ1 | 2 |ψ(ζ)| 2 dζ Ce ωt ε 2(1-s) (2π) 2n .
From the above estimate we deduce (2.22). To prove (2.25), we may proceed as in the proof of (2.23) above.

We are now in a position to prove Theorem 2.3 and Theorem 2.4.

Proof of Theorem 2.3. The proof is by contradiction. Let ω ω 0 . We assume that the system (2.1) is exponentially stabilizable in H with an exponential decay rate ω ω 0 . Then according to Proposition 2.6, there exist β ∈ (0, 1) and τ > 0, such that the observability inequality (2.8) holds for any (σ 0 , v 0 ) ∈ H. In particular, there exist β ∈ (0, 1), τ > 0 and C 0 such that, for any (σ 0 , v 0 ) ∈ H, the solution of (2.7) satisfies the following observability estimate

σ(τ, •) L 2 (Ω) -β σ 0 L 2 (Ω) + v(τ, •) L 2 (Ω) n C σ L 2 (0,τ ;L 2 (O 1 )) + v L 2 (0,τ ;L 2 (O 2 )) n + β v 0 L 2 (Ω) n . (2.29)
Let us fix such β and τ, and we take k ∈ N be such that k > 1 + n 4 . We are going to construct (σ 0 , v 0 ) such that the corresponding solution of (2.7) does not satisfy the above observability estimate. We choose x 0 and η > 0 such that

B(x 0 ; η) ⊂ Ω and B(x 0 ; η) ∩ O 1 = ∅.
We also fix m ∈ N, m 1 and c 0 > 1 in Theorem 2.8 such that

c 4 0 β 1-1/m < 1. (2.30)
With the above choice of parameters, let (σ , v ) be constructed as in Theorem 2.8. We define

h(t, x) := v (t, x) t ∈ [0, τ ], x ∈ ∂Ω.
By Theorem 2.8, there exists ε 0 > 0 such that, for all ε < ε 0 , we have

w 0 [H 1 (Ω)\B(x 0 ,η)] n Cε k-1-n/4
(2.31)

σ 0 = div w 0 in Ω, σ 0 L 2 (Ω) 1 (2π) n , (2.32) σ (τ, •) L 2 (Ω) β 1/m c 3 0 (2π) n e (m-1)(ω-ω 0 )τ m β 1/m c 3 0 (2π) n , (2.33) v 0 L 2 (Ω) n Cε, v (τ, •) L 2 (Ω) n Cε, (2.34) σ L 2 (0,τ ;L 2 (O 1 )) Cε k-n/4 , v L 2 (0,τ ;L 2 (O 2 )) n
Cε, (2.35)

h H 1 (0,τ ;H 3/2 (∂Ω)) Cε k-1-n/4 , (2.36)
for some constant C independent of ε. Let ζ be a smooth function in Ω such that 0 ζ 1, ζ = 0 in B(x 0 , η) and ζ = 1 on ∂Ω. Next, we consider the following system:

               ∂ t σ † -ωσ † -aγ ργ-2 divv † = 0 in (0, τ ) × Ω, ∂ t v † -ωv † - µ ρ ∆v † - µ + α ρ ∇(divv † ) -ρ∇σ † = 0 in (0, τ ) × Ω, v † = h on (0, τ ) × ∂Ω, σ † (0) = div(ζw 0 ), v † (0) = 0 in Ω.
(2.37)

The above system admits a unique solution (σ † , v † ) ∈ C([0, τ ]; H). Moreover, using (2.31) and (2.36), for ε < ε 0 , we have

(σ † , v † ) C([0,τ ];H) C h H 1 (0,τ ;H 3/2 (∂Ω)) + div(ζw 0 ) L 2 (Ω) Cε k-1-n/4 , (2.38)
for some constant C independent of ε. We set

σ(t, x) = σ (t, x) -σ † (t, x), v(t, x) = v (t, x) -v † (t, x) t ∈ [0, τ ], x ∈ Ω.
Then (σ, v) satisfies the system (2.7) with the initial data

σ 0 (x) = σ 0 (x) -σ † (0, x) = div(w 0 (x) -ζw 0 (x)), v 0 (x) = v 0 (x). ( 2
> σ(τ, •) L 2 (Ω) -β σ 0 L 2 (Ω) > 1 (2π) n β 1/m c 4 0 -β > 0, whereas R.H.S of (2.29) Cε.
This is a contradiction to (2.29). Therefore, the system (2.1) is not exponentially stabilizable in H with an exponential decay rate ω ω 0 .

Proof of Theorem 2.4. The proof of Theorem 2.4 is similar to the proof of Theorem 2.3 given above. The main difference is that we now have to construct initial data (σ 0 , v 0 ) ∈ H m instead of H. However, this follows from (2.39). In fact, integrating σ 0 over Ω and using the fact that ζ = 1 on ∂Ω we have

Ω σ 0 (x) dx = ∂Ω (w 0 (x) -ζw 0 (x)) • ν dγ = 0.
Thus the initial data (σ 0 , v 0 ) constructed in (2.39) belongs to H m . The rest of the proof is similar to that of Theorem 2.3.

Other one-dimensional ODE-parabolic coupled models

In this section, we consider two other linear compressible type models in one space dimension. The first one is compressible Navier-Stokes-Fourier system linearized around constant trajectories. This is a linear system coupled with two parabolic equations and an ODE. The second one is a linearized blood flow model. This system is similar to (2.1) in one space dimension with some lower order terms in the parabolic component.

3.1. Linearized Navier-Stokes-Fourier system. We consider the following linear control problem

                       ∂ t ρ + ρ∂ x u = 1 O 1 f 1 (t, x) ∈ (0, ∞) × (0, L), ∂ t u - α + 2µ ρ ∂ xx u + R θ ρ ∂ x ρ + R∂ x ϑ = 1 O 2 f 2 (t, x) ∈ (0, ∞) × (0, L), ∂ t ϑ - κ ρc v ∂ xx ϑ + R θ c v ∂ x u = 1 O 3 f 3 (t, x) ∈ (0, ∞) × (0, L), u(t, 0) = u(t, L) = ϑ(t, 0) = ϑ(t, L) = 0 t ∈ (0, ∞), ρ(0, x) = ρ 0 (x), u(0, x) = u 0 (x), ϑ(0, x) = ϑ 0 (x) x ∈ (0, L), (3.1)
where (α + 2µ), ρ, R, θ, κ and c v are positive constants, and f 1 , f 2 , f 3 are interior controls. Null controllability of the above system was studied in [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]. More precisely, if O 1 ⊂ (0, L) and O 2 , O 3 ⊆ (0, L) the system (3.1) is not null controllable in L 2 (0, L) 3 at any time τ > 0 (see for instance [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]Theorem 1.1]). Here we prove the following result

Theorem 3.1. Let O 1 ⊂ (0, L), O 2 , O 3 ⊆ (0, L), ω 0 := Rρ θ 2µ + α . (3.2)
The system (3.1) is not exponentially stabilizable in L 2 (0, L) 3 with an exponential decay rate ω ω 0 , by interior controls

f 1 ∈ L 2 (0, ∞; L 2 (O 1 )), f 2 ∈ L 2 (0, ∞; L 2 (O 2 )) and f 3 ∈ L 2 (0, ∞; L 2 (O 3 )).
This result can be proved in a similar manner as we proved Theorem 2.3. At first, we obtain suitable observability inequality which is equivalent to the exponential stabilizability. Next, we construct suitable Gaussian beam solutions to show that observability inequality does not hold. In fact, the Gaussian beam solution of the adjoint of system (3.1) was already constructed in [9, Section 2.2].The proof is left to the reader.

3.2.

Linear blood flow type model. Let us consider the following linear control system

         ∂ t a + ∂ x q = 1 O 1 f 1 (t, x) ∈ (0, ∞) × (0, L), ∂ t q -µ 0 ∂ xx q + µ 1 ∂ x q + µ 2 q + α 1 ∂ x a + α 2 a = 1 O 2 f 2 (t, x) ∈ (0, ∞) × (0, L), q(t, 0) = q(t, L) = 0 t ∈ (0, ∞), a(0, x) = a 0 (x), q(0, x) = q 0 (x) x ∈ (0, L), (3.3)
where µ 0 , µ 1 , µ 2 , α 1 , α 2 are constants and α 1 , µ 0 > 0. This system can be obtained by linearising system (1.3) of [START_REF] Maity | Existence and uniqueness of maximal strong solution of a 1d blood flow in a network of vessels[END_REF], which models one dimensional blood flow in a vessel with viscoelastic walls, around a constant steady state. Note that, the system (2.1) in dimension one is a special case of the above system.

The system (3.3) is also not null controllable at any time τ > 0 by the interior controls f 1 and f 2 (see for instance [START_REF] Ahamed | Lack of null controllability of one dimensional linear coupled transport-parabolic system with variable coefficients[END_REF]Theorem 1.4]). We have the following result regarding the lack of stabilizability

Theorem 3.2. Let O 1 ⊂ (0, L), O 2 ⊆ (0, L), ω 0 := α 1 µ 0 . (3.4)
The system (3.3) is not exponentially stabilizable in L 2 (0, L) 2 with an exponential decay rate ω ω 0 , by interior controls

f 1 ∈ L 2 (0, ∞; L 2 (O 1 )) and f 2 ∈ L 2 (0, ∞; L 2 (O 2 )).
Let us take H 0 = L 2 σ (Ω) × H 1 σ,0 (Ω). We define an unbounded operator (A 0 , D(A 0 )) on H 0 :

D(A 0 ) = (u, v) ∈ H 1 σ,0 (Ω) × H 1 σ,0 (Ω) | µ 0 u + v ∈ D(A 0 ) , A 0 u v = 1 ρ A 0 (µ 0 u + v) µ 2 u -µ 1 v . (4.5)
Lemma 4.3. The operator (A 0 , D(A 0 )) is the infinitesimal generator of a strongly continuous semigroup on H 0 .

We also determine the adjoint operator of A 0 :

Proposition 4.4. The adjoint of (A 0 , D(A 0 )) in H 0 is defined by

D(A * 0 ) = (w, z) ∈ H 1 σ,0 (Ω) × H 1 σ,0 (Ω) | µ 0 w -µ 2 z ∈ D(A 0 ) , A * 0 w z = A 0 (µ 0 w -µ 2 z) -w -µ 1 z . (4.6)
In view of Poincaré and Korn's inequalities, to prove Theorem 4.2, it is enough to prove the following result Theorem 4.5. Assume the hypothesis of Theorem 4.2. The system (4.4) is not exponentially stabilizable in H with an exponential decay rate ω ω 0 , by interior control f ∈ L 2 (0, ∞; L 2 (O)) n .

The controllability of linear viscoelastic flows has been studied only in few articles. For example, the lack of null controllability of the one-dimensional model has been proved in [START_REF] Renardy | A note on a class of observability problems for pdes[END_REF], whereas the lack of null controllability of the higher dimensional model has been proved in [START_REF] Maity | Lack of null controllability of viscoelastic flows[END_REF]. As far as we know, the stabilizability of the system (4.1) or (4.4) has not been studied rigorously. In this context, we mention the work [START_REF] Mitra | Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body[END_REF] where the exponential stabilizability with a decay ω < ω 0 , of linear viscoelastic flows with a rigid structure was proved.

The rest of this section is devoted towards the proof of Theorem 4.5. The proof is similar to the proof of Theorem 2.3 given in Section 2. First, we derive observability inequality equivalent to the exponential stabilizability of the system (4.4). Let ω 0. We consider the adjoint system

              
ρ∂ t w -ρωw -µ 0 ∆w + ∇q + µ 2 ∆z = 0 (t, x) ∈ (0, τ ) × Ω,

∂ t z -ωz + µ 1 z + w = 0 (t, x) ∈ (0, τ ) × Ω, div w = 0 = div z (t, x) ∈ (0, τ ) × Ω, w = 0 = z (t, x) ∈ (0, τ ) × ∂Ω, w(0) = w 0 , z(0) = z 0 x ∈ Ω. (4.7)
We have following equivalence result Proposition 4.6. Let ω > 0. The system (4.4) is exponentially stabilizable with a decay rate ω, by interior controls f ∈ L 2 (0, ∞; L 2 (O)) n , if and only if, there exist β ∈ (0, 1), τ > 0 and C 0 such that, for any (w 0 , z 0 ) ∈ H 0 , the solution of (4.7) satisfies the following observability estimate

w(τ, •) L 2 (Ω) n + z(τ, •) H 1 (Ω) n C w L 2 (0,τ ;L 2 (O)) n + β w 0 L 2 (Ω) + z 0 H 1 (Ω) n . (4.8)
As before we shall construct special solutions violating the observability inequality (4.8). The construction here is slightly different due to divergence free conditions, and different order coupling between ODE and parabolic equations. We just provide the Gaussian beam construction. The rest of the proof of Theorem 4.5 is similar to the proof of Theorem 2.3, and thus left to the reader. Consider the following system in (0, τ

) × R n          ρ∂ t w -ω ρw -µ 0 ∆w + µ 2 ∆z = 0 (t, x) ∈ (0, τ ) × R n , ∂ t z -ωz + µ 1 z + w = 0 (t, x) ∈ (0, τ ) × R n , div w = 0 = div z (t, x) ∈ (0, τ ) × R n , w(0) = w 0 := curl a 0 , z(0) = z 0 := curl b 0 x ∈ R n .
(4.9)

In the above, we use the same notation of "curl" in both two and three dimensions. If n = 2, the functions a 0 and b 0 are scalars and curl a 0 = ∇ ⊥ a 0 . If n = 3, they are vectors in R 3 and curl a 0 = ∇ × a 0 . The initial datas are chosen as "curl" some function so that they are automatically divergence free. Applying Fourier transform to (4.9), we obtain

d dt w z = ( A 0 (ξ) + ωI n ) w z , w z (0) = w 0 z 0 = iξ × a 0 iξ × b 0 t ∈ (0, τ ),
where

A 0 (ξ) = - µ 0 ρ |ξ| 2 I n µ 2 ρ |ξ| 2 I n -I n -µ 1 I n .
If n = 2, the term ξ × a 0 to be understood as ξ ⊥ a 0 . We have the following lemma Lemma 4.7. The following holds

(1) The eigenvalues of A 0 (ξ) are

λ 1 (ξ) = • • • = λ n (ξ) = λ(ξ) = -(µ 1 ρ + µ 0 |ξ| 2 ) -(µ 1 ρ + µ 0 |ξ| 2 ) 2 -4ρ(µ 1 µ 0 + µ 2 )|ξ| 2 2ρ δ 1 (ξ) = • • • = δ n (ξ) = δ(ξ) = -(µ 1 ρ + µ 0 |ξ| 2 ) + (µ 1 ρ + µ 0 |ξ| 2 ) 2 -4ρ(µ 1 µ 0 + µ 2 )|ξ| 2 2ρ .
(2) There exists ξ 0 > 0, such that for all |ξ| ξ 0 , all the eigenvalues are real. They satisfy

lim |ξ|→∞ λ k (ξ) |ξ| 2 = - µ 0 ρ , lim |ξ|→∞ δ k (ξ) = -µ 1 + µ 2 µ 0 := -ω 0 , k = 1, 2, . . . , n.
(3) For |ξ| ξ 0 , we have

-2ω 0 < δ(ξ) < -ω 0 , δ(ξ 2 ) > δ(ξ 1 ) for |ξ 2 | > |ξ 1 | > ξ 0 , and
∆ k ξ δ(ξ) C, (k ∈ N),
for some positive constant C independent of ξ. (4) The eigenfunction of A 0 (ξ) corresponding to δ(ξ) is (-

(µ 1 + δ(ξ))1 n , 1 n ) , where 1 n = (1, • • • , 1) .
Regarding the construction of the Gaussian beam solutions for (4.9), we have the following result: Theorem 4.8. Let β ∈ (0, 1), τ > 0, and ω ω 0 . Let ξ = ( 1 c 0 , 0, • • • , 0) ∈ R n with c 0 > 1, and x 0 ∈ R n . For any ε > 0, let ψ ε be defined as in Theorem 2.8, and we define

n = 2, a 0 = -ε 2 (µ 1 + δ(ξ))ψ ε , b 0 = ε 2 ψ ε , w 0 = iξ ⊥ a 0 , z 0 = iξ ⊥ b 0 , n = 3, a 0 = -ε 2 (µ 1 + δ(ξ))ψ ε e 3 , b 0 = ε 2 ψ ε e 3 , w 0 = iξ × a 0 , z 0 = iξ × b 0 ,
where e 3 = (0, 0, 1) . Then (w, z) defined by w(t, x) = F -1 ξ w 0 (ξ)e (δ(ξ)+ω)t , z(t, x) = F -1 ξ z 0 (ξ)e (δ(ξ)+ω)t , satisfies the system (4.9), with

a 0 (x) = F -1 ξ ( a 0 (ξ)) , b 0 (x) = F -1 ξ b 0 (ξ) , w 0 (x) = curl a 0 = F -1 ξ ( w 0 (ξ)) , z 0 (x) = curl b 0 = F -1 ξ ( z 0 (ξ))
. Moreover, there exists ε 0 > 0 such that, for all 0 < ε < ε 0 , we have (i) There exists a positive constant C depending on ε 0 , but independent of ε such that

a 0 H s (R n ) Cε 2-s (2π) n , s ∈ [0, 1], b 0 H 1 (R n ) Cε (2π) n β 1/m c 4 0 (2π) n e (m-1)(ω-ω 0 )τ m z(τ, •) H 1 (R n ) 1 (2π) n e (ω-ω 0 )τ , m ∈ N, w(t, •) L 2 (R n ) n Cε (2π) n , t ∈ [0, τ ].
(ii) For any η > 0, there exists a constant C, depending on ω, τ, η and ε 0 , but independent of ε such that

z(t, •) H 1 (|x-x 0 | η) Cε k-n/4 , k ∈ N, t ∈ [0, τ ], b 0 H 2 (|x-x 0 | η)) n Cε k-1-n/4 , k ∈ N, (w, z) H 1 (0,τ ;H 2 (|x-x 0 | η)) n Cε k-1-n/4 , k ∈ N.
Proof. The proof is similar to that of Theorem 2.8. Now we can proceed as the proof of Theorem 2.3 in Section 2 above to prove Theorem 4.5.

Conclusion and open problems

The main results in this article concern the lack of exponential stabilizability of some coupled ODE-parabolic systems. We show these systems are not exponentially stabilizable with arbitrary decay rate by localized interior controls. As a consequence, we recover the previously known results that these systems are not null at any time. In view of our results, several open questions seem natural.

Optimality of the decay rate: In all of the examples above, we are able to identify the decay rate ω 0 such that the system are not exponentially stabilizable with decay rate ω ω 0 . Thus the most natural question is whether these decay rates are optimal or not. In particular, it would be interesting to know whether these systems are exponentially stabilizable with a decay ω < ω 0 . As mentioned earlier, linearized compressible Navier-Stokes system i.e., system (2.1) in dimension one, is exponentially stabilizable in H m with a decay rate ω < ω 0 , ω 0 defined in (2.6), by a localized parabolic control only (see for instance [START_REF] Chowdhury | Local stabilization of the compressible Navier-Stokes system, around null velocity, in one dimension[END_REF]Proposition 3.4]). In [START_REF] Mitra | Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body[END_REF]Theorem 1.1], it was proved that linear viscoelastic fluid-structure model is exponentially stabilizable with a decay ω < ω 0 , with ω 0 defined as in (4.3). In view of all these results, we think, all the systems considered here are exponentially stablizable with decay ω < ω 0 . More precisely, we conjecture that, the decay rate ω 0 for the systems (2.1), (3.1), (3.3) and (4.4) is optimal, in the sense that, the corresponding systems are exponentially stabilizable with decay ω < ω 0 . In [START_REF] Chowdhury | Local stabilization of the compressible Navier-Stokes system, around null velocity, in one dimension[END_REF][START_REF] Mitra | Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body[END_REF], the exponential stabilizability was proved by checking Hautus criteria for exponential stabilizability, which requires the knowledge of the spectrum of the operator. However, in view of Theorem 1.6, perhaps Carleman type estimates may be helpful to prove the optimality of the decay rate.

Coupled ODE-parabolic system with variable coefficients: In this article, we have only considered constant coefficient operators. Recently, lack of null controllability of coupled ODE-parabolic system with coefficients depending on both space and time, was proved in [START_REF] Ahamed | Lack of null controllability of one dimensional linear coupled transport-parabolic system with variable coefficients[END_REF]. Thus it would be interesting to see whether similar result can be proved for variable coefficient operator. The Fourier transform technique used here to construct Gaussian beam solutions would not be possible. However, the ideas used in [START_REF] Ralston | Gaussian beams and the propagation of singularities[END_REF] and [START_REF] Ahamed | Lack of null controllability of one dimensional linear coupled transport-parabolic system with variable coefficients[END_REF] to construct Gaussian beam solutions for variable coefficient operators may be useful.

Debayan Maity was partially supported by INSPIRE faculty fellowship (IFA18-MA128) and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520. Debanjana Mitra acknowledges the support from an INSPIRE faculty fellowship, RD/0118-DSTIN40-001.

The proof this theorem can also be obtained by following same philosophy as the proof of Theorem 2.3. The construction of Gaussian beam solutions of the adjoint of system (3.3) is similar to Theorem 2.8 above and [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF]Theorem 2.9]. Thus the details are left to the reader.

Linear viscoelastic Jeffreys system

In this section, we consider linear viscoelastic flows in a smooth bounded domain Ω ⊂ R n , n = 2, 3. More precisely, we consider linear single mode Jeffreys system (see for instance [START_REF] Maity | Lack of null controllability of viscoelastic flows[END_REF]Eq. 1.3]):

where

and µ 0 , µ 1 and µ 2 are positive constants. In the above system u(t,

and S(t, •) : Ω → R n×n represent the fluid velocity, pressure and stress tensor respectively. Let us consider the spaces

, where ν is the outward unit normal to ∂Ω. Let P be the Leray projector from

n×n , and consider the operator A : D(A) → H defined by

We have the following result Lemma 4.1. The operator (A, D(A)) is the infinitesimal generator of a strongly continuous semigroup on H.

Proof. This lemma can also be proved by using Lumer-Phillips theorem.

Our main of result of this section is

The system (4.1) is not exponentially stabilizable in H with an exponential decay rate ω ω 0 , by interior control

In order to prove the above result, we assume that S is of the form S(t,