Null-controllability of underactuated linear parabolic-transport systems with constant coefficients

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Null-controllability of underactuated linear

parabolic-transport systems with constant coefficients

Armand Koenig

*

, Pierre Lissy

^†

January 1, 2023

Abstract

The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one- dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small time, such kind of systems are never controllable in appropriate Sobolev spaces, whereas in large time, null-controllability holds, for sufficiently regular initial data, if and and only if a spectral Kalman rank condition is verified. We also prove that initial data that are not regular enough are not controllable. Positive results are obtained by using the so-called fictitious control method together with an algebraic solvability argument, whereas the negative results are obtained by using an appropriate WKB construction of approximate solutions for the adjoint system associated to the control problem. As an application to our general results, we also investigate into details the case of 2 × 2systems (i.e., one pure transport equation and one parabolic equation).

MSC Classification93B05, 93B07, 93C20, 35M30.

KeywordsParabolic-transport systems, null-controllability, observability.

1 Introduction

1.1 Context and state of the art

Controllability properties of coupled systems of PDEs has attracted a lot of attention this last two decades, due to their link with real-life models and also the specific mathematical difficulties arising in this context. An important part of the literature is devoted to systems where all components of the equations have the same qualitative behaviour (meaning that they are for instance all parabolic, or all hyperbolic, etc.). However, the case where different dynamics are mixed has been less studied, despite its mathematical interest. Indeed, in this context, the controllability properties of each equation

*IMT, Université de Toulouse, CNRS, Université Toulouse III-Paul Sabatier (UPS), Toulouse, France ([email protected])

Armand Koenig is supported by the ANR LabEx CIMI (under grant ANR-11-LABX-0040) within the French State Programme

“Investissements d’Avenir”.

†CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL, 75016 Paris, France ([email protected]).

Pierre Lissy is supported by the Agence Nationale de la Recherche, Project TRECOS, under grant ANR-20-CE40-0009.

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taken separately might be totally different (for instance, the heat equation with distributed control is controllable in arbitrary small time from any open subset [29,22], whereas the wave equation with distributed control is controllable in large time and under some geometric conditions [6]), so that the controllability properties of the final coupled system might be difficult to guess. Moreover, when we are considering underactuated systems (in the sense that there are less controls than equations) as in the present article, additional mathematical difficulties are appearing, due notably to the algebraic and analytic effects of the coupling terms, that become predominant in the understanding of the controllability or observability properties of the system under study. Here, in the present article, we aim to study the indirect controllability properties of a model of coupled parabolic-transport equations as introduced in [7].

Let us mention that many realistic models already studied in the literature can be reformulated in terms of coupled parabolic-transport equations, notably the wave equation with structural damping [37,34,10,24], the heat equation with memory [26,23], the 1D-Linearized compressible Navier-Stokes equations [20,13,12,8], or the Benjamin-Bona-Mahony equation [38]. For more details, we also refer to [7, §1.4]. This justifies the interest of studying a general version of coupled parabolic-transport systems as in the present article, that can be seen as an attempt to find a unified framework in order to encompass many existing results of the literature and to generalize them. Other results of interest, related to the present work, are [2], where the authors study a one-dimensional system of one transport equation and one parabolic equation, for which they prove a non-controllability result in small time by a WKB approach, and [11], where the authors prove a controllability result in large time for a one-dimensional system of one transport equation and one elliptic equation.

1.2 Presentation of the parabolic-transport system under study

Let𝑇 > 0some final time ,𝕋 = ℝ/(2𝜋ℤ)the one-dimensional torus,𝜔an nonempty open subset of𝕋, 𝑑 ∈ ℕ^∗(which represents the number of equations in our system) ,𝑚 ∈ {1, … , 𝑑}(which represents the number of controls in our system),𝐴, 𝐵, 𝐾 ∈ ℳ𝑑(ℝ)(that are some constant coupling matrices), and𝑀 ∈ ℳ𝑑,𝑚(ℝ)(that is a constant control operator). Our goal is to study the controllability properties of the following coupled system of parabolic-transport equations:

{ 𝜕^𝑡𝑓 − 𝐵𝜕𝑥²𝑓 + 𝐴𝜕𝑥𝑓 + 𝐾𝑓 = 𝑀𝑢1𝜔 in(0, 𝑇) × 𝕋,

𝑓(0, ⋅) = 𝑓0 in𝕋. (Sys)

Here, the state is𝑓∶ [0, 𝑇] × 𝕋 → ℝ^𝑑, and the control is𝑢∶ [0, 𝑇] × 𝕋 → ℝ^𝑚. The exact regularity chosen for𝑓and𝑢will be made more precise later on.

We assume that

𝑑 = 𝑑h+ 𝑑pwith1 ≤ 𝑑h< 𝑑, 1 ≤ 𝑑p< 𝑑, (H.1) 𝐵 = (0 00 𝐷) , with𝐷 ∈ ℳ𝑑p(ℝ), (H.2)

ℜ(Sp(𝐷)) ⊂ (0, +∞). (H.3)

𝑑h represents the number of purely hyperbolic equations, whereas𝑑p represents the number of parabolic equations.

Notice that (H.3) is necessary to ensure that the matrix operator𝜕𝑡− 𝐷Δis parabolic is the sense of Petrovskii ([28, Chapter 7, Definition 2]). Introducing the similar block decomposition for the𝑑 × 𝑑 matrix𝐴 = (_𝐴^𝐴₂₁^′ ^𝐴_𝐴¹²₂₂), we make the following hypothesis on the matrix𝐴^′∈ ℳ𝑑_ℎ(ℝ)

𝐴^′is diagonalizable with Sp(𝐴^′) ⊂ ℝ. (H.4)

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Notice that it is well-known that (H.4) is necessary (and sufficient, see [7, §2.2]) to ensure the well- posedness of (Sys).

1.3 Main results

To state our results, we need to introduce the following notations:

ℓ(𝜔) ≔sup{|𝐼|; 𝐼connected component of𝕋 ⧵ 𝜔}, (1) 𝜇∗≔min{|𝜇|; 𝜇 ∈Sp(𝐴^′)},

and

𝑇^∗= 𝑇^∗(𝜔) ≔ { ^ℓ(𝜔)^𝜇^∗ if𝜇∗> 0,

+∞ if𝜇∗= 0. (2)

For𝑛 ∈ ℤ, we also set

𝐵𝑛≔ −𝑛²𝐵 −i𝑛𝐴 − 𝐾 (3)

and [𝐵𝑛|𝑀] ≔ (𝑀 𝐵𝑛𝑀 … 𝐵𝑛^𝑑−1𝑀) . (4)

Our main result is the following one.

Theorem 1. Assume that the hypotheses(H.1)–(H.4)hold, that𝑇 > 𝑇∗.

Then, the spectral Kalman rank conditionrank([𝐵𝑛|𝑀]) = 𝑑holds for all𝑛 ∈ ℤif and only if for every𝑓0 ∈ 𝐻^{4𝑑(𝑑−1)}(𝕋)^𝑑, there exists a control𝑢 ∈ 𝐿²([0, 𝑇] × 𝜔)^𝑚such that the solution𝑓of the parabolic-transport system(Sys)with initial condition𝑓0satisfies𝑓(𝑇, ⋅) = 0.

Remark2. • Recall that the Kalman rank condition is necessary for the control of ODE systems [14, Theorem 1.16]. Therefore, writing the parabolic-transport system in Fourier, we imme- diately find that for every𝑇 > 0, the spectral Kalman-rank condition∀𝑛 ∈ ℤ, rank([𝐵𝑛|𝑀]) = 𝑑 is necessary for the null-controllability of every𝐻^𝑘initial conditions in time𝑇.

• Actually, we prove two slightly stronger versions of this theorem, namely theorems9and12, that are useful in order to obtain some controllability results under some constraints on Fourier coefficients of the hyperbolic part of the initial condition (see proposition20, proposition21, proposition22).

• One can refine a little bit the regularity stated in theorem1, as follows. Assume that𝑇 > 𝑇∗

and that for all𝑛 ∈ ℤ, the spectral Kalman rank conditionrank([𝐵𝑛|𝑀]) = 𝑑holds. Then:

1. for every𝑓0∈ 𝐻^{4𝑑(𝑑−1)}(𝕋)^𝑑^ℎ× 𝐻^{4𝑑(𝑑−1)−1}(𝕋)^𝑑^𝑝, there exists a control𝑢 ∈ 𝐿²([0, 𝑇] × 𝜔)^𝑚 such that the solution𝑓of the parabolic-transport system (Sys) with initial condition𝑓0

satisfies𝑓(𝑇, ⋅) = 0.

2. if𝐴12 = 0, for every𝑓0 ∈ 𝐻^{4𝑑(𝑑−1)}(𝕋)^𝑑^ℎ× 𝐻^{4𝑑(𝑑−1)−2}(𝕋)^𝑑^𝑝, there exists a control𝑢 ∈ 𝐿²([0, 𝑇] × 𝜔)^𝑚such that the solution𝑓of the parabolic-transport system (Sys) with initial condition𝑓0satisfies𝑓(𝑇, ⋅) = 0.

Indeed, by letting evolve the system freely on a short interval of time, we can show using the method of lemma23that the parabolic component becomes𝐻^{4𝑑(𝑑−1)}(𝕋)^𝑑^𝑝, so that theorem1 can be applied, taking into account that the condition𝑇 > 𝑇^∗is open and that the system is time-invariant.

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• The spectral Kalman rank conditionrank([𝐵𝑛|𝑀]) = 𝑑was first introduced in [5] for coupled systems of heat equations with diagonalizable diffusions (see also [33] for non-diagonalizable diffusions).

Theorem 3. Let𝜇 ∈Sp(𝐴^′),𝑁 ∈ ℕand𝑇 > 0. Assume that every initial condition𝑓0 ∈ 𝐿²(𝕋)^𝑑∩ {∑_|𝑛|>𝑁𝑋𝑛e^i𝑛𝑥}is steerable to0in time𝑇with control in𝐿²((0, 𝑇)×𝜔). Then, there exists𝑉0∈ker(𝐴^′∗+𝜇)

such that𝑀^∗(^𝑉₀⁰) ≠ 0.

Remark4. Theorems1,9and12only ensures null-controllability of smooth enough initial conditions.

Theorem3proves that such a regularity condition is needed in general: even if the time is large enough and if the Kalman rank condition is satisfied for every𝑛, it might happen that some𝐿²initial condition cannot be steered to0with a𝐿²control.

1.4 Precise scope and organization of the article

This article can be seen as a continuation of [7], insofar as we generalize the results of the above- mentioned article, since we are able to treat any matrices𝐴, 𝐵, 𝐾, 𝑀without any restrictions on their structure. Indeed, in [7], the authors treated the case where𝑀 = 𝐼𝑑(where no Kalman rank condition is needed), or particular cases where only the parabolic or the hyperbolic parts are controlled, under strong restrictions on the structure of the coupling matrices𝐴, 𝐵and𝐾and also on the diffusion matrix𝐵.

Let us mention that our results are sharp in terms of the controllability conditions we obtain.

However, it is very likely that the initial state space (whose choice is determined by technical reasons coming from the specific strategy we use, that is consuming in terms of regularity, see Section3.2) is almost never sharp and depends strongly on the structure of the coupling terms. Finding the exact

“good” state space remains an open problem that seems to be difficult to solve in all generality.

The article is organized as follows. In section2, we give some notations and we gather some existing results that will be used in our proof. Section3is devoted to proving that the condition rank([𝐵𝑛|𝑀]) = 𝑑is sufficient in order to obtain our desired controllability result in large time The argument is based on a fictitious control argument detailed in section3.1, where we first prove an auxiliary controllability result, in the case𝑀 = 𝐼𝑑, with regular enough controls for regular enough initial data. Then, in section3.2, we explain how to obtain a control in the range on𝑀by performing algebraic manipulations. Notice that the method of fictitious control plus algebraic solvability, that has been introduced in [16] in the context of the controllability of PDEs, has been successfully used for various problems [4,18,19,32,15,39,40,17]. One of the main novelties here is that the algebraic solvability is not directly performed on the system (or its adjoint as in [17]) but on a projected version of the system on its Fourier components. Section4is devoted to proving some necessary conditions of controllability. Section4.1is devoted to constructing WKB solutions. These solutions are used to disprove controllability in small time in section4.2and to prove theorem3in section4.3. Section5 aims to give an application of our results to the particular case of2 × 2systems together with some considerations about the sharpness of our regularity assumptions in this precise setting. To conclude, appendixAproves a general result about a “control up to a finite-dimensional space plus unique continuation” strategy that is used in section3.1, in the spirit of [30,7].

2 Some notations and preliminary results

We will rely on some basic results on the parabolic-transport system (Sys) that are already known, see [7]. For the reader convenience, we collect here the notations and results we will use most often, and we will recall some others along the way as they are used.

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Letℒbe the unbounded operator on𝐿²(𝕋)^𝑑with domain𝐻¹(𝕋)^𝑑^h× 𝐻²(𝕋)^𝑑^pdefined by ℒ𝑓 = −𝐵𝜕𝑥²𝑓 + 𝐴𝜕𝑥𝑓 + 𝐾𝑓.

The operator −ℒgenerates a strongly continuous semigroup of bounded operators of 𝐿²(𝕋)^𝑑 [7, Proposition 11]. Every𝐻^𝑘(𝕋)^𝑑is stable bye^−𝑡ℒ, and the restriction ofe^−𝑡ℒon𝐻^𝑘(𝕋)^𝑑is a strongly continuous semigroup of bounded operators [7, Remark 13]. We denote by𝑆(𝑇, 𝑓0, 𝑢)the solution at time𝑇of the parabolic-transport system (Sys) with control matrix𝑀 = 𝐼𝑑(the identity matrix of size 𝑑,i.e., we control every component with a different control), initial condition𝑓0and control𝑢.

Let𝑛0∈ ℕto be chosen large enough later on. We denote by𝑒𝑛∶ 𝑥 ∈ 𝕋 ↦e^i𝑛𝑥. We also denote by𝐸∶ ℂ → ℳ𝑑(ℂ)the following function:

𝐸(𝑧) = 𝐵 + 𝑧𝐴 − 𝑧²𝐾.

Let𝑟 > 0small enough. For|𝑧| < 𝑟, let𝑃^h(𝑧)be the eigenprojection on the sum of eigenspaces of𝐸(𝑧)associated to the set of eigenvalues𝜆(𝑧) ∈ Sp(𝐸(𝑧))such that|𝜆(𝑧)| < 𝑟. According to [7, Proposition 5],𝑃^h(𝑧)satisfies:

• 𝑃^h(0) = (0 0^{𝐼 0});

• 𝑧 ↦ 𝑃^h(𝑧)is holomorphic;

• 𝑃^h(𝑧)is a projection that commutes with𝐸(𝑧);

• 𝑃^h(𝑧)𝐸(𝑧) = 𝑂(𝑧)as𝑧 → 0.

We also set𝑃^p(𝑧) = 𝐼 − 𝑃^h(𝑧). This projection𝑃^p(𝑧)satisfies similar properties as𝑃^h(𝑧)([7, Proposi- tions 6]).

Following [7, Proposition 18], we denote by𝐹⁰the space of frequencies less than𝑛0 and by 𝐹^h(respectively𝐹^p) the space of hyperbolic frequencies greater than𝑛0(respectively the space of parabolic frequencies greater than𝑛0),i.e.

𝐹⁰= ⨁_{|𝑛|≤𝑛}₀Span(𝑒𝑛);

𝐹^p= ⨁_|𝑛|>𝑛₀Range(𝑃^p(i/𝑛))𝑒𝑛; 𝐹^h= ⨁_|𝑛|>𝑛₀Range(𝑃^h(i/𝑛))𝑒𝑛. By [7, Proposition 18], we notably have

𝐿²(𝕋)^𝑑= 𝐹⁰⊕ 𝐹^p⊕ 𝐹^h.

The space𝐹^pis stable by the semigroupe^−𝑡ℒ(see the definition of 𝑃^p[7, Proposition 5] and the definition of𝐹^p[7, Proposition 18]). We denote byℒ^pthe restriction ofℒto𝐹^p.

Similarly, the space𝐹^his stable by the semigroupe^−𝑡ℒ. We denote byℒ^hthe restriction ofℒto𝐹^h, and−ℒ^hgenerates a strongly continuousgroupof bounded operators on𝐹^h[7, Proposition 19].

LetΠ⁰,Π^p,Π^handΠbe the projections defined by 𝐿²(𝕋)^𝑑= 𝐹⁰ ⊕ 𝐹^p⊕ 𝐹^h;

Π⁰= 𝐼𝐹⁰ + 0 + 0;

Π^p= 0 + 𝐼𝐹^p + 0;

Π^h= 0 + 0 + 𝐼𝐹^h;

Π = 0 + 𝐼𝐹^p + 𝐼𝐹^h = Π^p+ Π^h.

These projections are bounded operators on𝐿²(𝕋)^𝑑[7, Proposition 18] (and also on every𝐻^𝑘(𝕋)^𝑑, as one can readily convince by following the proof of [7, Proposition 18]).

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3 Null controllability of regular initial conditions

3.1 Regular controls for regular initial conditions

As a technical preparation for the proof of theorem1, we need some results regarding the regularity of controls, when the control matrix is𝑀 = 𝐼𝑑.

Proposition 5. Assume that𝑇 > 𝑇∗ (as defined in eq.(2)) and that𝑀 = 𝐼𝑑. Let𝑘, ℓ ∈ ℕ. For every𝑓0 ∈ 𝐻^𝑘(𝕋)^𝑑, there exists𝑢 ∈ 𝐻0^𝑘((0, 𝑇) × 𝜔)^𝑑^h× 𝐻0^ℓ((0, 𝑇) × 𝜔)^𝑑^psuch that the solution of the parabolic-transport system(Sys)with initial condition𝑓0and control𝑢satisfies𝑓(𝑇, ⋅) = 0.

We adapt the proof of the corresponding result when𝑘 = 0[7, Theorem 2]. First, we prove the following adaptation of [7, Proposition 21].

Proposition 6. Let𝑇^′∈ (𝑇^∗, 𝑇)and𝑘 ∈ ℕ. If𝑛0(in the definition of𝐹⁰, see [7, Eq. (40–42)]) is large enough, there exists a continuous operator

𝒰^h∶ 𝐻^𝑘(𝕋)^𝑑× 𝐻^𝑘0((𝑇^′, 𝑇) × 𝜔)^𝑑^p→ 𝐻^𝑘0((0, 𝑇^′) × 𝜔)^𝑑^h (𝑓0, 𝑢p) ↦ 𝑢h,

such that for every(𝑓0, 𝑢p) ∈ 𝐻^𝑘(𝕋)^𝑑× 𝐻0^𝑘((𝑇^′, 𝑇) × 𝜔)^𝑑^p(where𝑢pis extended by0on(0, 𝑇^′)and𝑢his extended by0on(𝑇^′, 𝑇)),

Π^h𝑆(𝑇; 𝑓0, (𝒰^h(𝑓0, 𝑢p), 𝑢p)) = 0.

Proof. As in [7, §4.3.1], the conclusion of proposition6is equivalent to the exact controllability of the system𝜕𝑡𝑓 + ℒ^h𝑓 = Π^h(𝑢, 0)at time𝑇^′. Since−ℒ^hgenerates a strongly continuousgroup, the exact controllability at time𝑇^′is equivalent to the null-controllability at time𝑇^′, which is what we are going to prove.

When𝑘 = 0, [7, Proposition 23] is the claimed result. To extend this result to𝑘 > 0, we use a general result of Ervedoza and Zuazua concerning the regularity of controls for regular initial data in the context of groups of operators [21, Theorem 1.4]. Let˜𝜔an open subset of𝕋such that ˜𝜔 ⊂ 𝜔and 𝑇^∗(˜𝜔) < 𝑇^′. Let𝜒 ∈ 𝐶𝑐^∞(𝜔)such that𝜒 = 1on˜𝜔. Let𝜂 ∈ 𝐶0^∞(0, 𝑇^′). Let𝑧0∈ 𝐻^𝑘(𝕋)^𝑑be an initial condition. Let𝑌𝑇^′as defined by [21, Proposition 1.3] and define the control as

𝑉(𝑡) = 𝜂(𝑡)𝜒(𝑥)𝑀^∗𝑌(𝑡), where𝑌is the solution to

𝜕𝑡𝑌 − 𝐵^∗𝜕𝑥²𝑌 − 𝐴^∗𝜕𝑥𝑌 + 𝐾^∗𝑌 = 0

associated to the initial condition𝑌(𝑇^′) = 𝑌𝑇^′. According to [21, Proposition 1.3],𝑉(𝑡)is a control that steers𝑧0to0at time𝑇^′. According to [21, Theorem 1.4],𝑌𝑇^′∈ 𝐻^𝑘(𝕋)^𝑑(hence𝑉 ∈ 𝐿²(0, 𝑇^′; 𝐻^𝑘(𝜔)^𝑑)) and𝑉 ∈ 𝐻^𝑘(0, 𝑇^′; 𝐿²(𝜔)^𝑑), with estimates of the form

‖𝑉‖²_𝐿²_(0,𝑇^′_;𝐻^𝑘_(𝜔)^𝑑₎+ ‖𝑉‖²_𝐻^𝑘_(0,𝑇^′_;𝐿²_(𝜔)^𝑑₎≤ 𝐶𝑘‖𝑧0‖²_𝐻^𝑘_(𝕋)^𝑑.

We claim that𝐿²(0, 𝑇^′; 𝐻^𝑘0(𝜔)) ∩ 𝐻^𝑘0(0, 𝑇^′; 𝐿²(𝜔)) ⊂ 𝐻^𝑘((0, 𝑇^′) × 𝜔). Indeed, for every𝜏 ∈ ℝand 𝜉 ∈ ℝ,

(1 + 𝜏²+ 𝜉²)^𝑘≤ 𝐶𝑘((1 + 𝜏²)^𝑘+ (1 + 𝜉²)^𝑘).

Hence, integrating in Fourier space,

‖𝑓‖²_𝐻^𝑘_(ℝ²₎≤ 𝐶𝑘(‖𝑓‖²_𝐿²_(ℝ;𝐻^𝑘_(ℝ))+ ‖𝑓‖²_𝐻^𝑘_(ℝ;𝐿²_(ℝ))).

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Recall that forΩ ⊂ ℝ^𝑛convex¹, 𝐻0^𝑘(Ω)is the set of functions whose extension by zero outside Ωare𝐻^𝑘(ℝ^𝑛). Hence,𝐿²(0, 𝑇^′; 𝐻0^𝑘(𝜔)) ∩ 𝐻^𝑘0(0, 𝑇^′; 𝐿²(𝜔)) ⊂ 𝐻^𝑘((0, 𝑇^′) × 𝜔)as claimed, so that 𝑉 ∈ 𝐻^𝑘((0, 𝑇^′) × 𝜔)^𝑑.

Since𝜂 ∈ 𝐶^∞(0, 𝑇^′)and𝜒 ∈ 𝐶0^∞(𝜔), we conclude that𝑉 ∈ 𝐻0^𝑘((0, 𝑇^′) × 𝜔)^𝑑. For the proof of proposition5, we will also use:

Proposition 7([7], proposition 22). Let𝑇^′∈ (𝑇^∗, 𝑇)and𝑘 ∈ ℕ. If𝑛0is large enough, there exists a continuous operator

𝒰^p∶ 𝐿²(𝕋)^𝑑× 𝐿²((0, 𝑇^′) × 𝜔)^𝑑^h→ 𝐶𝑐^∞((𝑇^′, 𝑇) × 𝜔)^𝑑^p (𝑓0, 𝑢h) ↦ 𝑢p,

(in the sense that for any𝑠 ∈ ℕ,𝒰^p∶ 𝐿²(𝕋)^𝑑× 𝐿²((0, 𝑇^′) × 𝜔)^𝑑^h→ 𝐻0^𝑠(𝑇^′, 𝑇) × 𝜔)^𝑑^pis continuous for the natural topologies associated to these spaces) such that for every(𝑓0, 𝑢h) ∈ 𝐿²(𝕋)^𝑑× 𝐿²((0, 𝑇^′) × 𝜔)^𝑑^h,

Π^p𝑆(𝑇; 𝑓0, (𝑢h, 𝒰^p(𝑓0, 𝑢h)) = 0.

We can now prove proposition5by mimicking the proof of the case𝑘 = 0[7, Proposition 20 &

§4.5].

Proof of proposition5. Step 1: Control up to final dimensional space. — We claim that there exists a closed finite codimensional space𝒢of𝐻^𝑘(𝕋)^𝑑and a continuous operator𝒰∶ 𝒢 → 𝐻^𝑘0((0, 𝑇^′)×𝜔)^𝑑^h× 𝐶𝑐^∞((𝑇^′, 𝑇) × 𝜔)^𝑑^p (in the sense that for any𝑠 ∈ ℕ,𝒰∶ 𝒢 → 𝐻0^𝑘((0, 𝑇^′) × 𝜔)^𝑑^h× 𝐻0^𝑠(𝑇^′, 𝑇) × 𝜔)^𝑑^p is continuous for the natural topologies associated to these spaces) such that for every 𝑓0 ∈ 𝒢, Π𝑆(𝑇, 𝑓0, 𝒰𝑓0) = 0.

The propertyΠ𝑆(𝑇, 𝑓0, (𝑢h, 𝑢p)) = 0holds if

{𝑢h= 𝒰^h(𝑓0, 𝑢p) = 𝒰1^h(𝑓0) + 𝒰2^h(𝑢p),

𝑢p= 𝒰^p(𝑓0, 𝑢h) = 𝒰1^p(𝑓0) + 𝒰2^p(𝑢h). (5) Set𝒞 = 𝒰1^p+ 𝒰²p𝒰1^h. Then, the previous relations hold if

𝒞𝑓0= (𝐼 − 𝒰2^p𝒰^h2)𝑢p. (6)

Since𝒰2^pis continuous from𝐻0^𝑘((𝑇^′, 𝑇) × 𝜔)^𝑑^pinto𝐶𝑐((𝑇^′, 𝑇) × 𝜔)^𝑑^p, we deduce that the operator 𝒞∶ 𝐻0^𝑘((𝑇^′, 𝑇) × 𝜔)^𝑑^p → 𝐻0^𝑘((𝑇^′, 𝑇) × 𝜔)^𝑑^pis compact. Thus, according to Fredholm’s alternative, the relation (6) holds on a closed finite codimensional space𝒢.

Step 2: Conclusion. — Dealing with the finite (co)dimensional spaces𝐹⁰and𝒢is a straightforward adaptation of [7, §4.5]; more specifically, we use proposition25proved in AppendixAwith𝐻 = 𝑉 = 𝐻^𝑘(𝕋)^𝑑,𝑈𝑇= 𝐻0^𝑘((0, 𝑇) × 𝜔)^𝑑^h× 𝐻0^ℓ((0, 𝑇) × 𝜔),𝐴 = −ℒ,𝐵 =1𝜔,𝒢 = 𝒢andℱ = 𝐹⁰. The control up to a finite dimensional space hypothesis is satisfied according to the previous step. The unique continuation hypothesis is satisfied because every generalised eigenvector is a finite sum of elements of the form𝑋𝑛e^i𝑛𝑥(𝑋𝑛∈ ℂ^𝑑), and finite linear combinations of𝑋𝑛e^i𝑛𝑥have the unique continuation property thanks to,e.g., Jerison-Lebeau’s spectral inequality (see [30, Theorem 3], or [7, Eq. (90)] for our specific case).

For technical reasons, we will need the control to be in the form𝑃(𝜕𝑥)𝑢, where𝑃(𝜕𝑥)is a constant coefficients differential operator to be chosen later on.

1More generally, satisfiying thesegment condition, see[1, Definition 3.21 & Theorem 5.29].

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Proposition 8. Assume that𝑇 > 𝑇∗(as defined in(2)) and that𝑀 = 𝐼𝑑. Let𝑘, ℓ ∈ ℕ. Let𝑃be a nonzero polynomial with complex coefficients. Assume thatℓ ⩾deg(𝑃). Let𝑓0∈ 𝐻^𝑘(𝕋)^𝑑be such that for every 𝑛 ∈ ℤ,𝑃(i𝑛) = 0 ⟹ 𝑐𝑛(𝑓0) = 0. Then, there exists𝑢 ∈ 𝐻0^{𝑘+deg(𝑃)}((0, 𝑇) × 𝜔)^𝑑^h× 𝐻0^ℓ((0, 𝑇) × 𝜔)^𝑑^p such that the solution of the parabolic-transport system(Sys)with initial condition𝑓0and control𝑃(𝜕𝑥)𝑢 satisfies𝑓(𝑇, ⋅) = 0.

Proof. 𝑘, ℓ ∈ ℕwithℓ ⩾ deg(𝑃).Let𝑓0 ∈ 𝐻^𝑘(𝕋)^𝑑 be such that for every𝑛 ∈ ℤ,𝑃(i𝑛) = 0 ⟹ 𝑐𝑛(𝑓0) = 0. We define𝑓˜0 ≔ 𝑃(𝜕𝑥)⁻¹𝑓0by 𝑐𝑛( ˜𝑓0) ≔ 𝑃(i𝑛)⁻¹𝑐𝑛(𝑓0)if𝑃(i𝑛) ≠ 0and𝑐𝑛( ˜𝑓0) ≔ 0if 𝑃(i𝑛) = 0. Note that𝑃(𝜕𝑥) ˜𝑓0 = 𝑓0and that𝑓˜0 ∈ 𝐻0^{𝑘+deg(𝑃)}(𝜔)^𝑑. Then, applying proposition5to𝑓˜0

leads to the fact that there exists ̃𝑢 ∈ 𝐻0^{𝑘+deg(𝑃)}((0, 𝑇) × 𝜔)^𝑑^h× 𝐻0^ℓ((0, 𝑇) × 𝜔)^𝑑^psuch that the solution 𝑓˜of the parabolic-transport system (Sys) with initial condition𝑓˜0and control ̃𝑢satisfies𝑓(𝑇, ⋅) = 0˜ . Moreover, since𝑓˜0∈ 𝐻0^{𝑘+deg(𝑃)}(𝜔)^𝑑and ̃𝑢 ∈ 𝐻0^{𝑘+deg(𝑃)}((0, 𝑇)×𝜔)^𝑑^h×𝐻0^ℓ((0, 𝑇)×𝜔)^𝑑^pwithℓ ⩾deg(𝑃), we notably have𝑓 ∈ 𝐿˜ ²((0, 𝑇); 𝐻^{𝑘+deg(𝑃)}(𝕋)). Hence, setting𝑓 = 𝑃(𝜕𝑥) ˜𝑓and𝑢 = 𝑃(𝜕𝑥) ˜𝑓, and using that𝑃(𝜕𝑥)has constant coefficients (so that it commutes with the operator𝜕𝑡− 𝐵𝜕𝑥²+ 𝐴𝜕𝑥+ 𝐾𝐼𝑑)) ensures that𝑓verifies (Sys) with initial condition𝑓0and control𝑃(𝜕𝑥)𝑢. Moreover, since𝑓(𝑇, ⋅) = 0˜ , we also have𝑓(𝑇, ⋅) = 𝑃(𝜕𝑥) ˜𝑓(𝑇, ⋅) = 0, which leads to the desired result.

3.2 Algebraic solvability

For𝑘 ∈ ℕ, we define

[𝐵𝑛|𝑀]𝑘≔ (𝑀 𝐵𝑛𝑀 … 𝐵𝑛^𝑘−1𝑀) . (7) We prove the following variant of theorem1.

Theorem 9. Assume that the hypotheses(H.1)–(H.4)hold, and that𝑇 > 𝑇∗. Let𝑘 ∈ ℕ. Assume that for all|𝑛| ∈ ℕlarge enough, the Kalman rank conditionrank([𝐵𝑛|𝑀]𝑘) = 𝑑holds. Define the following space of functions

𝐸 ≔ {𝑓 ∈ 𝐿²(𝕋)^𝑑∶ ∀𝑛 ∈ ℤ, 𝑐𝑛(𝑓) ∈Range([𝐵𝑛|𝑀])}.

Set, when it is defined,

[𝐵𝑛|𝑀]⁺𝑘 = (𝐿^h𝑛,1 𝐿^p𝑛,1

⋮ ⋮

𝐿^h𝑛,𝑘 𝐿^p𝑛,𝑘

) ,

where the𝐿^h𝑛,𝑗are of size𝑚×𝑑hand the𝐿^p𝑛,𝑗are of size𝑚×𝑑p. Considering the𝐿^h𝑛,𝑗as rational functions of𝑛, and denoting their degree bydeg(𝐿^h𝑛,𝑗), set

𝑝 ≔ max

1≤𝑗≤𝑘deg(𝑛^𝑗−1𝐿^h𝑛,𝑗) = max

1≤𝑗≤𝑘(𝑗 − 1 +deg(𝐿^h𝑛,𝑗)).

Then, for every𝑓0∈ 𝐻^𝑝(𝕋)^𝑑∩ 𝐸, there exists a control𝑢 ∈ 𝐿²([0, 𝑇] × 𝜔)such that the solution𝑓of the parabolic-transport system(Sys)with initial condition𝑓0satisfies𝑓(𝑇, ⋅) = 0.

The idea of the proof is to first choose a “fictitious” control that acts on every components. Then, we look at the Fourier coefficients of𝑓. This transforms the control system (Sys) into a family of finite-dimensional control systems. On each of these finite-dimensional system, we perform some algebraic manipulations, called algebraic solvability, that transform the fictitious control (that acted on every component) into an “actual” control (that acts only onRange(𝑀)).

We begin with the algebraic solvability result we will use, which is essentially taken from [32,

§2.1].

(9)

Lemma 10. Let𝑘 ∈ ℕ^∗. Let˜𝐵 ∈ ℳ𝑑(ℂ)and𝑀 ∈ ℳ˜ 𝑚,𝑑(ℝ). Let𝑋0∈ ℂ^𝑑and𝑤 ∈ 𝐻0^𝑘−1(0, 𝑇; ℂ^𝑚𝑘). Write𝑤by blocks as

𝑤 = (𝑤1

𝑤⋮𝑘

) ,

where𝑤𝑗∈ 𝐻0^𝑘−1(𝕋; ℂ^𝑚), and set𝑢 = 𝑤1+ 𝑤^′2+ ⋯ + 𝑤^(𝑘−1)𝑘 . Let𝑋, ˜𝑋 ∈ 𝐶⁰(0, 𝑇; ℂ^𝑑)be the solutions of 𝑋^′= ˜𝐵𝑋 + [˜𝐵| ˜𝑀]𝑘𝑤, ˜𝑋^′= ˜𝐵𝑋 + ˜𝑀𝑢, 𝑋(0) = ˜𝑋(0) = 𝑋0,

where

[˜𝐵| ˜𝑀]𝑘≔ ( ˜𝑀 ˜𝐵𝑀 … ˜𝐵^𝑘−1𝑀) .˜ Then𝑋(𝑇) = ˜𝑋(𝑇).

Proof. Considerℳ˜𝑘the operator matrix with𝑑 + 𝑚rows and𝑘𝑚columns defined by blocks as ℳ˜𝑘≔ ( 0 −𝑀 ⋯ − ∑˜ ^𝑘−2_𝑗=0𝜕^𝑗𝑡˜𝐵^{𝑘−2−𝑗}𝑀˜

−𝐼 −𝜕𝑡 ⋯ −𝜕𝑡^𝑘−1

) = (ℳ˜𝑘,1

ℳ˜𝑘,2) . Set also

𝒫∶ (𝑋, 𝑊) ∈ 𝐻0¹(0, 𝑇; ℂ^𝑑) × 𝐿²(0, 𝑇; ℂ^𝑚) → 𝜕𝑡𝑋 − ˜𝐵𝑋 − ˜𝑀𝑊 ∈ 𝐿²(0, 𝑇; ℂ^𝑑).

We claim that

𝒫 ∘ ˜ℳ𝑘= [˜𝐵| ˜𝑀]𝑘. (8)

Indeed, we have by blocks𝒫 ∘ ˜ℳ𝑘= (𝐶0 ⋯ 𝐶𝑘−1)with 𝐶ℓ= −(𝜕𝑡− ˜𝐵)^ℓ−1∑

𝑗=0

𝜕𝑡^𝑗˜𝐵^{ℓ−1−𝑗}𝑀 + ˜˜ 𝑀𝜕𝑡^ℓ.

Then, remarking that this is a telescoping sum, 𝐶ℓ= −∑^ℓ

𝑗=1

𝜕𝑡^𝑗˜𝐵^ℓ−𝑗𝑀 +˜ ^ℓ−1∑

𝑗=0

𝜕𝑡^𝑗˜𝐵^ℓ−𝑗𝑀 + ˜˜ 𝑀𝜕𝑡^ℓ

= −𝜕𝑡^ℓ𝑀 + ˜𝐵˜ ^ℓ𝑀 − ˜˜ 𝑀𝜕𝑡^ℓ, which proves the claimed formula (8).

Now, plug eq. (8) into the differential equation𝑋^′= ˜𝐵𝑋 + [˜𝐵| ˜𝑀]𝑘𝑤, which gives 𝑋^′= ˜𝐵𝑋 + (𝜕𝑡− ˜𝐵)˜ℳ𝑘,1𝑤 − ˜𝑀 ˜ℳ𝑘,2𝑤.

With𝑌 ≔ 𝑋 − ˜ℳ𝑘,1𝑤, and remarking thatℳ˜𝑘,2𝑤 = −𝑢, this can be written as𝑌^′= ˜𝐵𝑌 + ˜𝑀𝑢. Since 𝑤 ∈ 𝐻0^𝑘−1(0, 𝑇; ℂ^𝑚𝑘),ℳ˜𝑘,1𝑤(0) = ˜ℳ𝑘,1𝑤(𝑇) = 0. Hence𝑌(0) = 𝑋(0) = ˜𝑋(0)and𝑌(𝑇) = 𝑋(𝑇). Thus𝑌solves the same Cauchy problem as˜𝑋. This proves that𝑌 = ˜𝑋, hence˜𝑋(𝑇) = 𝑌(𝑇) = 𝑋(𝑇).

We can now prove theorem9.

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Proof of theorem9. Let𝑓0 ∈ 𝐻^𝑝(𝕋)^𝑑. Set𝑋𝑛(𝑡) = 𝑐𝑛(𝑓(𝑡, ⋅))and𝑢𝑛(𝑡) = 𝑐𝑛(𝑢(𝑡, ⋅)). The desired conclusion𝑓(𝑇, ⋅) = 0reads in Fourier as:∀𝑛 ∈ ℤ,𝑋𝑛(𝑇) = 0. Moreover,𝑋𝑛satisfies

{ 𝑋^𝑛^′(𝑡) = 𝐵𝑛𝑋𝑛(𝑡) + 𝑀𝑢𝑛(𝑡), 𝑡 ∈ (0, 𝑇),

𝑋𝑛(0) = 𝑐𝑛(𝑓0). (9)

First, let us give the idea of the proof: if𝑣steers𝑓0to0when𝑀 = 𝐼, we want to define𝑤𝑛by 𝑐𝑛(𝑣(𝑡, ⋅)) = [𝐵𝑛|𝑀]𝑘𝑤𝑛(this is possible for𝑛large enough) and choose𝑢𝑛≔ 𝑤𝑛1+ 𝑤^′𝑛2+ ⋯ + 𝑤𝑛𝑘^(𝑘−1). Then, according to lemma10, the function𝑢𝑛steers𝑋𝑛from𝑐𝑛(𝑓0)to0. There are two problems with this crude choice of𝑢𝑛: this construction only works for𝑛large enough, and more importantly, we have no guarantee that the support of∑ 𝑢𝑛e^i𝑛𝑥is included in[0, 𝑇] × 𝜔.

The control strategy is to first bring frequencies less than𝑛0to0in time𝜖for some𝑛0> 0large enough to be chosen later and𝜖 > 0small enough so that𝑇 > 𝑇∗+ 2𝜖, and second use a refined version of the construction outlined above.

Step 1: Control of a finite number of frequencies. — Recall thatΠis the projection on frequencies larger than𝑛0and that𝐸was defined in the statement of theorem9. We claim that for any𝑛0∈ ℕ^∗, 𝜖 > 0and𝑓0∈ 𝐸there exists𝑢 ∈ 𝐿²(0, 𝜖; 𝐻0^𝑝(𝜔))^𝑚such that(1 − Π)𝑆(𝜖, 𝑓0, 𝑀𝑢) = 0.

This property is equivalent to the null-controllability of the system (Sys) projected on frequencies less or equal than𝑛0. The observability inequality associated with this problem [14, Theorem 2.44] is:

∃𝐶 > 0, ∀𝑔0∈ (1 − Π)𝐸, ‖e^−𝜖ℒ^∗𝑔0‖²_𝐻^−𝑝_(𝕋)^𝑑≤ 𝐶 ∫^𝜖

0

‖𝑀^∗e^−𝑡ℒ^∗𝑔0‖²_𝐿²_(𝜔)^𝑚d𝑡.

Since(1 − Π)𝐸is finite dimensional, this is equivalent to the unique continuation property

∀𝑔0∈ (1 − Π)𝐸, (𝑀^∗e^−𝑡ℒ^∗𝑔0(𝑥) = 0for(𝑡, 𝑥) ∈ (0, 𝜖) × 𝜔) ⟹ 𝑔0= 0.

Let us prove this property. Let𝑔0∈ (1 − Π)𝐸such that𝑀^∗e^−𝑡ℒ^∗𝑔0(𝑥) = 0for(𝑡, 𝑥) ∈ (0, 𝜖) × 𝜔. Since finite sums ofe^i𝑛𝑥have the unique continuation property, we have for every0 < 𝑡 < 𝜖and

|𝑛| ≤ 𝑛0,

𝑐𝑛(𝑀^∗e^−𝑡ℒ^∗𝑔0) = 0.

We can rewrite this as

𝑀^∗e^−𝑡𝐵^𝑛^∗𝑐𝑛(𝑔0) = 0.

Differentiatingℓtimes in𝑡and evaluating at𝑡 = 0, we get that for allℓ ∈ ℕand|𝑛| ≤ 𝑛0, 𝑀^∗(𝐵𝑛^∗)^ℓ𝑐𝑛(𝑔0) = 0.

Step 2: Construction of𝑢𝑛. — We set𝑇^′= 𝑇∗+ 𝜖 = 𝑇 − 𝜖.

Let us write [𝐵𝑛|𝑀]⁺𝑘 = 𝑄(i𝑛)/𝑃(i𝑛)where𝑄is a polynomial with matrix coefficients,𝑃is a polynomial (with scalar coefficients). If we denote the adjugate matrix of a matrix𝐶byAdj(𝐶), note that we may take

𝑄(i𝑛) = [𝐵𝑛|𝑀]^∗𝑘Adj([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘);

𝑃(i𝑛) =det([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘).

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Increasing𝑛0if necessary, we may assume that for every|𝑛| > 𝑛0,𝑃(i𝑛) ≠ 0. We first apply a control as in step1: for any𝑓0∈ 𝐸, there exists𝑢 ∈ 𝐿²(0, 𝜖; 𝐻0^𝑝(𝜔))^𝑚such that(1−Π)𝑆(𝜖, 𝑓0, 𝑀𝑢) = 0. Then, the resulting solution𝑓(𝜖, ⋅)is such that𝑃(i𝑛) = 0 ⟹ 𝑐𝑛(𝑓(𝜖, ⋅)) = 0, since𝑃(i𝑛) ≠ 0for

|𝑛| > 𝑛0and𝑐𝑛(𝑓(𝜖, ⋅)) = 0for|𝑛| ⩽ 𝑛0.

We consider this𝑓(𝜖, ⋅)as our new initial condition, that we denote by𝑓𝜖, and we have to steer it to0in time𝑇^′. Note that since𝑓0∈ 𝐻^𝑝(𝕋)and𝑢 ∈ 𝐿²(0, 𝜖; 𝐻^𝑝0(𝜔))^𝑑, according to Duahmel’s formula and the fact that the semigroupe^−𝑡ℒis strongly continuous on𝐻^𝑝(𝕋)^𝑑, the state𝑓𝜖also belongs to 𝐻^𝑝(𝕋)^𝑑.

Letℓ ∈ ℕlarge enough. According to proposition8, there exists 𝑣 ∈ 𝐻^𝑝+deg𝑃0 ((0, 𝑇^′) × 𝜔)^𝑑^h× 𝐻0^ℓ((0, 𝑇^′) × 𝜔)^𝑑^p such that𝑆(𝑇^′, 𝑓𝜖, 𝑃(𝜕𝑥)𝑣) = 0. Write𝑄(i𝑛)by blocks as:

𝑄(i𝑛) = (𝑄1(i𝑛)

𝑄𝑘⋮(i𝑛)) = (𝑄1^h(i𝑛) 𝑄1^p(i𝑛)

⋮ ⋮

𝑄𝑘^h(i𝑛) 𝑄𝑘^p(i𝑛)) .

where the𝑄𝑗(i𝑛)are of size𝑚 × 𝑑, the𝑄𝑗^h(i𝑛)are of size𝑚 × 𝑑hand𝑄𝑗^p(i𝑛)are of size𝑚 × 𝑑p. Notice that the𝐿^h𝑛,𝑗defined in the statement of theorem9are𝐿^h𝑛,𝑗= 𝑄𝑗^h(i𝑛)/𝑃(i𝑛). Set also

𝑤𝑛(𝑡) ≔ 𝑄(i𝑛)𝑐𝑛(𝑣(𝑡, ⋅)).

Write it by blocks as

𝑤𝑛(𝑡) = (𝑤𝑛,1(𝑡)

𝑤𝑛,𝑘⋮(𝑡)) = (𝑄1(i𝑛)𝑐𝑛(𝑣(𝑡, ⋅)) 𝑄𝑘(i𝑛)𝑐⋮𝑛(𝑣(𝑡, ⋅))) .

Finally, set

𝑢𝑛(𝑡) ≔ 𝑤𝑛,1(𝑡) + 𝑤𝑛,2^′ (𝑡) + ⋯ + 𝑤^(𝑘−1)𝑛,𝑘 (𝑡).

Step 3: Conclusion. — Remark that for every𝑛 ∈ ℤ, [𝐵𝑛|𝑀]𝑘𝑤𝑛(𝑡) = [𝐵𝑛|𝑀]𝑘𝑄(i𝑛)𝑐𝑛(𝑣(𝑡, ⋅))

= [𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘Adj([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘)𝑐𝑛(𝑣(𝑡, ⋅))

=det([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘)𝑐𝑛(𝑣(𝑡, ⋅))

= 𝑃(i𝑛)𝑐𝑛(𝑣(𝑡, ⋅)).

Moreover, since𝑆(𝑇^′, 𝑓𝜖, 𝑃(𝜕𝑥)𝑣) = 0, the control 𝑛̃𝑣 (𝑡) ≔ 𝑃(i𝑛)𝑐𝑛(𝑣(𝑡, ⋅))steers𝑐𝑛(𝑓𝜖)to0for the system𝑋𝑛^′ = 𝐵𝑛𝑋𝑛+ ̃𝑣𝑛in time𝑇^′. That is to say,𝑤𝑛steers𝑐𝑛(𝑓𝜖)to0for the system𝑋𝑛^′ = 𝐵𝑛𝑋𝑛+ [𝐵𝑛|𝑀]𝑘𝑤𝑛in time𝑇^′. Thus, according to lemma10,𝑢𝑛steers𝑐𝑛(𝑓𝜖)to0for the system (9) in time𝑇^′. Thus, the control𝑢formally defined by𝑢 ≔ ∑_𝑛∈ℤ𝑢𝑛𝑒𝑛is such that𝑆(𝑓𝜖, 𝑇^′, 𝑀𝑢) = 0. Notice that the previous sum is well-defined in𝐿²(0, 𝑇^′; 𝐿²(𝕋)). Remark that, if we define𝑢in the sense of distributions,

𝑢 = (𝑄1(𝜕𝑥) + 𝜕𝑡𝑄2(𝜕𝑥) + ⋯ + 𝜕𝑡^𝑘−1𝑄𝑘(𝜕𝑥))𝑣.

Since𝑣is supported on[0, 𝑇^′] × 𝜔, so is𝑢. Consider the differential operator𝒬 ≔ 𝑄1(𝜕𝑥) + 𝜕𝑡𝑄2(𝜕𝑥) +

⋯ + 𝜕𝑡^𝑘−1𝑄𝑘(𝜕𝑥). We have𝑢 = 𝒬𝑤. Write this operator by blocks as𝒬 = (𝒬^h 𝒬^p). In other words, 𝒬^h≔ 𝑄1^h(𝜕𝑥) + 𝜕𝑡𝑄2^h(𝜕𝑥) + ⋯ + 𝜕𝑡^𝑘−1𝑄𝑘^h(𝜕𝑥).

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The order of the differential operator𝒬^his at most Order(𝒬^h) ≤ max

1≤𝑗≤𝑘(𝑗 − 1 +deg(𝑄𝑗^h)).

Since𝐿^h𝑛,𝑗= 𝑄𝑗^h(i𝑛)/𝑃(i𝑛), according to the definition of𝑝(see theorem9),Order(𝒬^h) ≤ 𝑝 +deg(𝑃). Moreover, recall that𝑣 ∈ 𝐻^{𝑝+deg(𝑃)}0 ((0, 𝑇^′)×𝜔)^𝑑^h×𝐻0^ℓ((0, 𝑇^′)×𝜔)^𝑑^p. Thus, if we chooseℓ ≥Order(𝒬^p), 𝑢 ∈ 𝐿²((0, 𝑇^′) × 𝜔).

3.3 Upper bound on the loss of regularity

Theorem9requires initial condition to be𝐻^𝑝for some𝑝. In this section, we provide an elementary upper bound on𝑝.

Proposition 11. Assume that for|𝑛|large enough, the Kalman rank conditionrank([𝐵𝑛|𝑀]) = 𝑑holds.

Let 𝑘(𝑛) ≔inf{𝑘∶ rank([𝐵𝑛|𝑀]𝑘) = 𝑑} ∈ {−∞} ∩ ℕ.

Then, the sequence(𝑘(𝑛))𝑛∈ℤis eventually constant when|𝑛| → +∞. We will denote𝑘0≔lim_{|𝑛|→+∞}𝑘(𝑛). Proof. The rank conditionrank([𝐵𝑛|𝑀]𝑘) = 𝑑is equivalent todet([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘) ≠ 0. Let𝑃𝑘(𝑛) = det([𝐵𝑛|𝑀]𝑘[𝐵𝑛|𝑀]^∗𝑘). 𝑃𝑘 is a polynomial in𝑛, hence if𝑃𝑘(𝑛0) ≠ 0for some𝑛0, then𝑃𝑘(𝑛) ≠ 0for every large enough|𝑛|. Thus, for every𝑛0, there exists𝑛1such that𝑘(𝑛) ≤ 𝑘(𝑛0)whenever|𝑛| ≥ 𝑛1. Since𝑘(𝑛)is integer valued, it is eventually constant.

Then, we have the following version of theorem9.

Theorem 12. Assume that the hypotheses(H.1)–(H.4)hold, that𝑇 > 𝑇∗and that for all|𝑛| ∈ ℕlarge enough, the Kalman rank conditionrank([𝐵𝑛|𝑀]) = 𝑑holds. Let𝑘0as in proposition11. Let𝐸as in theorem9.

Then, for every𝑓0∈ 𝐻^4𝑑(𝑘⁰⁻¹⁾(𝕋)^𝑑∩𝐸, there exists a control𝑢 ∈ 𝐿²([0, 𝑇]×𝜔)such that the solution 𝑓of the parabolic-transport system(Sys)with initial condition𝑓0satisfies𝑓(𝑇, ⋅) = 0.

The sufficient part of theorem1, as stated in the introduction is a special case of this theorem, since we always have𝑘0⩽ 𝑑. Here is the main lemma that allows us to bound the𝑝of theorem9(see also [5, Theorem 2.1] for similar considerations).

Lemma 13. Let𝐴 ∈ ℳ𝑑(ℂ)𝑝[𝑋]a polynomial of degree at most𝑝with𝑑 × 𝑑matrices coefficients.

Assume that for some𝑧0∈ ℂ,𝐴(𝑧0)is invertible. Then,𝐴⁻¹∈ ℂ^𝑑×𝑑_{𝑝(𝑑−1)}(𝑋), i.e., the coefficients of(𝐴(𝑧))⁻¹ are rational functions of𝑧of degree at most𝑝(𝑑 − 1).

Proof. Write

𝐴(𝑧)⁻¹= 1

det(𝐴(𝑧))Adj(𝐴(𝑧)),

whereAdj(𝐴(𝑧))is the adjugate matrix of𝐴(𝑧).det(𝐴(𝑧))andAdj(𝐴(𝑧))are nonzero polynomials in 𝑧. Moreover, the coefficients ofAdj(𝐴(𝑧))are sums of products on𝑑 − 1coefficients of𝐴(𝑧). Hence, they are polynomials of degree at most(𝑑 − 1)𝑝.

The case we are interested in is:

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Proof. We have[𝐵𝑛|𝑀]𝑘0 ∈ ℂ^{𝑑×𝑚𝑘}2(𝑘0−1)⁰[𝑋], hence

[𝐵𝑛|𝑀]𝑘0[𝐵𝑛|𝑀]^∗𝑘0∈ ℂ^𝑑×𝑑4(𝑘0−1). According to the previous lemma,

([𝐵𝑛|𝑀]𝑘0[𝐵𝑛|𝑀]^∗𝑘0)⁻¹ ∈ ℂ^𝑑×𝑑4(𝑘0−1)(𝑑−1)(𝑋).

Hence[𝐵𝑛|𝑀]⁺𝑘0∈ ℂ^𝑑×𝑑𝑘 (𝑋)with𝑘 = 4(𝑘0− 1)(𝑑 − 1) + 2(𝑘0− 1) = 2(𝑘0− 1)(2𝑑 − 1).

Proof of theorem12. According to theorem9, every initial condition in𝐸 ∩ 𝐻^𝑝(𝕋)^𝑑can be steered to0, where𝑝 = deg([𝐵𝑛|𝑀]⁺𝑘0) + 𝑘0 − 1(degree as a rational function of 𝑛). But according to corollary14,deg([𝐵𝑛|𝑀]⁺𝑘0) ≤ 2(𝑘0− 1)(2𝑑 − 1). Thus𝑝 ≤ 4𝑑(𝑘0− 1). Hence, every initial condition in𝐸 ∩ 𝐻^4𝑑(𝑘⁰⁻¹⁾(𝕋)^𝑑can be steered to0.

4 Necessary conditions for null-controllability

4.1 Construction of WKB solutions

We will give other necessary conditions of null-controllability using so calledWKB solutions, that we construct here. Using these kind of approximate solutions is standard for wave equation (see, e.g., [25, pp. 426–428] or [31, Appendix B] for a more elementary presentation) or Schrödinger equation (see, e.g., [35, pp. 16–17]). WKB solutions were also used to disprove observability of some2 × 2 parabolic-transport system with variable coefficients [2, §3] (see also [3, §3] for a Navier-Stokes system with Maxwell’s law). Our construction is a generalization of their construction for system of arbitrary size, which brings a few difficulties. For the sake of clarity, we construct WKB solutions only for systems with constant coefficients, which is enough for our purposes. But it is likely that such a construction could be adapted to a large class of variable-coefficients parabolic-transport systems of arbitrary sizes.

To disprove the observability inequality, these WKB solutions ought to be constructed for the adjoint system. But the parabolic-transport system (Sys) and its adjoint have the same structure, so, in order to lighten the notations, we construct the WKB solutions for the system (Sys).

Let𝜙 ∈ 𝐶^∞([0, 𝑇] × 𝕋; ℂ)such thatℑ(𝜙) ≥ 0and𝜕𝑥𝜙never vanishes. We search approximate solutions𝑔^WKBℎ (𝑡, 𝑥)of the parabolic-transport system (Sys) with the following ansatz, whereℎ > 0is assumed to be small:

{𝑔^WKBℎ (𝑡, 𝑥) = 𝑋ℎ(𝑡, 𝑥)e^{i𝜙(𝑡,𝑥)/ℎ}, 𝑋ℎ(𝑡, 𝑥) ∼ ∑

𝑗≥0

ℎ^𝑗𝑌𝑗(𝑡, 𝑥). (10)

We have

𝜕𝑥𝑔^WKBℎ = (𝜕𝑥𝑋ℎ+ i

ℎ𝜕^𝑥𝜙𝑋ℎ)e^i𝜙/ℎ,

𝜕𝑡𝑔^WKBℎ = (𝜕𝑡𝑋ℎ+ i

ℎ𝜕^𝑡𝜙𝑋ℎ)e^i𝜙/ℎ,

𝜕𝑥²𝑔^WKBℎ = (𝜕𝑥²𝑋ℎ+ 2i

ℎ 𝜕^𝑥𝜙𝜕𝑥𝑋ℎ− 1ℎ²(𝜕𝑥𝜙)²𝑋ℎ+ i

ℎ𝜕^𝑥²𝜙𝑋ℎ)e^i𝜙/ℎ.