Perturbations of controlled systems

TL;DR

This paper demonstrates that the Fattorini criterion ensures exact controllability of perturbed linear systems and applies this to achieve controllability results for PDE systems, including coupled wave and heat equations.

Contribution

It introduces a perturbation approach based on the Fattorini criterion, enabling new controllability results for complex PDE systems with fewer controls and in smaller times.

Findings

Exact controllability of perturbed systems established.

Controllability of coupled wave equations with fewer controls.

Null controllability of coupled heat equations in small time.

Abstract

Using a compactness-uniqueness approach, we show that the Fattorini criterion implies the exact controllability of general compactly perturbed controlled linear systems. We then apply this perturbation result to obtain new controllability results for systems governed by partial differential equations. Notably, we combine it with the fictitious control method to establish the exact controllability of a cascade system of coupled wave equations by a reduced number of controls and with the same control time as the one for a single wave equation. We also combine this perturbation result with transmutation techniques to prove the null controllability in arbitrarily small time of a one-dimensional non diagonalizable system of coupled heat equations with as many controls as equations.

Equations

$${\int }_0^T{\Vert {\mathcal{B}}^{\ast }{S}_{\mathcal{A}}(t{)}^{\ast }z\Vert }_U^2\le C{\Vert z\Vert }_H^2,\forall z\in D({\mathcal{A}}^{\ast }).$$

$$\left\{\begin{array}[]{rcll}\displaystyle\frac{d}{dt}y&=&\mathcal{A}y+\mathcal{B}u,&t\in(0,T),\\ y(0)&=&y^{0},&\end{array}\right.$$

$$y\in C^0([0,T];H),$$

$${\Vert z\Vert }_H^2\le C{\int }_0^T{\Vert {\mathcal{B}}^{\ast }{S}_{\mathcal{A}}(t{)}^{\ast }z\Vert }_U^{2}dt,\forall z\in H,$$

$$\Big{(}\mathcal{B}^{*}S_{\mathcal{A}}(t)^{*}z=0,\quad\mathrm{a.e.}\,\,t\in(0,T)\Big{)}\Longrightarrow z=0,\quad\forall z\in H.$$

$$\ker (\lambda -{\mathcal{A}}_{\mathcal{K}}^{\ast })\cap \ker {\mathcal{B}}^{\ast }=\left\{0\right\},\forall \lambda \in \mathbb{C},$$

$$\alpha {\Vert z\Vert }_{H_1}\le {\Vert Lz\Vert }_{H_2}+{\Vert Kz\Vert }_{H_3},\forall z\in H_1.$$

$$\beta {\Vert z\Vert }_{H_1}\le {\Vert Lz\Vert }_{H_2},\forall z\in H_1.$$

$${\Vert z\Vert }_H^2\le C{\int }_0^T{\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_0}(t{)}^{\ast }z\Vert }_U^{2}dt,$$

$${\Vert z\Vert }_H^2\le 2C\left({\int }_0^T{\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z\Vert }_U^{2}dt+{\int }_0^T{\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_0}(t{)}^{\ast }z-{\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z\Vert }_U^{2}dt\right).$$

$$\begin{array}[]{rlll}L_{T}:&H&\longrightarrow&L^{2}(0,T;U)\\ &z&\longmapsto&\mathcal{B}^{*}S_{\mathcal{A}_{\mathcal{K}}}(\cdot)^{*}z,\end{array}\qquad\begin{array}[]{rlll}K_{T}:&H&\longrightarrow&L^{2}(0,T;U)\\ &z&\longmapsto&\mathcal{B}^{*}S_{\mathcal{A}_{0}}(\cdot)^{*}z-\mathcal{B}^{*}S_{\mathcal{A}_{\mathcal{K}}}(\cdot)^{*}z.\end{array}$$

$$S_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z=S_{{\mathcal{A}}_0}(t{)}^{\ast }z+{\int }_0^t{S}_{{\mathcal{A}}_0}(t-s{)}^{\ast }{\mathcal{K}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(s{)}^{\ast }zds,$$

$$K_{T}z(t)=-{\mathcal{B}}^{\ast }{\int }_0^t{S}_{{\mathcal{A}}_0}(t-s{)}^{\ast }{\mathcal{K}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(s{)}^{\ast }zds,$$

$$\exists C>0,{\Vert {\mathcal{B}}^{\ast }{\int }_0^t{S}_{{\mathcal{A}}_0}(t-s{)}^{\ast }f(s)ds\Vert }_{L^2(0,T;U)}\le C{\Vert f\Vert }_{L^2(0,T;H)},\forall f\in H^1(0,T;H).$$

$${\Vert K_{T}z\Vert }_{L^2(0,T;U)}\le C{\Vert {\mathcal{K}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(\cdot {)}^{\ast }z\Vert }_{L^2(0,T;H)},$$

$${\mathcal{K}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(\cdot {)}^{\ast }{z}_n\underset{n\to +\infty }{\to }0\text{mbox}stronglyin{L}^2(0,T;H).$$

$$S_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }{z}_n\underset{n\to +\infty }{\to }0\text{mbox}weaklyinH,\forall t\in [0,T].$$

$${\mathcal{K}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }{z}_n\underset{n\to +\infty }{\to }0\text{mbox}stronglyinH,\forall t\in [0,T].$$

$${\Vert z\Vert }_H^2\le C{\int }_0^{T^{\ast }}{\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z\Vert }_U^{2}dt,\forall z\in H.$$

$$\dim \ker L_T<+\infty ,\forall T\ge T^{\ast }.$$

$$\ker L_T\subset D({{\mathcal{A}}_{\mathcal{K}}}^{\ast }).$$

$$u_n=\frac{\left(S_{{\mathcal{A}}_{\mathcal{K}}}(t_n{)}^{\ast }z-z\right)}{t_n}$$

$${\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(\cdot {)}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t_n{)}^{\ast }z\Vert }_{L^2(0,T-\epsilon ;U)}={\Vert {\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(\cdot {)}^{\ast }z\Vert }_{L^2(t_n,t_n+T-\epsilon ;U)},$$

$$u_n\in \ker L_{T-\epsilon },\forall n\ge N.$$

$${\Vert z\Vert }_{-1}={\Vert (\mu -{{\mathcal{A}}_{\mathcal{K}}}^{\ast }{)}^{-1}z\Vert }_H.$$

$$(\mu -{{\mathcal{A}}_{\mathcal{K}}}^{\ast }{)}^{-1}{u}_n=\frac{S_{{\mathcal{A}}_{\mathcal{K}}}(t_n{)}^{\ast }-\mathrm{Id}}{t_n}(\mu -{{\mathcal{A}}_{\mathcal{K}}}^{\ast }{)}^{-1}z\underset{n\to +\infty }{\to }{{\mathcal{A}}_{\mathcal{K}}}^{\ast }(\mu -{{\mathcal{A}}_{\mathcal{K}}}^{\ast }{)}^{-1}z\text{mbox}inH.$$

$$\ker L_T\subset \ker {\mathcal{B}}^{\ast }.$$

$$\frac{d}{dt}{\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z={\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }{\mathcal{A}}_{\mathcal{K}}^{\ast }z,a.e.t\in (0,T).$$

$${\int }_0^T{\phi }^{\prime }(t){\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }zdt=-{\int }_0^T\phi (t){\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }{\mathcal{A}}_{\mathcal{K}}^{\ast }zdt.$$

$${\mathcal{B}}^{\ast }{S}_{{\mathcal{A}}_{\mathcal{K}}}(t{)}^{\ast }z=0,a.e.t\in (0,T).$$

Full text

Keywords. Compactness-uniqueness, Fattorini criterion, exact controllability.

1 Introduction and main result

In this work, we study the exact controllability property of general compactly perturbed controlled linear systems using a compactness-uniqueness approach. This technique has been introduced for the very first time in the pioneering work [25] to establish the exponential decay of the solution to some hyperbolic equations. On the other hand, the first controllability results using this method were obtained in [28] for a plate equation and then in [29] for a wave equation perturbed by a bounded potential. Wether one wants to establish a stability result or a controllability result, one is lead in both cases to prove estimates, energy estimates or observability inequalities. For a perturbed system, a general procedure is to start by the known estimate satisfied by the unperturbed system and to try to derive the desired estimate, up to some "lower order terms" that we would like to remove. The compactness-uniqueness argument then reduces the task of absorbing these additional terms to a unique continuation property for the perturbed system. We should point out that, surprisingly enough, and despite the numerous applications of this flexible method to successfully establish the controllability of systems governed by partial differential equations (see e.g. [28, 29, 5, 8, 19], etc.), no systematic treatment has been provided so far, by which we mean that there is no abstract result available in the literature that covers all type of systems, regardless the nature of the PDE we are considering (wave, plate, etc.). This will be the first point of the present paper to fill this gap, see Theorem 1.1 below. Then, more importantly, we improve this result by showing that the so-called Fattorini criterion - a far weaker kind of unique continuation property - is actually sufficient to ensure the exact controllability of the perturbed system, see Theorem 1.2 below. The proofs of these two results are based on the Peetre Lemma, introduced in [24], which is in fact the root of compactness-uniqueness methods.

Then, in the rest of the paper, we show that this perturbation result can be applied to many systems governed by partial differential equations. We will limit ourselves to three applications but this result can obviously be used to deal with plenty of other control problems. The first application concerns the controllability of a first-order perturbed hyperbolic equation that has been studied in [7, 10]. This is a basic application of our perturbation result which nevertheless leads to new results. As a second application, we study a system of coupled wave equations controlled by only one control. We generalize the previous results of [20] to the case of space varying couplings. The proof is based on the so-called fictitious control method, that consists in a first step to control the system by as many controls as equations, and then, in a second step, to reduce the number of controls. The first step is achieved thanks to our perturbation result and the reduction of the number of controls is performed with the strategy of algebraic solvability introduced in a PDE framework in [11]. Finally, in a third and last application, we will consider a system of coupled heat equations with as many controls as equations. Usually, for this kind of problems, the main tools are the Carleman estimates. However, in the case of non diagonalizable matrix of diffusion, this technique can not be applied a priori anymore (see e.g. [15]). As a result, we propose to apply our perturbation result on an auxiliary system of coupled wave equations and then to transfer the obtained controllability properties to the initial parabolic system by using the so-called transmutation technique introduced in [22].

Let us now introduce some notations and recall some basic facts about the controllability of abstract linear evolution equations. Let $H$ and $U$ be two (real or complex) Hilbert spaces, let $\mathcal{A}:D(\mathcal{A})\subset H\longrightarrow H$ be the generator of a $C_{0}$-semigroup $(S_{\mathcal{A}}(t))_{t\geq 0}$ on $H$ and let $\mathcal{B}\in\mathcal{L}(U,D(\mathcal{A}^{*})^{\prime})$. We assume that $\mathcal{B}$ is admissible for $\mathcal{A}$, which means that for every $T>0$ there exists $C>0$ such that

[TABLE]

This inequality shows that the map $z\in D(\mathcal{A}^{*})\longmapsto\mathcal{B}^{*}S_{\mathcal{A}}(\cdot)^{*}z\in L^{2}(0,T;U)$ has a unique continuous extension to $H$ since $D(\mathcal{A}^{*})$ is dense. We shall keep the same notation to denote this extension. However, the reader should keep in mind that this is only a notation, which may lead to technical developments in the sequel (notably, in the proofs of Theorem 1.1 and Theorem 1.2 below).

Let us now consider the abstract evolution system

[TABLE]

where $T>0$ is the time of control, $y^{0}\in H$ is the initial data, $y$ is the state and $u\in L^{2}(0,T;U)$ is the control. Since $\mathcal{B}$ is admissible for $\mathcal{A}$, system (1.1) is well-posed: for every $y^{0}\in H$ and every $u\in L^{2}(0,T;U)$, there exists a unique solution (by transposition) to system (1.1)

[TABLE]

(see e.g. [9, Theorem 2.37]). This regularity allows us to consider control problems for system (1.1). We say that system (1.1) or $(\mathcal{A},\mathcal{B})$ is:

• •

exactly controllable in time $T$ if, for every $y^{0},y^{1}\in H$, there exists $u\in L^{2}(0,T;U)$ such that the corresponding solution $y$ to system (1.1) satisfies $y(T)=y^{1}$.

• •

null controllable in time $T$ if the above property holds for $y^{1}=0$.

• •

approximately controllable in time $T$ if, for every $\varepsilon>0$ and every $y^{0},y^{1}\in H$, there exists $u\in L^{2}(0,T;U)$ such that the corresponding solution $y$ to system (1.1) satisfies $\left\|y(T)-y^{1}\right\|_{H}\leq\varepsilon$.

Clearly, exact controllability in time $T$ implies null and approximate controllability at the same time.

It is also well-known that the controllability has a dual concept named observability. More precisely, $(\mathcal{A},\mathcal{B})$ is exactly controllable in time $T$ if, and only if, there exists $C>0$ such that

[TABLE]

and $(\mathcal{A},\mathcal{B})$ is approximately controllable in time $T$ if, and only if,

[TABLE]

A similar dual characterization holds as well for the null controllability, but it will not be needed in this paper. We refer the reader to [9, Theorem 2.42 and 2.43] for a proof and more details.

Let us now state the two main results of this paper.

Theorem** 1.1****.**

Let $H$ and $U$ be two (real or complex) Hilbert spaces. Let $\mathcal{A}_{0}:D(\mathcal{A}_{0})\subset H\longrightarrow H$ be the generator of a $C_{0}$-semigroup on $H$ and let $\mathcal{B}\in\mathcal{L}(U,D(\mathcal{A}_{0}^{*})^{\prime})$ be an admissible control operator for $\mathcal{A}_{0}$. Let $\mathcal{K}\in\mathcal{L}(H)$ and let us form the unbounded operator $\mathcal{A}_{\mathcal{K}}=\mathcal{A}_{0}+\mathcal{K}$ with $D(\mathcal{A}_{\mathcal{K}})=D(\mathcal{A}_{0})$ 111 $\mathcal{A}_{\mathcal{K}}$ is then the generator of a $C_{0}$-semigroup on $H$ and $\mathcal{B}$ is also admissible for $\mathcal{A}_{\mathcal{K}}$, see e.g. [27, Theorem 5.4.2]. . We assume that:

1. (i)

There exists $T^{*}>0$ such that $(\mathcal{A}_{0},\mathcal{B})$ is exactly controllable in time $T^{*}$. 2. (ii)

$\mathcal{K}$* is compact.* 3. (iii)

$(\mathcal{A}_{\mathcal{K}},\mathcal{B})$* is approximately controllable in time $T^{*}$.*

Then, $(\mathcal{A}_{\mathcal{K}},\mathcal{B})$ is exactly controllable in time $T^{*}$.

The second result shows that the approximate controllability assumption iii can be weakened to the Fattorini criterion:

Theorem** 1.2****.**

Under the assumptions of Theorem 1.1, if $H$ and $U$ are complex Hilbert spaces and iii is replaced by the Fattorini criterion for $(\mathcal{A}_{\mathcal{K}},\mathcal{B})$, namely,

[TABLE]

then $(\mathcal{A}_{\mathcal{K}},\mathcal{B})$ is exactly controllable in time $T$ for every $T>T^{*}$.

Remark** 1.3****.**

In many applications the spaces $H$ and $U$ are real Hilbert spaces. To apply Theorem 1.2 in such a framework, we first introduce the complexified spaces $\hat{H}=H+iH$ and $\hat{U}=U+iU$ and we define the complexified operators $\hat{\mathcal{A}}_{\mathcal{K}}$ and $\hat{\mathcal{B}}$ by $\hat{\mathcal{A}}_{\mathcal{K}}(y_{1}+iy_{2})=\mathcal{A}_{\mathcal{K}}y_{1}+i\mathcal{A}_{\mathcal{K}}y_{2}$ for $y_{1},y_{2}\in D(\mathcal{A}_{\mathcal{K}})$ and $\hat{\mathcal{B}}(u_{1}+iu_{2})=\mathcal{B}u_{1}+i\mathcal{B}u_{2}$ for $u_{1},u_{2}\in U$. Splitting up the evolution system described by $(\hat{\mathcal{A}}_{\mathcal{K}},\hat{\mathcal{B}})$ into real and imaginary parts, we readily see that $(\hat{\mathcal{A}}_{\mathcal{K}},\hat{\mathcal{B}})$ is (exactly, approximately or null) controllable in time $T$ if, and only if, so is $(\mathcal{A}_{\mathcal{K}},\mathcal{B})$. Then, we check the Fattorini criterion for $(\hat{\mathcal{A}}_{\mathcal{K}},\hat{\mathcal{B}})$. In the sequel we shall keep the same notation to denote the operators and their extensions.

Theorem 1.2 shows that, in order to prove the exact controllability of a compactly perturbed system which is known to be exactly controllable, it is (necessary and) sufficient to only check the Fattorini criterion (1.4). This result has been established in a particular case in [8, Theorem 5] for a perturbed Euler-Bernoulli equation with distributed controls. The Fattorini criterion appears for the very first time in [14, Corollary 3.3] and it is also sometimes misleadingly known as the Hautus test in finite dimension, despite it has been introduced earlier by Fattorini, moreover in a much larger setting. In a complete abstract control theory framework, it is the sharpest sufficient condition one can hope for since it is always a necessary condition for the exact, null or approximate controllability, to hold in some time. This is easily seen through the dual characterizations (1.2) or (1.3) since $S_{\mathcal{A}}(t)^{*}z=e^{\lambda t}z$ for $z\in\ker(\lambda-\mathcal{A}^{*})$. It is also nowadays well-known that this condition characterizes the approximate controllability of a large class of systems generated by analytic semigroups (see [14, 4, 23]). Surprisingly enough, Theorem 1.2 shows that it may as well characterize the exact controllability property for some systems. In practice, the Fattorini criterion can be checked by various techniques, such as elliptic Carleman estimates (as it will be done in the present paper, see also [8]) or through a spectral analysis when this later technique cannot be used at all (see e.g. [23, 6]).

Let us mention that it is not clear wether the Fattorini criterion (1.4) is sufficient or not to obtain the exact controllability of the perturbed system in the optimal time $T^{*}$. Therefore, both Theorem 1.1 and Theorem 1.2 are important. Obviously, Theorem 1.2 is a stronger result if we do not look for the best time. However, it may very well happen that the optimal time is required to apply some other results, as for instance in [10] where the authors fundamentally need it to stabilize a perturbed hyperbolic equation.

Finally, let us point out that we do not request any spectral properties whatsoever on the operators $\mathcal{A}_{0}$ or $\mathcal{A}_{\mathcal{K}}$, contrary to the papers [17, 21] where the existence of a Riesz basis of generalized eigenvectors or related spectral properties are required.

2 Proofs of the main results

The proofs of Theorem 1.1 and Theorem 1.2 rely on the Peetre Lemma (see [24, Lemma 3 and 4]):

Lemma** 2.1****.**

Let $H_{1},H_{2},H_{3}$ be three Banach spaces. Let $L\in\mathcal{L}(H_{1},H_{2})$ and $K\in\mathcal{L}(H_{1},H_{3})$ be two linear bounded operators. We assume that $K$ is compact and that there exists $\alpha>0$ such that

[TABLE]

Then,

1. (i)

$\ker L$* is finite dimensional.* 2. (ii)

If, moreover, $\ker L=\left\{0\right\}$, then there exists $\beta>0$ such that

[TABLE]

Let us denote by $(S_{\mathcal{A}_{0}}(t))_{t\geq 0}$ (resp. $(S_{\mathcal{A}_{\mathcal{K}}}(t))_{t\geq 0}$) the $C_{0}$-semigroup generated by $\mathcal{A}_{0}$ (resp. $\mathcal{A}_{\mathcal{K}}$). If $(\mathcal{A}_{0},\mathcal{B})$ is exactly controllable in time $T^{*}$, then there exists $C>0$ such that, for every $T\geq T^{*}$ and every $z\in H$,

[TABLE]

so that

[TABLE]

Therefore, we would like to apply Lemma 2.1 to the operators

[TABLE]

Note that $L_{T}$ and $K_{T}$ are bounded linear operators since $\mathcal{B}$ is admissible for $\mathcal{A}_{0}$ and $\mathcal{A}_{\mathcal{K}}$. To apply Lemma 2.1, we have to check that $K_{T}$ is compact.

Lemma** 2.2****.**

The operator $K_{T}$ is compact for every $T>0$.

Proof of Lemma 2.2.

Since $H$ is a Hilbert space, we will prove that, if $(z_{n})_{n}\subset H$ is such that $z_{n}\to 0$ weakly in $H$ as $n\to+\infty$, then $K_{T}z_{n}\to 0$ strongly in $H$ as $n\to+\infty$. Let us compute $K_{T}$. To this end, we first recall the integral equation satisfied by semigroups of boundedly perturbed operators:

[TABLE]

valid for every $z\in H$ and $t\in[0,T]$. Now observe that $\int_{0}^{t}S_{\mathcal{A}_{0}}(t-s)^{*}f(s)\,ds\in D(\mathcal{A}_{0}^{*})$ for every $f\in H^{1}(0,T;H)$ and $t\in[0,T]$. This shows that each term of the previous identity belongs to $D(\mathcal{A}_{0}^{*})$ if $z\in D(\mathcal{A}_{0}^{*})=D(\mathcal{A}_{\mathcal{K}}^{*})$. Therefore, we can apply $\mathcal{B}^{*}$ to obtain that

[TABLE]

for every $z\in D(\mathcal{A}_{0}^{*})$ and $t\in[0,T]$. Now the key point is the following estimate, which is actually used to prove the admissibility of $\mathcal{B}$ for $\mathcal{A}_{\mathcal{K}}$:

[TABLE]

This estimate holds because $\mathcal{B}$ is admissible for $\mathcal{A}_{0}$, assuming in addition that $(S_{\mathcal{A}_{0}}(t))_{t\geq 0}$ is stable (which can be done without any consequences for the controllability properties by considering $\mathcal{A}_{0}-\lambda$, with $\lambda>0$ large enough, instead of $\mathcal{A}_{0}$). For a proof of (2.1) we refer to the first step of the proof of [27, Theorem 5.4.2] (with $A=\mathcal{A}_{0}^{*}$, $B=\mathrm{Id}$ and $C=\mathcal{B}^{*}$). It follows that there exists $C>0$ such that

[TABLE]

for every $z\in D(\mathcal{A}_{\mathcal{K}}^{*})$, and thus for every $z\in H$ by density. Applying this estimate to the sequence $(z_{n})_{n}$ we see that it only remains to show that

[TABLE]

Since $z_{n}\to 0$ weakly in $H$ as $n\to+\infty$, using the strong (and therefore weak) continuity of semigroups on $H$, we obtain

[TABLE]

Since $\mathcal{K}^{*}$ is compact, we obtain

[TABLE]

On the other hand, by the classical semigroup estimate, $(\mathcal{K}^{*}S_{\mathcal{A}_{\mathcal{K}}}(t)^{*}z_{n})_{n}$ is clearly uniformly bounded in $H$ with respect to $t$ and $n$. Therefore, the Lebesgue’s dominated convergence theorem applies, so that (2.2) holds and $K_{T}z_{n}\to 0$ strongly in $L^{2}(0,T;H)$ as $n\to+\infty$. This shows that $K_{T}$ is compact. ∎

The proof of Theorem 1.1 is now easy.

Proof of Theorem 1.1.

The assumptions of Lemma 2.1 are satisfied for $L_{T^{*}}$ and $K_{T^{*}}$. Moreover, the assumption iii of Theorem 1.1 exactly means that $\ker L_{T^{*}}=\left\{0\right\}$ (see (1.3)). Therefore, item ii of Lemma 2.1 shows that there exists $C>0$ such that

[TABLE]

∎

The proof of Theorem 1.2 requires a longer development.

Proof of Theorem 1.2.

The assumptions of Lemma 2.1 are satisfied for $L_{T}$ and $K_{T}$ for every $T\geq T^{*}$. Therefore, item i of Lemma 2.1 gives that

[TABLE]

Let us now prove that $\ker L_{T}=\left\{0\right\}$ for every $T>T^{*}$. We follow the ideas of the proof of [8, Theorem 5]. From now on, $T$ is fixed such that $T>T^{*}$. Let $\varepsilon\in(0,T-T^{*}]$ so that $T-\varepsilon\geq T^{*}$. The key point is to establish that

[TABLE]

Let $z\in\ker L_{T}$. We have to show that, for any sequence $t_{n}>0$ with $t_{n}\to 0$ as $n\to+\infty$, the sequence

[TABLE]

converges in $H$ as $n\to+\infty$. Let $N\in\mathbb{N}$ be large enough so that $t_{n}<\varepsilon$ for every $n\geq N$. Observe that

[TABLE]

(this is true for $z\in D(\mathcal{A}_{\mathcal{K}}^{*})$ and thus for $z\in H$ by density and admissibility of $\mathcal{B}$). Since $z\in\ker L_{T}$ and $t_{n}+T-\varepsilon<T$, this shows that

[TABLE]

Let $\mu\in\rho({\mathcal{A}_{\mathcal{K}}}^{*})$ be fixed and let us introduce the following norm on $\ker L_{T-\varepsilon}$:

[TABLE]

Since $(\mu-{\mathcal{A}_{\mathcal{K}}}^{*})^{-1}z\in D({\mathcal{A}_{\mathcal{K}}}^{*})$, we have

[TABLE]

Therefore, $(u_{n})_{n\geq N}$ is a Cauchy sequence in $\ker L_{T-\varepsilon}$ for the norm $\left\|\cdot\right\|_{-1}$. Since $T-\varepsilon\geq T^{*}$, by the first step of the proof, we know that $\ker L_{T-\varepsilon}$ is finite dimensional. Thus, all the norms are equivalent on $\ker L_{T-\varepsilon}$ and $(u_{n})_{n\geq N}$ is then a Cauchy sequence for the usual norm $\left\|\cdot\right\|_{H}$ as well and, as a result, converges for this norm. This shows that $z\in D({\mathcal{A}_{\mathcal{K}}}^{*})$.

Next, observe that

[TABLE]

Indeed, if $z\in\ker L_{T}$, then $z\in D({\mathcal{A}_{\mathcal{K}}}^{*})$ as we have just seen, so that the map $t\in(0,T)\mapsto\mathcal{B}^{*}S_{\mathcal{A}_{\mathcal{K}}}(t)^{*}z\in U$ is continuous and we can take $t=0$ in the definition of $\ker L_{T}$ to obtain that $\mathcal{B}^{*}z=0$.

Let us now prove that $\ker L_{T}$ is stable by ${\mathcal{A}_{\mathcal{K}}}^{*}$. Firstly, we show that, for every $z\in D(\mathcal{A}_{\mathcal{K}}^{*})$, the map $t\mapsto\mathcal{B}^{*}S_{\mathcal{A}_{\mathcal{K}}}(t)^{*}z$ belongs to $H^{1}(0,T;U)$ with

[TABLE]

This identity is clearly true for $z\in D(\left(\mathcal{A}_{\mathcal{K}}^{*}\right)^{2})$. In particular, for every $\varphi\in C^{\infty}_{c}(0,T)$ and $z\in D(\left(\mathcal{A}_{\mathcal{K}}^{*}\right)^{2})$ we have

[TABLE]

By density of $D(\left(\mathcal{A}_{\mathcal{K}}^{*}\right)^{2})$ in $D(\mathcal{A}_{\mathcal{K}}^{*})$ and admissibility of $\mathcal{B}$ for $\mathcal{A}_{\mathcal{K}}$, this identity remains valid for every $z\in D(\mathcal{A}_{\mathcal{K}}^{*})$. Let now $z\in\ker L_{T}$ and let us show that $\mathcal{A}_{\mathcal{K}}^{*}z\in\ker L_{T}$. We have

[TABLE]

By the previous step, we can differentiate this identity to obtain

[TABLE]

that is $\mathcal{A}_{\mathcal{K}}^{*}z\in\ker L_{T}$. This shows that $\ker L_{T}$ is stable by ${\mathcal{A}_{\mathcal{K}}}^{*}$.

Consequently, the restriction of ${\mathcal{A}_{\mathcal{K}}}^{*}$ to $\ker L_{T}$ is a linear operator from the finite dimensional space $\ker L_{T}$ into itself and, if $\ker L_{T}\neq\left\{0\right\}$, therefore possesses at least one complex eigenvalue (here we use that $H$ is a complex Hilbert space). Since in addition $\ker L_{T}\subset\ker\mathcal{B}^{*}$, this shows that there exist $\lambda\in\mathbb{C}$ and $\phi\in D(\mathcal{A}_{\mathcal{K}}^{*})$ with $\phi\neq 0$ such that

[TABLE]

which is in contradiction with the Fattorini criterion (1.4). Thus, we must have $\ker L_{T}=\left\{0\right\}$. Applying item ii of Lemma 2.1, we obtain that there exists $C>0$ such that

[TABLE]

∎

3 Controllability of an integral transport equation

Let us start with a simple application of our main results. In [18, 7, 10], the authors investigated the stabilization properties of the following hyperbolic equation:

[TABLE]

where $T>0$ is the time of control, $L>0$ is the length of the domain, $y^{0}\in L^{2}(0,L)$ is the initial data and $y$ is the state, $k\in L^{2}((0,L)\times(0,L))$ is a given kernel function and, finally, $u\in L^{2}(0,T)$ is the boundary control.

In [10], the authors gave a necessary and sufficient condition for the stabilization in finite time of system (3.1), that is the property of wether there exists or not a feedback $F\in\mathcal{L}(L^{2}(0,L),\mathbb{R})$ such that the closed-loop system (3.1) with $u(t)=Fy(t)$ satisfies, for some $T>0$, $y(t)=0$ for every $t\geq T$. More precisely, they proved that (3.1) is stabilizable in finite time $T=L$ if, and only if, system (3.1) is exactly controllable in time $T=L$ (see [10, Theorem 1.1]) and then, in a second part, they studied the exact controllability of system (3.1) in time $T=L$.

To see that our main results apply in this framework, we recast system (3.1) in the abstract form (1.1). The operator $\mathcal{A}_{\mathcal{K}}:D(\mathcal{A}_{\mathcal{K}})\subset L^{2}(0,L)\longrightarrow L^{2}(0,L)$ is given by

[TABLE]

and the operator $\mathcal{B}\in\mathcal{L}(\mathbb{R},D(\mathcal{A}_{\mathcal{K}}^{*})^{\prime})$ is

[TABLE]

Note that $\mathcal{B}$ is well-defined since $\mathcal{B}u$ is continuous on $H^{1}(0,L)$ (by the trace theorem $H^{1}(0,L)\hookrightarrow C^{0}([0,L])$) and since $\left\|\cdot\right\|_{D(\mathcal{A}_{\mathcal{K}}^{*})}$ and $\left\|\cdot\right\|_{H^{1}(0,L)}$ are equivalent norms on $D(\mathcal{A}_{\mathcal{K}}^{*})$. Clearly, $\mathcal{A}_{\mathcal{K}}=\mathcal{A}_{0}+\mathcal{K}$, where $\mathcal{A}_{0}:D(\mathcal{A}_{0})\subset L^{2}(0,L)\longrightarrow L^{2}(0,L)$ is given by

[TABLE]

and $\mathcal{K}:L^{2}(0,L)\longrightarrow L^{2}(0,L)$ is given by

[TABLE]

It is well-known that $\mathcal{A}_{0}$ generates a $C_{0}$-semigroup on $L^{2}(0,L)$ and that $\mathcal{K}$ is compact. On the other hand, using the multiplier method it is not difficult to prove that $\mathcal{B}$ is admissible for $\mathcal{A}_{0}$. Finally, we recall that $(\mathcal{A}_{0},\mathcal{B})$ is exactly controllable in time $T$ if, and only if,

[TABLE]

(see e.g. [9, Theorem 2.6]). Therefore, all the assumptions of Theorems 1.1 and 1.2 are satisfied and this immediately gives the following new result:

Theorem** 3.1****.**

Equation (3.1) is exactly controllable:

1. (i)

in time $T=L$ if (and only if) it is approximately controllable in time $T=L$. 2. (ii)

in time $T$ for every $T>L$ if (and only if)

[TABLE]

As said in Remark 1.3, we recall that, in (3.2), the operators $\mathcal{A}_{\mathcal{K}}$ and $\mathcal{B}$ actually denote the complexified operators.

Wether the Fattorini criterion (3.2) characterizes or not the exact controllability in the optimal time $T=L$ for the equation (3.1) remains an open problem so far. It has been proved in [10] that this is indeed true in some particular cases (see [10, Theorem 1.2]). Note that this can not be obtained as a consequence of our results and that it is important in their work in view of their stabilization result that we mentioned above. Finally, the authors also provided easy checkable conditions on the kernel $k$ to see wether the Fattorini criterion (3.2) is satisfied or not. We summarize their results in the following proposition (see [10, Remark 4 and Proposition 3.3]), which completes item (ii) of Theorem 3.1:

Proposition** 3.2****.**

1. (i)

The Fattorini criterion (3.2) is satisfied if one of the following condition holds:

1. (a)

$\left\|k\right\|_{L^{2}((0,L)\times(0,L))}<\sqrt{2}/L$. 2. (b)

$k=0$* in $\mathcal{T}_{-}$ or $k=0$ in $\mathcal{T}_{+}$, where*

[TABLE] 2. (ii)

If $k$ depends only on its first variable, that is $k(x,\xi)=k(x)$, then (3.2) is equivalent to

[TABLE]

where $\lambda_{0}=\int_{0}^{L}k(x)\,dx$ and $\lambda_{n}=i\frac{2n\pi}{L}$ for $n\neq 0$.

4 Controllability of a system of wave equations

In this section, we will establish a new result concerning the controllability of systems of coupled wave equations. We will first apply Theorem 1.2 to obtain the exact controllability with as many controls as equations. We will then reduce the number of controls using the fictitious control method, with the help of the results of [13] to obtain the necessary improved regularity on the controls to make this method works. This gives Theorem 4.1 below, which generalizes the result [20, Theorem 7] (see also Corollary 4.10 below).

Let $T>0$, let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ regular enough (for example of class $C^{\infty}$). We consider the following linear system of $n$ coupled wave equations:

[TABLE]

where $(y^{0},\dot{y}^{0})$ is the initial data222Here and in the rest of this article, $\dot{y}^{0},\dot{y}^{1}$ and $\dot{y}$ are only a notation., $y=(y_{1},\ldots,y_{n})$ is the state, $u$ is the control and $\omega\subset\Omega$ is the part of the domain where we can act. We recall that $\mathds{1}_{\omega}$ denotes the function that is equal to $1$ in $\omega$ and [math] outside. In system (4.1), $A=(a_{ij})_{1\leq i,j\leq n}$ is a space dependent coupling matrix with entries $a_{ij}\in L^{\infty}(\Omega)$, and $B\in\mathbb{R}^{n}$ is a constant vector.

For $T>0$ and a non empty open subset $\omega\subset\Omega$, we say that the couple $(T,\omega)$ satisfies the Geometric Control Condition, in short (GCC) , if every ray of geometric optic in $\Omega$ with a velocity equal to one enters in $\omega$ in a time smaller that $T$. We shall also say that $\omega$ satisfies (GCC) if there exists $T>0$ such that $(T,\omega)$ satisfies (GCC) .

Finally, for $k\in\mathbb{N}$ we introduce the spaces

[TABLE]

where $\lceil\alpha\rceil$ denotes the smallest integer such that $\alpha\leq\lceil\alpha\rceil$.

The main result of this section is the following:

Theorem** 4.1****.**

Let $n\geq 2$. Let $T^{*}>0$ and $\omega\subset\subset\Omega$ be a nonempty open subset such that $(T^{*},\omega)$ satisfies (GCC) . Assume that $A$ and $B$ have the following structure:

[TABLE]

Assume moreover that $a_{ij}\in C^{2(n-3)}(\omega)$ for every $i,j\in\{1,...,n\}$ and that, for every $i\in\left\{2,\ldots,n\right\}$,

[TABLE]

Then, system (4.1) is exactly controllable in time $T$ for every $T>T^{*}$, in the following sense: for every $(y^{0},\dot{y}^{0}),(y^{1},\dot{y}^{1})\in H_{(0)}^{2n-1}(\Omega)^{n}\times H_{(0)}^{2n-2}(\Omega)^{n}$, there exists $u\in L^{2}(0,T;L^{2}(\Omega))$ such that the corresponding solution $y\in C^{0}([0,T];H^{1}_{0}(\Omega)^{n})\cap C^{1}([0,T];L^{2}(\Omega)^{n})$ to system (4.1) satisfies

[TABLE]

Assumption (4.2) is an assumption on the structure of system (4.1), namely that $A$ and $B$ are in cascade. Note that in the particular case $n=2$ no structural assumption is imposed on $A$. Theorem 4.1 is the complete analoguous result of [16, Theorem 1.2] for parabolic systems. We recall that this latter is based on parabolic Carleman estimates.

Let us emphasize that, unlike to the results of [1, 26, 2], we obtain the optimal time $T^{*}$ which is the one provided for a single equation. Therefore, Theorem 4.1 must be seen as a result which gives sufficient conditions on the structure of system (4.1) to ensure that the optimal time $T^{*}$ for a single wave equation remains the same for a whole system of coupled wave equations. Evidently, this also has some drawbacks. Firstly, we require more regularity on the initial data (see also Remark 4.2 below). Secondly, we also assume that the supports of the coupling terms contain the control domain $\omega$. This is important since otherwise the time $T^{*}$ could not be preserved for the system (see the results of [12]). Finally, it is also worth mentioning that, contrary to [1, 26, 2, 12], we do not make any sign assumption on $a_{i\,i-1}$ outside the control domain $\omega$.

Remark** 4.2****.**

As in [20], we prove in Theorem 4.1 the exact controllability of system (4.1) for some regular initial data. Therefore, it is natural to ask if the exact controllability still holds for less regular initial data. In [19] the authors showed that the Neumann boundary controllability of the system

[TABLE]

in the natural space $H^{1}(\Omega)^{n}\times L^{2}(\Omega)^{n}$ is impossible if $\mathrm{rank}\,B<n$, whatever the time $T$ is (see [19, Theorem 4.2]). The problem comes from the fact that all the components of the initial data are in the same energy space, that is $y^{0}_{i}\in H^{1}(\Omega)$ and $\dot{y}^{0}_{i}\in L^{2}(\Omega)$ for every $i\in\left\{1,\ldots,n\right\}$. On the other hand, the exact controllability becomes possible if the components of the initial data are allowed to lie in different energy spaces but we do not investigate this question in this paper (see e.g. [1, 26, 2, 12]).

4.1 Controllability of $n$ coupled equations by $n$ controls

As previously mentioned, the first step to establish Theorem 4.1 is to control the system with as many controls as equations. This is for this particular step that we are going to use our perturbation result Theorem 1.2. Note that, all along Section 4.1, the only assumption that we make is that $(T^{*},\omega)$ satisfies (GCC) . Therefore, in this section, we consider

[TABLE]

where $A\in L^{\infty}(\Omega)^{n\times n}$ is any matrix and, this time, $u=(u_{1},\ldots,u_{n})$ are $n$ controls. The goal of this section is to establish the following result:

Proposition** 4.3****.**

Let $T^{*}>0$ and $\omega\subset\subset\Omega$ be a nonempty open subset such that $(T^{*},\omega)$ satisfies (GCC) . Then, for every $T>T^{*}$, system (4.4) is exactly controllable in time $T$, that is, for every $(y^{0},\dot{y}^{0}),(y^{1},\dot{y}^{1})\in H^{1}_{0}(\Omega)^{n}\cap L^{2}(\Omega)^{n}$, there exists $u\in L^{2}(0,T;L^{2}(\Omega)^{n})$ such that the corresponding solution $y\in C^{0}([0,T];H^{1}_{0}(\Omega)^{n})\cap C^{1}([0,T];L^{2}(\Omega)^{n})$ to system (4.4) satisfies

[TABLE]

To apply Theorem 1.2 we recast (4.4) as a first-order abstract evolution system. The state space $H$ and the control space $U$ are

[TABLE]

The operator $\mathcal{A}_{\mathcal{K}}:D(\mathcal{A}_{\mathcal{K}})\subset H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}\longrightarrow H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}$ is

[TABLE]

and the control operator $\mathcal{B}:L^{2}(\Omega)^{n}\longrightarrow H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}$ is

[TABLE]

Clearly, $\mathcal{A}_{\mathcal{K}}=\mathcal{A}_{0}+\mathcal{K}$, where $\mathcal{A}_{0}:D(\mathcal{A}_{0})\subset H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}\longrightarrow H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}$ is given by

[TABLE]

and $\mathcal{K}:H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}\longrightarrow H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}$ is given by

[TABLE]

It is well-known that $\mathcal{A}_{0}$ is the generator of a $C_{0}$-group on $H^{1}_{0}(\Omega)^{n}\times L^{2}(\Omega)^{n}$. On the other hand, by the compact embedding $H^{1}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$, it is clear that $\mathcal{K}$ is compact. Finally, observe that $\mathcal{B}$ is bounded and thus admissible. Therefore, the proof of Proposition 4.3 will simply consists in checking the exact controllability of the unperturbed system and then the Fattorini criterion for the perturbed system. To prove the latter, we will need the following unique continuation property for elliptic systems:

Lemma** 4.4****.**

Let $\alpha_{ij}\in L^{\infty}(\Omega;\mathbb{C})$, $i,j\in\left\{1,\ldots,n\right\}$. Let $y_{1},\ldots,y_{n}\in H^{2}(\Omega;\mathbb{C})\cap H^{1}_{0}(\Omega;\mathbb{C})$ be such that

[TABLE]

If $y_{1}=\ldots=y_{n}=0$ in $\omega$, then $y_{1}=\ldots=y_{n}=0$ in $\Omega$.

This Lemma can be proved by applying elliptic Carleman estimates to each equation and then adding them up. Note that it is enough to prove it only for real valued functions by splitting up the system into real and imaginary parts, up to increase the number of equations.

Proof of Proposition 4.3.

Let $T>T^{*}$. Since $(T^{*},\omega)$ satisfies (GCC) by assumption, we know from the results of [5] that the wave equation is exactly controllable in time $T^{*}$. Therefore, the following uncoupled system of wave equations:

[TABLE]

is also exactly controllable in time $T^{*}$. This shows that the first hypothesis of Theorem 1.2 is satisfied. To apply Theorem 1.2, we only have to check the Fattorini criterion. Let $\lambda\in\mathbb{C}$, $\theta\in H^{2}(\Omega;\mathbb{C})^{n}\cap H^{1}_{0}(\Omega;\mathbb{C})^{n}$ and $\xi\in H^{1}_{0}(\Omega;\mathbb{C})^{n}$ be such that

[TABLE]

Therefore, $\theta\in H^{2}(\Omega;\mathbb{C})^{n}\cap H^{1}_{0}(\Omega;\mathbb{C})^{n}$ satisfies

[TABLE]

with $M=A^{*}-\lambda^{2}$. Using Lemma 4.4, we obtain $\theta=0$ in $\Omega$ and it follows that $\xi=0$ in $\Omega$ as well. As a result, Theorem 1.2 applies and (4.4) is exactly controllable in time $T$.

∎

4.2 Controllability of $n$ coupled equations by $1$ control

The goal of this section is to prove Theorem 4.1 by means of the method of algebraic solvability. The first step is to improve the regularity of the controls in order to be able to take their derivatives in the sequel. This is the reason why we need to take more regular initial data.

Proposition** 4.5****.**

Let $T^{*}>0$ and $\omega\subset\subset\Omega$ be a nonempty open subset such that $(T^{*},\omega)$ satisfies (GCC) . Then, for every $T>T^{*}$, for every $(y^{0},\dot{y}^{0}),(y^{1},\dot{y}^{1})\in H_{(0)}^{2n-1}(\Omega)^{n}\times H_{(0)}^{2n-2}(\Omega)^{n}$ there exists

[TABLE]

with

[TABLE]

such that solution to

[TABLE]

satisfies

[TABLE]

The proof of Proposition 4.5 is a consequence of some results of [13] (see Theorem 1.4 and the proof of Corollary 1.5), summarized in the following lemma:

Lemma** 4.6****.**

Let $H$ and $U$ be two Hilbert spaces. Let $\mathcal{A}:D(\mathcal{A})\subset H\longrightarrow H$ be the generator of a $C_{0}$-group on $H$ and let $\mathcal{B}\in\mathcal{L}(U,H)$. Assume that there exists $T_{0}>0$ such that $(\mathcal{A},\mathcal{B})$ is exactly controllable in time $T_{0}$. Let $\delta>0$ and set $T_{\delta}=T_{0}+2\delta$. Let $\eta\in C^{\infty}(\mathbb{R})$ be a cut-off function in time satisfying

[TABLE]

Then, for every $y^{0},y^{1}\in D(\mathcal{A}^{s})$ ($s\in\mathbb{N}$), there exists

[TABLE]

such that the solution $y$ to

[TABLE]

satisfies $y(T_{\delta})=y^{1}$.

We will also need the following lemma:

Lemma** 4.7****.**

Let $T^{*}>0$ and $\omega\subset\subset\Omega$ be a nonempty open subset such that $(T^{*},\omega)$ satisfies (GCC) . Then, for every $\delta>0$, there exists a non empty open subset $\widetilde{\omega}\subset\subset\omega$ such that $(T^{*}+\delta,\widetilde{\omega})$ satisfies (GCC) .

Proof of Proposition 4.5.

Let $T>T^{*}$ and set $\delta=(T-T^{*})/4$ and $T_{0}=T^{*}+2\delta$, so that $T_{\delta}=T$. Using Lemma 4.7, there exists a non empty open subset $\widetilde{\omega}\subset\subset\omega$ such that $(T_{0}-\delta,\widetilde{\omega})$ satisfies (GCC) . Applying Proposition 4.3, the system

[TABLE]

is exactly controllable in time $T_{0}$. Let $\theta\in C^{\infty}(\mathbb{R}^{N})$ be cut-off function in space satisfying

[TABLE]

Clearly, the exact controllability of system (4.8) in time $T_{0}$ implies that the system

[TABLE]

is also exactly controllable in time $T_{0}$. Therefore, applying Lemma 4.6 with $\mathcal{B}=\theta$ and $s=2n-2$, we see that the function $\widehat{u}=\eta\theta^{2}Y$ is a control that possesses all the desired properties. ∎

The second and final step is the algebraic solvability. Here we finally use structure assumption (4.2) and the crucial condition (4.3).

Proposition** 4.8****.**

Let $f\in H^{2n-2}(0,T;L^{2}(\Omega)^{n})\cap L^{2}(0,T;H^{2n-2}(\Omega)^{n})$ with $\operatorname{supp}f\subset\subset(0,T)\times\omega$. Assume that $a_{ij}\in C^{2(n-3)}(\omega)$ for every $i,j\in\{1,...,n\}$ and that (4.2) and (4.3) hold. Then, there exist $\overline{y}\in H^{2}(0,T;L^{2}(\Omega)^{n})\cap L^{2}(0,T;H^{2}(\Omega)^{n})$ and $\overline{u}\in L^{2}(0,T;L^{2}(\Omega))$ such that

[TABLE]

Proof of Proposition 4.8.

Thanks to the structure assumption (4.2), the equation in (4.9) reads as

[TABLE]

Using assumption (4.3), this is easily solved by taking

[TABLE]

Note that the conditions of support in (4.9) are satisfied since $\overline{y}$ and $\overline{u}$ are only linear combinations of derivatives of $f$ and $\operatorname{supp}f\subset\subset(0,T)\times\omega$ by assumption. On the other hand, the claimed regularities of $\overline{y}$ and $\overline{u}$ follows by remarking that, to compute $\overline{y}$ (resp. $\overline{u}$), we apply $n-2$ times (resp. $n-1$ times) an operator of order two in time and two in space.

∎

The proof of Theorem 4.1 is now a simple consequence of Propositions 4.5 and 4.8.

Proof of Theorem 4.1.

Let $\widehat{u}$ be provided by Proposition 4.5 and let $\widehat{y}$ be the corresponding solution to system (4.5). Applying now Proposition 4.8 to $f=-\widehat{u}$, we obtain the existence of $\overline{y}$ and $\overline{u}$ that satisfy (4.9). Then, taking $u=\overline{u}$ we see that the corresponding solution $y$ to system (4.1) writes $y=\widehat{y}+\overline{y}$ and, thanks to (4.6) and to the condition on the supports in (4.9), it satisfies

[TABLE]

∎

Remark** 4.9****.**

We can easily see from the proof of Theorem 4.1 (see, especially, Section 4.2) that it can be generalized to the case of systems with a structure of cascade in bloc. More precisely, the conclusion of Theorem 4.1 remains true if we replace the assumption (4.2) by the more general one

[TABLE]

where each couple $(A_{ii},B_{i})$, $i\in\left\{1,\ldots,n^{\prime}\right\}$, has the form (4.2) and satisfies (4.3).

4.3 The case of constant matrices

As an immediate consequence of Theorem 4.1, we can treat the case of general but constant matrices $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$ with $m$ controls $u=(u_{1},\ldots,u_{m})$.

Corollary** 4.10****.**

Let $T^{*}>0$ and $\omega\subset\subset\Omega$ be a nonempty open subset such that $(T^{*},\omega)$ satisfies (GCC) . Assume that $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$ are constant and satisfy

[TABLE]

Then, we get the same conclusion as in Theorem 4.1.

We recall that this result is not new and that it has already been obtained in [20, Theorem 7], but the point of view we bring here is slightly different. Indeed, in the present paper, the idea is to first perform a change of variable furnished by the Kalman rank condition (4.11) and then to solve algebraically the resulting system, which actually turns out to be a very simple task to do (see Proposition 4.8 above).

Proof of Corollary 4.10.

Let us prove it for $m=1$. Let $K=(B|AB|A^{2}B|\cdots|A^{n-1}B)\in\mathbb{R}^{n\times n}$. By assumption, $K$ is invertible. Let $z=K^{-1}y$. Then $z$ solves

[TABLE]

with

[TABLE]

where $\widetilde{a}_{in}$, $i\in\left\{1,\ldots,n\right\}$, are such that $A^{n}=\widetilde{a}_{1n}+\widetilde{a}_{2n}A+\ldots+\widetilde{a}_{nn}A^{n-1}$ (using the Cayley-Hamilton theorem). Therefore, Theorem 4.1 applies to system (4.12).

For the general case $m\in\mathbb{N}^{*}$, using the Kalman condition (4.11) we can extract a basis $K$ from $(B|AB|A^{2}B|\cdots|A^{n-1}B)$ such that, in this new basis, $A$ and $B$ have the cascade in bloc structure (4.10) (see e.g. [3, Lemma 3.1]) and we conclude with Remark 4.9. ∎

5 Controllability of a non diagonalizable parabolic system

In this section, we will establish a new controllability result for coupled linear parabolic systems. We will show that the null controllability holds in arbitrarily small time for some non diagonalizable coupled parabolic systems when we have at our disposal as many controls as equations. The proof combines our perturbation result Theorem 1.2 with the so-called transmutation technique, introduced for the first time in a control framework in [22], that allows to transfer some controllability properties of wave processes to heat processes.

The class of parabolic systems that we consider here is the following:

[TABLE]

where $y^{0}$ is the initial data, $y=(y_{1},\ldots,y_{n})$ is the state, $u=(u_{1},\ldots,u_{m})$ are the controls and $\omega\subset\Omega$ is the domain of control. In (5.1), $D\in\mathbb{R}^{n\times n}$ is a constant matrix such that, for some $\alpha>0$, we have

[TABLE]

where $\cdot$ denotes the scalar product in $\mathbb{R}^{n}$, $A\in L^{\infty}(\Omega)^{n\times n}$ is a space dependent coupling matrix and $B\in\mathbb{R}^{n\times m}$ is a constant matrix. We recall that, under these assumptions, system (5.1) is well-posed: for every $y^{0}\in L^{2}(\Omega)^{n}$ and $u\in L^{2}(0,T;L^{2}(\Omega)^{m})$, there exists a unique (weak) solution $y\in C^{0}([0,T];L^{2}(\Omega)^{n})\cap L^{2}(0,T;H^{1}_{0}(\Omega)^{n})$ to system (5.1). As in the introduction, we say that system (5.1) is null controllable in time $T$ if, for every $y^{0}\in L^{2}(\Omega)^{n}$, there exists $u\in L^{2}(0,T;L^{2}(\Omega)^{m})$ such that the corresponding solution $y\in C^{0}([0,T];L^{2}(\Omega)^{n})$ to system (5.1) satisfies $y(T,\cdot)=0$.

When the matrix $D$ is a diagonal matrix, or more generally a diagonalizable matrix, using parabolic Carleman estimates on each equation of the adjoint system and adding them up, it is easy to show that for every $T>0$ system (5.1) is null controllable in time $T$ if there are as many controls as equations in system (5.1), that is if

[TABLE]

When the matrix $D$ is a more general matrix (for instance, a Jordan block), then Carleman estimates can still be used, up to some extent though. Indeed, because of the new couplings of order 2 that appear, there is a technical restriction on number of equations of system (5.1) to use this method, namely, that $n$ has to be less than or equal to $4$. We refer to [15], especially Theorem 1.1, for more details. In the present paper, we will show that, as expected, this condition on the number of equations was only the consequence of the technique used and that it can actually be removed.

The main result of this section is the following.

Theorem** 5.1****.**

Assume that $B\in\mathbb{R}^{n\times m}$ satisfies (5.3) and that $D\in\mathbb{R}^{n\times n}$ satisfies (5.2) and possesses only real eigenvalues. Let $\omega\subset\Omega$ be a non empty open subset satisfying (GCC) . Then, the system (5.1) is null controllable in time $T$ for every $T>0$.

Obviously, (GCC) is not a natural assumption for parabolic systems and it is probably true that we can remove this assumption in Theorem 5.1. Nevertheless, this completely solves this problem at least in dimension one since we recall that any non empty open subset $\omega$ satisfies (GCC) in this case.

Proof of Theorem 5.1.

By assumption on $D$, there exists a triangular matrix $\mathcal{T}\in\mathbb{R}^{n\times n}$

[TABLE]

with $\tau_{ii}>0$ for every $i\in\left\{1,\ldots,n\right\}$, and there exists an invertible matrix $P\in\mathbb{R}^{n\times n}$ such that

[TABLE]

Let $\tilde{A}=P^{-1}AP$ and $\tilde{B}=P^{-1}B$. Clearly, system (5.1) is null controllable in time $T$ if, and only if, so is the following system:

[TABLE]

Note also that (5.3) becomes $\mathrm{rank}\,\tilde{B}=n$. Next, observe that it is enough to prove that system (5.5) is null controllable for $\tilde{B}=\mathrm{Id}_{\mathbb{R}^{n}}$. Indeed, since $\mathrm{rank}\,\tilde{B}=n$, there exists a right inverse $C\in\mathbb{R}^{m\times n}$, i.e. $\tilde{B}C=\mathrm{Id}_{\mathbb{R}^{n}}$, and we can take $Cu$ as control.

The next step is to control the following system of wave equations in some time $S>0$:

[TABLE]

(we denote by $s$ the time variable for systems of wave equations). Using Theorem 1.2 we are going to prove that system (5.6) is exactly controllable in time $S$ for every $S>nS^{*}$ where $S^{*}$ is the minimal time such that $(S^{*},\omega)$ satisfies (GCC) . Firstly, let us consider the system

[TABLE]

Thanks to the particular structure (5.4) we see that only the first component $z_{1}$ of $z$ is involved in the first equation of system (5.7). Since $(S^{*},\omega)$ satisfies (GCC) by assumption, we know from the results of [5] that there exists $v_{1}\in L^{2}(0,S^{*};L^{2}(\Omega))$ such that

[TABLE]

It follows that only the second component $z_{2}$ of $z$ is involved in the second equation of system (5.7) on $(S^{*},S)\times\Omega$. Therefore, we can repeat the previous argument and obtain that

[TABLE]

Repeating the same argument over and over, we obtain in the end that system (5.7) is null controllable in time $nS^{*}$. Since null and exact controllability are equivalent for system (5.7) (as it is reversible in time), the first hypothesis of Theorem 1.2 is satisfied. Let us now check the Fattorini criterion for the perturbed system (5.6). Let $\lambda\in\mathbb{C}$, $\theta\in H^{2}(\Omega;\mathbb{C})^{n}\cap H^{1}_{0}(\Omega;\mathbb{C})^{n}$ and $\xi\in H^{1}_{0}(\Omega;\mathbb{C})^{n}$ be such that

[TABLE]

Since $\mathcal{T}$ is invertible, we have

[TABLE]

with $M=\mathcal{T}^{-1}\tilde{A}^{*}-\lambda^{2}\mathcal{T}^{-1}$. Using once again Lemma 4.4, we obtain that $\theta=0$ in $\Omega$ and it follows that $\xi=0$ in $\Omega$ as well. Therefore, Theorem 1.2 applies and system (5.6) is null controllable in time $S$ for every $S>nS^{*}$.

To conclude the proof of Theorem 5.1, we apply the transmutation technique to derive the null controllability of the parabolic system (5.5) (with $\tilde{B}=\mathrm{Id}_{\mathbb{R}^{n}}$) in arbitrarily small time (but still assuming that $\omega$ satisfies (GCC) though). This procedure is by now standard but let us give a proof for a sake of completeness. Let $T>0$ and $y^{0}\in L^{2}(\Omega)^{n}$ be fixed. In the previous step of the proof, we have obtained that there exist $S>0$ large enough and $v\in L^{2}(0,S;L^{2}(\Omega)^{n})$ such that the solution $z\in C^{0}([0,S];H^{1}_{0}(\Omega)^{n})\cap C^{1}([0,S];L^{2}(\Omega)^{n})$ to the system of wave equations (5.6) satisfies

[TABLE]

Let us extend $z$ and $v$ by symmetry to $[-S,S]$:

[TABLE]

Then, $\bar{z}$ and $\bar{v}$ solve the same system, now posed on $[-S,S]\times\Omega$. Observe also that, thanks to (5.8),

[TABLE]

Let us denote by $\delta_{0}$ the Dirac mass at $s=0$ on $(-S,S)$. Since $\delta_{0}\in H^{-1}(-S,S)$ (recall that $H^{1}_{0}(-S,S)\subset C^{0}([-S,S])$), using the method of moments, we know that there exists a control $w\in L^{2}(0,T)$ such that the corresponding solution (by transposition) $k\in L^{2}((0,T)\times(-S,S))\cap C^{0}([0,T];H^{-1}(-S,S))$ to the one-dimensional heat equation

[TABLE]

satisfies

[TABLE]

Then, we define the following kernel transforms of $\bar{z}$ and $\bar{v}$:

[TABLE]

Since $k\in L^{2}((0,T)\times(-S,S))$ and $\bar{z}\in L^{2}(-S,S;H^{1}_{0}(\Omega)^{n})$, we have

[TABLE]

On the other hand, using (5.9), we have, for every $t\geq 0$ and $\varphi\in L^{2}(\Omega)^{n}$,

[TABLE]

where $\cdot$ denotes the scalar product in $\mathbb{R}^{n}$. Since $k\in C^{0}([0,T];H^{-1}(-S,S))$, this shows that

[TABLE]

It is also clear from (5.10) and (5.11) that

[TABLE]

Finally, it is slightly tedious but we can check that $y$ is indeed the solution to the system of heat equations (5.5). ∎