Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order

TL;DR

This paper investigates the controllability of coupled linear parabolic systems with fewer controls than equations, providing necessary and sufficient conditions for null and approximate controllability in small time.

Contribution

It introduces new controllability criteria for coupled parabolic systems with minimal controls, using algebraic and Carleman estimate techniques.

Findings

Necessary and sufficient conditions for controllability with constant coefficients.

Generic sufficient conditions for systems with variable coefficients in one dimension.

Application of the fictitious control method combined with algebraic and Carleman techniques.

Abstract

This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain in R^N, we consider a system of m linear parabolic equations (m > 2) with coupling terms of first and zero order, and m-1 controls localized in some arbitrary nonempty open subset. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m = 2 and N = 1, we also give a generic sufficient condition to obtain the null or approximate controllability in arbitrary small time for general coefficients depending on the space and times variables, provided that the supports of the coupling terms intersect the control domain. The results are obtained thanks to the fictitious control method together with an algebraic method and some appropriate Carleman estimates.

Figures

Equations

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\operatorname{div}(d_{1}\nabla y_{1})+\sum_{i=1}^{m}g_{1i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{1i}y_{i}+\mathds{1}_{\omega}u_{1}&\mathrm{in}~{}Q_{T},\\ \partial_{t}y_{2}=\operatorname{div}(d_{2}\nabla y_{2})+\sum_{i=1}^{m}g_{2i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{2i}y_{i}+\mathds{1}_{\omega}u_{2}&\mathrm{in}~{}Q_{T},\\ \vdots\\ \partial_{t}y_{m-1}=\operatorname{div}(d_{m-1}\nabla y_{m-1})+\sum_{i=1}^{m}g_{(m-1)i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{(m-1)i}y_{i}+\mathds{1}_{\omega}u_{m-1}&\mathrm{in}~{}Q_{T},\\ \partial_{t}y_{m}=\operatorname{div}(d_{m}\nabla y_{m})+\sum_{i=1}^{m}g_{mi}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{mi}y_{i}&\mathrm{in}~{}Q_{T},\\ y_{1}=\ldots=y_{m}=0&\mathrm{on}~{}\Sigma_{T},\\ y_{1}(0,\cdot)=y_{1}^{0},\ldots,~{}y_{m}(0,\cdot)=y_{m}^{0}&\mathrm{in}~{}\Omega,\end{array}\right.$$

$$\mathrm{div}(d_l\nabla )=\sum _{i,j=1}^N{\partial }_i(d_l^{ij}{\partial }_j),$$

$$\left\{\begin{array}[]{l}d^{ij}_{l}\in W^{1}_{\infty}(Q_{T}),\\ d^{ij}_{l}=d^{ji}_{l}\mathrm{~{}in~{}}Q_{T},\end{array}\right.$$

$$\sum _{i,j=1}^N{d}_l^{ij}{\xi }_i{\xi }_j⩾d_0|\xi {|}^2\mathrm{in}{Q}_T,\forall \xi \in {\mathbb{R}}^N,$$

$$D:=\mathrm{diag}(d_1,\dots ,d_m),$$

$$A:=(a_{ij}{)}_{1⩽i,j⩽m}\in {\mathcal{M}}_m(\mathbb{R}),$$

$$\left\{\begin{array}[]{ll}\partial_{t}y=\operatorname{div}(D\nabla y)+G\cdot\nabla y+Ay+\mathds{1}_{\omega}Bu&\mbox{ in }Q_{T},\\ y=0&\mbox{ on }\Sigma_{T},\\ y(0,\cdot)=y^{0}&\mbox{ in }~{}\Omega.\end{array}\right.$$

$$W(0,T):=L^2(0,T;H_0^1(\Omega ))\cap H^1(0,T;H^{-1}(\Omega ))\hookrightarrow {\mathcal{C}}^0([0,T];L^2(\Omega )).$$

$$W_2^{2,1}(Q_T):=L^2(0,T;H^2(\Omega )\cap H_0^1(\Omega ))\cap H^1(0,T;L^2(\Omega ))\hookrightarrow {\mathcal{C}}^0([0,T];H_0^1(\Omega )).$$

$$y(T)\equiv 0\mathrm{in}\Omega .$$

$$\Vert y(T)-y_T{\Vert }_{L^2(\Omega {)}^m}^2⩽\epsilon .$$

$$\begin{array}[]{c}g_{mi_{0}}\neq 0~{}~{}\mathrm{or}~{}~{}a_{mi_{0}}\neq 0.\end{array}$$

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\operatorname{div}(d_{1}\nabla y_{1})+g_{11}\cdot\nabla y_{1}+g_{12}\cdot\nabla y_{2}+a_{11}y_{1}+a_{12}y_{2}+\mathds{1}_{\omega}u&\mathrm{in}~{}Q_{T},\\ \partial_{t}y_{2}=\operatorname{div}(d_{2}\nabla y_{2})+g_{21}\cdot\nabla y_{1}+g_{22}\cdot\nabla y_{2}+a_{21}y_{1}+a_{22}y_{2}&\mathrm{in}~{}Q_{T},\\ y_{1}=y_{2}=0&\mathrm{on}~{}\Sigma_{T},\\ y_{1}(0,\cdot)=y_{1}^{0},~{}y_{2}(0,\cdot)=y_{2}^{0},&\mathrm{in}~{}\Omega,\end{array}\right.$$

$$\left.\begin{array}[]{ll}g_{21}=0\mathrm{~{}and~{}}a_{21}\not=0&\mathrm{~{}in~{}}(a,b)\times{\mathcal{O}}.\\ \end{array}\right.$$

$$\begin{array}[]{l}|\mathrm{det}(H(t,x))|>C\mbox{ for every }(t,x)\in(a,b)\times{\mathcal{O}},\end{array}$$

$$H:=\left(\begin{array}[]{ccccccc}-a_{21}+\partial_{x}g_{21}&g_{21}&0&0&0&0\\ -\partial_{x}a_{21}+\partial_{xx}g_{21}&-a_{21}+2\partial_{x}g_{21}&0&g_{21}&0&0\\ -\partial_{t}a_{21}+\partial_{tx}g_{21}&\partial_{t}g_{21}&-a_{21}+\partial_{x}g_{21}&0&g_{21}&0\\ -\partial_{xx}a_{21}+\partial_{xxx}g_{21}&-2\partial_{x}a_{21}+3\partial_{xx}g_{21}&0&-a_{21}+3\partial_{x}g_{21}&0&g_{21}\\ -a_{22}+\partial_{x}g_{22}&g_{22}-\partial_{x}d_{2}&-1&-d_{2}&0&0\\ -\partial_{x}a_{22}+\partial_{xx}g_{22}&-a_{22}+2\partial_{x}g_{22}-\partial_{xx}d_{2}&0&g_{22}-2\partial_{x}d_{2}&-1&-d_{2}\end{array}\right).$$

$$\begin{array}[]{l}d_{i}\in\mathcal{C}^{1}((a,b),\mathcal{C}^{2}(\mathcal{O})),~{}g_{ij}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{2}(\mathcal{O})),~{}a_{ij}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{1}(\mathcal{O}))\end{array}$$

$$\left\{\begin{array}[]{l}d_{1}\in\mathcal{C}^{1}((a,b),\mathcal{C}^{4}(\mathcal{O}))\cap\mathcal{C}^{2}((a,b),\mathcal{C}^{1}(\mathcal{O})),\\ g_{11},\mbox{ }g_{12}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{4}(\mathcal{O}))\cap\mathcal{C}^{1}((a,b),\mathcal{C}^{1}(\mathcal{O})),\\ a_{11},\mbox{ }a_{12}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{3}(\mathcal{O}))\cap\mathcal{C}^{1}((a,b),\mathcal{C}^{0}(\mathcal{O})),\\ g_{21}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{7}(\mathcal{O}))\cap\mathcal{C}^{3}((a,b),\mathcal{C}^{1}(\mathcal{O})),\\ d_{2},\mbox{ }g_{22}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{6}(\mathcal{O}))\cap\mathcal{C}^{2}((a,b),\mathcal{C}^{2}(\mathcal{O})),\\ a_{22}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{5}(\mathcal{O}))\cap\mathcal{C}^{2}((a,b),\mathcal{C}^{1}(\mathcal{O})),\\ a_{21}\in\mathcal{C}^{0}((a,b),\mathcal{C}^{6}(\mathcal{O}))\cap\mathcal{C}^{3}((a,b),\mathcal{C}^{0}(\mathcal{O})),\\ \end{array}\right.$$

$$\begin{array}[]{l}\mathrm{det}(H)=2\frac{\partial a_{21}}{\partial x}\frac{\partial d_{2}}{\partial x}g_{21}^{2}-4\frac{\partial a_{21}}{\partial x}{d_{2}}\frac{\partial g_{21}}{\partial x}{g_{21}}+\frac{\partial^{2}a_{21}}{\partial x^{2}}{d_{2}}g_{21}^{2}+2{a_{21}}\frac{\partial a_{21}}{\partial x}{d_{2}}{g_{21}}-\frac{\partial a_{21}}{\partial x}g_{21}^{2}{g_{22}}+\frac{\partial a_{21}}{\partial t}g_{21}^{2}\\ -4{a_{21}}\frac{\partial d_{2}}{\partial x}\frac{\partial g_{21}}{\partial x}{g_{21}}+{a_{21}}\frac{\partial^{2}d_{2}}{\partial x^{2}}g_{21}^{2}+a_{21}^{2}\frac{\partial d_{2}}{\partial x}{g_{21}}-3{a_{21}}{d_{2}}\frac{\partial^{2}g_{21}}{\partial x^{2}}{g_{21}}+6{a_{21}}{d_{2}}(\frac{\partial g_{21}}{\partial x})^{2}-2a_{21}^{2}{d_{2}}\frac{\partial g_{21}}{\partial x}+{a_{21}}\frac{\partial g_{21}}{\partial x}{g_{21}}{g_{22}}\\ -{a_{21}}\frac{\partial g_{21}}{\partial t}{g_{21}}-{a_{21}}g_{21}^{2}\frac{\partial g_{22}}{\partial x}+\frac{\partial a_{22}}{\partial x}g_{21}^{3}-\frac{\partial^{2}d_{2}}{\partial x^{2}}\frac{\partial g_{21}}{\partial x}g_{21}^{2}-2\frac{\partial d_{2}}{\partial x}\frac{\partial^{2}g_{21}}{\partial x^{2}}g_{21}^{2}+3\frac{\partial d_{2}}{\partial x}(\frac{\partial g_{21}}{\partial x})^{2}{g_{21}}-{d_{2}}\frac{\partial^{3}g_{21}}{\partial x^{3}}g_{21}^{2}\\ +5{d_{2}}\frac{\partial g_{21}}{\partial x}\frac{\partial^{2}g_{21}}{\partial x^{2}}{g_{21}}-4{d_{2}}(\frac{\partial g_{21}}{\partial x})^{3}+\frac{\partial g_{21}}{\partial x}g_{21}^{2}\frac{\partial g_{22}}{\partial x}+\frac{\partial g_{21}}{\partial x^{2}}g_{21}^{2}{g_{22}}-\frac{\partial g_{21}}{\partial x}^{2}{g_{21}}{g_{22}}-\frac{\partial g_{21}}{\partial x\partial t}g_{21}^{2}+\frac{\partial g_{21}}{\partial x}\frac{\partial g_{21}}{\partial t}{g_{21}}\\ -g_{21}^{3}\frac{\partial g_{22}}{\partial x^{2}}.\end{array}$$

$$g_{21}\ne 0\mathrm{in}(a,b)\times \mathcal{O}.$$

$$\left\{\begin{array}[]{ll}g_{21}\equiv\kappa\in\mathbb{R}^{*}&\mathrm{~{}in~{}}(a,b)\times{\mathcal{O}},\\ a_{21}\equiv 0&\mathrm{~{}in~{}}(a,b)\times{\mathcal{O}},\\ \partial_{x}a_{22}\neq\partial_{xx}g_{22}&\mathrm{~{}in~{}}(a,b)\times{\mathcal{O}}.\end{array}\right.$$

$$\left.\begin{array}[]{ll}g_{21}(t)\partial_{t}a_{21}(t)\neq a_{21}(t)\partial_{t}g_{21}(t)&\mathrm{~{}in~{}}(a,b).\end{array}\right.$$

$$\Vert u{\Vert }_{H^1(\Omega )}⩽C\Vert (g_{21}\cdot \nabla +a_{21}{)}^{\ast }u{\Vert }_{L^2(\Omega {)}^N},$$

$$\begin{array}[]{c}g_{21}\equiv 0\mathrm{~{}in~{}}q_{T}\\ \mathrm{and}\\ (a_{21}>a_{0}\mathrm{~{}in~{}}q_{T}\mathrm{~{}~{}~{}or~{}~{}~{}}a_{21}<-a_{0}\mathrm{~{}in~{}}q_{T}),\end{array}$$

$$\left\{\begin{array}[]{l}\mathrm{there~{}exists~{}an~{}nonempty~{}open~{}subset}~{}\gamma~{}\mathrm{of}~{}\partial\omega\cap\partial\Omega,\\ \exists x_{0}\in\gamma\mathrm{~{}s.t.~{}}g_{21}(t,x_{0})\cdot\nu(x_{0})\neq 0\mathrm{~{}~{}for~{}all~{}}t\in[0,T],\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\partial_{t}z=\operatorname{div}(D\nabla z)+G\cdot\nabla z+Az+\mathcal{N}(\mathds{1}_{\widetilde{\omega}}v)&\mathrm{in}~{}Q_{T},\\ z=0&\mathrm{on}~{}\Sigma_{T},\\ z(0,\cdot)=y^{0},~{}z(T,\cdot)=0&\mathrm{in}~{}\Omega,\end{array}\right.$$

$$\begin{array}[]{c}\displaystyle\int_{\Omega}|\psi(0,x)|^{2}dx\leqslant C_{obs}\displaystyle\iint_{q_{T}}\rho|\mathcal{N}^{*}\psi(t,x)|^{2}\,dxdt,\end{array}$$

$$\left\{\begin{array}[]{ll}-\partial_{t}\psi=\operatorname{div}(D\nabla\psi)-G^{*}\cdot\nabla\psi+A^{*}\psi&\mathrm{in}~{}Q_{T},\\ \psi=0&\mathrm{on}~{}\Sigma_{T},\\ \psi(T,\cdot)=\psi^{0}&\mathrm{in}~{}\Omega.\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\partial_{t}\widehat{z}=\operatorname{div}(D\nabla\widehat{z})+G\cdot\nabla\widehat{z}+A\widehat{z}+B\widehat{v}+\mathcal{N}f&\mathrm{in}~{}Q_{T},\\ \widehat{z}=0&\mathrm{on}~{}\Sigma_{T},\\ \widehat{z}(0,\cdot)=\widehat{z}(T,\cdot)=0&\mathrm{in}~{}\Omega,\end{array}\right.$$

$$\mathcal{L}(\hat{z},\hat{v})=\mathcal{N}f,$$

$$\mathcal{L}(\widehat{z},\widehat{v}):=\begin{array}[]{ll}\partial_{t}\widehat{z}-\operatorname{div}(D\nabla\widehat{z})-G\cdot\nabla\widehat{z}-A\widehat{z}-B\widehat{v}.\end{array}$$

Full text

††journal: Journal de Mathématiques Pures et Appliquées

**Résumé

**Cet article est consacré à l’étude de la contrôlabilité à zéro et approchée d’une classe de systèmes paraboliques linéaires couplés avec moins de contrôles que d’équations. Plus précisément, pour un domaine donné borné $\Omega$ de $\mathbb{R}^{N}$ ($N\in\mathbb{N}^{*}$), nous considérons un système de $m$ équations ($m\geqslant 2$) avec des termes de couplages d’ordre un et zéro, et $m-1$ contrôles localisés dans un sous-ensemble ouvert non-vide arbitraire $\omega$ de $\Omega$. Dans le cas de coefficients de couplage constants, nous fournissons une condition nécessaire et suffisante pour obtenir la contrôlabilité à zéro ou approchée en temps arbitrairement petit. Dans le cas $m=2$ et $N=1$, nous donnons également une condition générique suffisante pour obtenir la contrôlabilité à zéro ou approchée en temps arbitrairement petit pour des coefficients généraux dépendants des variables de temps et d’espace, lorsque le support des termes de couplage intersecte le domaine de contrôle $\omega$. Les résultats sont obtenus par combinaison de la méthode par contrôle fictif avec une méthode algébrique et des estimations de Carleman appropriées.

1 Introduction

1.1 Presentation of the problem and main results

Let $T>0$, let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ ($N\in\mathbb{N}^{*}$) supposed to be regular enough (for example of class $\mathcal{C}^{\infty}$), and let $\omega$ be an arbitrary nonempty open subset of $\Omega$. Let $Q_{T}:=(0,T)\times\Omega$, $q_{T}:=(0,T)\times\omega$, $\Sigma_{T}:=(0,T)\times\partial\Omega$ and $m\geqslant 2$. We consider the following system of $m$ parabolic linear equations, where the $m-1$ first equations are controlled:

[TABLE]

where $y^{0}:=(y_{1}^{0},\ldots,y_{m}^{0})\in L^{2}(\Omega)^{m}$ is the initial condition and $u:=(u_{1},\ldots,u_{m-1})\in L^{2}(Q_{T})^{m-1}$ is the control. The zero and first order coupling terms $(a_{ij})_{1\leqslant i,j\leqslant m}$ and $(g_{ij})_{1\leqslant i,j\leqslant m}$ are assumed to be respectively in $L^{\infty}(Q_{T})$ and in $L^{\infty}(0,T;W^{1}_{\infty}(\Omega)^{N})$. Given some $l\in\{1,...,m\}$, the second order elliptic self-adjoint operator $\operatorname{div}(d_{l}\nabla)$ is given by

[TABLE]

with

[TABLE]

where the coefficients $d^{ij}_{l}$ satisfy the uniform ellipticity condition

[TABLE]

for a constant $d_{0}>0$.

In order to simplify the notation, from now on, we will denote by

[TABLE]

so that we can write System (1.1) as

[TABLE]

It is well-known (see for instance [2, Th. 3 & 4, p. 356-358]) that for any initial data $y^{0}\in L^{2}(\Omega)^{m}$ and $u\in L^{2}(Q_{T})^{m-1}$, System (1.2) admits a unique solution $y$ in $W(0,T)^{m}$, where

[TABLE]

Moreover, one can prove (see for instance [2, Th. 5, p. 360]) that if $y^{0}\in H^{1}_{0}(\Omega)^{m}$ and $u\in L^{2}(Q_{T})^{m-1}$, then the solution $y$ is in $W^{2,1}_{2}(Q_{T})^{m}$, where

[TABLE]

The main goal of this article is to analyse the null controllability and approximate controllability of System (1.1). Let us recall the definition of these notions. It will be said that

• $\bullet$

System (1.1) is null controllable at time $T$ if for every initial condition $y^{0}\in L^{2}(\Omega)^{m}$, there exists a control $u\in L^{2}(Q_{T})^{m-1}$ such that the solution $y$ in $W(0,T)^{m}$ to System (1.1) satisfies

[TABLE]

• $\bullet$

System (1.1) is approximately controllable at time $T$ if for every $\varepsilon>0$, every initial condition $y^{0}\in L^{2}(\Omega)^{m}$ and every $y_{T}\in L^{2}(\Omega)^{m}$, there exists a control $u\in L^{2}(Q_{T})^{m-1}$ such that the solution $y$ in $W(0,T)^{m}$ to System (1.1) satisfies

[TABLE]

Let us remark that if System (1.1) is null controllable on the time interval $(0,T)$, then it is also approximately controllable on the time interval $(0,T)$ (this is an easy consequence of usual results of backward uniqueness concerning parabolic equations as given for example in [3]).

Our first result gives a necessary and sufficient condition for null (or approximate) controllability of System (1.1) in the case of constant coefficients.

Theorem** 1****.**

Let us assume that $D$, $G$ and $A$ are constant in space and time. Then System (1.1) is null (resp. approximately) controllable at time $T>0$ if and only if there exists $i_{0}\in\{1,...,m-1\}$ such that

[TABLE]

This condition is the natural one that can be expected, since it means that the last equation is coupled with at least one of the others, and is clearly a necessary condition (otherwise one cannot act on the last component $y_{n}$ which evolves freely).

Our second result concerns the case of general coefficients depending on space and time variables, in the particular case of two equations (i.e. $m=2$), and gives a controllability result under some technical conditions on the coefficients (see (1.7) and (1.8)) coming from the algebraic solvability (see Section $3.1$). To understand why this kind of condition appears here, we refer to the simple example given in [4, Ex. 1, Sec. 1.3]. Let us emphasize that the second point is only valid for $N=1$ and under Condition (1.8), which is clearly technical since it does not even cover the case of constant coefficients. However, Condition (1.8) is generic as soon as we restrict to the coupling coefficients verifying $g_{21}\not=0$ on $(0,T)\times\omega$, in the following sense: it only requires some regularity on the coefficients and a given determinant, involving some coefficients and their derivatives, to be non-zero. Since this condition may seem a little bit intricate, we will give in Remark 1 some particular examples that clarify the scope of Theorem $2$.

Theorem** 2****.**

Consider the following system:

[TABLE]

where $y^{0}=(y^{0}_{1},y^{0}_{2})\in L^{2}(\Omega)^{2}$ is the initial condition.

Then System (1.1) is null (resp. approximately) controllable at time $T$ if there exists an open subset $(a,b)\times{\mathcal{O}}\subseteq q_{T}$ where one of the following conditions is verified:

1. (i)

Coefficients of System (1.6) satisfy $d_{i}\in\mathcal{C}^{1}((a,b),\mathcal{C}^{2}(\mathcal{O})^{N^{2}})$, $g_{ij}\in\mathcal{C}^{1}((a,b),\mathcal{C}^{2}(\mathcal{O})^{N})$, $a_{ij}\in\mathcal{C}^{1}((a,b),\mathcal{C}^{2}(\mathcal{O}))$ for $i,j=1,2$ and

[TABLE] 2. (ii)

$N=1$* and coefficients of System (1.6) satisfy $d_{i},~{}g_{ij},~{}a_{ij}\in\mathcal{C}^{3}((a,b),\mathcal{C}^{7}(\mathcal{O}))$ for $i=1,2$ and*

[TABLE]

where

[TABLE]

Remark* 1**.*

1. (a)

The first part of Theorem 2 has already been proved in [5] and [6] (see the point $4.$ of Section $9$ in [6]) with less regularity on the coefficients, and is not a new result. 2. (b)

We will see during the proof of Theorem 2 that, in Item (ii) of Theorem 2, taking into account the derivatives of the appearing in (1.9), (3.5) and (3.10) and the regularity needed for the control in Proposition 3.4, we only need the following regularity for the coefficients:

[TABLE]

for all $i,j\in\{1,2\}$ in the first case (i) and

[TABLE]

in the second case $(ii)$. 3. (c)

One can easily compute explicitly the determinant of matrix $H$ appearing in (1.8):

[TABLE] 4. (d)

We remark that Condition (1.8) implies in particular that

[TABLE]

Our conjecture is that, as in the case of constant coefficients, either the first line of (1.7) or (1.12) is sufficient as soon as we restrict to the class of coupling terms that intersect the control region, since it is the minimal conditions one can expect (as in the case of constant coefficients, this only means that the last equation is coupled with one of the others). 5. (e)

Even though Condition (1.8) seems complicated, it can be simplified in some cases. Indeed, for example, System (1.6) is null controllable at time $T$ if there exists an open subset $(a,b)\times{\mathcal{O}}\subseteq q_{T}$ such that

[TABLE]

In the case $\partial_{x}a_{22}=\partial_{xx}g_{22}$, we do not know if the controllability holds and we are not able to prove it using the same techniques as in this paper.

Another simple situation is the case where the coefficients depend only on the time variable. In this case, it is easy to check that Condition (1.8) becomes simply: there exists an open interval $(a,b)$ of $(0,T)$ such that

[TABLE] 6. (f)

In fact, it is likely that one could obtain a far more general result than the one obtained in Theorem 2. By using the same reasoning, one would be able to obtain a result of controllability for arbitrary $m$ and $N$, however the generic Condition (1.8) would be far more complicated and in general impossible to write down explicitly. That is the reason why we chose to treat only the case $m=2$ and $N=1$.

This paper is organized as follows. In Section $1.2$, we recall some previous results and explain precisely the scope of the present contribution. In Section $1.3$, we give some natural perspectives of this work. In Section $1.4$ we present the main method used here, that is to say the fictitious control method together with some algebraic method. Section 2 is devoted to the proof of Theorem 1. We finish with the proof of Theorem 2 in Section $3$.

1.2 State of the art

The study of what is called the indirect controllability for linear or non-linear parabolic coupled systems has been an intensive subject of interest these last years. The main issue is to try to control many equations with less controls than equations (and ideally only one control if possible), with the hope that one can act indirectly on the equations that are not directly controlled thanks to the coupling terms. For a recent survey concerning this kind of control problems, we refer to [7]. Here, we will mainly present the results related to this work, that is to say the case of the null or approximate controllability of linear parabolic systems with distributed controls.

First of all, in the case of zero order coupling terms, a necessary and sufficient algebraic condition is proved in [8] for the controllability of parabolic systems, for constant coefficients and diffusion coefficients $d_{i}$ that are equal. This condition is similar to the usual algebraic Kalman rank condition for finite-dimensional systems. These results were then extended in [9], where a necessary and sufficient condition is given for constant coefficients but different diffusion coefficients $d_{i}$ (the Laplace operator $\Delta$ can also be replaced by some general time-independent elliptic operator). Moreover, in [8], some results are obtained in the case of time-dependent coefficients under a sharp sufficient condition which is similar to the sharp Silverman-Meadows Theorem in the finite-dimensional case.

Concerning the case of space-varying coefficients, there is currently no general theory. In the case where the support of the coupling terms intersect the control domain, the most general result is proved in [5] for parabolic systems in cascade form with one control force (and possibly one order coupling terms). We also mention [10], where a result of null controllability is proved in the case of a system of two equations with one control force, with an application to the controllability of a non-linear system of transport-diffusion equations. In the case where the coupling regions do not intersect the control domain, only few results are known and in general there are some technical and geometrical restrictions (see for example [11], [12] or [13]). These restrictions come from the use of the transmutation method that requires a controllability result on some related hyperbolic system. Let us also mention [14], where the authors consider a system of two equations in one space dimension and obtain a minimal time for null controllability, when the supports of the coupling terms do not intersect the control domain.

Concerning the case of first order coupling terms, there are also only few results. The first one is [15], where the author studies notably the case of $N+1$ coupled heat equations with $N$ control forces (we recall that $N$ is the dimension of $\Omega$) and obtains the null controllability of System (1.1) at any time when the following estimate holds:

[TABLE]

for all $u\in H^{1}_{0}(\Omega)$. Let us emphasize that inequality (1.13) is very restrictive, because it notably implies that $g_{21}$ has to be non-zero on each of its components (due to the $H^{1}-$norm appearing in the left-hand side). Moreover, in the case of two equations, the result given in [15] is true only in the dimension one. Another case is the null controllability at any time $T>0$ of $m$ equations with one control force, which is studied in [5], under many assumptions: the coupling matrix $G$ has to be upper triangular (except on the controlled equation) and many coefficients of $A$ have to be non identically equal to zero on $\omega$, notably, in the $2\times 2$ case, we should have

[TABLE]

for a constant $a_{0}>0$. The last result concerning first order coupling terms is the recent work [16], where the case of $2\times 2$ and $3\times 3$ systems with one control force is studied under some technical assumptions. Notably, in the $2\times 2$ case, the authors assume that

[TABLE]

where $\nu$ represents the exterior normal unit vector to the boundary $\partial\Omega$. Under these technical restrictions on the control domain and the coupling terms, System (1.1) is null controllable at any time $T>0$.

Here we detail how our results differ from the existing ones:

In the case of constant coefficients, we are able to obtain a necessary and sufficient condition in the case of $m$ equations, $m-1$ controls and coupling terms of order [math] or $1$, which is the main new result. Moreover, the diffusion coefficients can be different, we are able to treat the case of as many equations as wanted and we use an inequality similar to Condition (1.13) but with the $L^{2}-$norm in the left-hand side (see Lemma 2.4). To finish, we do not need the control domain to extend up to the boundary as in Condition (1.15). The main restriction is that all the coefficients of System (1.1) must be constant. 2. 2.

In the case $N=1$ and $m=2$, we are able to obtain the controllability in arbitrary small time under some generic condition which is purely technical. In Theorem 2, the geometric condition (1.15) is not necessary, which is satisfying.

1.3 Related open problems and perspectives

Let us describe briefly some related open problems and the difficulties that prevent us to go further in the present paper:

As explained in Remark $1$, we believe that the condition involving the determinant of $H$ is purely technical, which means that either the first line of (1.7) or (1.12) is sufficient as soon as we restrict to the class of coupling terms that intersect the control region. This conjecture includes the particular cases mentioned in item (e) of Remark $1$ (for instance the condition appearing in the case of time-dependent coefficients should not be necessary). However, we were not able to treat the general case because it is crucial in the proof of Theorem $1$ that the coefficients are constant, at least at two levels:

1. (a)

Lemma 2.4 is not true anymore for variable coefficients (maybe in the case of time-varying coefficients only this can be adapted though). 2. (b)

During the proof of the Carleman estimate (2.15), it is crucial that the operator $\mathcal{N}$ introduced in (2.2) has constant coefficients, so that it commutes with all the terms of the equation. This enables us to obtain an equation like (2.17), hence we apply directly Lemma (2.2), which would not be possible in the case of non-constant coefficients (there will be some remaining terms that cannot be absorbed).

• 2.

Another important limitation of our study is that we are restricted to the case of $m-1$ controls. One natural extension would then be to try to remove more controls. However, even in the case of $m$ equations and $m-2$ controls ($m\geqslant 3$) and constant coefficients, the situation is much more intricate because we do not have any idea on what would be a “natural condition” similar to the one of Theorem $1$ that will be enough to ensure the null controllability. Moreover, using the fictitious control method is not totally straightforward, because it is likely that the operator $\mathcal{N}$ needed in (2.2) has to be of order $2$ in space, which would lead to important technical issues since it would require to prove a Carleman estimate similar to (2.15) with a local term involving derivatives of order $2$.

• 3.

To prove the algebraic solvability of system (1.6) (see (1.21) for the definition of this notion), we introduce a differential operator $\mathcal{Q}$ in (3.7) which is a key issue of the proof of the second item in Theorem 2. The matrix $H$ given in (1.9) is then deduced from $\mathcal{Q}$. The operator $\mathcal{Q}$ has been found with the help of formal calculations. The authors have not been able to find simpler expressions. It could be interesting to improve it and notably to investigate if looking at higher derivatives would give some different (and simpler) conditions.

1.4 Strategy

The method described in this section is sometimes called fictitious control method and has already been used for instance in [6], [17] and [18]. One important limitation of this method is that it will never be useful to treat the case where the support of the coupling terms do not intersect the control region, because, in what we call the algebraic resolution, we have to work locally on the control region.

Roughly, the method is the following: we first control the equations with $m$ controls (one on each equation) and we try to eliminate the control on the last equation thanks to algebraic manipulations. Let us be more precise and decompose the problem into two different steps:

Analytic problem:

Find a solution $(z,v)$ in an appropriate space to the control problem by $m$ controls which are regular enough and are in the range of a differential operator. More precisely, solve

[TABLE]

where $\mathcal{N}$ is some differential operator to be chosen later and $\widetilde{\omega}$ is strongly included in $\omega$. Solving Problem (1.16) is easier than solving the null controllability at time $T$ of System (1.1), because we control System (1.16) with a control on each equation. The important points (and somehow different from the usual methods) are that:

The control has to be of a special form (it has to be in the range of a differential operator $\mathcal{N}$), 2. 2.

The control has to be regular enough, so that it can be differentiated a certain amount of times with respect to the space and/or time variables (see the next section about the algebraic resolution).

If we look for a control $v$ in the weighted Sobolev space $L^{2}(Q_{T},\rho^{-1/2})^{m}$ for some weight $\rho$, it is well known (see, e.g. [19, Th. 2.44, p. 56–57]) that the null controllability at time $T$ of System (1.16) is equivalent to the following observability inequality:

[TABLE]

where $\psi$ is the solution to the dual system

[TABLE]

Inequalities like (1.17) can be proved thanks to some appropriate Carleman estimates. The weight $\rho$ can be chosen to be exponentially decreasing at times $t=0$ and $t=T$, which will be useful later. In fact, we will have to adapt the usual HUM duality method to ensure that one can find such controls.

Algebraic problem:

For $f:=\mathds{1}_{\omega}v$, find a pair $(\widehat{z},\widehat{v})$ (where $\widehat{v}$ now acts only on the first $m-1$ equations) in an appropriate space satisfying the following control problem:

[TABLE]

and such that the spatial support of $\widehat{v}$ is strongly included in $\omega$. We will solve this problem using the notion of algebraic solvability of differential systems, which is based on ideas coming from [20, Section 2.3.8] and was already widely used in [17] and [18]. The idea is to write System (1.18) as an underdetermined system in the variables $\widehat{z}$ and $\widehat{v}$ and to see $\mathcal{N}f$ as a source term, so that we can write Problem (1.18) under the abstract form

[TABLE]

where

[TABLE]

The goal will be then to find a partial differential operator $\mathcal{M}$ satisfying

[TABLE]

When (1.21) is satisfied, we say that System (1.18) is algebraically solvable. This exactly means that one can find a solution $(\widehat{z},\widehat{v})$ to System (1.18) which can be written as a linear combination of some derivatives of the source term $\mathcal{N}f$. The main advantage of this method is that one can only work locally on $q_{T}$, because the solution depends locally on the source term and then has the same support as the source term (to obtain a solution which is defined everywhere on $Q_{T}$, one just extends it by [math]). This part will be explained in more details in Sections $2.1$ and $3.1$.

**Conclusion:

**If we can solve the analytic and algebraic problems, then it is easy to check that $(y,u):=(z-\widehat{z},-\widehat{v})$ will be a solution to System (1.1) in an appropriate space and will satisfy $y(T)\equiv 0$ in $\Omega$ (for more explanations, see [17, Prop. 1] or Sections 2.4 and 3.3 of the present paper).

2 Proof of Theorem 1

Let us remind that in this case, $D$, $G$ and $A$ are constant. In Section 2.1, 2.2 and 2.3, we will always consider some $i_{0}\in\{1,...,m-1\}$ such that

[TABLE]

We will follow the strategy described in Section 1.4, and we first begin with finding some appropriate operator $\mathcal{N}$.

2.1 Algebraic resolution

We will here explain how to choose the differential operator $\mathcal{N}$ used in the next section. We will assume from now on that all differential operators of this section are defined in $\mathcal{C}^{\infty}(Q_{T})^{m}$. The appropriate spaces will be specified in Section 2.4. We consider $\mathcal{N}$ as the operator defined for all $f:=(f_{1},\ldots,f_{m})$ by

[TABLE]

Let us recall that the definition of $\mathcal{L}$ is given in (1.20).

As explained in Section 1.4, we want find a differential operator $\mathcal{M}$ such that

[TABLE]

We have the following proposition:

Proposition** 2.1****.**

Let $\mathcal{N}$ be defined as in (2.2). Then there exists a differential operator $\mathcal{M}$ of order $1$ in time and $2$ in space, with constant coefficients, such that (2.3) is verified.

Proof of Proposition 2.1. We can remark that equality (2.3) is equivalent to

[TABLE]

The adjoint $\mathcal{L}^{*}$ of the operator $\mathcal{L}$ is given for all $\varphi\in\mathcal{C}^{\infty}(Q_{T})^{m}$ by

[TABLE]

the $m-1$ last lines coming from the particular form of our control operator $B$. Now we apply $g_{mi_{0}}\cdot\nabla-a_{mi_{0}}$ to the $(m+i)^{th}$ line for $i\in\{1,...,m-1\}$ and we add $(\partial_{t}+\operatorname{div}(d_{i_{0}}\nabla))\mathcal{L}^{*}_{m+i_{0}}\varphi+\sum_{j=1}^{m-1}(-g_{ji_{0}}\cdot\nabla+a_{ji_{0}})\mathcal{L}^{*}_{m+j}\varphi$ to the $(i_{0})^{th}$ line. Hence, remarking that $\mathcal{L}^{*}_{m+i}\varphi=\varphi_{i}$ ($i\in\{1,...,m-1\}$), we obtain

[TABLE]

Thus if we define $\mathcal{M}^{*}$ for $\psi:=(\psi_{1},...,\psi_{2m-1})\in\mathcal{C}^{\infty}(Q_{T})^{2m-1}$ by

[TABLE]

then equality (2.4) is satisfied and hence equality (2.3) also. Moreover, the coefficients of $\mathcal{M}^{*}$ are constant, hence it is also the case for the coefficients of $\mathcal{M}$.

Remark* 2**.*

Looking carefully at the proof of Proposition 2.1, we remark that one could also have constructed some differential operator $\widetilde{\mathcal{M}}$ such that $\mathcal{L}\circ\widetilde{\mathcal{M}}=\widetilde{\mathcal{N}}$ with $\widetilde{\mathcal{N}}_{1}=\ldots=\widetilde{\mathcal{N}}_{m-1}=Id$ and $\widetilde{\mathcal{N}}_{m}=-g_{mi_{0}}.\nabla-a_{mi_{0}}$. However, it is more convenient for the proof of (2.15) to work with a differential operator of same order on each line of $\mathcal{N}$ and that is the reason why we copied $-g_{mi_{0}}.\nabla-a_{mi_{0}}$ on each line.

2.2 An appropriate Carleman estimate

Let us consider the following dual system associated to System (1.16)

[TABLE]

The two main results of this section are Propositions 2.2 and 2.3, which are respectively some Carleman estimate and observability inequality. The particularity of theses inequalities is that the observation will not be directly the $L^{2}-$norm of the solution $\psi$ to System (2.7) on the subset $\omega$, but it will be the $L^{2}-$norm of some linear combination of $\psi$ and its derivatives of first order on the subset $\omega$. This particular form will be used in the next section to construct a solution to the analytic control Problem (1.16).

Let $\omega_{0}$, $\omega_{1}$ and $\omega_{2}$ be three nonempty open subsets included in $\omega$ satisfying

[TABLE]

Before stating the Carleman estimate, let us introduce some notations. For $s,\lambda>0$, let us define

[TABLE]

where

[TABLE]

Here, $\eta^{0}\in\mathcal{C}^{2}(\overline{\Omega})$ is a function satisfying

[TABLE]

with $\kappa>0$. The proof of the existence of such a function $\eta^{0}$ can be found in [21, Lemma 1.1, Chap. 1] (see also [19, Lemma 2.68, Chap. 2]). We will use the two notations

[TABLE]

for all $t\in(0,T)$.

2.2.1 Some auxiliary results

Let us now give some useful auxiliary results that we will need in our proofs. The first one is a Carleman estimate which holds for solutions of the heat equation with non-homogeneous Neumann boundary conditions:

Lemma** 2.1****.**

Let us assume that $d>0$, $u^{0}\in L^{2}(\Omega)$, $f_{1}\in L^{2}(Q_{T})$ and $f_{2}\in L^{2}(\Sigma_{T})$. Then there exists a constant $C:=C(\Omega,\omega_{2})>0$ such that the solution to the system

[TABLE]

satisfies

[TABLE]

for all $\lambda\geqslant C$ and $s\geqslant C(T^{5}+T^{10})$.

The proof of Lemma 2.1 can essentially be found in [22]. In fact, in this article, the weights are a little bit different ($t(T-t)$ instead of $t^{5}(T-t)^{5}$), but the proof just needs to be slightly adapted to obtain the present result.

From Lemma 2.1, one can deduce the following result:

Lemma** 2.2****.**

Let $h\in L^{2}(\Sigma_{T})^{m}$. Then there exists a constant $C:=C(\Omega,\omega_{2})>0$ such that for every $\varphi^{0}\in L^{2}(\Omega)^{m}$, the solution $\varphi$ to the system

[TABLE]

satisfies

[TABLE]

for every $\lambda\geqslant C$ and $s\geqslant s_{0}=C(T^{5}+T^{10})$.

The proof of Lemma 2.2 is standard and is left to the reader (one just have to apply Lemma 2.1 separately to all equations of System (2.11), sum of all the Carleman estimates and absorb the remaining lower-order terms thanks to the left-hand side).

In this section, we will use also the following estimate.

Lemma** 2.3****.**

Let $r\in\mathbb{R}$. Then there exists $C:=C(r,\omega_{2},\Omega)>0$ such that, for every $T>0$ and every $u\in L^{2}(0,T;H^{1}(\Omega))$,

[TABLE]

for every $\lambda\geqslant C$ and $s\geqslant C(T^{5}+T^{10})$.

The proof of this lemma can be found for example in [23, Lemma 3]. Our next Lemma is some Poincaré-type inequality involving the differential operator $\mathcal{N}^{*}$.

Lemma** 2.4****.**

There exists a constant $C:=C(\Omega)>0$ such that for every $u\in H_{0}^{1}(\Omega)$, the following estimate holds:

[TABLE]

where $\mathcal{N}^{*}:=g_{mi_{0}}\cdot\nabla-a_{mi_{0}}$.

Lemma 2.4 is obvious if $g_{mi_{0}}=0$. If $a_{mi_{0}}=0$, this is exactly the usual Poincaré inequality. The case $a_{mi_{0}}\not=0$ and $g_{mi_{0}}\not=0$ can be reduced to the previous case by considering

[TABLE]

In order to deal with more regular solutions, one needs the following lemma:

Lemma** 2.5****.**

Let $z_{0}\in H^{1}_{0}(\Omega)^{m}$ and $f\in L^{2}(Q_{T})^{m}$. Let us denote by $\mathcal{R}:=-\operatorname{div}(D\nabla\cdot)-G\cdot\nabla-A$ and consider $z$ the solution in $W^{2,1}_{2}(Q_{T})^{m}$ to the system

[TABLE]

Let $d\in\mathbb{N}$. Let us assume that $z_{0}\in H^{2d+1}(\Omega)^{m}$, $f\in L^{2}(0,T;H^{2d}(\Omega)^{m})\cap H^{d}(0,T;L^{2}(\Omega)^{m})$ and satisfy the following compatibility conditions:

[TABLE]

Then $z\in L^{2}(0,T;H^{2d+2}(\Omega)^{m})\cap H^{d+1}(0,T;L^{2}(\Omega)^{m})$ and we have the estimate

[TABLE]

It is a classical result that can be easily deduced for example from [2, Th. 6, p. 365].

2.2.2 Carleman inequality

We are now able to prove the following inequality:

Proposition** 2.2****.**

There exists a constant $C:=C(\omega_{0},\Omega)>0$ such that for every $\psi^{0}\in L^{2}(\Omega)^{m}$, the corresponding solution $\psi$ to System (2.7) satisfies

[TABLE]

for every $\lambda\geqslant C$ and $s\geqslant s_{0}=C(T^{5}+T^{10})$.

Remark* 3**.*

It may be quite surprising that one can put so much derivatives at the left-hand-side of equality (2.15), because the initial condition $\psi^{0}$ is only supposed to be $L^{2}$, hence $\psi$ is only assumed to be in $W(0,T)$ (see (1.3)). However, because of the fact that the exponential weight $e^{-2s\alpha}$ is strong enough to absorb the singularity that only exists at initial time $t=0$, it is quite easy to prove that all the integrals appearing in the left-hand side of (2.15) exist (this can notably be deduced for example from inequalities like (2.26), (2.27) or (2.28)).

Proof of Proposition 2.2.

The proof is inspired by [23]. The main difference here is that we keep $\mathcal{N}^{*}\psi$ at the right-hand side, which complicates a little bit the proof. Let us denote by

[TABLE]

We can assume without loss of generality that

[TABLE]

(The general case follows from a density argument). Thus, using Lemma 2.5, the solution $\psi$ to System (2.7) is an element of $L^{2}(0,T;H^{6}(\Omega)^{m})\cap H^{3}(0,T;L^{2}(\Omega)^{m})$. First of all, let us apply the differential operator

[TABLE]

to System (2.7) satisfied by $\psi$. Thus, if we call $\phi:=(\phi_{ij})_{1\leqslant i,j\leqslant N}$ with $\phi_{ij}:=\partial_{i}\partial_{j}\mathcal{N}^{*}\psi$, then one observes that $\phi$ is a solution of the following system:

[TABLE]

By applying Lemma 2.2 to $\phi$, we have

[TABLE]

for every $\lambda\geqslant C$ and $s\geqslant C(T^{5}+T^{10})$.

The proof will be divided into three steps :

In the first step, we will estimate the boundary term in the right-hand side of inequality (2.18) with some global interior term involving $\psi$ that will be absorbed later.

• 2.

In the second step, we will compare $I(s,\lambda,\phi)$ with the left-hand side of inequality (2.15).

• 3.

Finally, in the last step, we will estimate the local term of high order appearing in inequality (2.18) thanks to some local terms that will be absorbed in the left-hand side of inequality (2.18) and also thanks to the local term of the right-hand side of inequality (2.15).

Step 1: Let us consider a function $\theta\in\mathcal{C}^{2}(\overline{\Omega})$ such that

[TABLE]

After an integration by parts of the boundary term, we obtain

[TABLE]

Using successively Cauchy-Schwarz inequality and Young’s inequality, we deduce that

[TABLE]

Let us introduce $\widehat{\psi}:=\rho\psi$ with $\rho\in\mathcal{C}^{\infty}([0,T])$ defined by

[TABLE]

for some $a\in\mathbb{R}$ to be chosen later.

One remark that $\rho$ verifies $\partial^{i}_{t}\rho(0)=0$ for all $i\in\mathbb{N}$. Then $\widehat{\psi}$ is solution to the system

[TABLE]

Lemma 2.5 gives for $\widehat{\psi}$ the estimate

[TABLE]

for $d\in\{0,1,2\}$. Using the definitions of $\xi^{*}$ and $\alpha^{*}$ given in (2.10) and the particular form of $\rho$ chosen, we have

[TABLE]

[TABLE]

and

[TABLE]

Using inequality (2.21) with $\rho:=e^{-s\alpha^{*}}(s\xi^{*})^{4/5}$ and $d=1$, we obtain

[TABLE]

Applying now inequality (2.21) with $\rho:=\partial_{t}(e^{-s\alpha^{*}}(s\xi^{*})^{4/5})$ and $d=0$, we get

[TABLE]

Using (2.23) with $a=4/5$ together with (2.26) and (2.25), we deduce

[TABLE]

Using exactly the same proof and taking into account (2.24), one can also prove that

[TABLE]

Using inequalities (2.27) and (2.28), the interpolation inequality

[TABLE]

and the Cauchy-Schwarz inequality, we deduce that

[TABLE]

Thus inequalities (2.18), (2.19), (2.27) and (2.29) lead to

[TABLE]

for every $s\geqslant C(T^{5}+T^{10})$ and $\lambda\geqslant C$. Hence, since

[TABLE]

we have

[TABLE]

for every $s\geqslant C(T^{5}+T^{10})$ and $\lambda\geqslant C$.

Step 2: We apply Lemma 2.3 successively to $\mathcal{N}^{*}\psi$ with $r=5$, then to $\nabla\mathcal{N}^{*}\psi$ with $r=3$, and we obtain

[TABLE]

and

[TABLE]

for every $\lambda\geqslant C$ and $s\geqslant C(T^{5}+T^{10})$. A combination of inequalities (2.30)-(2.32) gives

[TABLE]

Step 3: Let us consider $\theta_{1}\in\mathcal{C}^{2}(\overline{\Omega})$ such that

[TABLE]

Then, after an integration by parts,

[TABLE]

Using the definition of $\xi$ and $\alpha$ given in (2.9), we deduce that

[TABLE]

which, combined with Young’s inequality, leads, for every $\varepsilon>0$, to

[TABLE]

where $C_{\varepsilon}$ depends only on $\varepsilon$. Thus, thanks to inequalities (2.33) and (2.34), one can absorb (by taking $\varepsilon$ small enough) the local terms involving $|\nabla\nabla\mathcal{N}^{*}\psi|^{2}$ and $|\nabla\nabla\nabla\mathcal{N}^{*}\psi|^{2}$ into the right-hand side of inequality (2.36) and obtain

[TABLE]

Using exactly the same reasoning we just performed for the term

[TABLE]

one can also absorb the term $s^{5}\lambda^{6}\iint_{(0,T)\times\omega_{1}}e^{-2s\alpha}\xi^{5}|\nabla\mathcal{N}^{*}\psi|^{2}dxdt$ in the right-hand side of (2.37) and we obtain

[TABLE]

Applying now Lemma 2.4, and using the definitions of $\alpha^{*}$ and $\xi^{*}$ given in (2.10), we obtain the following inequalities:

[TABLE]

The two last inequalities (2.38) and (2.39) give

[TABLE]

Hence, since $34/5<7$, one can absorb the global term of the right-hand side by taking $s$ large enough and obtain inequality (2.15).

Thanks to our Carleman inequality, we can deduce the following observability inequality:

Proposition** 2.3****.**

Then for every $\psi^{0}\in L^{2}(\Omega)^{m}$, the solution $\psi$ in $W(0,T)^{m}$ to System (2.7) satisfies

[TABLE]

where $C_{obs}:=Ce^{C(1+T+1/T^{5})}$.

The proof of Proposition 2.3 is very classical and is mainly based on dissipation estimates and the fact that the weights are bounded from below by some positive constant as soon as we are far from [math] and $T$ (for example on $(T/4,3T/4)$, together with the fact that Lemma 2.4 leads to the inequality

[TABLE]

2.3 Analytic resolution

This section is devoted to constructing a solution to the analytic control problem (1.16), with a control regular enough belonging to the range of the differential operator $\mathcal{N}$. We recall that the definition of $\mathcal{N}$ is given in (2.2). Let us consider $\theta\in\mathcal{C}^{2}(\overline{\Omega})$ such that

[TABLE]

Proposition** 2.4****.**

Let us assume that Condition (2.1) holds. Consider the system

[TABLE]

Then System (2.42) is null controllable at time $T$, i.e. for every $y^{0}\in L^{2}(\Omega)^{m}$ , there exists a control $v\in L^{2}(Q_{T})^{m}$ such that the solution $z$ to System (2.42) satisfies $z(T)\equiv 0$ in $\Omega$. Moreover, for every $K\in(0,1)$, we have $e^{Ks_{0}\alpha^{*}}v\in W^{2,1}_{2}(Q_{T})^{m}$ (the definition of $W^{2,1}_{2}(Q_{T})$ is given in (1.4)) and

[TABLE]

Proof of Proposition 2.4.

We will use the usual duality method developed by Fursikov and Imanuvilov in [21] in the spirit of what was done in [24] to obtain more regular controls. Let $y^{0}\in L^{2}(\Omega)^{m}$ and $\rho$ be the weight defined by

[TABLE]

Let $k\in\mathbb{N}^{*}$ and let us consider the following optimal control problem

[TABLE]

where $z$ is the solution in $W(0,T)^{m}$ to System (2.42).

The functional $J_{k}:L^{2}(Q_{T},\rho^{-1/2})^{m}\rightarrow\mathbb{R}^{+}$ is differentiable, coercive and strictly convex on the space $L^{2}(Q_{T},\rho^{-1/2})^{m}$. Therefore there exists a unique solution to the control optimal problem (2.44) (see [25, p. 128]) and the optimal control $v_{k}$ is characterized thanks to the solution $z_{k}$ of the primal system

[TABLE]

the solution $\varphi_{k}$ to the dual system

[TABLE]

and the relation

[TABLE]

The rest of the proof is divided into two steps. In the first step, we will prove that the sequence $(v_{k})_{k\in\mathbb{N}^{*}}$ converges to a control $v\in L^{2}(Q_{T},\rho^{-1/2})^{m}$ with an associated solution $z$ to System (2.42) satisfying $z(T)\equiv 0$ in $\Omega$. Then, in the second step, we will establish (2.43).

Step 1:

Firstly, the characterization (2.45), (2.46) and (2.47) of the minimizer $v_{k}$ of $J_{k}$ in $L^{2}(Q_{T},\rho^{-1/2})^{m}$ leads to the following computations

[TABLE]

Moreover, using the definition of $J_{k}$, the definition of $\theta$, our observability inequality (2.40), the expression (2.48) and the Cauchy-Schwarz inequality, we infer

[TABLE]

from which we deduce

[TABLE]

Then, using (2.48) and (2.49), we deduce

[TABLE]

Furthermore, we have (see [25])

[TABLE]

where $C$ does not depend on $y^{0}$ and $k$. Then, using inequalities (2.50) and (2.51), we deduce that there exist subsequences, which are still denoted $v_{k}$, $z_{k}$, such that the following weak convergences hold:

[TABLE]

Passing to the limit in $k$, $z$ is solution to System (2.42). Moreover, using the expression of $J_{k}$ given in (2.44) and inequality (2.50), we deduce by letting $k$ going to $\infty$ that $z(T)\equiv 0$ in $\Omega$. Thus the solution $z$ to System (2.42) with control $v\in L^{2}(Q_{T},\rho^{-1/2})^{m}$ satisfies $z(T)\equiv 0$ in $\Omega$ and using (2.50), we obtain the inequality

[TABLE]

Step 2: One remarks that for every $K\in(0,1)$, there exists a constant $C>0$ such that

[TABLE]

This inequality and estimate (2.50) imply that

[TABLE]

We recall that $v_{k}$ is defined in (2.47), moreover, thanks to the definitions of $\xi$ and $\alpha$ given in (2.9), one has, for every $\eta\geqslant 0$ ,

[TABLE]

These above inequalities, for $\eta:=7$, lead to the fact that

[TABLE]

[TABLE]

and

[TABLE]

For every $\eta,\nu>0$ there exists a constant $C_{\eta,\nu}$ such that

[TABLE]

Combining the last inequality with (2.53)-(2.56), we deduce that

[TABLE]

Using (2.15), we obtain that $e^{Ks_{0}\alpha^{*}}\nabla v_{k}$, $e^{Ks_{0}\alpha^{*}}\Delta v_{k}$, $\partial_{t}(e^{Ks_{0}\alpha^{*}}v_{k})\in L^{2}(Q_{T})^{m}$, and that

[TABLE]

The estimate (2.50) of $v_{k}$ gives

[TABLE]

We conclude by letting $k\rightarrow+\infty$ (after extracting an adequate subsequence) in the inequalities above.

2.4 End of the proof of Theorem 1

Let us assume that Condition 1.5 holds. We will prove first the null controllability at time $T$ of System (1.1). Let $y^{0}\in L^{2}(\Omega)^{m}$. We follow the method explained in Section 1.4. Let us remind that $\theta$ is defined in (2.41). Using Proposition 2.4, the following system:

[TABLE]

is null controllable at time $T$, thus there exists a control $v\in L^{2}(Q_{T})^{m}$ such that the solution $z$ in $W(0,T)^{m}$ to System (2.57) satisfies

[TABLE]

Moreover

[TABLE]

Taking into account Proposition 2.1 and the definition of $\mathcal{M}^{*}$ given in (2.6), one has (2.3) with the operator

[TABLE]

defined by

[TABLE]

Let $(\widehat{z},\widehat{v})$ be defined by

[TABLE]

Using (2.58) and the fact that $\mathcal{M}$ is a differential operator of order $1$ in time and $2$ in space (see Proposition 2.1) with bounded coefficients, we obtain that $(\widehat{z},\widehat{v})\in L^{2}(Q_{T})^{m}\times L^{2}(Q_{T})^{m-1}$. Moreover, using (2.58), we have $\widehat{z}(0,\cdot)=\widehat{z}(T,\cdot)=0$ in $\Omega$ and we remark that $(\widehat{z},\widehat{v})$ is a solution to the control problem

[TABLE]

in particular $\mathcal{L}\circ\mathcal{M}=\mathcal{N}$. Finally, $(\widehat{z},\widehat{v})\in W(0,T)^{m}\times L^{2}(Q_{T})^{m-1}$ thanks to the usual parabolic regularity. Thus the pair $(y,u):=(z-\widehat{z},-\widehat{v})$ is a solution to System (1.1) in $W(0,T)^{m}\times L^{2}(Q_{T})^{m-1}$ and satisfies

[TABLE]

3 Proof of Theorem 2

Let us remind that in this case, we have only $2$ equations.

3.1 Algebraic resolution

We will assume in this section that all differential operators are defined in $\mathcal{C}^{\infty}(Q_{T})^{2}$. We recall that $\mathcal{N}$ is simply the identity operator and $\mathcal{L}$ is given in (1.20). We want to find a differential operator $\mathcal{M}$ satisfying

[TABLE]

Let us emphasize that when coefficients are depending on time and space, we prove equality (1.21) with the identity operator in the right-hand side (and not $\mathcal{N}$ as defined in (2.2)), that is equality (3.1), because Proposition 2.4 holds only for constant coefficients and does not seem to be adapted to the case of non-constant coefficients. We have the following proposition:

Proposition** 3.1****.**

One has

1. (i)

Under Conditions (1.7) and (1.10), there exists a differential operator $\mathcal{M}$ of order at most $1$ in time and $2$ in space, with bounded coefficients on $(a,b)\times{\mathcal{O}}$, such that equality (3.1) holds. 2. (ii)

If $N=1$ and under Condition (1.8) and (1.11), there exists a differential operator $\mathcal{M}$ of order at most $2$ in time, $4$ in space, and $1-2$ respectively in crossed space-time, with bounded coefficients on $(a,b)\times{\mathcal{O}}$, such that equality (3.1) holds.

Proof of Proposition 3.1.

Equality (3.1) is equivalent to

[TABLE]

Taking into account the definition of $\mathcal{L}$ given in (1.20), the adjoint $\mathcal{L}^{*}$ of the operator $\mathcal{L}$ is given by

[TABLE]

We remark first that

[TABLE]

1. (i)

Let us consider some open subset $\widetilde{\mathcal{O}}$ included in $\mathcal{O}$ on which we have $|a_{21}|\geqslant C>0$ (such an open subset exists thanks to (1.7)). Since $g_{21}=0$ in $(a,b)\times{\mathcal{O}}$, one can just consider $\mathcal{M}^{*}$ defined for every $\psi:=(\psi_{1},\psi_{2},\psi_{3})\in\mathcal{C}^{\infty}(Q_{T})^{3}$, locally on $(a,b)\times{\widetilde{\mathcal{O}}}$, by

[TABLE]

so that equality (3.2) is satisfied. Moreover, the coefficients of $\mathcal{M}^{*}$ (and hence of $\mathcal{M}$) are bounded. 2. (ii)

If $N=1$ and under Condition (1.8), one can proceed as follows: we consider the operator

[TABLE]

i.e.

[TABLE]

It is easy to see that there are only $7$ different derivatives of $\varphi$ appearing in (3.6), which are the following ones:

[TABLE]

Hence, we can see the operator $\mathcal{Q}$ as a matrix $M$ acting on these derivatives (see [17, Sec. 3.2]), so that $M$ is a square matrix of size $7\times 7$. More precisely, one has

[TABLE]

with

[TABLE]

Matrix $M$ is invertible since Condition (1.8) is verified. Let us call $P$ the projection on the two first components

[TABLE]

Then, by definition of the inverse, we have

[TABLE]

The expression of $\mathcal{Q}$ given in (3.8) leads to

[TABLE]

If we denote by $\mathcal{R}_{1}:=\partial_{t}+\partial_{x}(d_{1}\partial_{x}\cdot)-\partial_{x}(g_{11}\cdot)+a_{11}$ and $\mathcal{R}_{2}:=a_{12}-\partial_{x}(g_{12}\cdot)$, using (3.6), we remark that the previous equality can be rewritten as

[TABLE]

where

[TABLE]

Hence equality (3.2) is satisfied for

[TABLE]

Moreover, thanks to Conditions (1.8), the coefficients of $\mathcal{M}^{*}$ (and hence of $\mathcal{M}$) are bounded.

Remark* 4**.*

The most important point in the construction of $\mathcal{Q}$ is to differentiate enough time the equations as in (3.6) in order to obtain the same number equations than “unknowns”, the unknowns being $\varphi$ and its derivatives seen as independent algebraic variables (see notably [17, Section 3.2.2] for further explanations). One can check that this cannot be performed by differentiating the equations less times.

3.2 Analytic resolution

Let $\omega_{1}$ be a nonempty open subsets included in $\omega$ satisfying

[TABLE]

Let us consider $\theta\in\mathcal{C}^{2}(\overline{\Omega})$ such that

[TABLE]

We are going to explain what are the main differences with Subsection 2.2 in order to obtain a Carleman inequality. First of all, we need to find an equivalent to Lemma 2.1, which is the following:

Lemma** 3.1****.**

Let us assume that $u^{0}\in L^{2}(\Omega)$, $f_{1}\in L^{2}(Q_{T})$ and $d\in W^{1}_{\infty}(Q_{T})$ such that $d>C>0$ in $Q_{T}$. Then there exists a constant $C:=C(\Omega,\omega_{2})>0$ such that the solution to the system

[TABLE]

satisfies

[TABLE]

for all $\lambda\geqslant C$ and $s\geqslant C(T^{5}+T^{10})$.

The proof of Lemma 3.1 can be easily obtained by using the method of [22] together with the Carleman estimate proved in [26]. Let us consider the backward system

[TABLE]

where $\psi^{0}:=(\psi_{1}^{0},\psi_{2}^{0})\in L^{2}(\Omega)^{2}$. From the last Lemma, one can deduce:

Proposition** 3.2****.**

One has

1. (i)

Under Condition (1.7) and (1.10), there exists a constant $C:=C(\omega_{1},\Omega)>0$ such that for every $\psi^{0}:=(\psi_{1}^{0},\psi_{2}^{0})\in L^{2}(\Omega)^{2}$, the corresponding solution $\psi:=(\psi_{1},\psi_{2})$ of the backward problem (3.13) satisfies

[TABLE] 2. (ii)

If $N=1$ and under Condition (1.8) and (1.11), we obtain the same conclusion as in item (i) by replacing estimate ((i)) by

[TABLE]

The proof is very similar to the proof of Proposition $2.2$, the only difference is the beginning of the proof, we apply the operator $\nabla$ to the equation (3.13) in the case (1.7) and the operators $\nabla\nabla\nabla$ in the case (1.8). After that we exactly follow the steps $1$,$2$, and $3$ of the proof of Proposition $2.2$.

As a consequence, we also can derive the following observability inequality, whose proof is very classical (see also Proposition 2.3):

Proposition** 3.3****.**

Under assumptions (1.7) and (1.10) or under assumptions (1.8) and (1.11), for every $\psi^{0}\in L^{2}(\Omega)^{2}$, the solution to System (3.13) satisfies

[TABLE]

where $s_{0}:=C(T^{5}+T^{10})$ and $C_{obs}:=e^{C(1+T+1/T^{5})}$.

To conclude, one can obtain the following controllability result:

Proposition** 3.4****.**

Consider the following system:

[TABLE]

Under Conditions (1.7) and (1.10) or if $N=1$ and under Condition (1.8) and (1.11), System (3.16) is null controllable at time $T$, that is for every $y^{0}\in L^{2}(\Omega)^{2}$ there exists a control $v:=(v_{1},v_{2})\in L^{2}(Q_{T})^{2}$ such that the solution $z$ to System (3.16) satisfies $z(T)\equiv 0$ in $\Omega$. Moreover for every $K\in(0,1)$, we have $e^{Ks_{0}\alpha^{*}}v\in\mathcal{X}^{2}$ where:

1. (i)

Under Conditions (1.7) and (1.11), $\mathcal{X}$ is defined by

[TABLE] 2. (ii)

If $N=1$ and under Conditions (1.8) and (1.11), $\mathcal{X}$ is defined by

[TABLE]

Moreover in the both cases, we have the estimate

[TABLE]

One more time, the proof is very similar to the proof of Proposition $2.3$, notably and one can recover easily estimates on the derivatives of order $1$ and $2$ in time of the control by using equation (3.13) verified by $\varphi$ and estimates similar to (2.56).

3.3 Proof of Theorem 2

The proof is totally similar to the one of Theorem 1. Let us assume that one of the two Conditions (1.7) or (1.8) holds, and let us prove that System (1.6) is null controllable at time $T$ (which will imply the approximate controllability at time $T$). Let $y^{0}\in L^{2}(\Omega)^{2}$. Using Proposition 3.4, System (3.16) is null controllable at time $T$, more precisely there exists a control $v\in L^{2}(Q_{T})^{2}$ such that the solution $z$ in $W(0,T)^{2}$ to System (3.16) satisfies

[TABLE]

Moreover

[TABLE]

Let us remind that $\theta$ was defined in (3.12). Let $(\widehat{z}_{1},\widehat{z}_{2},\widehat{v})$ be defined by

[TABLE]

with $\mathcal{M}:\mathcal{X}^{2}\rightarrow L^{2}(Q_{T})^{2}\times L^{2}(Q_{T})$ given as the adjoint of $\mathcal{M}^{*}$ defined in (3.5) and (3.11). Thanks to the definition of $\theta$ given in (3.12), the fact the coefficients of $\mathcal{M}$ are necessarily at least in $L^{\infty}((a,b)\times\omega_{0})$, the definition of $\mathcal{X}$ given in (3.17)-(3.18) and the fact that $\mathcal{M}$ is of order $1$ in time and $2$ in space under Condition (1.7) and is of order $2$ in time, $4$ in space and $1-2$ in crossed time-space (which is an interpolation space between $L^{2}((0,T),H^{4}(\Omega))$ and $H^{2}((0,T),L^{2}(\Omega))$ thanks to [27, 13.2, P. 96]) under Condition (1.8), we obtain $(\widehat{z}_{1},\widehat{z}_{2},\widehat{v})\in L^{2}(Q_{T})^{2}\times L^{2}(Q_{T})$. Moreover, using (3.20), we remark that $(\widehat{z}_{1},\widehat{z}_{2},\widehat{v})$ is a solution to the control problem

[TABLE]

Finally, $\widehat{z}\in W(0,T)^{2}$ thanks to the usual parabolic regularity. Thus the pair $(y,u):=(z-\widehat{z},-\widehat{v})$ is a solution to System (1.6) in $W(0,T)^{2}\times L^{2}(Q_{T})$ and satisfies

[TABLE]

which concludes the proof of Theorem 2.

Acknowledgements

The authors would like to thank the anonymous referees for their numerous comments and corrections that greatly contributed to improving the final version of the paper.