Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms

TL;DR

This paper investigates the controllability of coupled parabolic equations with zero and first order terms, establishing new conditions for null controllability and providing examples where approximate controllability fails.

Contribution

It introduces a new sufficient condition for null controllability and presents the first example of a system that is not approximately controllable under certain coupling conditions.

Findings

Null controllability achieved under a new condition

First example of non-approximate controllability with intersecting support

Insights into the influence of coupling terms on controllability

Abstract

This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of R^N (N>1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain.

Figures

Equations

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\operatorname{div}(d_{1}\nabla y_{1})+g_{11}\cdot\nabla y_{1}+g_{12}\cdot\nabla y_{2}+a_{11}y_{1}+a_{12}y_{2}+\mathds{1}_{\omega}u&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{2}=\operatorname{div}(d_{2}\nabla y_{2})+g_{21}\cdot\nabla y_{1}+g_{22}\cdot\nabla y_{2}+a_{21}y_{1}+a_{22}y_{2}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y=0&\mbox{on }\Sigma_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0,\cdot)=y^{0}&\mbox{in }\Omega,\end{array}\right.$$

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

$$\left\{\begin{array}[]{l}d^{ij}_{l}\in W^{1}_{\infty}(Q_{T}),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr d^{ij}_{l}=d^{ji}_{l}\mbox{ in }Q_{T},\end{array}\right.$$

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

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

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

$$y(T,\cdot )=0\text{mbox}in\Omega .$$

$$g_{21}\ne 0\text{mbox}in(t_0,t_1)\times {\omega }_0.$$

$$g_{21}\cdot \nabla +a_{21}={\partial }_{x_1}\text{mbox}on{\mathcal{O}}_T:=(t_0,t_1)\times \mathcal{O}.$$

$$\left\{\begin{array}[]{l}\widetilde{a}_{22}\mbox{ is not an element of the }\mathcal{C}^{0}_{t,x_{2},...,x_{N}}(\overline{\omega}_{T})\mbox{-module }\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left\langle 1,\widetilde{g}_{22}^{2},...,\widetilde{g}_{22}^{N},d_{2}^{22},...,d_{2}^{NN}\right\rangle_{\mathcal{C}^{0}_{t,x_{2},...,x_{N}}(\overline{\omega}_{T})},\end{array}\right.$$

$$\left\{\begin{array}[]{l}\widetilde{g}_{22}^{i}:=g_{22}^{i}-\sum\limits_{j=1}^{N}\partial_{x_{j}}d_{22}^{ij},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\widetilde{a}_{22}:=-a_{22}+\operatorname{div}(g_{22}).\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\operatorname{div}(d_{1}\nabla y_{1})+\sum_{i=1}^{m}g_{1i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{1i}y_{i}+\mathds{1}_{\omega}u_{1}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{2}=\operatorname{div}(d_{2}\nabla y_{2})+\sum_{i=1}^{m}g_{2i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{2i}y_{i}+\mathds{1}_{\omega}u_{2}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\vdots\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{m-1}=\operatorname{div}(d_{m-1}\nabla y_{m-1})+\sum_{i=1}^{m}g_{(m-1)i}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{(m-1)i}y_{i}+\mathds{1}_{\omega}u_{m-1}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{m}=\operatorname{div}(d_{m}\nabla y_{m})+\sum_{i=1}^{m}g_{mi}\cdot\nabla y_{i}+\sum_{i=1}^{m}a_{mi}y_{i}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y_{1}=\ldots=y_{m}=0&\mbox{on }\Sigma_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y_{1}(0,\cdot)=y_{1}^{0},\ldots,\leavevmode\nobreak\ y_{m}(0,\cdot)=y_{m}^{0}&\mbox{in }\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{l}\widetilde{a}_{mm}\mbox{ is not an element of the }\mathcal{C}^{0}_{t,x_{2},...,x_{N}}(\overline{\omega}_{T})\mbox{-module }\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left\langle 1,\widetilde{g}_{mm}^{2},...,\widetilde{g}_{mm}^{N},d_{2}^{22},...,d_{2}^{NN}\right\rangle_{\mathcal{C}^{0}_{t,x_{2},...,x_{N}}(\overline{\omega}_{T})},\end{array}\right.$$

$$\left\{\begin{array}[]{l}\widetilde{g}_{mm}^{i}:=g_{mm}^{i}-\sum\limits_{j=1}^{N}\partial_{x_{j}}d_{mm}^{ij},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\widetilde{a}_{mm}:=-a_{mm}+\operatorname{div}(g_{mm}).\end{array}\right.$$

$$\left\{\begin{array}[]{l}\theta=1\mbox{ in }\omega,\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\operatorname{Supp}(\theta)\subset\overline{\omega},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\theta>0\mbox{ in }\omega_{1}\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\Delta y+\mathds{1}_{\omega}u&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{2}=\Delta y_{2}+ay_{2}+\partial_{x_{1}}(\theta y_{1})&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y=0&\mbox{on }\Sigma_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0,\cdot)=y^{0}&\mbox{in }\Omega\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\partial_{t}y_{1}=\partial_{xx}y+\mathds{1}_{\omega}u&\mbox{in }(0,T)\times(0,\pi),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}y_{2}=\partial_{xx}y_{2}+ay_{2}+\partial_{x}y_{1}&\mbox{in }(0,T)\times(0,\pi),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(\cdot,0)=y(\cdot,\pi)=0&\mbox{on }(0,T),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0,\cdot)=y^{0}&\mbox{in }(0,\pi).\end{array}\right.$$

$${\partial }_t{\tilde{y}}_2=\mathrm{div}({\tilde{d}}_2\nabla {\tilde{y}}_2)+{\tilde{g}}_{22}\cdot \nabla {\tilde{y}}_2+{\tilde{a}}_{22}{\tilde{y}}_2+{\partial }_{x_1}\widetilde{y_1}+{\tilde{a}}_{21}{\tilde{y}}_1\text{mbox}in{U}_{\epsilon }.$$

$$F:s_0:=(s_2,\dots ,s_N)\in U\subset {\mathbb{R}}^{N-1}\mapsto F(s_0)\in \gamma ,$$

$$\left\{\begin{array}[]{ll}\frac{d}{ds}\Phi(t,s,\sigma)=g^{T}_{21}(\Phi(t,s,\sigma)),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\Phi(t,0,\sigma)=(t,\sigma).\end{array}\right.$$

$$\Lambda :(t,s,z)\in (t_0,t_1)\times (0,\epsilon )\times U\mapsto \Phi (t,s,F(z)).$$

$${\partial }_t{\tilde{y}}_i(t,s,z)=({\partial }_t{y}_i)\circ \Lambda (t,s,z)\text{mbox}fori=1,2\text{leavevmode}\text{mbox}and{\partial }_s{\tilde{y}}_2(t,s,z)=(g_{21}\cdot \nabla y_2)\circ \Lambda (t,s,z),$$

$${\overline{y}}_1(t,x):={\theta }^{-1}(t,x){\tilde{y}}_1(t,x)$$

$${\overline{y}}_2(t,x):={\theta }^{-1}(t,x){\tilde{y}}_2(t,x),$$

$${\partial }_t{\overline{y}}_2=\mathrm{div}({\tilde{d}}_2\nabla {\overline{y}}_2)+{\partial }_{x_1}{\overline{y}}_1+{\overline{g}}_{22}\cdot \nabla {\overline{y}}_2+{\overline{a}}_{22}{\overline{y}}_2\text{mbox}in{\mathcal{O}}_T.$$

$$\left\{\begin{array}[]{l}\overline{y}_{1}(t,x):=\theta^{-1}(t,x)\widetilde{y}_{1}(t,x),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\overline{y}_{2}(t,x):=\theta^{-1}(t,x)\widetilde{y}_{2}(t,x).\end{array}\right.$$

$${\partial }_t{\overline{y}}_2=\mathrm{div}({\tilde{d}}_2\nabla {\overline{y}}_2)+{\overline{g}}_{22}\cdot \nabla {\overline{y}}_2+{\overline{a}}_{22}{\overline{y}}_2+{\partial }_{x_1}{\overline{y}}_1+{\theta }^{-1}({\partial }_{x_1}\theta +{\tilde{a}}_{21}\theta ){\overline{y}}_1,$$

$$\left\{\begin{array}[]{ll}\partial_{t}\widehat{y}_{1}=\operatorname{div}(d_{1}\nabla\widehat{y}_{1})+g_{11}\cdot\nabla\widehat{y}_{1}+g_{12}\cdot\nabla\widehat{y}_{2}+a_{11}\widehat{y}_{1}+a_{12}\widehat{y}_{2}+\widehat{u}_{1}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}\widehat{y}_{2}=\operatorname{div}(d_{2}\nabla\widehat{y}_{2})+g_{21}\cdot\nabla\widehat{y}_{1}+g_{22}\cdot\nabla\widehat{y}_{2}+a_{21}\widehat{y}_{1}+a_{22}\widehat{y}_{2}+\widehat{u}_{2}&\mbox{in }Q_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\widehat{y}=0&\mbox{on }\Sigma_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\widehat{y}(0,\cdot)=y^{0},\leavevmode\nobreak\ \widehat{y}(T,\cdot)=0&\mbox{in }\Omega,\end{array}\right.$$

$$\Vert u{\Vert }_{{\mathcal{C}}^k(Q_T{)}^2}⩽C_k\Vert y^0{\Vert }_{L^2(\Omega {)}^2}.$$

$$\left\{\begin{array}[]{ll}\partial_{t}z_{1}=\operatorname{div}(d_{1}\nabla z_{1})+g_{11}\cdot\nabla z_{1}+g_{12}\cdot\nabla z_{2}+a_{11}z_{1}+a_{12}z_{2}+\widehat{u}_{1}+v&\mbox{in }\omega_{T},\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\partial_{t}z_{2}=\operatorname{div}(d_{2}\nabla z_{2})+\partial_{x_{1}}z_{1}+g_{22}\cdot\nabla z_{2}+a_{22}z_{2}+\widehat{u}_{2}&\mbox{in }\omega_{T},\end{array}\right.$$

Full text

Keywords: Controllability; Parabolic systems; Fictitious control method; Algebraic solvability.

MSC Classification: 93B05; 93B07; 35K40.

1 Introduction

1.1 Presentation of the problem and main results

Let $T>0$, let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$ ($N\in\mathbb{N}^{*}$) of class $\mathcal{C}^{2}$ and let $\omega$ be an arbitrary nonempty open subset of $\Omega$. Let $Q_{T}:=(0,T)\times\Omega$, $q_{T}:=(0,T)\times\omega$ and $\Sigma_{T}:=(0,T)\times\partial\Omega$. We consider the following system of two parabolic linear equations with variable coefficients and coupling terms of order zero and one

[TABLE]

where $y^{0}\in L^{2}(\Omega)^{2}$ is the initial condition and $u\in L^{2}(Q_{T})$ is the control.

The zero and first order coupling terms $(a_{ij})_{1\leqslant i,j\leqslant 2}$ and $(g_{ij})_{1\leqslant i,j\leqslant 2}$ are assumed (for the moment) to be in $L^{\infty}(Q_{T})$ and in $L^{\infty}(0,T;W^{1}_{\infty}(\Omega))^{N}$, respectively. For $l\in\{1,2\}$, 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 $d_{0}>0$.

It is well-known (see for instance [21, Th. 3, p. 356-358]) that for every initial data $y^{0}\in L^{2}(\Omega)^{2}$ and every control $u\in L^{2}(Q_{T})$, System (1.1) admits a unique solution $y$ in $W(0,T)^{2}$, where

[TABLE]

In this article, we are concerned with the approximate or null controllability of System (1.1). Let us recall the precise definitions of these notions. We say that System (1.1) is

• $\bullet$

approximately controllable on the time interval $(0,T)$ if for every initial condition $y^{0}\in L^{2}(\Omega)^{2}$, every target $y^{T}\in L^{2}(\Omega)^{2}$ and every $\varepsilon>0$, there exists a control $u\in L^{2}(Q_{T})$ such that the corresponding solution $y$ to System (1.1) satisfies

[TABLE]

• $\bullet$

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

[TABLE]

It is well-known that if a parabolic system like (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 for parabolic equations as given for example in [11]).

Since the case $a_{21}\neq 0$ and $g_{21}=0$ in $(t_{0},t_{1})\times\omega_{0}\subset q_{T}$ has already been studied in [24], we will always work under the following assumption.

Assumption** 1.1****.**

There exists $t_{0}<t_{1}\in(0,T)$ and a nonempty open subset $\omega_{0}$ of $\omega$ such that

[TABLE]

Moreover, as we will see in Section 2, it is possible, with the help of appropriate changes of variables and unknowns (we lose a little bit of regularity on the coefficients though, see Section 2), to replace the coupling operator $g_{21}\cdot\nabla+a_{21}$ by the simpler coupling operator $\partial_{x_{1}}$ (where $x_{1}$ is the first direction in space), at least locally on some subset of $q_{T}$.

Hence, without loss of generality, we can also work under the following assumption.

Assumption** 1.2****.**

There exists a nonempty open subset $\mathcal{O}_{T}$ of $\omega_{0}$ such that

[TABLE]

For a nonempty set $\omega_{T}\subset\mathbb{R}^{N+1}$, let us denote by $\mathcal{C}^{0}_{t,x_{2},...,x_{N}}(\overline{\omega}_{T})$ the subset of $\mathcal{C}^{0}(\overline{\omega}_{T})$ composed by the functions depending only on the variables $t,\leavevmode\nobreak\ x_{2},\leavevmode\nobreak\ x_{3},...,x_{N}$. Let us now introduce the following condition, which will be crucial in our following results, and which is closely related to the particular form for the coupling term given in Assumption 1.2 (removing this assumption would make Condition 1.1 impossible to write down explicitly).

Condition** 1.1****.**

There exists a nonempty open set $\omega_{T}\subset(t_{0},t_{1})\times\mathcal{O}$ such that

[TABLE]

where

[TABLE]

Our first main result is the following:

Theorem** 1****.**

Let $d_{i}^{kl},g_{ij}^{k}\in\mathcal{C}^{N^{2}+3}(\overline{\omega}_{T})$ and $a_{ij}\in\mathcal{C}^{N^{2}+2}(\overline{\omega}_{T})$ for every $i,j\in\{1,2\}$ and $k,l\in\{1,...,N\}$. Assume that Assumptions 1.1, 1.2 and Condition 1.1 hold. Then System (1.1) is null controllable at any time $T>0$.

Remark* 1**.*

Theorem 1 is stated and will be proved in the case of two coupled parabolic equations and one control. However, as in [19], it is possible to extend Theorem 1 to systems of $m$ parabolic equations controlled by $m-1$ controls for arbitrary $m\geqslant 2$. More precisely, consider the system

[TABLE]

where $y^{0}:=(y_{1}^{0},\ldots,y_{m}^{0})\in L^{2}(\Omega)^{m}$ is the initial data and $u:=(u_{1},\ldots,u_{m-1})\in L^{2}(Q_{T})^{m-1}$ is the control. Let us suppose that there exists $i\in\{1,...,m\}$, $t_{0}<t_{1}\in(0,T)$ and a nonempty open subset $\omega_{0}$ of $\omega$ such that $g_{mi}(t,x)\neq 0$ on $q_{T}:=(t_{0},t_{1})\times\omega_{0}$. As explained in Section 2, we can suppose that the operator $g_{mi}\cdot\nabla+a_{mi}$ is equal to $\partial_{x_{1}}$ in $(t_{0},t_{1})\times\mathcal{O}\subset q_{T}$. Assume that there exists an open set $\omega_{T}\subset(t_{0},t_{1})\times\mathcal{O}$ such that

[TABLE]

where

[TABLE]

Then we can adapt the proof of Theorem 1 to prove that System (1.4) is null controllable on the time interval $(0,T)$ under suitable regularity conditions on the coefficients.

Remark* 2**.*

Condition 1.1 is clearly technical since it does not even cover the case of constant coefficients proved in [19], the general case given in [12] (under some assumption on the control domain) or the one-dimensional result given in [18]. However, Theorem 3 implies that one cannot expect the null controllability to be true in general without extra assumptions on the coefficients. We do not know what would be a reasonable necessary and sufficient condition on the coupling terms for the null controllability of System (1.1).

The second main result of the present paper is the following surprising result.

Theorem** 2****.**

Let us assume that $\omega\subset\subset\Omega$. Let $\omega_{1}$ be a nonempty regular open set satisfying $\omega\subset\subset\omega_{1}\subset\subset\Omega$. and consider a function $\theta\in\mathcal{C}^{\infty}(\overline{\Omega})$ satisfying

[TABLE]

Then there exists $a\in\mathcal{C}^{\infty}(\overline{\Omega})$ such that the system

[TABLE]

is not approximately controllable (hence not null controllable) on the time interval $(0,T)$.

In other words, Theorem 2 tells us that for every control set $\omega$ strongly included in $\Omega$, there exists a potential $a$ for which approximate controllability of (1.5) does not hold, in any space dimension. We may improve a bit this result on the one-dimensional case, where we are able to obtain the following result, which expresses that for some well-constructed potential $a$, that there exists one control domain on which System (1.6) is not approximately controllable (hence not null controllable) and another control domain on which System (1.6) is null controllable (hence approximatively controllable), highlighting the surprising fact that some geometrical conditions on the control domain has to be imposed in order to obtain a controllability result.

Theorem** 3****.**

Consider the following system

[TABLE]

There exists a coefficient $a\in\mathcal{C}^{\infty}([0,\pi])$ such that:

There exists an open interval $(a,b)\subset\subset(0,\pi)$ such that, for all $T>0$, System (1.6) is null controllable (then approximatively controllable) at time $T$. 2. 2.

There exists an open interval $(a,b)\subset\subset(0,\pi)$ such that, for all $T>0$, System (1.6) is not approximatively controllable (then not null controllable) at time $T$.

Remark* 3**.*

Let us mention that Theorems 2 and 3 are the first negative result for the controllability of System (1.1) when the support of the first order coupling term intersects the control domain in the case of distributed controls. The authors want to highlight that the coupling operator is constant in the whole domain and nevertheless the system can be controllable or not following the localisation of the control domain, which is an unexpected phenomenon.

1.2 State of the art

Many models of interest involve (linear or non-linear) coupled equations of parabolic systems, notably in fluid mechanics, medicine, chemistry, ecology, geology, etc., and this explains why during the past years, the study of the controllability properties of linear or nonlinear parabolic systems has been an increasing subject of interest (see for example the survey [7]). The main issue is what is called the indirect controllability, that is to say one wants to control many equations with less controls than equations, by acting indirectly on the equations where no control term appears thanks to the coupling terms appearing in the system. This notion is fondamental for real-life applications, since in some complex systems only some quantities can be effectively controlled. Here, we will concentrate on the previous results concerning the null or approximate controllability of linear parabolic systems with distributed controls, but there are also many other results concerning boundary controls or other classes of systems like hyperbolic systems.

First of all, in the case of zero order coupling terms, the case of constant coefficients is now completely treated and we refer to [5] and [6] for parabolic systems having constant coupling coefficients (with diffusion coefficients that may depend on the space variable though) and for some results in the case of time-dependent coefficients. In the case of zero and one order coupling terms and constant coefficients, a necessary and sufficient condition in the case of $m$ equations and $m-1$ controls for constant coefficients is provided in [19] by the authors.

The case of space-varying coefficients remains still widely open despite many new partial results these last years. In the case where the support of the coupling terms intersects the control domain, a general result is proved in [24] for parabolic systems in cascade form with one control force (and possibly one order coupling terms). We also mention [4], 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 nonlinear system of transport-diffusion equations. In the situation where the coupling regions do not intersect the control domain, the situation is still not very well-understood and we have partial results, in general under technical and geometrical restrictions, notably on the control domain (see for example [1], [3], [29] and [8]). Let us mention that in this case, there might appear a minimal time for the null controllability of System (1.1) (see [9]), which is a very surprising phenomenon for parabolic equations, because of the infinite speed of propagation of the information.

Concerning the case of first order coupling terms, we mention [24] which gives some controllability results when the coefficient $g_{21}$ is equal to zero on the control domain. Let us also mention the recent work [12], which concerns the small systems in small dimension, that is to say $2\times 2$ and $3\times 3$ systems. The authors of [12] suppose that the control domain contains a part of the boundary $\partial\Omega$. Recently, in [18], the first author studied a particular cascade system with space dependent coefficients and in dimension one thanks to the moment method, and obtained necessary and sufficient conditions on the coupling terms of order [math] and $1$ for the null controllability. To conclude, let us also mention another result given in [19] by the authors, which provides a sufficient condition for null controllability in dimension one for space and time-varying coefficients under some technical conditions on the coefficients, which turns out to be exactly equivalent to Condition 1.1 under Assumption 1.2 (but with more regularity than in Assumption 1.1). Hence, Theorem 1 can be seen as a generalization in the multi-dimensional case of the one-dimensional result given in [19]. For a more detailed state of the art concerning this problem, we refer to [19].

Hence, the present paper improves the previous results in the following sense:

• •

Contrary to [12, 27, 18, 19], we prove in Theorem 1 the null controllability of System (1.1) with a condition on $a_{22}$ but for space/time dependent coefficients, in any space dimension and without any condition on the control domain.

• •

In the previous results, it was surprising to have some very different sufficient conditions for the null controllability of System (1.1) in the case of first order coupling terms, for example on one hand constant coupling coefficients and on the other hand a region of control which intersects the boundary of the domain. Through the example of a not approximately controllable system given in Theorem 2 and 3, we can now better understand why such different conditions appeared since the expected general condition for the null controllability of System (1.1) with space and time-varying coefficients (i.e. it is sufficient that the control and coupling region intersect) may be false in general if $\omega\subset\subset\Omega$.

2 Simplification of the coupling term

In this section, we will prove that it is possible to replace locally the coupling operator $g_{21}\cdot\nabla+a_{21}$ by $\partial_{x_{1}}$, where $x_{1}$ is the first direction in space. This kind of simplification has already been used in [12, Lemma 2.6] for example, and we refer to this article for a more detailed proof (see also [18]). Let us remark that the regularities stated in Lemma 2.1 are higher than the one stated in Theorem 1 due to technical reasons appearing in the proofs of Lemmas 2.1 and 2.2.

Lemma** 2.1****.**

Let $d_{i}^{kl},\leavevmode\nobreak\ g_{ij}^{k},\leavevmode\nobreak\ a_{ij}\in\mathcal{C}^{N^{2}+4}([t_{0},t_{1}]\times\overline{\omega}_{0})$ for every $i,j\in\{1,2\}$ and $k,l\in\{1,...,N\}$. Suppose that Assumption 1.1 is verified. Then, there exist a nonempty open subset $U$ of $\mathbb{R}^{N-1}$, a positive real number $\varepsilon$ and a $\mathcal{C}^{N^{2}+3}$-diffeomorphism $\Lambda$ from $U_{\varepsilon}:=(t_{0},t_{1})\times(0,\varepsilon)\times U$ to an open set $(t_{0},t_{1})\times\mathcal{O}\subset(t_{0},t_{1})\times\omega_{0}$ that keeps $t$ invariant and such that if we call $\widetilde{y}_{1}:=y_{1}\circ\Lambda$ and $\widetilde{y}_{2}:=y_{2}\circ\Lambda$, then there exist a matrix $\widetilde{d}_{2}\in\mathcal{M}_{N}(\mathcal{C}^{N^{2}+3}(U_{\varepsilon}))$, a vector $\widetilde{g}_{22}\in(\mathcal{C}^{N^{2}+3}(U_{\varepsilon}))^{N}$ and coefficients $\widetilde{a}_{21},\leavevmode\nobreak\ \widetilde{a}_{22}\in\mathcal{C}^{N^{2}+3}(U_{\varepsilon})$ such that locally on $U_{\varepsilon}$ one has

[TABLE]

**Proof of Lemma 2.1

**Let us consider some open hyper-surface $\gamma$ of class $\mathcal{C}^{\infty}$ included in $\omega_{0}$ on which $g_{21}\cdot\nu<0$, where $\nu$ is the normalized outward normal on $\gamma$ (this can always be done since $g_{21}\not=0$ on $(t_{0},t_{1})\times\omega_{0}$ and is at least continuous), small enough such that it can be parametrized by a local diffeomorphism

[TABLE]

where $U$ is a nonempty open set. We call $\gamma_{T}:=(t_{0},t_{1})\times\gamma$. Let us consider some $\mathcal{C}^{N^{2}+4}$ extension of $g_{21}$ (that exists thanks to the regularity of $\gamma$ and $g_{21}$) that we denote by $g^{T}_{21}:(t,x)\in\mathbb{R}^{N+1}\mapsto(0,g_{21}(t,x))\in\mathbb{R}^{N+1}$. Using the Cauchy-Lipschitz Theorem, we infer that for every $(t,\sigma)\in\gamma_{T}$, there exists a unique global solution to the Cauchy Problem

[TABLE]

Since $\Phi$ is continuous and $g_{21}\cdot\nu<0$ on $\gamma_{T}$, we deduce that there exists some $\varepsilon>0$ such that $\Phi(t,s,\sigma)\in(t_{0},t_{1})\times\omega_{0}$ for every $s\in(0,\varepsilon)$ and every $(t,\sigma)\in\gamma_{T}$. We define

[TABLE]

Then, by the inverse mapping theorem, $\Lambda$ is a $\mathcal{C}^{N^{2}+4}$-diffeomorphism from $U_{\varepsilon}$ to $(t_{0},t_{1})\times\mathcal{O}:=\Lambda(U_{\varepsilon})$ with $\mathcal{O}\subset\omega_{0}$. Let us call $\widetilde{y}_{1}(t,s,z):=y_{1}(\Lambda(t,s,z))$ and $\widetilde{y}_{2}(t,s,z):=y_{2}(\Lambda(t,s,z))$, then it is clear that

[TABLE]

and hence we obtain (2.1) and the regularities wished for the new coefficients by writing down the equation verified by $\widetilde{y}$.

Let us now perform a second useful reduction.

Lemma** 2.2****.**

There exists an open subset$\mathcal{O}_{T}$ of $U_{\varepsilon}$ and a function $\theta\in\mathcal{C}^{N^{2}+4}(\Omega)$ such that $|\theta(x)|\geqslant C$ for some constant $C>0$ and if

[TABLE]

and

[TABLE]

then there exists some coefficients $\overline{a}_{22}\in\mathcal{C}^{N^{2}+2}(\mathcal{O}_{T})$ and $\overline{g}_{22}\in\mathcal{C}^{N^{2}+3}(\mathcal{O}_{T})^{N}$ such that locally on $\mathcal{O}_{T}$ one has

[TABLE]

**Proof of Lemma 2.2

**Let us consider some $\theta\in\mathcal{C}^{N^{2}+4}(\overline{\Omega})$ such that $|\theta(x)|\geqslant C$ for some constant $C>0$, and consider the change of unknowns

[TABLE]

Using equation (2.1), we infer that $\overline{y}_{2}$ verifies

[TABLE]

where $\overline{g}_{22}:=2\theta^{-1}\widetilde{d}_{2}\nabla\theta+\widetilde{g}_{22}$ and $\overline{a}_{22}:=\theta^{-1}\operatorname{div}(\widetilde{d}_{2}\nabla\theta)+\theta^{-1}\widetilde{g}_{22}\nabla\theta+\widetilde{a}_{22}$. Hence, if we choose $\theta\in\mathcal{C}^{N^{2}+4}(\overline{\Omega})$ satisfying $\partial_{x_{1}}\theta+\widetilde{a}_{21}\theta=0$ and $|\theta(x)|\geqslant C$ in $Q_{T}$, which is always possible, then $\overline{y}_{1}$ and $\overline{y}_{2}$ verify (2.2) and we have $\overline{a}_{22}\in\mathcal{C}^{N^{2}+2}(\mathcal{O}_{T})$ and $\overline{g}_{22}\in\mathcal{C}^{N^{2}+3}(\mathcal{O}_{T})^{N}$ .

3 Proof of Theorem 1

During all this Section, we will always assume that Assumptions 1.1 and 1.2 are satisfied.

3.1 Strategy : Fictitious control method

The fictitious control method has already been used for instance in [25], [17], [2], [16] and [19]. Roughly, the method is the following: we first control the equations with two controls (one on each equation) and we try to eliminate the control on the last equation thanks to algebraic manipulations locally on the control domain. For more details, see for example [19, Section 1.3]. Let us be more precise and decompose the problem into three different steps:

• (i)

**Analytic Problem: Null controllability by two forces

**Find a solution $(\widehat{y},\widehat{u})$ in an appropriate space to the control problem by two controls

[TABLE]

where the controls $\widehat{u}_{1}$ and $\widehat{u}_{2}$ are regular enough and with a support strongly included in $\omega_{T}$ (remind that $\omega_{T}$ was introduced in Condition 1.1). Solving Problem (3.1) is easier than solving the null controllability on the time interval $(0,T)$ of System (1.1), because we control System (3.1) with one control on each equation. The important point is that 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).

Proposition** 3.1****.**

Let $k\in\mathbb{N}^{*}$. Suppose that $d_{i}^{kl},\leavevmode\nobreak\ g_{ij}^{k}\in\mathcal{C}^{k+2}(\overline{\omega}_{T})$ and $a_{ij}\in\mathcal{C}^{k+1}(\overline{\omega}_{T})$ for every $i,j\in\{1,2\}$ and $k,l\in\{1,...,N\}$. Then there exists two constants $K>0$ and $C_{k}$ such that for every initial condition $y^{0}\in L^{2}(\Omega)^{2}$ one can find a control $u\in\mathcal{C}^{k}(Q_{T})^{2}$ verifying moreover $\operatorname{Supp}(u)\subset\subset\omega_{T}$ for which the solution to System (3.1) is equal to zero at time $T$ and the following estimate holds:

[TABLE]

The controllability of parabolic systems with regular controls is nowadays well-known. For a proof of Proposition 3.1, one can adapt the strategy developed in [13, 14, 15, 25] where the authors prove the controllability of parabolic systems with $L^{\infty}$ controls thanks to the fictitious control method and the local regularity of parabolic equations. For more details, we refer to [20, Chap. I, Sec. 2.4]. It is also possible to use the Carleman estimates (see for instance [10] and [19, Section 2.3]), however this will impose the coefficients of System (3.1) to be regular in the whole space $Q_{T}$ (and would require higher regularity on $\Omega$).

• (ii)

**Algebraic Problem: Null controllability by one force

**For given $\widehat{u}_{1},\widehat{u}_{2}$ with $\operatorname{Supp}(\widehat{u}_{1},\widehat{u}_{2})$ strictly included in $\omega_{T}$, find $(z,v)$, in an appropriate space, satisfying the following control problem:

[TABLE]

with $\operatorname{Supp}(z,v)$ strictly included in $\omega_{T}$, which impose the initial and final data and the boundary conditions. We recall that $g_{21}\cdot\nabla+a_{21}$ is equal to $\partial_{x_{1}}$ in $\omega_{T}$. We will solve this problem using the notion of algebraic resolvability of differential systems, which is based on ideas coming from [26, Section 2.3.8] and was already used in some different contexts in [17], [2], [19] or [16]. The idea is to write System (3.3) as an underdetermined system in the variables $z$ and $v$ and to see $\widehat{u}$ as a source term. More precisely, we remark that System (3.3) can be rewritten as

[TABLE]

where $f:=\widehat{u}$ and

[TABLE]

The goal in Section 3.2 will be then to find a partial differential operator $\mathcal{M}$ satisfying

[TABLE]

Thus to solve control problem (3.3), it suffices to take

[TABLE]

When (3.5) is satisfied, we say that System (3.4) is algebraically solvable.

• (iii)

**Conclusion

**If we are able to solve the analytic and algebraic problems, then it is easy to check that $(y,u):=(\widehat{y}-z,-v)$ will be a solution to System (1.1) in an appropriate space and will satisfy $y(T,\cdot)\equiv 0$ in $\Omega$ (for more explanations, see [17, Prop. 1] and the proof of Theorem 1 on pages 11-12).

3.2 Algebraic solvability of the linear control problem

The goal of this section is to solve algebraic problem (3.4). We will use the following lemma:

Lemma** 3.1****.**

Let $\omega$ be a nonempty open subset of $\mathbb{R}^{n}$ ($n\geqslant 1$)* and let $R\in\mathbb{N}^{*}$. Consider two differential operators $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ defined for every $\varphi\in\mathcal{C}^{\infty}(\overline{\omega})$ by*

[TABLE]

where, for $\alpha_{i}=(\alpha_{i}^{2},...,\alpha_{i}^{n})$, $D^{\alpha_{i}}:=\partial_{x_{2}}^{\alpha_{i}^{2}}\cdots\partial_{x_{n}}^{\alpha_{i}^{n}}$. If $a_{i}\in\mathcal{C}^{M}(\overline{\omega})$ for every $i\in\{0,...,R\}$ where

[TABLE]

and $a_{0}$ is not an element of the $\mathcal{C}^{0}_{x_{2},...,x_{n}}(\overline{\widetilde{\omega}})$-module

[TABLE]

for a nonempty open subset $\widetilde{\omega}$ of $\omega$, then there exists two differential operators $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ such that

[TABLE]

**Proof of Lemma 3.1

**The goal is to apply some differential operators $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ to $\mathcal{L}_{1}\varphi$ and $\mathcal{L}_{2}\varphi$ in order to obtain $\varphi$. So, since $\varphi$ is not appearing in $\mathcal{L}_{1}\varphi$, we would like to eliminate all the derivatives $D^{\alpha_{i}}\varphi$ in the expression of $\mathcal{L}_{2}\varphi$ by differentiations and linear combinations.

If $a_{0}\not=0$ and $a_{i}=0$ in $\omega$ for every $i\in\{1,....,R\}$, we define

[TABLE]

If not, let $k_{1}$ be the smallest number of $\{1,....,R\}$ such that there exists a nonempty open subset $\omega_{1}$ of $\omega$ where $|a_{k_{1}}|>\delta>0$. Then we consider $\mathcal{L}_{3}$ the commutator of $\mathcal{L}_{1}$ and $a_{k_{1}}^{-1}\mathcal{L}_{2}$:

[TABLE]

Again, if for every $i\in\{k_{1}+1,....,R\}$, we have $\partial_{x_{1}}\left(\frac{a_{i}}{a_{k_{1}}}\right)=0$ in $\omega$, we define

[TABLE]

If not, let $k_{2}$ be the smallest number of $\{k_{1}+1,....,R\}$ such that there exists a nonempty open subset $\omega_{2}$ of $\omega_{1}$ where $|\partial_{x_{1}}\left(\frac{a_{k_{2}}}{a_{k_{1}}}\right)|>\delta>0$. Then we consider $\mathcal{L}_{4}$ the commutator of $\mathcal{L}_{1}$ and $\left[\partial_{x_{1}}\left(\frac{a_{k_{2}}}{a_{k_{1}}}\right)\right]^{-1}\mathcal{L}_{3}$:

[TABLE]

Again, if, for every $i\in\{k_{2}+1,....,R\}$, we have $\partial_{x_{1}}\left(\frac{\partial_{x_{1}}\left(\frac{a_{i}}{a_{k_{1}}}\right)}{\partial_{x_{1}}\left(\frac{a_{k_{2}}}{a_{k_{1}}}\right)}\right)=0$ in $\omega_{2}$, we define

[TABLE]

If not, we continue the same reasoning that will stop at some point since there is only a finite order of derivatives $R$. Hence, we obtain, for a $m\in\{1,...,R\}$, a nonempty open subset $\widetilde{\omega}$ of $\omega$ and an operator

[TABLE]

Moreover, $\mathcal{N}$ is obtained by making iterated commutators of operators involving only $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. Hence it is clear that there exists two linear partial differential operators $\widetilde{\mathcal{M}}_{1}$ and $\widetilde{\mathcal{M}}_{2}$ such that

[TABLE]

Hence, in view of (3.8), we will have the desired conclusion as soon as the coefficient in the right-hand side in (3.8) is different from zero. Let us explain into more details what this condition exactly means. For the sake of clarity, let us assume that $m=3$ (but the following reasoning can be extended to any $m\in\{1,\ldots,R\}$). We remark that

[TABLE]

holds only if, for some $\lambda_{3}\in\mathcal{C}^{0}_{x_{2},...,x_{n}}(\overline{\widetilde{\omega}})$, we have

[TABLE]

The last expression can be rewritten as

[TABLE]

Again, (3.10) holds only if, for some $\lambda_{2}\in\mathcal{C}^{0}_{x_{2},...,x_{n}}(\overline{\widetilde{\omega}})$, we have

[TABLE]

or, equivalently,

[TABLE]

Thus (3.9) is satisfied only if, for some $\lambda_{1},\leavevmode\nobreak\ \lambda_{2},\leavevmode\nobreak\ \lambda_{3}\in\mathcal{C}^{0}_{x_{2},...,x_{n}}(\overline{\widetilde{\omega}})$, we have

[TABLE]

Hence, we find back condition (1.1) and the proof of Lemma 3.1 is achieved.

We are now able to prove the algebraic solvability of (3.4).

Proposition** 3.2****.**

Suppose that $d_{i}^{kl},\leavevmode\nobreak\ g_{ij}^{k},\leavevmode\nobreak\ a_{ij}\in\mathcal{C}^{N^{2}}(\overline{\omega}_{T})$ for every $i,j\in\{1,2\}$ and $k,l\in\{1,...,N\}$. Then, under Condition 1.1, System (3.4) is algebraically solvable with an operator $\mathcal{M}$ of order $N^{2}$.

**Proof of Proposition 3.2

**Let us remark that the first equation of System (3.4) can be rewritten locally on $\omega_{T}$ as

[TABLE]

hence one can always solve algebraically first the second equation of System (3.4), $v$ will then be given with respect to $z_{1}$, $z_{2}$ and $f_{1}$. Hence, solving (3.4) is equivalent to solving

[TABLE]

where

[TABLE]

Hence, finding a differential operator $\mathcal{M}$ such that (3.5) is satisfied is now equivalent to finding a differential operator $\mathcal{M}_{0}$ such that

[TABLE]

We can remark that equality (3.12) is formally equivalent to

[TABLE]

where the formal adjoint $\mathcal{L}_{0}^{*}$ of the operator $\mathcal{L}_{0}$ is given for every $\varphi\in\mathcal{C}^{\infty}(\overline{\omega}_{T})$ by

[TABLE]

Operator $\mathcal{L}_{2}$ can be rewritten as

[TABLE]

where $\widetilde{g}_{22}^{i}$ and $\widetilde{a}_{22}$ are given in (1.3). Let us first consider the following linear combination of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$:

[TABLE]

Lemma 3.1 leads to the algebraic resolvability of System (3.4) under Condition 1.1.

Concerning the order of $\mathcal{M}$, if we follow the proof of Lemma 3.1 step by step, we apply at most $N\times(N-1)/2$ operators of order two to eliminate the terms $d_{2}^{ij}\partial_{x_{i}x_{j}}$ with $i,j\in\{2,...,N\}$ (thanks to the symmetry property of $d_{2}$), then at most $N-1$ operators of order one for the term $\widetilde{g}_{22}^{i}\partial_{x_{i}}$ with $i\in\{2,...,N\}$ and finally an operator of order at most one for $\partial_{t}$. Thus the operator $\mathcal{M}$ is of order at most $N\times(N-1)+(N-1)+1=N^{2}.$

We are now ready for the proof of Theorem 1.

**Proof of Theorem 1.

**We apply Proposition 3.1 with $k=N^{2}+1$ and obtain the existence of two constants $K>0$ and $C>0$ such that for every initial condition $y^{0}\in L^{2}(\Omega)^{2}$ one can find a control $\widehat{u}\in\mathcal{C}^{N^{2}+1}(\overline{Q_{T}})$ verifying $\operatorname{Supp}(\widehat{u})\subset\subset\omega_{T}$ for which the solution $\widehat{y}$ to System (3.1) is equal to zero at time $T$ and the following estimate holds:

[TABLE]

Now, using Proposition 3.2, locally on $\omega_{T}$ there exists a solution $(z,v)\in\mathcal{C}^{1}(\overline{Q}_{T})^{3}\subset W(0,T)^{2}\times L^{2}(Q_{T})$ to the following control problem:

[TABLE]

with $(\widehat{u}_{1},\widehat{u}_{2}):=\widehat{u}$. Moreover, since $\operatorname{Supp}(z)\subset\subset\omega_{T}$, we have $z(0,\cdot)=z(T,\cdot)=0$ in $\Omega$.

We conclude by remarking that $(y,u):=(\widehat{y}-z,-v)$ is a solution to System (1.1) which satisfies $y(T,\cdot)\equiv 0$ in $\Omega$.

4 Proof of Theorem 2

Let $\omega_{1}$ be a nonempty regular open set satisfying $\omega\subset\subset\omega_{1}\subset\subset\Omega$. Let $\theta$ be a function of $\mathcal{C}^{\infty}(\overline{\Omega})$ satisfying

[TABLE]

Consider the following system

[TABLE]

where $u\in L^{2}(Q_{T})$ is the control and $a\in L^{\infty}(\Omega)$ will be specified later. If we can control approximately System (4.1), then it implies that we are also able to control approximately the following equation:

[TABLE]

where $v\in L^{2}((0,T),H^{1}(\Omega))$ is the control. Since $\theta>0$ on $\omega_{1}$, the approximate controllability on the time interval $(0,T)$ of System (4.1) is equivalent to the following property, called the Fattorini criterion (see [28, Theorem 1 & Section 3]):

Theorem** 4****.**

System (4.2) is approximately controllable on the time interval $(0,T)$, if and only if for every $s\in\mathbb{C}$ and every $\varphi\in D(\Delta)$, we have

[TABLE]

Since $\omega_{1}\subset\subset\Omega$, Then there exists a open set $\omega_{2}$ such that $\omega_{1}\subset\subset\omega_{2}\subset\subset\Omega$. The first eigenfunction $\varphi_{1}$ of $-\Delta$ is well-known to be positive in $\Omega$, so we can define a function $\varphi\in\mathcal{C}^{\infty}(\overline{\Omega})$ satisfying

[TABLE]

For instance, if $\Omega:=(0,\pi)$ and $\omega_{1}:=(2\pi/5,3\pi/5)$, as in Figure 1, we may construct a function $\varphi\in\mathcal{C}^{2}([0,\pi])$ satisfying

[TABLE]

Consider

[TABLE]

Thanks to the definition of $\varphi$, is well defined in $\overline{\Omega}$ and is an element of $\mathcal{C}^{\infty}(\overline{\Omega})$. Thus $\varphi$ satisfies

[TABLE]

Using Theorem 4, System (4.2) is not approximately controllable on the time interval $(0,T)$.

Remark* 4**.*

Let us emphasize that in this case, as expected, Condition 1.1 is not verified: on $\omega$ we have by definition $a_{22}=-1$, $g_{22}=0$ and $d^{ii}_{2}=0$ for every $i\in\{2,\ldots,N\}$, which implies that $\tilde{a}_{22}=-1$ on $\omega$ and $\widetilde{g}_{22}=0$, hence

[TABLE]

This will also be the case for the potential constructed in the first part of the proof of Theorem 3.

5 Proof of Theorem 3

Let $\Omega:=(0,\pi)$ and $\omega:=(7\pi/15,8\pi/15)$. Consider the following system

[TABLE]

where $u\in L^{2}(Q_{T})$ is the control and $a\in C^{\infty}(\overline{\Omega})$ will be specified later.

As in the previous section, it is well-known that the approximate controllability on the time interval $(0,T)$ of System (4.2) is equivalent to the following property:

Theorem** 5****.**

System (5.1) is approximately controllable on the time interval $(0,T)$, if and only if for every $s\in\mathbb{C}$ and every $\varphi\in D(\Delta)$, we have

[TABLE]

Let us construct three functions $\varphi$, $\psi$, $a\in\mathcal{C}^{\infty}(\overline{\Omega})$ satisfying

[TABLE]

The idea will be to construct the function $\psi$ as a perturbation of $x\mapsto\sin(3x)$. Consider $\psi$ a function of $\mathcal{C}^{\infty}(\overline{\Omega})\cap D(\Delta)$ satisfying

[TABLE]

where $\theta_{1},\leavevmode\nobreak\ \theta_{2},\leavevmode\nobreak\ \theta_{3}$ are three nontrivial functions of $\mathcal{C}^{\infty}(\overline{\Omega})$ satisfying

[TABLE]

$\varepsilon>0$ small enough and $C_{1},\leavevmode\nobreak\ C_{2},\leavevmode\nobreak\ C_{2}$ are three positive constants to determined (See Figure 2 for some examples of function $\psi$). Let us remark that, for a constant $\alpha\in\mathbb{R}$ to determined, the function $\varphi\in\mathcal{C}^{\infty}(\overline{\Omega})$ defined for all $x\in\overline{\Omega}$ by

[TABLE]

is solution to the first equation of (5.2). In order to apply Theorem 5, let us first prove that $C_{1}$ and $\alpha$ can be chosen such that $\varphi=0$ in $\omega$. Since $\psi=\sin(7\pi/5)$ in $\omega$,

[TABLE]

for all $x\in\omega$. Since $\cos(3x)>0,\leavevmode\nobreak\ \sin(3x)>0$ for all $x$ in $(\pi/12,\pi/6)$ and

[TABLE]

then, according to the last line of (5.3), for $\epsilon$ small enough, it is possible to choose $C_{1}>0$ in order to obtain

[TABLE]

Thus, for $\alpha$ given by

[TABLE]

we obtain $\varphi=0$ in $\omega$. By definition of $\varphi$, we have $\varphi(0)=0$. Let us now prove that for some appropriate $C_{2}$ and $C_{3}$, we have $\varphi(\pi)=0$. We remark that

[TABLE]

Let us distinguish two cases:

If

[TABLE]

is negative, then, using the fact that $\sin(3x),\leavevmode\nobreak\ \cos(3x)>0$ for all $x\in(9\pi/12,5\pi/6)$, one can choose $C_{3}:=0$ and find some some $C_{2}>0$ such that $\varphi(\pi)=0$. 2. 2.

If now the quantity (5.5) is positive, since $\sin(3x)>0$ and $\cos(3x)<0$ for all $x\in(5\pi/6,11\pi/12)$, one can choose $C_{2}:=0$ and find some some $C_{3}>0$ such that $\varphi(\pi)=0$.

The function $\psi$ will have one of the two following forms

To satisfy the second equality in (5.2), we define the function $a\in\mathcal{C}^{\infty}(\overline{\Omega})$ as follows

[TABLE]

This function $a$ is bounded since at each point where $\psi$ is null, i.e. at [math], $\pi/3$, $2\pi/3$ and $\pi$, there exists a neighbourhood in which $\psi(x)$ is equal to $\sin(3x)$. Thus the constructed $\varphi$, $\psi$ and $a$ verify (5.2). Using Theorem 5, System (5.1) is not approximately controllable on the time interval $(0,T)$.

Let us now prove the second item of Theorem 3. We remark that it is possible to chose $\theta_{1}=\mbox{exp}$ in $\omega_{1}\subset(\pi/12,\pi/6)$ with $\omega_{1}$ small enough. Then $a$ is defined in $\omega_{1}$ for all $x\in\omega_{1}$ by

[TABLE]

Thus $a$ satisfies Condition 1.1 for $\omega:=\omega_{1}$, that is $a$ is non-constant in the space variable on $\omega_{1}$. We conclude applying Theorem 1 for $\omega:=\omega_{1}$.

Funding

Pierre Lissy is partially supported by the project IFSMACS funded by the french Agence Nationale de la Recherche, 2015-2019 (Reference: ANR-15-CE40-0010).

Conflict of Interest

The authors declare that they have no conflict of interest.