# Observer-based feedback-control for the stabilization of a class of parabolic systems

Imene Aicha Djebour, Karim Ramdani, Julie Valein

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Imene Aicha Djebour, Karim Ramdani, Julie Valein. Observer-based feedback-control for the stabilization of a class of parabolic systems. 2023. hal-04155214

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# Observer-based feedback-control for the stabilization of a class of parabolic systems

Imene Aicha Djebour, Karim Ramdani, Julie Valein

► To cite this version:

Imene Aicha Djebour, Karim Ramdani, Julie Valein. Observer-based feedback-control for the stabilization of a class of parabolic systems. Journal of Optimization Theory and Applications, 2024, 202 (3), pp.1217-1241. 10.1007/s10957-024-02496-1. hal-04155214v2

HAL Id: hal-04155214

https://hal.science/hal-04155214v2

Submitted on 26 Aug 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Observer-based feedback-control for the stabilization of a class of parabolic systems

Imene Aicha Djebour¹, Karim Ramdani¹, and Julie Valein¹

¹Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

July 7, 2023

## Abstract

We consider the stabilization of a class of linear evolution systems $z' = Az + Bv$ under the observation $y = Cz$ by means of a finite dimensional control $v$. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator $A$ is supposed to generate an analytical semigroup with compact resolvent and the operators $B$ and $C$ are unbounded operators whereas in the finite dimensional case, $A$ is assumed to be a self-adjoint operator with compact resolvent, $B$ and $C$ are supposed to be bounded operators. In both cases, we show that if $(A, B)$ and $(A, C)$ verify the Fattorini-Hautus Criterion, then we can construct an observer-based control $v$ of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of $A$) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the $N$ dimensional convection-diffusion system with Dirichlet boundary control and an internal observation.

**Keywords:** Parabolic systems, Feedback control, Stabilization, Luenberger observers.

**2010 Mathematics Subject Classification** 93B53, 93D15, 93C20.

## Contents

### 1 Introduction and main results

<table><tr><td>1.1</td><td>Infinite-Dimensional Observer (IDO)</td><td>3</td></tr><tr><td>1.2</td><td>Finite-Dimensional Observer (FDO)</td><td>4</td></tr><tr><td>1.3</td><td>Related works</td><td>4</td></tr><tr><td>1.4</td><td>Outline</td><td>5</td></tr></table>

### 2 Infinite dimensional observer

<table><tr><td>2.1</td><td>Spectral decomposition of the system</td><td>5</td></tr><tr><td>2.2</td><td>Construction of an infinite dimensional observer based control</td><td>6</td></tr><tr><td>2.3</td><td>Stability of the closed-loop system</td><td>8</td></tr></table>

### 3 Finite dimensional observer

<table><tr><td>3.1</td><td>Spectral decomposition of the system</td><td>10</td></tr><tr><td>3.2</td><td>Finite dimensional observer based control and stability of the closed-loop system</td><td>11</td></tr></table>

### 4 Stabilization of the reaction-diffusion equation

<table><tr><td>2</td></tr><tr><td>3</td></tr><tr><td>4</td></tr><tr><td>4</td></tr><tr><td>5</td></tr><tr><td>5</td></tr><tr><td>6</td></tr><tr><td>8</td></tr><tr><td>10</td></tr><tr><td>10</td></tr><tr><td>11</td></tr></table>
# Observer-based feedback-control for the stabilization of a class of parabolic systems

Imene Aicha Djebour*¹, Karim Ramdani†¹, and Julie Valein‡¹

¹Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France.

July 2, 2024

## Abstract

We consider the stabilization of a class of linear evolution systems $z' = Az + Bv$ under the observation $y = Cz$ by means of a finite dimensional control $v$. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator $A$ is supposed to generate an analytical semigroup with compact resolvent and the operators $B$ and $C$ are unbounded operators whereas in the finite dimensional case, $A$ is assumed to be a self-adjoint operator with compact resolvent, $B$ and $C$ are supposed to be bounded operators. In both cases, we show that if $(A, B)$ and $(A, C)$ verify the Fattorini-Hautus Criterion, then we can construct an observer-based control $v$ of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of $A$) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the diffusion system.

**Keywords:** Parabolic systems, Feedback control, Stabilization, Luenberger observers.

2010 Mathematics Subject Classification 93B53, 93D15, 93C20.

## 1 Introduction and main results

Given $b \in L^2(0,1)$, consider the one-dimensional controlled heat equation

$$ egin{cases} 
\partial_t z(t, x) = \partial_{xx} z(t, x) + b(x)v(t), & t > 0, \quad x \in (0,1), \\ 
\partial_x z(t, 0) = 0, & z(t, 1) = 0, \\ 
z(0) = z^0. 
\end{cases} 
\quad (1.1) $$

Obviously, the open-loop system (i.e. for $v = 0$) is exponentially stable, with a decay rate defined by the smallest eigenvalue of the underlying operator describing the free dynamics (namely the positive definite self-adjoint operator $-\partial_{xx}$ with Neumann boundary condition at $x = 0$ and Dirichlet boundary condition at $x = 1$). Based on the observation

$$ y(t) = \int_{0}^{1} c(x)z(t, x) \, dx, \quad (1.2) $$

<footnote>*E-mail: imene.djebour@univ-lorraine.fr</footnote>

<footnote>†E-mail: karim.ramdani@inria.fr</footnote>

<footnote>‡E-mail: julie.valein@univ-lorraine.fr</footnote>
# 1 Introduction and main results

Given $b \in L^2(0, 1)$, consider the one-dimensional controlled heat equation

$$ egin{cases} 
\partial_t z(t, x) = \partial_{xx} z(t, x) + b(x)v(t), & t > 0, \quad x \in (0, 1), \\
\partial_x z(t, 0) = 0, & z(t, 1) = 0, \\
z(0) = z^0.
\end{cases} 
\quad (1.1) $$

Obviously, the open-loop system (i.e. for $v = 0$) is exponentially stable, with a decay rate defined by the smallest eigenvalue of the underlying operator describing the free dynamics (namely the positive definite self-adjoint operator $-\partial_{xx}$ with Neumann boundary condition at $x = 0$ and Dirichlet boundary condition at $x = 1$).

Based on the observation

$$ y(t) = \int_{0}^{1} c(x)z(t, x) \, dx, \quad (1.2) $$

where $c \in L^2(0, 1)$, a natural question that arises is to know whether it is possible to design a finite dimensional feedback control $v$, such that the closed-loop system (1.1) is exponentially stable with an arbitrary prescribed decay rate $\sigma > 0$. It is a recent work, a positive and constructive answer to this question has been proposed by Katz and Fridman [11], using an observer-based feedback control. More precisely, the authors proposed feasible design conditions for the construction of such controls for a more general 1D reaction-diffusion equation with variable coefficients (i.e. for a free dynamics described by an operator of the form $\partial_x (p(x)\partial_{x^\cdot}) - q(x)\cdot$).

In this paper, our objective is to generalize this result to a large class of parabolic systems, possibly multi-dimensional and involving unbounded control and/or observation operators. More precisely, given three Hilbert spaces $\mathbb{H}$ (the state space), $\mathcal{U}$ (the control space) and $\mathcal{Y}$ (the observation space), consider the linear infinite dimensional system

$$ egin{cases} 
z'(t) = Az(t) + Bv(t), \\ 
z(0) = z^0, \\ 
y(t) = Cz(t), 
\end{cases} 
\quad (1.3) $$

where $A : \mathcal{D}(A) \to \mathbb{H}$ is an unbounded operator, $B \in \mathcal{L}(\mathcal{U}, (\mathcal{D}(A^*))')$ and $C \in \mathcal{L}(\mathcal{D}(A), \mathcal{Y})$. Given $\sigma > 0$, the goal of this paper is to prove the existence of an observer-based control $v$ such that the solution of (1.3) is exponentially stable, with a decay rate $-\sigma$:

$$ \|z(t)\|_{\mathbb{H}} \leq Me^{-\sigma t} \|z^0\|_{\mathbb{H}}. $$

We will investigate two classes of systems, depending on whether the observer used is infinite-dimensional (IDO case) or finite-dimensional (FDO case). We will make the following assumptions on $A$, $B$ and $C$ in these two cases (below, $\rho(A)$ denote the resolvent set of $A$):

## • Infinite-Dimensional Observer (IDO)

A is an analytic operator with compact resolvent on $\mathbb{H}$, (H1.A)

($\mu_0 \mathrm{Id} - A)^{-\gamma} B \in \mathcal{L}(\mathcal{U}, \mathbb{H})$ is a linear bounded operator for some $\gamma \in [0, 1)$ and $\mu_0 \in \rho(A)$, (H1.B)

$C(\mu_0 \mathrm{Id} - A)^{-\hat{\gamma}} \in \mathcal{L}(\mathbb{H}, \mathcal{Y})$ is a linear bounded operator for some $\hat{\gamma} \in [0, 1)$ and $\mu_0 \in \rho(A)$, (H1.C)

$$ egin{cases} 
\forall \varepsilon \in \mathcal{D}(A^*), \forall \lambda \in \mathbb{C}, \mathrm{Re} \, \lambda \geq -\sigma, \quad A^*\varepsilon = \bar{\lambda}\varepsilon \quad \text{and} \quad B^*\varepsilon = 0 \quad \implies \varepsilon = 0, \\ 
\forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{C}, \mathrm{Re} \, \lambda \geq -\sigma, \quad A\varepsilon = \lambda\varepsilon \quad \text{and} \quad C\varepsilon = 0 \quad \implies \varepsilon = 0. 
\end{cases} 
\quad (H1.D) $$

## • Finite-Dimensional Observer (FDO)

A is a self-adjoint operator with compact resolvent, (H2.A)
where $c \in L^2(0, 1)$, a natural question that arises is to know whether it is possible to design a finite dimensional feedback control $v$, such that the closed-loop system (1.1) is exponentially stable with an arbitrary prescribed decay rate $\sigma > 0$. In a recent work, a positive and constructive answer to this question has been proposed by Katz and Fridman [13], using an observer-based feedback control. More precisely, the authors proposed feasible design conditions for the construction of such controls for a more general 1D reaction-diffusion equation with variable coefficients (i.e. for a free dynamics described by an operator of the form $\partial_x (p(x)\partial_x \cdot) - q(x) \cdot$).

In this paper, our objective is to generalize this result to a large class of parabolic systems, possibly multi-dimensional and involving unbounded control and/or observation operators. More precisely, given three complex Hilbert spaces $\mathbb{H}$ (the state space), $\mathbb{U}$ (the control space) and $\mathbb{Y}$ (the observation space), consider the linear infinite dimensional system

$$ egin{cases} z'(t) = Az(t) + Bv(t), \\ z(0) = z^0, \\ y(t) = Cz(t), \end{cases} \qquad (1.3) $$

where $A : \mathcal{D}(A) \to \mathbb{H}$ is an unbounded operator, $B \in \mathcal{L}(\mathbb{U}, (\mathcal{D}(A^*))')$ and $C \in \mathcal{L}(\mathcal{D}(A), \mathbb{Y})$. Given $\sigma > 0$, the goal of this paper is to prove the existence of an observer-based control $v$ such that the solution of (1.3) is exponentially stable, with a decay rate $-\sigma$:

$$ \|z(t)\|_{\mathbb{H}} \leq Me^{-\sigma t} \|z^0\|_{\mathbb{H}}. $$

We will investigate two classes of infinite dimensional systems of the form (1.3). For the first one, the observer-based feedback is constructed using an infinite-dimensional observer (IDO). For the second one, under strongest assumptions on $A$, $B$ and $C$, we design a finite dimensional observer (FDO) to construct the feedback control. Of course, a finite dimensional observer can be considered as a particular case of infinite dimensional observer, but the former is much more interesting for computational purposes (and, clearly, more challenging to design). For instance, if the system (1.3) describes an evolution PDE (like the reaction-diffusion equation of Section 4), constructing an infinite dimensional observer requires to solve an evolution PDE, corresponding to the observer (1.7). On the contrary, the finite dimensional observer described by (1.10) is simply a set of evolution ODE. As one might expect, the proofs to get these two results use different techniques.

We will consider the two following sets of assumptions on $A$, $B$ and $C$ (below, $\rho(A)$ denotes the resolvent set of $A$):

### Infinite-Dimensional Observer (IDO)

A is an operator with compact resolvent generating an analytic semigroup on $\mathbb{H}$, (H1.A)

$(\mu_0 \mathrm{Id} - A)^{-\gamma} B \in \mathcal{L}(\mathbb{U}, \mathbb{H})$ is a linear bounded operator for some $\gamma \in [0, 1)$ and $\mu_0 \in \rho(A)$, (H1.B)

$C(\mu_0 \mathrm{Id} - A)^{-\hat{\gamma}} \in \mathcal{L}(\mathbb{H}, \mathbb{Y})$ is a linear bounded operator for some $\hat{\gamma} \in [0, 1)$ and $\mu_0 \in \rho(A)$, (H1.C)

$$ \left\{ \begin{array}{ll} \forall \varepsilon \in \mathcal{D}(A^*), \forall \lambda \in \mathbb{C}, \mathrm{Re} \, \lambda \geq -\sigma, & A^*\varepsilon = \bar{\lambda}\varepsilon \quad \text{and} \quad B^*\varepsilon = 0 \quad \implies \varepsilon = 0, \\ \forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{C}, \mathrm{Re} \, \lambda \geq -\sigma, & A\varepsilon = \lambda\varepsilon \quad \text{and} \quad C\varepsilon = 0 \quad \implies \varepsilon = 0. \end{array} \right. \qquad (H1.D) $$

### Finite-Dimensional Observer (FDO)

A is a self-adjoint operator with compact resolvent, (H2.A)

$B \in \mathcal{L}(\mathbb{U}, \mathbb{H})$, (H2.B)

$C \in \mathcal{L}(\mathbb{H}, \mathbb{Y})$, (H2.C)

$$ \left\{ \begin{array}{ll} \forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{R}, \lambda \geq -\sigma, & A\varepsilon = \lambda\varepsilon \quad \text{and} \quad B^*\varepsilon = 0 \quad \implies \varepsilon = 0, \\ \forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{R}, \lambda \geq -\sigma, & A\varepsilon = \lambda\varepsilon \quad \text{and} \quad C\varepsilon = 0 \quad \implies \varepsilon = 0. \end{array} \right. \qquad (H2.D) $$
1

$$B \in \mathcal{L}(\mathbb{U}, \mathbb{H}), \qquad (\text{H2.B})$$

2

$$C \in \mathcal{L}(\mathbb{H}, \mathbb{Y}), \qquad (\text{H2.C})$$

3

$$\left\{ \begin{array}{l} \forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{R}, \lambda \ge -\sigma, \quad A\varepsilon = \lambda\varepsilon \quad \text{et} \quad B^*\varepsilon = 0 \implies \varepsilon = 0, \\ \forall \varepsilon \in \mathcal{D}(A), \forall \lambda \in \mathbb{R}, \lambda \ge -\sigma, \quad A\varepsilon = \lambda\varepsilon \quad \text{et} \quad C\varepsilon = 0 \implies \varepsilon = 0. \end{array} \right. \qquad (\text{H2.D})$$

4

5

For every $\nu > 0$, we set

6

$$\Sigma_{\nu}^{+} := \{\lambda_j \in \sigma(A) ; \text{Re} \, \lambda_j \ge -\nu\}, \quad \Sigma_{\nu}^{-} := \{\lambda_j \in \sigma(A) ; \text{Re} \, \lambda_j < -\nu\}, \qquad (1.4)$$

7

8

where $\sigma(A)$ is the spectrum of $A$. Condition (H1.A) in the IFD case and (H2.A) in the FDO case imply that $\Sigma_{\nu}^{+}$ describes a finite set. We define the projection

$$P_{\nu}^{+} = -\frac{1}{2i\pi} \int_{\Gamma_{\nu}^{+}} (\zeta \mathrm{Id} - A)^{-1} d\zeta, \quad P_{\nu}^{-} = \mathrm{Id} - P_{\nu}^{+}, \qquad (1.5)$$

where $\Gamma_{\nu}^{+}$ is a curve enclosing $\Sigma_{\nu}^{+}$ but no other point of the spectrum of $A$ and oriented counterclockwise (see [10, V.5, p.272]). We set

$$z_{\nu}^{\pm} := P_{\nu}^{\pm} z, \quad \forall z \in \mathbb{H}.$$

We also introduce the finite dimensional operators

$$A_{\nu}^{\pm} := A P_{\nu}^{\pm}, \quad B_{\nu}^{\pm} := P_{\nu}^{\pm} B, \quad C_{\nu}^{\pm} := C I_{\nu}^{\pm},$$

9

10

where $I_{\nu}^{\pm}$ is the embedding operator from $\mathbb{H}_{\nu}^{\pm} := P_{\nu}^{\pm} \mathbb{H}$ to $\mathbb{H}$. Finally, we denote by $Q_{\nu}^{+}$ the orthogonal projection from $\mathbb{Y}$ onto $\mathbb{Y}_{\nu}^{+} := CP_{\nu}^{+} \mathbb{H}$.

11

We are now in position to state our main results in the (IDO) and (FDO) cases.

12

## 1.1 Infinite-Dimensional Observer (IDO)

13

14

15

**Theorem 1.1.** Let $\sigma > 0$ be given and assume that assumptions (H1.A), (H1.B), (H1.C) and (H1.D) hold true. Then, there exist two operators $K_{\sigma}^{+} \in \mathcal{L}(\mathbb{H}, B^{*}(P_{\sigma}^{+})^{*}\mathbb{H})$ and $L_{\sigma}^{+} \in \mathcal{L}(CP_{\sigma}^{+}\mathbb{H}, P_{\sigma}^{+}\mathbb{H})$ such that the observer-based feedback control defined by

$$v(t) = K_{\sigma}^{+} \hat{z}(t), \qquad (1.6)$$

16

where the infinite dimensional observer $\hat{z}$ solves

$$\left\{ \begin{array}{l} \hat{z}'(t) = A\hat{z}(t) + Bv(t) + L_{\sigma}^{+}Q_{\sigma}^{+}(C\hat{z}(t) - y(t)), \\ \hat{z}(0) = 0, \end{array} \right. \qquad (1.7)$$

ensures that for any $z^0 \in \mathbb{H}$, the solution $z$ of the closed loop (1.3)-(1.6)-(1.7), that is

$$\left\{ \begin{array}{l} z'(t) = Az(t) + BK_{\sigma}^{+}\hat{z}(t), \\ z(0) = z^0, \end{array} \right. $$

17

satisfies

$$\|z(t)\|_{\mathbb{H}} \le Me^{-\sigma t}\|z^0\|_{\mathbb{H}}, \quad \forall t > 0. \qquad (1.8)$$

To prove this result, we introduce the error $e := z - \hat{z}$ and we check that systems (1.3) and (1.7) yield

$$\begin{pmatrix} z \\ e \end{pmatrix}' = \begin{pmatrix} A + BK_{\sigma}^{+} & -BK_{\sigma}^{+} \\ 0 & A + L_{\sigma}^{+}Q_{\sigma}^{+}C \end{pmatrix} \begin{pmatrix} z \\ e \end{pmatrix}.$$

18

19

20

The result follows then by choosing the operators $K_{\sigma}^{+}$ and $L_{\sigma}^{+}$ in such a way that both $A+BK_{\sigma}^{+}$ and $A+L_{\sigma}^{+}Q_{\sigma}^{+}C$ generate analytic semigroups with decay rate less than $-\sigma$. This is achieved by solving two Riccati equations, using the method proposed by Badra-Takahashi [1] (for $K_{\sigma}^{+}$) combined to a duality argument (for $L_{\sigma}^{+}$).
In this work, we provide a generalization of the result obtained in [13] for a class of parabolic systems. First, in Section 2, we consider the case of an infinite dimensional observer (IDO). More precisely, we prove Theorem 1.1: we split the system (1.3) into two parts in the same spirit as [2, 3], one part corresponds to the unstable modes of A that defines a finite dimensional system and the other part is the infinite dimensional system corresponding to the stable modes of A. Using (H1.D), we construct a Luenberger observer in the spirit of [16]. We prove the stability of the closed loop system since the state, the observer and the error define all together a triangular system. In Section 3, we deal with the case of the finite dimensional observer (FDO) and we prove Theorem 1.2: here, A is supposed to be a selfadjoint operator with compact resolvent. Hence, the projection operator defined in (1.5) becomes orthogonal. The first step will be also to split the system and to construct a finite dimensional observer using (H2.D) in the spirit of [13]. The challenge here consists to prove the stability of the observer, the error and the state that define a strongly coupled system this time. To deal with this issue, we show that it involves a Volterra type equation that can be solved by using a fixed point argument, the mapping defines a contraction if the observer dimension is large enough.

It is worth mentioning that assumption (H1.D) (and its counterpart (H2.D) in the self-adjoint case) is the well-known Fattorini-Hautus criterion for exponential stabilization (see [8], [10] and [3]).

For every $\nu > 0$, we set

$$ \Sigma_{\nu}^{+} := \{\lambda_j \in \sigma(A) \mid \mathrm{Re} \, \lambda_j \ge -\nu\}, \quad \Sigma_{\nu}^{-} := \{\lambda_j \in \sigma(A) \mid \mathrm{Re} \, \lambda_j < -\nu\}, \quad (1.4) $$

where $\sigma(A)$ is the spectrum of A. Condition (H1.A) in the IDO case and (H2.A) in the FDO case imply that $\Sigma_{\nu}^{+}$ describes a finite set. We define the projection

$$ P_{\nu}^{+} = -\frac{1}{2i\pi} \int_{\Gamma_{\nu}^{+}} (\zeta \mathrm{Id} - A)^{-1} d\zeta, \quad P_{\nu}^{-} = \mathrm{Id} - P_{\nu}^{+}, \quad (1.5) $$

where $\Gamma_{\nu}^{+}$ is a curve enclosing $\Sigma_{\nu}^{+}$ but no other point of the spectrum of A and oriented counterclockwise (see [12, V.5, p.272]). We set

$$ z_{\nu}^{\pm} := P_{\nu}^{\pm} z, \quad \forall z \in \mathbb{H}. $$

We also introduce the finite dimensional operators

$$ A_{\nu}^{\pm} := AP_{\nu}^{\pm}, \quad B_{\nu}^{\pm} := P_{\nu}^{\pm} B, \quad C_{\nu}^{\pm} := C_{\nu}^{\pm}, $$

where $\iota_{\nu}^{\pm}$ is the embedding operator from $\mathbb{H}_{\nu}^{\pm} := P_{\nu}^{\pm} \mathbb{H}$ to $\mathbb{H}$. Finally, we denote by $Q_{\nu}^{+}$ the orthogonal projection from $\mathbb{Y}$ onto $\mathbb{Y}_{\nu}^{+} := CP_{\nu}^{+} \mathbb{H}$.

We are now in position to state our main results in the (IDO) and (FDO) cases.

## 1.1 Infinite-Dimensional Observer (IDO)

**Theorem 1.1.** Let $\sigma > 0$ be given and assume that assumptions (H1.A), (H1.B), (H1.C) and (H1.D) hold true. Then, there exist two operators $K_{\sigma}^{+} \in \mathcal{L}(\mathbb{H}, B^{*}(P_{\sigma}^{+})^{*}\mathbb{H})$ and $L_{\sigma}^{+} \in \mathcal{L}(CP_{\sigma}^{+}\mathbb{H}, P_{\sigma}^{+}\mathbb{H})$ such that the observer-based feedback control defined by

$$ v(t) = K_{\sigma}^{+} \hat{z}(t), \quad (1.6) $$

where the infinite dimensional observer $\hat{z}$ solves

$$ \begin{cases} \dot{\hat{z}}(t) = A\hat{z}(t) + Bv(t) + L_{\sigma}^{+}Q_{\sigma}^{+}(C\hat{z}(t) - y(t)), \\ \hat{z}(0) = 0, \end{cases} \quad (1.7) $$

ensures that for any $z^{0} \in \mathbb{H}$, the solution $z$ of the closed loop (1.3)-(1.6)-(1.7), that is

$$ \begin{cases} z'(t) = Az(t) + BK_{\sigma}^{+}\hat{z}(t), \\ z(0) = z^{0}, \end{cases} \quad (1.8) $$

satisfies

$$ \|z(t)\|_{\mathbb{H}} \le Me^{-\sigma t} \|z^{0}\|_{\mathbb{H}}, \quad \forall t > 0. \quad (1.8) $$
## 1.2 Finite-Dimensional Observer (FDO)

In this case, the stabilizing control is based on a finite dimensional observer of size $\sigma^* > 0$, where $\sigma^* > \sigma$ and

need to be chosen large enough.

**Theorem 1.2.** Let $\sigma > 0$ be given and assume that assumptions (H2.A), (H2.B), (H2.C) and (H2.D) hold true.

Let $K_{\sigma}^{+} \in L(\mathbb{H}, B^{*}P_{\sigma}^{+}\mathbb{H})$ and $L_{\sigma}^{+} \in \mathcal{L}(CP_{\sigma}^{+}\mathbb{H}, P_{\sigma}^{+}\mathbb{H})$ be the operators defined in Theorem 1.1. Then, there exists

$\sigma^* > \sigma$ such that the observer-based feedback control defined by

$$v(t) = K_{\sigma}^{+} \hat{z}_{\star}(t) \in B^{*} P_{\sigma}^{+} \mathbb{H} \subset B^{*} P_{\sigma^{*}}^{+} \mathbb{H}, \quad (1.9)$$

where the finite dimensional observer $\hat{z}_{\star} \in P_{\sigma^{*}}^{+}\mathbb{H}$ solves

$$\begin{cases} \hat{z}'_{\star}(t) = A_{\sigma^{*}}^{+} \hat{z}_{\star}(t) + B_{\sigma^{*}}^{+} v(t) + L_{\sigma}^{+} Q_{\sigma}^{+} (C_{\sigma^{*}}^{+} \hat{z}_{\star}(t) - y(t)), \\ \hat{z}_{\star}(0) = 0, \end{cases} \quad (1.10)$$

ensures that for any $z^0 \in \mathbb{H}$, the solution $z$ of the closed loop (1.3)-(1.9)-(1.10), that is

$$\begin{cases} z'(t) = Az(t) + BK_{\sigma}^{+} \hat{z}_{\star}(t), \\ z(0) = z^{0}, \end{cases}$$

satisfies

$$\|z(t)\|_{\mathbb{H}} \leq Me^{-\sigma t}\|z^0\|_{\mathbb{H}}, \quad \forall t > 0. \quad (1.11)$$

To prove this result, we proceed as follows. Introducing the auxiliary variables

$$e := z_{\sigma^{*}}^{+} - \hat{z}_{\star} \qquad \mathcal{X} = \begin{pmatrix} \hat{z}_{\star} \\ e \end{pmatrix},$$

we show that the equations satisfied by the state $z$ and the observer $\hat{z}_{\star}$ yield

$$\begin{cases} \mathcal{X}'(t) = A\mathcal{X}(t) + \mathbb{L}(z_{\sigma^{*}}^{-}(t)), \\ (z_{\sigma^{*}}^{-})'(t) = A_{\sigma^{*}}^{-} z_{\sigma^{*}}^{-}(t) + B_{\sigma^{*}}^{-} \mathbb{K}_{\sigma}^{+} \mathcal{X}(t), \end{cases} \quad (1.12)$$

where

$$A = \begin{pmatrix} A_{\sigma^{*}}^{+} + B_{\sigma^{*}}^{+} K_{\sigma}^{+} & -L_{\sigma}^{+} Q_{\sigma}^{+} C_{\sigma^{*}}^{+} \\ 0 & A_{\sigma^{*}}^{+} + L_{\sigma}^{+} Q_{\sigma}^{+} C_{\sigma^{*}}^{+} \end{pmatrix}, \quad \mathbb{L}(z_{\sigma^{*}}^{-}) = \begin{pmatrix} -L_{\sigma}^{+} Q_{\sigma}^{+} C z_{\sigma^{*}}^{-} \\ L_{\sigma}^{+} Q_{\sigma}^{+} C z_{\sigma^{*}}^{-} \end{pmatrix}, \quad \mathbb{K}_{\sigma}^{+} = (K_{\sigma}^{+}, 0).$$

We prove then that the matrix $\exp(tA)$ is exponentially stable with a decay rate less than $-\sigma$. Next, thanks to the first equation in (1.12), we use Duhamel's formula to express $\mathcal{X}$ in terms of $z_{\sigma^{*}}^{-}$. Plugging the obtained relation in the second equation of (1.12), we obtain an integral equation for $z_{\sigma^{*}}^{-}$. We use a fixed point argument to prove the well-posedness of this integral equation in the weighted space

$$L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-}) := \{f \in L^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-}) \text{ such that } e^{\sigma(\cdot)}f(\cdot) \in L^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-})\}.$$

This provides the expected result, that is the exponential decay of the controlled system with a decay rate less than $-\sigma$.

## 1.3 Related works

As already mentioned, the closest reference to our work is [11], in which the authors considered the case of a one dimensional heat equation. Their strategy is based on a (modal) splitting of the system into two parts: a finite dimensional unstable one and a stable infinite dimensional one. A Luenberger observer of large enough dimension is then constructed and the stability of the closed loop system is proved using a Lyapunov function.
To prove this result, we introduce the error $e := z - \hat{z}$ and we check that systems (1.3) and (1.7) yield

$$ \begin{pmatrix} z \\ e \end{pmatrix}' = \begin{pmatrix} A + BK_{\sigma}^{+} & -BK_{\sigma}^{+} \\ 0 & A + L_{\sigma}^{+}Q_{\sigma}^{+}C \end{pmatrix} \begin{pmatrix} z \\ e \end{pmatrix}. $$

The result follows then by choosing the operators $K_{\sigma}^{+}$ and $L_{\sigma}^{+}$ in such a way that both $A+BK_{\sigma}^{+}$ and $A+L_{\sigma}^{+}Q_{\sigma}^{+}C$ generate analytic semigroups with decay rate less than $-\sigma$. This is achieved by solving two Riccati equations, using the method proposed by Badra-Takahashi [2] (for $K_{\sigma}^{+}$) combined to a duality argument (for $L_{\sigma}^{+}$).

## 1.2 Finite-Dimensional Observer (FDO)

In this case, the stabilizing control is based on a finite dimensional observer of size $\sigma^* > 0$, where $\sigma^* > \sigma$ and need to be chosen large enough (see (3.19)).

**Theorem 1.2.** Let $\sigma > 0$ be given and assume that assumptions (H2.A), (H2.B), (H2.C) and (H2.D) hold true. Let $K_{\sigma}^{+} \in \mathcal{L}(\mathbb{H}, B^{*}P_{\sigma}^{+}\mathbb{H})$ and $L_{\sigma}^{+} \in \mathcal{L}(CP_{\sigma}^{+}\mathbb{H}, P_{\sigma}^{+}\mathbb{H})$ be the operators defined in Theorem 1.1. Then, there exists $\sigma^* > \sigma$ such that the observer-based feedback control defined by

$$ v(t) = K_{\sigma}^{+} \hat{z}_{\star}(t) \in B^{*} P_{\sigma}^{+} \mathbb{H} \subset B^{*} P_{\sigma}^{+} \mathbb{H}, \quad (1.9) $$

where the finite dimensional observer $\hat{z}_{\star} \in P_{\sigma}^{+}\mathbb{H}$ solves

$$ \begin{cases} \hat{z}'(t) = A_{\sigma^*}^{+} \hat{z}_{\star}(t) + B_{\sigma^*}^{+} v(t) + L_{\sigma}^{+} Q_{\sigma}^{+} (C_{\sigma^*}^{+} \hat{z}_{\star}(t) - y(t)), \\ \hat{z}_{\star}(0) = 0, \end{cases} \quad (1.10) $$

ensures that for any $z^0 \in \mathbb{H}$, the solution $z$ of the closed loop (1.3)-(1.9)-(1.10), that is

$$ \begin{cases} z'(t) = Az(t) + BK_{\sigma}^{+} \hat{z}_{\star}(t), \\ z(0) = z^0, \end{cases} $$

satisfies

$$ \|z(t)\|_{\mathbb{H}} \leq Me^{-\sigma t} \|z^0\|_{\mathbb{H}}, \quad \forall t > 0. \quad (1.11) $$

To prove this result, we proceed as follows. Introducing the auxiliary variables

$$ e := z_{\sigma^*}^{+} - \hat{z}_{\star}, \qquad \mathbb{X} = \begin{pmatrix} \hat{z}_{\star} \\ e \end{pmatrix}, $$

we show that the equations satisfied by the state $z$ and the observer $\hat{z}_{\star}$ yield

$$ \begin{cases} \mathbb{X}'(t) = A\mathbb{X}(t) + \mathbb{L}(z_{\sigma^*}^{-}(t)), \\ (z_{\sigma^*}^{-})'(t) = A_{\sigma^*}^{-} z_{\sigma^*}^{-}(t) + B_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} \mathbb{X}(t), \end{cases} \quad (1.12) $$

where

$$ A = \begin{pmatrix} A_{\sigma^*}^{+} + B_{\sigma^*}^{+} K_{\sigma}^{+} & -L_{\sigma}^{+} Q_{\sigma}^{+} C_{\sigma^*}^{+} \\ 0 & A_{\sigma^*}^{+} + L_{\sigma}^{+} Q_{\sigma}^{+} C_{\sigma^*}^{+} \end{pmatrix}, \quad \mathbb{L}(z_{\sigma^*}^{-}) = \begin{pmatrix} -L_{\sigma}^{+} Q_{\sigma}^{+} C z_{\sigma^*}^{-} \\ L_{\sigma}^{+} Q_{\sigma}^{+} C z_{\sigma^*}^{-} \end{pmatrix}, \quad \mathbb{K}_{\sigma}^{+} = (K_{\sigma}^{+}, 0). $$

We prove then that the matrix $\exp(tA)$ is exponentially stable with a decay rate less than $-\sigma$. Next, thanks to the first equation in (1.12), we use Duhamel's formula to express $\mathbb{X}$ in terms of $z_{\sigma^*}^{-}$. Plugging the obtained relation in the second equation of (1.12), we obtain an integral equation for $z_{\sigma^*}^{-}$. We use a fixed point argument to prove the well-posedness of this integral equation in the weighted space

$$ L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-}) := \{f \in L^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-}) \text{ such that } e^{\sigma(t)} f(t) \in L^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})\}. $$

This provides the expected result, that is the exponential decay of the controlled system with a decay rate less than $-\sigma$.
Contrarily to the proof proposed here, the arguments used in [11] are valid only in dimension one and heavily rely on the type of the considered equation. Let us also mention that in [12], the authors used a similar approach to prove the stabilization of a one dimension convection diffusion equation in the case of a boundary control. The use of modal splitting for the stabilization of infinite dimensional systems has also been achieved in some specific settings, like Burgers equations [20, 4, 19], Navier-Stokes system [1, 2, 7, 16, 18], semi-linear wave equation [5] and populations dynamics [14, 15].

## 1.4 Outline

In Section 2, we prove Theorem 1.1, which provides the stabilizing observer-based feedback-control through an infinite dimensional observer. In Section 3, we construct a finite dimensional observer to design a similar feedback-control. For this case, we need to assume that the operator A is self-adjoint and the control and observation operators are bounded. Finally, in Section 4, these abstract results are applied to obtain a stabilizing control for a reaction-diffusion system.

# 2 Infinite dimensional observer

## 2.1 Spectral decomposition of the system

In this section, we suppose that assumptions (H1.A), (H1.B), (H1.C) and (H1.D) hold true. We consider below a classical modal decomposition (it has been used, for instance, in [1, 2, 7, 18]) that we recall it for the sake of completeness. Let $\sigma > 0$. We first separate the spectrum of A into “unstable” and “stable” modes using the projection $P_{\sigma}^{+}$ defined in (1.5). We set

$$ \mathbb{H}_{\sigma}^{+} = P_{\sigma}^{+} \mathbb{H}, \quad \mathbb{H}_{\sigma}^{-} = (\mathrm{Id} - P_{\sigma}^{+}) \mathbb{H}, \quad \mathbb{H} = \mathbb{H}_{\sigma}^{+} \oplus \mathbb{H}_{\sigma}^{-}. $$

According to this projection, we set

$$ A_{\sigma}^{+} := A_{|\mathbb{H}_{\sigma}^{+}} : \mathbb{H}_{\sigma}^{+} \to \mathbb{H}_{\sigma}^{+}, \quad A_{\sigma}^{-} := A_{|\mathbb{H}_{\sigma}^{-}} : \mathcal{D}(A) \cap \mathbb{H}_{\sigma}^{-} \to \mathbb{H}_{\sigma}^{-}. $$

Then the spectrum of $A_{\sigma}^{+}$ is exactly $\Sigma_{\sigma}^{+}$ and the spectrum of $A_{\sigma}^{-}$ is exactly $\Sigma_{\sigma}^{-}$ where $\Sigma_{\sigma}^{+}$ and $\Sigma_{\sigma}^{-}$ are defined in (1.4). We denote by $A^{*}$ the adjoint operator of A and we define similarly the projection $(P_{\sigma}^{+})^{*}$ such that

$$ (P_{\sigma}^{+})^{*} = -\frac{1}{2i\pi} \int_{\Gamma_{\sigma}^{+}} (\zeta \mathrm{Id} - A^{*})^{-1} d\zeta. \qquad (2.1) $$

The projection (2.1) provides also the following spaces

$$ (\mathbb{H}_{\sigma}^{+})^{*} = (P_{\sigma}^{+})^{*} \mathbb{H}, \quad (\mathbb{H}_{\sigma}^{-})^{*} = (\mathrm{Id} - (P_{\sigma}^{+})^{*}) \mathbb{H}, \quad \mathbb{H} = (\mathbb{H}_{\sigma}^{+})^{*} \oplus (\mathbb{H}_{\sigma}^{-})^{*}, \qquad (2.2) $$

with

$$ (A_{\sigma}^{+})^{*} := A_{(\mathbb{H}_{\sigma}^{+})^{*}}^{*} : (\mathbb{H}_{\sigma}^{+})^{*} \to (\mathbb{H}_{\sigma}^{+})^{*}, \quad (A_{\sigma}^{-})^{*} := A_{(\mathbb{H}_{\sigma}^{-})^{*}} : \mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*} \to (\mathbb{H}_{\sigma}^{-})^{*}. $$

**Lemma 2.1.** There exist $\varepsilon > 0$ and $M > 0$ such that for any $\delta \ge 0$, $t > 0$

$$ \begin{aligned} \|e^{A_{\sigma}^{-}t}\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\le Me^{-(\sigma+\varepsilon)t}, & \| (\mu_0 \mathrm{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-}t} \|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\le \frac{M}{t^{\delta}} e^{-(\sigma+\varepsilon)t}, \\ \|e^{(A_{\sigma}^{-})^{*}t}\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\le Me^{-(\sigma+\varepsilon)t}, & \| (\mu_0 \mathrm{Id} - (A_{\sigma}^{-})^{*})^{\delta} e^{(A_{\sigma}^{-})^{*}t} \|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\le \frac{M}{t^{\delta}} e^{-(\sigma+\varepsilon)t}. \end{aligned} \qquad (2.3) $$

*Proof.* We detail the proof only for the operator $A_{\sigma}^{-}$, as the arguments for its adjoint are similar. The first inequality is obvious. Concerning the second one, we first note that

$$ (\mu_0 \mathrm{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-}t} = (\mu_0 \mathrm{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} (A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-}t}. $$
## 1.3 Related works

The output feedback stabilization of linear and non linear time-invariant systems has been extensively studied in the last decades. For finite dimensional systems, we refer the interested reader to the survey [21] where necessary and sufficient conditions are given. In the infinite dimensional case, sufficient conditions can be found for instance [7, Chapter 5] using Luenberger observers for the design of compensators. In particular, for bounded observation and control operators, such sufficient conditions are the exponential stabilizability and exponential detectability of the system (A, B, C). A complete list of references and new necessary and sufficient conditions can be found in the recent work [25]. As already mentioned, the closest reference to our work is [13], in which the authors considered the case of a one dimensional heat equation. Their strategy is based on a (modal) splitting of the system into two parts: a finite dimensional unstable one and a stable infinite dimensional one. A Luenberger observer of large enough dimension is then constructed and the stability of the closed loop system is proved using a Lyapunov function. Contrarily to the proof proposed here, the arguments used in [13] are valid only in dimension one and heavily rely on the type of the considered equation. Let us also mention that in [14], the authors used a similar approach to prove the stabilization of a one dimension convection diffusion equation in the case of a boundary control. The use of modal splitting for the stabilization of infinite dimensional systems has also been achieved in some specific settings, like Burgers equations [23, 5, 22], Navier-Stokes system [2, 3, 9, 18, 20], semi-linear wave equation [6] and population dynamics [16, 17].

## 1.4 Outline

In Section 2, we prove Theorem 1.1, which provides the stabilizing observer-based feedback-control through an infinite dimensional observer. In Section 3, we construct a finite dimensional observer to design a similar feedback-control. For this case, we need to assume that the operator A is self-adjoint and the control and observation operators are bounded. Finally, in Section 4 and in Section 5, these abstract results are applied to obtain a stabilizing control for reaction-diffusion systems. The paper ends with a brief concluding section.

# 2 Infinite dimensional observer

## 2.1 Spectral decomposition of the system

In this section, we suppose that assumptions (H1.A), (H1.B), (H1.C) and (H1.D) hold true. We consider below a classical modal decomposition (it has been used, for instance, in [2, 3, 9, 20]) that we recall it for the sake of completeness. Let $\sigma > 0$. We first separate the spectrum of A into “unstable” and “stable” modes using the projection $P_{\sigma}^{+}$ defined in (1.5). We set

$$ \mathbb{H}_{\sigma}^{+} = P_{\sigma}^{+} \mathbb{H}, \quad \mathbb{H}_{\sigma}^{-} = (\mathrm{Id} - P_{\sigma}^{+}) \mathbb{H}, \quad \mathbb{H} = \mathbb{H}_{\sigma}^{+} \oplus \mathbb{H}_{\sigma}^{-}. $$

According to this projection, we set

$$ A_{\sigma}^{+} := A_{|\mathbb{H}_{\sigma}^{+}} : \mathbb{H}_{\sigma}^{+} \to \mathbb{H}_{\sigma}^{+}, \quad A_{\sigma}^{-} := A_{|\mathbb{H}_{\sigma}^{-}} : \mathcal{D}(A) \cap \mathbb{H}_{\sigma}^{-} \to \mathbb{H}_{\sigma}^{-}. $$

Then the spectrum of $A_{\sigma}^{+}$ is exactly $\Sigma_{\sigma}^{+}$ and the spectrum of $A_{\sigma}^{-}$ is exactly $\Sigma_{\sigma}^{-}$ where $\Sigma_{\sigma}^{+}$ and $\Sigma_{\sigma}^{-}$ are defined in (1.4). We denote by $A^{*}$ the adjoint operator of A and we define similarly the projection $(P_{\sigma}^{+})^{*}$ such that

$$ (P_{\sigma}^{+})^{*} = -\frac{1}{2i\pi} \int_{\Gamma_{\sigma}^{+}} (\zeta \mathrm{Id} - A^{*})^{-1} d\zeta. \qquad (2.1) $$

The projection (2.1) provides also the following spaces

$$ (\mathbb{H}_{\sigma}^{+})^{*} = (P_{\sigma}^{+})^{*} \mathbb{H}, \quad (\mathbb{H}_{\sigma}^{-})^{*} = (\mathrm{Id} - (P_{\sigma}^{+})^{*}) \mathbb{H}, \quad \mathbb{H} = (\mathbb{H}_{\sigma}^{+})^{*} \oplus (\mathbb{H}_{\sigma}^{-})^{*}, \qquad (2.2) $$

with

$$ (A_{\sigma}^{+})^{*} := A_{|(\mathbb{H}_{\sigma}^{+})^{*}}^{*} : (\mathbb{H}_{\sigma}^{+})^{*} \to (\mathbb{H}_{\sigma}^{+})^{*}, \quad (A_{\sigma}^{-})^{*} := A_{|(\mathbb{H}_{\sigma}^{-})^{*}} : \mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*} \to (\mathbb{H}_{\sigma}^{-})^{*}. $$
Applying [13, Corollary 6.11] with $B = (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta}$, $A = A_{\sigma}^{-}$ and $x = (A_{\sigma}^{-})^{-\delta} y$, for $y \in \mathbb{H}_{\sigma}^{-}$, we obtain that for some constant positive $C$,

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} y \right\|_{\mathbb{H}} \le C \|y\|_{\mathbb{H}}, \quad \forall y \in \mathbb{H}_{\sigma}^{-}, $$

and thus

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \le C. $$

Consequently,

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-}t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \le \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \left\| (A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-}t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})}, $$

1 and the desired estimate follows then immediately from [13, Theorem 6.13]. □

We also define

$$ U_{\sigma}^{+} := B^{*}(\mathbb{H}_{\sigma}^{+})^{*}, \quad U_{\sigma}^{-} := B^{*}(\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}), $$

and

$$ p_{\sigma}^{+} : \mathbb{U} \to \mathbb{U}_{\sigma}^{+}, \quad p_{\sigma}^{-} : \mathbb{U} \to \mathbb{U}_{\sigma}^{-}, \quad i_{\sigma}^{+} : \mathbb{U}_{\sigma}^{+} \to \mathbb{U}, \quad i_{\sigma}^{-} : \mathbb{U}_{\sigma}^{-} \to \mathbb{U}, $$

2 the orthogonal projections and the inclusion maps. Note that we have the following relations for the above

3 maps:

$$ i_{\sigma}^{+} = (p_{\sigma}^{+})^{*}, \quad i_{\sigma}^{-} = (p_{\sigma}^{-})^{*}. \qquad (2.4) $$

From [18], we can extend $P_{\sigma}^{+}$ and $(\operatorname{Id} - P_{\sigma}^{+})$ as bounded operators

$$ P_{\sigma}^{+} \in \mathcal{L}(\mathcal{D}(A^{*})', \mathbb{H}_{\sigma}^{+}), \quad (\operatorname{Id} - P_{\sigma}^{+}) \in \mathcal{L}(\mathcal{D}(A^{*})', [\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}]'). $$

We can thus define

$$ B_{\sigma}^{+} := P_{\sigma}^{+} B i_{\sigma}^{+} \in \mathcal{L}(\mathbb{U}_{\sigma}^{+}, \mathbb{H}_{\sigma}^{+}), \quad B_{\sigma}^{-} := (\operatorname{Id} - P_{\sigma}^{+}) B i_{\sigma}^{-} \in \mathcal{L}(\mathbb{U}_{\sigma}^{-}, [\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}]'). $$

We show as in [1, 2, 18] that

$$ P_{\sigma}^{+} B = B_{\sigma}^{+} p_{\sigma}^{+}, \quad (\operatorname{Id} - P_{\sigma}^{+}) B = B_{\sigma}^{-} p_{\sigma}^{-}. $$

4 Using the projections $P_{\sigma}^{+}$ and $\operatorname{Id} - P_{\sigma}^{+}$, system (1.3) can be split into the two sub-systems (see [1, 2, 18]).

$$ (z_{\sigma}^{+})'(t) = A_{\sigma}^{+} z_{\sigma}^{+}(t) + B_{\sigma}^{+} p_{\sigma}^{+} v(t), \quad z_{\sigma}^{+}(0) = P_{\sigma}^{+} z^{0}, \qquad (2.5) $$

$$ (z_{\sigma}^{-})'(t) = A_{\sigma}^{-} z_{\sigma}^{-}(t) + B_{\sigma}^{-} p_{\sigma}^{-} v(t), \quad z_{\sigma}^{-}(0) = (\operatorname{Id} - P_{\sigma}^{+}) z^{0}. \qquad (2.6) $$

We also introduce the orthogonal projections $Q_{\sigma}^{+}$ from $\mathbb{Y}$ into $\mathbb{Y}_{\sigma}^{+} = C\mathbb{H}_{\sigma}^{+}$ and we define

$$ C_{\sigma}^{+} = C \iota_{\sigma}^{+}, $$

6 where $\iota_{\sigma}^{+}$ designates the injection operator from $\mathbb{H}_{\sigma}^{+}$ to $\mathbb{H}$. We are now in position to prove Theorem 1.1.

## 2.2 Construction of an infinite dimensional observer based control

8 Let us consider first the system

$$ \begin{cases} \dot{\tilde{z}}'(t) = A\tilde{z}(t) + Bu(t), \\ \tilde{z}(0) = \tilde{z}^0. \end{cases} \qquad (2.7) $$

We want to construct a finite dimensional vector $u$ such that the system (2.7) is exponentially stable. Let $N_{\sigma} \in \mathbb{N}^{*}$ and $(w_j)_{1 \le j \le N_{\sigma}} \subset \mathbb{U}$, and let us suppose that the control $u(t)$ is of the form

$$ u(t) = \sum_{j=1}^{N_{\sigma}} u_j(t)w_j, $$
**Lemma 2.1.** There exist $\varepsilon > 0$ and $M > 0$ such that for any $\delta \ge 0$, $t > 0$

$$ \begin{aligned} \left\| e^{A_{\sigma}^{-} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\leq M e^{-(\sigma+\varepsilon)t}, & \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\leq \frac{M}{t^{\delta}} e^{-(\sigma+\varepsilon)t}, \\ \left\| e^{(A_{\sigma}^{-})^{*} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\leq M e^{-(\sigma+\varepsilon)t}, & \left\| (\mu_0 \operatorname{Id} - (A_{\sigma}^{-})^{*})^{\delta} e^{(A_{\sigma}^{-})^{*} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} &\leq \frac{M}{t^{\delta}} e^{-(\sigma+\varepsilon)t}. \end{aligned} \quad (2.3) $$

*Proof.* We detail the proof only for the operator $A_{\sigma}^{-}$, as the arguments for its adjoint are similar. The first inequality is obvious. Concerning the second one, we first note that

$$ (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-} t} = (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} (A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-} t}. $$

Applying [15, Corollary 6.11] with $B = (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta}$, $A = A_{\sigma}^{-}$ and $x = (A_{\sigma}^{-})^{-\delta} y$, for $y \in \mathbb{H}_{\sigma}^{-}$, we obtain that for some constant positive $C$,

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} y \right\|_{\mathbb{H}} \le C \|y\|_{\mathbb{H}}, \quad \forall y \in \mathbb{H}_{\sigma}^{-}, $$

and thus

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \le C. $$

Consequently,

$$ \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \le \left\| (\mu_0 \operatorname{Id} - A_{\sigma}^{-})^{\delta} (A_{\sigma}^{-})^{-\delta} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})} \left\| (A_{\sigma}^{-})^{\delta} e^{A_{\sigma}^{-} t} \right\|_{\mathcal{L}(\mathbb{H}_{\sigma}^{-})}, $$

and the desired estimate follows then immediately from [15, Theorem 6.13]. □

We also define

$$ U_{\sigma}^{+} := B^{*}(\mathbb{H}_{\sigma}^{+})^{*}, \quad U_{\sigma}^{-} := B^{*}(\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}), $$

and

$$ p_{\sigma}^{+} : U \to U_{\sigma}^{+}, \quad p_{\sigma}^{-} : U \to U_{\sigma}^{-}, \quad i_{\sigma}^{+} : U_{\sigma}^{+} \to U, \quad i_{\sigma}^{-} : U_{\sigma}^{-} \to U, $$

the orthogonal projections and the inclusion maps. Note that we have the following relations for the above maps:

$$ i_{\sigma}^{+} = (p_{\sigma}^{+})^{*}, \quad i_{\sigma}^{-} = (p_{\sigma}^{-})^{*}. \quad (2.4) $$

From [20], we can extend $P_{\sigma}^{+}$ and $(\operatorname{Id} - P_{\sigma}^{+})$ as bounded operators

$$ P_{\sigma}^{+} \in \mathcal{L}(\mathcal{D}(A^{*})', \mathbb{H}_{\sigma}^{+}), \quad (\operatorname{Id} - P_{\sigma}^{+}) \in \mathcal{L}(\mathcal{D}(A^{*})', [\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}]')' $$

We can thus define

$$ B_{\sigma}^{+} := P_{\sigma}^{+} B i_{\sigma}^{+} \in \mathcal{L}(U_{\sigma}^{+}, \mathbb{H}_{\sigma}^{+}), \quad B_{\sigma}^{-} := (\operatorname{Id} - P_{\sigma}^{+}) B i_{\sigma}^{-} \in \mathcal{L}(U_{\sigma}^{-}, [\mathcal{D}(A^{*}) \cap (\mathbb{H}_{\sigma}^{-})^{*}]')' $$

We show as in [2, 3, 20] that

$$ P_{\sigma}^{+} B = B_{\sigma}^{+} p_{\sigma}^{+}, \quad (\operatorname{Id} - P_{\sigma}^{+}) B = B_{\sigma}^{-} p_{\sigma}^{-}. $$

Using the projections $P_{\sigma}^{+}$ and $\operatorname{Id} - P_{\sigma}^{+}$, system (1.3) can be split into the two sub-systems (see [2, 3, 20]):

$$ (z_{\sigma}^{+})'(t) = A_{\sigma}^{+} z_{\sigma}^{+}(t) + B_{\sigma}^{+} p_{\sigma}^{+} v(t), \quad z_{\sigma}^{+}(0) = P_{\sigma}^{+} z^{0}, \quad (2.5) $$

$$ (z_{\sigma}^{-})'(t) = A_{\sigma}^{-} z_{\sigma}^{-}(t) + B_{\sigma}^{-} p_{\sigma}^{-} v(t), \quad z_{\sigma}^{-}(0) = (\operatorname{Id} - P_{\sigma}^{+}) z^{0}. \quad (2.6) $$

We also introduce the orthogonal projections $Q_{\sigma}^{+}$ from $\mathbb{Y}$ into $\mathbb{Y}_{\sigma}^{+} = C\mathbb{H}_{\sigma}^{+}$ and we define

$$ C_{\sigma}^{+} = C_{\iota_{\sigma}^{+}}, $$

where $\iota_{\sigma}^{+}$ designates the injection operator from $\mathbb{H}_{\sigma}^{+}$ to $\mathbb{H}$. We are now in position to prove Theorem 1.1.
where $u_j(t) \in \mathbb{C}$, for $1 \le j \le N_\sigma$ and $t \ge 0$. It is natural to introduce the mapping

$$ \mathfrak{B}: \mathbb{C}^{N_{\sigma}} \to \mathcal{D}(A^{*})', \quad \mathbf{\Theta} = (\theta_1, \dots, \theta_{N_{\sigma}}) \mapsto \mathfrak{B}\mathbf{\Theta} = \sum_{j=1}^{N_{\sigma}} \theta_j B w_j, $$

in such a way that setting

$$ \mathbf{u}(t) := (u_1(t), \dots, u_{N_{\sigma}}(t)), $$

1. system (2.7) is equivalent to

$$ \begin{cases} \tilde{z}'(t) = A\tilde{z}(t) + \mathfrak{B}\mathbf{u}(t), \\ \tilde{z}(0) = \tilde{z}^0. \end{cases} \tag{2.8} $$

It is worth noticing that the adjoint $\mathfrak{B}^* \in \mathcal{L}(\mathcal{D}(A^*), \mathbb{C}^{N_\sigma})$ is given by

$$ \mathfrak{B}^* \psi = (\langle w_1, B^* \psi \rangle_{\mathbb{U}}, \dots, \langle w_{N_{\sigma}}, B^* \psi \rangle_{\mathbb{U}}). $$

2. Using the projection $P_\sigma^+$, we get that (2.8) is equivalent to

$$ \begin{cases} (\tilde{z}_{\sigma}^{+})'(t) = A_{\sigma}^{+} \tilde{z}_{\sigma}^{+}(t) + \mathfrak{B}_{\sigma}^{+} \mathbf{u}(t), & \tilde{z}_{\sigma}^{+}(0) = P_{\sigma}^{+} \tilde{z}^{0}, & \mathfrak{B}_{\sigma}^{+} = P_{\sigma}^{+} \mathfrak{B}, \\ (\tilde{z}_{\sigma}^{-})'(t) = A_{\sigma}^{-} \tilde{z}_{\sigma}^{-}(t) + \mathfrak{B}_{\sigma}^{-} \mathbf{u}(t), & \tilde{z}_{\sigma}^{-}(0) = P_{\sigma}^{-} \tilde{z}^{0}, & \mathfrak{B}_{\sigma}^{-} = P_{\sigma}^{-} \mathfrak{B}, \end{cases} \tag{2.9} $$

3. where $\tilde{z}_\sigma^\pm = P_\sigma^\pm \tilde{z}$. We need to show that the finite dimensional part $(2.9)_1$ is exactly controllable. Let

$$ N_{\sigma} \ge \max_{\mathrm{Re} \, \lambda_j \ge -\sigma} \ell_j, \tag{2.10} $$

4. where $\ell_j$ is the geometric multiplicity of the eigenvalue $\lambda_j$ of the operator $A$. From [1, Theorem 5] and the first condition in (H1.D), there exists a family $(w_j)_{1 \le j \le N_\sigma} \subset \mathbb{U}_\sigma^+ \subset \mathbb{U}$ such that $(2.9)_1$ is exactly controllable.

5. Moreover, it is proved that $\mathbf{u}$ is expressed by means of a linear feedback operator

$$ u = K_{\sigma}^{+} \tilde{z}, \quad K_{\sigma}^{+}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_j, B^{*} \Pi P_{\sigma}^{+}(\cdot) \rangle_{\mathbb{U}} w_j = - \sum_{j=1}^{N_{\sigma}} (\mathfrak{B}^{*}(\Pi P_{\sigma}^{+}(\cdot)))_j w_j, \tag{2.11} $$

7. where $\Pi \in \mathcal{L}(\mathbb{H}_\sigma^+, (\mathbb{H}_\sigma^+)^*)$ is the unique solution of the algebraic Riccati equation: for all $\xi, \zeta \in \mathbb{H}_\sigma^+$

$$ \left\{ \begin{array}{l} \langle \xi, \zeta \rangle_{\mathbb{H}} + \langle (A_{\sigma}^{+} + \sigma \mathrm{Id})\xi, \Pi \zeta \rangle_{\mathbb{H}} + \langle \Pi \xi, (A_{\sigma}^{+} + \sigma \mathrm{Id})\zeta \rangle_{\mathbb{H}} - \sum_{j=1}^{N_{\sigma}} \langle B^{*} \Pi \xi, w_j \rangle_{\mathbb{U}} \langle B^{*} \Pi \zeta, w_j \rangle_{\mathbb{U}} = 0, \\ \langle \Pi \xi, \zeta \rangle_{\mathbb{H}} = \langle \xi, \Pi \zeta \rangle_{\mathbb{H}}, \quad \text{and} \quad \forall \xi \ne 0, \langle \Pi \xi, \xi \rangle_{\mathbb{H}} > 0. \end{array} \right. \tag{2.12} $$

This choice ensures that the solution of the finite dimensional system (2.9)$_1$

$$ (\tilde{z}_{\sigma}^{+})'(t) = A_{\sigma}^{+} \tilde{z}_{\sigma}^{+}(t) - \mathfrak{B}_{\sigma}^{+} (\mathfrak{B}^{*} (\Pi P_{\sigma}^{+} \tilde{z}(t))) $$

is exponentially stable i.e.

$$ \| \tilde{z}_{\sigma}^{+}(t) \|_{\mathbb{H}} \le Me^{-(\sigma+\epsilon)t} \| \tilde{z}^{0} \|_{\mathbb{H}}, \quad t > 0. $$

8. It follows from Duhamel's formula that the whole system (2.9) is exponentially stable (see [1]). We can construct

9. $L_\sigma^+$ similarly considering the system

$$ \left\{ \begin{aligned} \tilde{z}_{\star}'(t) &= A^{*} \tilde{z}_{\star}(t) + C^{*} u_{\star}(t), \\ \tilde{z}_{\star}(0) &= \tilde{z}_{\star}^{0}. \end{aligned} \right. \tag{2.13} $$

Using similar arguments and the second condition in (H1.D), we show that there exists a family $(w_j^*)_{1 \le j \le N_\sigma} \subset \mathbb{Y}_\sigma^+$ such that

$$ u_{\star} = L_{\star} \tilde{z}_{\star}, \quad L_{\star}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_j^*, C \Pi_{\star} (P_{\sigma}^{+})^*(\cdot) \rangle_{\mathbb{Y}} w_j^* = - \sum_{j=1}^{N_{\sigma}} (\mathfrak{C}_{\star}^*(\Pi_{\star}(P_{\sigma}^{+})^*(\cdot)))_j w_j^*, $$
## 2.2 Construction of an infinite dimensional observer-based control

Let us consider first the system

$$ egin{cases} \dot{\tilde{z}}(t) = A\tilde{z}(t) + Bu(t), \\ \tilde{z}(0) = \tilde{z}^0. \end{cases} \qquad (2.7) $$

We want to construct a finite dimensional vector $u$ such that the system (2.7) is exponentially stable. Let $N_{\sigma} \in \mathbb{N}^*$ and $(w_j)_{1 \le j \le N_{\sigma}} \subset \mathbb{U}$, and let us suppose that the control $u(t)$ is of the form

$$ u(t) = \sum_{j=1}^{N_{\sigma}} u_j(t)w_j, $$

where $u_j(t) \in \mathbb{C}$, for $1 \le j \le N_{\sigma}$ and $t \ge 0$. It is natural to introduce the mapping

$$ \mathfrak{B}: \mathbb{C}^{N_{\sigma}} \to \mathcal{D}(A^*)', \quad \Theta = (\theta_1, \dots, \theta_{N_{\sigma}}) \mapsto \mathfrak{B}\Theta = \sum_{j=1}^{N_{\sigma}} \theta_j Bw_j, $$

in such a way that setting

$$ u(t) := (u_1(t), \dots, u_{N_{\sigma}}(t)), $$

system (2.7) is equivalent to

$$ egin{cases} \dot{\tilde{z}}(t) = A\tilde{z}(t) + \mathfrak{B}u(t), \\ \tilde{z}(0) = \tilde{z}^0. \end{cases} \qquad (2.8) $$

It is worth noticing that the adjoint $\mathfrak{B}^* \in \mathcal{L}(\mathcal{D}(A^*), \mathbb{C}^{N_{\sigma}})$ is given by

$$ \mathfrak{B}^* \psi = (\langle w_1, B^* \psi \rangle_{\mathbb{U}}, \dots, \langle w_{N_{\sigma}}, B^* \psi \rangle_{\mathbb{U}}). $$

Using the projection $P_{\sigma}^{+}$, we get that (2.8) is equivalent to

$$ egin{cases} (\tilde{z}_{\sigma}^{+})'(t) = A_{\sigma}^{+} \tilde{z}_{\sigma}^{+}(t) + \mathfrak{B}_{\sigma}^{+} u(t), & \tilde{z}_{\sigma}^{+}(0) = P_{\sigma}^{+} \tilde{z}^{0}, & \mathfrak{B}_{\sigma}^{+} = P_{\sigma}^{+} \mathfrak{B}, \\ (\tilde{z}_{\sigma}^{-})'(t) = A_{\sigma}^{-} \tilde{z}_{\sigma}^{-}(t) + \mathfrak{B}_{\sigma}^{-} u(t), & \tilde{z}_{\sigma}^{-}(0) = P_{\sigma}^{-} \tilde{z}^{0}, & \mathfrak{B}_{\sigma}^{-} = P_{\sigma}^{-} \mathfrak{B}, \end{cases} \qquad (2.9) $$

where $\tilde{z}_{\sigma}^{\pm} = P_{\sigma}^{\pm} \tilde{z}$. We need to show that the finite dimensional part (2.9)$_1$ is exactly controllable. Let

$$ N_{\sigma} \ge \max_{\text{Re } \lambda_j \ge -\sigma} \ell_j, \qquad (2.10) $$

where $\ell_j$ is the geometric multiplicity of the eigenvalue $\lambda_j$ of the operator $A$. From [2, Theorem 5] and the first condition in (H1.D), there exists a family $(w_j)_{1 \le j \le N_{\sigma}} \subset \mathbb{U}_{\sigma}^{+} \subset \mathbb{U}$ such that (2.9)$_1$ is exactly controllable. Moreover, it is proved that $u$ is expressed by means of a linear feedback operator

$$ u = K_{\sigma}^{+} \tilde{z}, \quad K_{\sigma}^{+}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_j, B^{*} \Pi P_{\sigma}^{+}(\cdot) \rangle_{\mathbb{U}} w_j = - \sum_{j=1}^{N_{\sigma}} (\mathfrak{B}^{*}(\Pi P_{\sigma}^{+}(\cdot)))_j w_j, \qquad (2.11) $$

where $\Pi \in \mathcal{L}(\mathbb{H}_{\sigma}^{+}, (\mathbb{H}_{\sigma}^{+})^{*})$ is the unique solution of the algebraic Riccati equation: for all $\xi, \zeta \in \mathbb{H}_{\sigma}^{+}$

$$ egin{cases} \langle \xi, \zeta \rangle_{\mathbb{H}} + \langle (A_{\sigma}^{+} + \sigma \text{Id})\xi, \Pi\zeta \rangle_{\mathbb{H}} + \langle \Pi\xi, (A_{\sigma}^{+} + \sigma \text{Id})\zeta \rangle_{\mathbb{H}} - \sum_{j=1}^{N_{\sigma}} \langle B^{*}\Pi\xi, w_j \rangle_{\mathbb{U}} \langle B^{*}\Pi\zeta, w_j \rangle_{\mathbb{U}} = 0, \\ \langle \Pi\xi, \zeta \rangle_{\mathbb{H}} = \langle \xi, \Pi\zeta \rangle_{\mathbb{H}}, \quad \text{and} \quad \forall \xi \ne 0, \langle \Pi\xi, \xi \rangle_{\mathbb{H}} > 0. \end{cases} \qquad (2.12) $$

It is well-known (see for instance [7, page 292]) that this Riccati operator is strongly related to the standard LQR problem with infinite horizon. In particular, this choice ensures that the solution of the finite dimensional system (2.9)$_1$

$$ (\tilde{z}_{\sigma}^{+})'(t) = A_{\sigma}^{+} \tilde{z}_{\sigma}^{+}(t) - \mathfrak{B}_{\sigma}^{+}(\mathfrak{B}^{*}(\Pi P_{\sigma}^{+} \tilde{z}(t))), $$
where

$$ \mathfrak{C}_* : \mathbb{C}^{N_\sigma} \to \mathcal{D}(A^*)', \quad \mathbf{\Theta} = (\theta_1, \dots, \theta_{N_\sigma}) \to \mathfrak{C}_* \mathbf{\Theta} = \sum_{j=1}^{N_\sigma} \theta_j C^* w_j^*, $$

1. where $\Pi_* \in \mathcal{L}((\mathbb{H}_\sigma^+)^*, \mathbb{H}_\sigma^+)$ is the unique solution of the algebraic Riccati equation: for all $\xi, \zeta \in (\mathbb{H}_\sigma^+)^*$
$$ \begin{cases} \langle \xi, \zeta \rangle_{\mathbb{H}} + \langle ((A_\sigma^+)^* + \sigma \mathrm{Id})\xi, \Pi_* \zeta \rangle_{\mathbb{H}} + \langle \Pi_* \xi, ((A_\sigma^+)^* + \sigma \mathrm{Id})\zeta \rangle_{\mathbb{H}} - \sum_{j=1}^{N_\sigma} \langle C \Pi_* \xi, w_j \rangle_{\mathbb{Y}} \langle C \Pi_* \zeta, w_j \rangle_{\mathbb{Y}} = 0, \\ \langle \Pi_* \xi, \zeta \rangle_{\mathbb{H}} = \langle \xi, \Pi_* \zeta \rangle_{\mathbb{H}}, \quad \text{and} \quad \forall \xi \neq 0, \langle \Pi_* \xi, \xi \rangle_{\mathbb{H}} > 0. \end{cases} \quad (2.14) $$

2. Hence, we define

$$ L_{\sigma}^{+}(\cdot) = L_{*}^{*}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_{j}^{*}, \cdot \rangle_{\mathbb{Y}} \chi_{j}, \quad (2.15) $$

3. with

$$ \chi_j = P_{\sigma}^{+} \Pi_{*} C^{*} w_j^{*} \in \mathbb{H}_{\sigma}^{+}. \quad (2.16) $$

4. With this choice, we get that $(A + L_{\sigma}^{+}C)^*$ and hence $A + L_{\sigma}^{+}Q_{\sigma}^{+}C$ is exponentially stable with decay rate less

5. than $-\sigma$. Finally, using $K_{\sigma}^{+}$ and $L_{\sigma}^{+}$ we construct the observer $\tilde{z}$ satisfying (1.6)-(1.7), that is

$$ \begin{cases} \tilde{z}'(t) = A\tilde{z}(t) + BK_{\sigma}^{+}\tilde{z}(t) + L_{\sigma}^{+}Q_{\sigma}^{+}(C\tilde{z}(t) - y(t)), \\ \tilde{z}(0) = 0. \end{cases} \quad (2.17) $$

## 2.3 Stability of the closed-loop system

7. We define the error $e = z - \tilde{z}$. Then, we obtain

$$ \left\{ \begin{array}{ll} e'(t) = (A + L_{\sigma}^{+}Q_{\sigma}^{+}C)e(t), & e(0) = z^{0}, \\ z'(t) = (A + BK_{\sigma}^{+})z(t) - BK_{\sigma}^{+}e(t), & z(0) = z^{0}. \end{array} \right. \quad (2.18) $$

8. We prove that $e$ is exponentially stable with decay rate $-\sigma$.

9. **Proposition 2.2.** *Systems (2.17) and (2.18) are exponentially stable with decay rate $-\sigma$.*

10. *Proof.* Since $(A + L_{\sigma}^{+}Q_{\sigma}^{+}C)$ is of negative type strictly less than $-\sigma$, then there exists $0 < \varepsilon'' < \varepsilon$ such that

$$ \|e(t)\|_{\mathbb{H}} \le M e^{-t(\sigma+\varepsilon'')} \|z^0\|_{\mathbb{H}}. \quad (2.19) $$

Going back to system (2.5) with the control given by (1.6), we have, since $K_{\sigma}^{+}z_{\sigma}^{-} = 0$,

$$ (z_{\sigma}^{+})'(t) = (A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})z_{\sigma}^{+}(t) - B_{\sigma}^{+}K_{\sigma}^{+}e(t), \quad z_{\sigma}^{+}(0) = P_{\sigma}^{+}z^{0}. $$

Moreover, there exists $\varepsilon' > 0$ with $\varepsilon'' < \varepsilon' < \varepsilon$ such that $(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})$ is exponentially stable with rate $-\sigma - \varepsilon'$. We have

$$ z_{\sigma}^{+}(t) = e^{t(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})} P_{\sigma}^{+} z^{0} - \int_{0}^{t} e^{(t-s)(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})} B_{\sigma}^{+} K_{\sigma}^{+} e(s) ds. $$

11. From (2.19), we see that

$$ \|z_{\sigma}^{+}(t)\|_{\mathbb{H}} \le M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}}. \quad (2.20) $$

We deal now with the infinite dimensional part $z_{\sigma}^{-}$ of the state. From (2.6) with the control given by (1.6), we have

$$ (z_{\sigma}^{-})'(t) = A_{\sigma}^{-} z_{\sigma}^{-}(t) + B_{\sigma}^{-} p_{\sigma}^{-} K_{\sigma}^{-} z_{\sigma}^{-}(t) - B_{\sigma}^{-} p_{\sigma}^{-} K_{\sigma}^{-} e(t), \quad z_{\sigma}^{-}(0) = (\mathrm{Id} - P_{\sigma}^{+}) z^{0}. $$
is exponentially stable i.e.

$$ \|\tilde{z}_{\sigma}^{+}(t)\|_{\mathbb{H}} \leq Me^{-(\sigma+\epsilon)t}\|\tilde{z}^{0}\|_{\mathbb{H}}, \quad t > 0. $$

It follows from Duhamel's formula that the whole system (2.9) is exponentially stable (see [2]). We can construct $L_{\sigma}^{+}$ similarly considering the system

$$ \begin{cases} \tilde{z}_{\star}^{\prime}(t) = A^{*}\tilde{z}_{\star}(t) + C^{*}u_{\star}(t), \\ \tilde{z}_{\star}(0) = \tilde{z}_{\star}^{0}. \end{cases} \qquad (2.13) $$

Using similar arguments and the second condition in (H1.D), we show that there exists a family $(w_j^*)_{1 \le j \le N_\sigma} \subset \mathbb{Y}_\sigma^+$ such that

$$ u_{\star} = L_{\star} \tilde{z}_{\star}, \quad L_{\star}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_j^*, C \Pi_{\star} (P_{\sigma}^{+})^*(\cdot) \rangle_{\mathbb{Y}} w_j^* = - \sum_{j=1}^{N_{\sigma}} (\mathfrak{C}_{\star}^*(\Pi_{\star}(P_{\sigma}^{+})^*(\cdot)))_j w_j^*, $$

where

$$ \mathfrak{C}_{\star}: \mathbb{C}^{N_{\sigma}} \to \mathcal{D}(A^{*})', \quad \Theta = (\theta_1, \dots, \theta_{N_{\sigma}}) \mapsto \mathfrak{C}_{\star} \Theta = \sum_{j=1}^{N_{\sigma}} \theta_j C^{*} w_j^{*}, $$

where $\Pi_{\star} \in \mathcal{L}((\mathbb{H}_{\sigma}^{+})^{*}, \mathbb{H}_{\sigma}^{+})$ is the unique solution of the algebraic Riccati equation: for all $\xi, \zeta \in (\mathbb{H}_{\sigma}^{+})^{*}$

$$ \left\{ \begin{array}{l} \langle \xi, \zeta \rangle_{\mathbb{H}} + \langle ((A_{\sigma}^{+})^{*} + \sigma \mathrm{Id})\xi, \Pi_{\star}\zeta \rangle_{\mathbb{H}} + \langle \Pi_{\star}\xi, ((A_{\sigma}^{+})^{*} + \sigma \mathrm{Id})\zeta \rangle_{\mathbb{H}} - \sum_{j=1}^{N_{\sigma}} \langle C\Pi_{\star}\xi, w_j \rangle_{\mathbb{Y}} \langle C\Pi_{\star}\zeta, w_j \rangle_{\mathbb{Y}} = 0, \\ \langle \Pi_{\star}\xi, \zeta \rangle_{\mathbb{H}} = \langle \xi, \Pi_{\star}\zeta \rangle_{\mathbb{H}}, \quad \text{and} \quad \forall \xi \neq 0, \langle \Pi_{\star}\xi, \xi \rangle_{\mathbb{H}} > 0. \end{array} \right. \qquad (2.14) $$

Hence, we define

$$ L_{\sigma}^{+}(\cdot) = L_{\star}^{*}(\cdot) = - \sum_{j=1}^{N_{\sigma}} \langle w_{j}^{*}, \cdot \rangle_{\mathbb{Y}} \chi_{j}, \qquad (2.15) $$

with

$$ \chi_j = P_{\sigma}^{+} \Pi_{\star} C^{*} w_j^{*} \in \mathbb{H}_{\sigma}^{+}. \qquad (2.16) $$

With this choice, we get that $(A + L_{\sigma}^{+}C)^*$ and hence $A + L_{\sigma}^{+}Q_{\sigma}^{+}C$ are exponentially stable with decay rate less than $-\sigma$. Finally, using $K_{\sigma}^{+}$ and $L_{\sigma}^{+}$ we construct the observer $\tilde{z}$ satisfying (1.6)-(1.7), that is

$$ \left\{ \begin{array}{l} \tilde{z}^{\prime}(t) = A\tilde{z}(t) + BK_{\sigma}^{+}\tilde{z}(t) + L_{\sigma}^{+}Q_{\sigma}^{+}(C\tilde{z}(t) - y(t)), \\ \tilde{z}(0) = 0. \end{array} \right. \qquad (2.17) $$

## 2.3 Stability of the closed-loop system

We define the error $e = z - \tilde{z}$. Then, we obtain

$$ \left\{ \begin{array}{l} e'(t) = (A + L_{\sigma}^{+}Q_{\sigma}^{+}C)e(t), \quad e(0) = z^{0}, \\ z'(t) = (A + BK_{\sigma}^{+})z(t) - BK_{\sigma}^{+}e(t), \quad z(0) = z^{0}. \end{array} \right. \qquad (2.18) $$

We prove that $e$ is exponentially stable with decay rate $-\sigma$.

**Proposition 2.2.** *Systems (2.17) and (2.18) are exponentially stable with decay rate $-\sigma$.*

*Proof.* Since $(A + L_{\sigma}^{+}Q_{\sigma}^{+}C)$ is of negative type strictly less than $-\sigma$, then there exists $0 < \varepsilon'' < \varepsilon$ such that

$$ \|e(t)\|_{\mathbb{H}} \leq Me^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}}. \qquad (2.19) $$
Using Duhamel's formula, we obtain that

$$z_{\sigma}^{-}(t) = e^{tA_{\sigma}^{-}} (\mathrm{Id} - P_{\sigma}^{+}) z^{0} + \int_{0}^{t} e^{(t-s)A_{\sigma}^{-}} B_{\sigma}^{-} p_{\sigma}^{-} K_{\sigma}^{+} (z_{\sigma}^{+}(s) - e(s)) \, ds.$$

1. We note that since the resolvent commutes with the projection $P_{\sigma}^{+}$ and $e^{(t-s)A_{\sigma}^{-}}$, we obtain for $\mu_{0} \in \rho(A)$ and

2. $\gamma \in [0, 1)$,

$$ \begin{aligned} e^{(t-s)A_{\sigma}^{-}} B_{\sigma}^{-} &= e^{(t-s)A_{\sigma}^{-}} (\mu_{0} \mathrm{Id} - A)^{\gamma} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-} \\ &= e^{(t-s)A_{\sigma}^{-}} (\mu_{0} \mathrm{Id} - A_{\sigma}^{-})^{\gamma} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-} \quad (2.21) \\ &= (\mu_{0} \mathrm{Id} - A_{\sigma}^{-})^{\gamma} e^{(t-s)A_{\sigma}^{-}} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-}. \end{aligned} $$

3. Using (H1.B), (2.3), (2.19), (2.20) and (2.21), we get

$$ \begin{aligned} \|z_{\sigma}^{-}(t)\|_{\mathbb{H}} &\le M \|z^{0}\|_{\mathbb{H}} \left( e^{-t(\sigma+\varepsilon)} + \int_{0}^{t} \frac{1}{(t-s)^{\gamma}} e^{-(t-s)(\sigma+\varepsilon)} e^{-s(\sigma+\varepsilon'')} \, ds \right) \\ &\le M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}} \left( 1 + \int_{0}^{t} \frac{1}{(t-s)^{\gamma}} e^{(t-s)(\varepsilon''-\varepsilon)} \, ds \right). \end{aligned} \quad (2.22) $$

4. Then, since $\varepsilon'' < \varepsilon$, we obtain

$$ \|z_{\sigma}^{-}(t)\|_{\mathbb{H}} \le M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}}. \quad (2.23) $$

5. Then from (2.19), (2.20) and (2.23), we obtain that $z, \hat{z}$ and the error $e$ are exponentially stable. $\square$

6. That concludes the proof of Theorem 1.1.

**Remark 2.3.** According to (1.6) and (2.11), the control reads

$$v(t) = \sum_{i=1}^{N_{\sigma}} K_{i} (\hat{z}_{\sigma}^{+}) w_{i},$$

with $K_{i} \in \mathcal{L}(\mathbb{H}_{\sigma}^{+}, \mathbb{C})$ and $w_{i} \in \mathbb{U}_{\sigma}^{+}$, $i = 1, \dots, N_{\sigma}$. From decomposition (2.2) and the fact that $(\mathbb{H}_{\sigma}^{+})^{\perp} = (\mathbb{H}_{\sigma}^{-})^{*}$, we have that if $\zeta \in (\mathbb{H}_{\sigma}^{+})^{*}$, then

$$ \forall \phi \in \mathbb{H}_{\sigma}^{+}, \quad \langle \phi, \zeta \rangle_{\mathbb{H}} = 0 \quad \Longrightarrow \quad \zeta = 0. $$

Since $\dim((\mathbb{H}_{\sigma}^{+})^{*}) = \dim(\mathcal{L}(\mathbb{H}_{\sigma}^{+}, \mathbb{C}))$, we infer that there exists a unique $\zeta_{i} \in (\mathbb{H}_{\sigma}^{+})^{*} \subset \mathcal{D}(A^{*})$ such that

$$ K_{i} (\hat{z}_{\sigma}^{+}) = \langle \hat{z}, \zeta_{i} \rangle_{\mathbb{H}}. $$

In other words, the control can also be written in the form

$$ v(t) = \sum_{i=1}^{N_{\sigma}} \langle \hat{z}, \zeta_{i} \rangle_{\mathbb{H}} w_{i}. $$

In the special case where there is only one unstable simple eigenvalue (with an eigenspace spanned by an eigen-function $\varepsilon_{1} \in \mathbb{H}$), the above relations take simpler forms. Indeed, we have then

$$ w_{1} = B^{*} \varepsilon_{1} $$

and for all $\varphi \in \mathbb{H}_{\sigma}^{+} = \mathrm{Span}\{\varepsilon_{1}\}$:

$$ K_{1}(\varphi) = -\langle B^{*} \varepsilon_{1}, B^{*} \Pi P_{\sigma}^{+} \varphi \rangle_{\mathbb{U}} = \langle \zeta_{1}, \varphi \rangle_{(\mathbb{H}_{\sigma}^{+})^{*}, \mathbb{H}_{\sigma}^{+}}, \quad \zeta_{1} = -(P_{\sigma}^{+})^{*} \Pi B B^{*} \varepsilon_{1}. $$
Going back to system (2.5) with the control given by (1.6), we have, since $K_{\sigma}^{+}z_{\sigma}^{-} = 0$,

$$ (z_{\sigma}^{+})'(t) = (A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})z_{\sigma}^{+}(t) - B_{\sigma}^{+}K_{\sigma}^{+}e(t), \quad z_{\sigma}^{+}(0) = P_{\sigma}^{+}z^{0}. $$

Moreover, there exists $\varepsilon' > 0$ with $\varepsilon'' < \varepsilon' < \varepsilon$ such that $(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})$ is exponentially stable with rate $-\sigma - \varepsilon'$. We have

$$ z_{\sigma}^{+}(t) = e^{t(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})} P_{\sigma}^{+} z^{0} - \int_{0}^{t} e^{(t-s)(A_{\sigma}^{+} + B_{\sigma}^{+}K_{\sigma}^{+})} B_{\sigma}^{+} K_{\sigma}^{+} e(s) ds. $$

From (2.19), we see that

$$ \|z_{\sigma}^{+}(t)\|_{\mathbb{H}} \leq M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}}. \qquad (2.20) $$

We deal now with the infinite dimensional part $z_{\sigma}^{-}$ of the state. From (2.6) with the control given by (1.6), we have

$$ (z_{\sigma}^{-})'(t) = A_{\sigma}^{-}z_{\sigma}^{-}(t) + B_{\sigma}^{-}p_{\sigma}^{-}K_{\sigma}^{+}z_{\sigma}^{+}(t) - B_{\sigma}^{-}p_{\sigma}^{-}K_{\sigma}^{+}e(t), \quad z_{\sigma}^{-}(0) = (\mathrm{Id} - P_{\sigma}^{+})z^{0}. $$

Using Duhamel's formula, we obtain that

$$ z_{\sigma}^{-}(t) = e^{tA_{\sigma}^{-}}(\mathrm{Id} - P_{\sigma}^{+})z^{0} + \int_{0}^{t} e^{(t-s)A_{\sigma}^{-}} B_{\sigma}^{-} p_{\sigma}^{-} K_{\sigma}^{+} (z_{\sigma}^{+}(s) - e(s)) \, ds. $$

We note that since the resolvent commutes with the projection $P_{\sigma}^{+}$ and $e^{(t-s)A_{\sigma}^{-}}$, we obtain for $\mu_{0} \in \rho(A)$ and $\gamma \in [0, 1)$,

$$ \begin{aligned} e^{(t-s)A_{\sigma}^{-}} B_{\sigma}^{-} &= e^{(t-s)A_{\sigma}^{-}} (\mu_{0} \mathrm{Id} - A)^{\gamma} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-} \\ &= e^{(t-s)A_{\sigma}^{-}} (\mu_{0} \mathrm{Id} - A_{\sigma}^{-})^{\gamma} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-} \qquad (2.21) \\ &= (\mu_{0} \mathrm{Id} - A_{\sigma}^{-})^{\gamma} e^{(t-s)A_{\sigma}^{-}} (\mathrm{Id} - P_{\sigma}^{+}) (\mu_{0} \mathrm{Id} - A)^{-\gamma} B i_{\sigma}^{-}. \end{aligned} $$

Using (H1.B), (2.3), (2.19), (2.20) and (2.21), we get

$$ \begin{aligned} \|z_{\sigma}^{-}(t)\|_{\mathbb{H}} &\leq M \|z^{0}\|_{\mathbb{H}} \left( e^{-t(\sigma+\varepsilon)} + \int_{0}^{t} \frac{1}{(t-s)^{\gamma}} e^{-(t-s)(\sigma+\varepsilon)} e^{-s(\sigma+\varepsilon'')} \, ds \right) \\ &\leq M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}} \left( 1 + \int_{0}^{t} \frac{1}{(t-s)^{\gamma}} e^{(t-s)(\varepsilon''-\varepsilon)} \, ds \right). \end{aligned} \qquad (2.22) $$

Then, since $\varepsilon'' < \varepsilon$, we obtain

$$ \|z_{\sigma}^{-}(t)\|_{\mathbb{H}} \leq M e^{-t(\sigma+\varepsilon'')} \|z^{0}\|_{\mathbb{H}}. \qquad (2.23) $$

Then from (2.19), (2.20) and (2.23), we obtain that $z, \hat{z}$ and the error $e$ are exponentially stable. $\square$

That concludes the proof of Theorem 1.1.

**Remark 2.3.** According to (1.6) and (2.11), the control reads

$$ v(t) = \sum_{i=1}^{N_{\sigma}} K_{i} (\hat{z}_{\sigma}^{+}) w_{i}, $$

with $K_{i} \in \mathcal{L}(\mathbb{H}_{\sigma}^{+}, \mathbb{C})$ and $w_{i} \in \mathbb{U}_{\sigma}^{+}$, $i = 1, \dots, N_{\sigma}$. From the decomposition (2.2) and the fact that $(\mathbb{H}_{\sigma}^{+})^{\perp} = (\mathbb{H}_{\sigma}^{-})^{*}$, we have that if $\zeta \in (\mathbb{H}_{\sigma}^{+})^{*}$, then

$$ \forall \phi \in \mathbb{H}_{\sigma}^{+}, \quad \langle \phi, \zeta \rangle_{\mathbb{H}} = 0 \quad \Longrightarrow \quad \zeta = 0. $$

Since $\dim((\mathbb{H}_{\sigma}^{+})^{*}) = \dim(\mathcal{L}(\mathbb{H}_{\sigma}^{+}, \mathbb{C}))$, we infer that there exists a unique $\zeta_{i} \in (\mathbb{H}_{\sigma}^{+})^{*} \subset \mathcal{D}(A^{*})$ such that

$$ K_{i}(\hat{z}_{\sigma}^{+}) = \langle \hat{z}, \zeta_{i} \rangle_{\mathbb{H}}. $$
## 3 Finite dimensional observer

### 3.1 Spectral decomposition of the system

In this section, we assume hypotheses (H2.A), (H2.B), (H2.C) and (H2.D) to hold true.

Consider $\nu > 0$ and let us introduce the projection operators $P_\nu^+$ as in (1.5) where in this case $\Gamma_\nu^+$ is a circle enclosing $\Sigma_\nu^+$ but no other point of the spectrum of A and oriented counterclockwise (see [10, V.5, p.272]). Since A is a self-adjoint operator, then $P_\nu^+$ is well defined. Moreover from the expression of the projections, it follows that

$$(P_\nu^+)^* = P_\nu^+.$$

Thus, $P_\nu^+$ is orthogonal projection of norm equal to 1. We set

$$\mathbb{H}_\nu^+ = P_\nu^+ \mathbb{H}, \quad \mathbb{H}_\nu^- = (\mathrm{Id} - P_\nu^+) \mathbb{H}, \quad \mathbb{H} = \mathbb{H}_\nu^+ \oplus \mathbb{H}_\nu^-,$$

and

$$A_\nu^+ := A_{|\mathbb{H}_\nu^+} : \mathbb{H}_\nu^+ \to \mathbb{H}_\nu^+, \quad A_\nu^- := A_{|\mathbb{H}_\nu^-} : \mathcal{D}(A) \cap \mathbb{H}_\nu^- \to \mathbb{H}_\nu^-.$$

We also define as before

$$\mathbb{U}_\nu^+ := B^* \mathbb{H}_\nu^+, \quad \mathbb{U}_\nu^- := B^* (\mathcal{D}(A) \cap \mathbb{H}_\nu^-),$$

and

$$p_\nu^+ : \mathbb{U} \to \mathbb{U}_\nu^+, \quad p_\nu^- : \mathbb{U} \to \mathbb{U}_\nu^-, \quad i_\nu^+ : \mathbb{U}_\nu^+ \to \mathbb{U}, \quad i_\nu^- : \mathbb{U}_\nu^- \to \mathbb{U},$$

the orthogonal projections and the inclusion maps. Note that we have the following relations for the above

maps:

$$i_\nu^+ = (p_\nu^+)^*, \quad i_\nu^- = (p_\nu^-)^* \qquad (3.1)$$

We can thus define

$$B_\nu^+ := P_\nu^+ B i_\nu^+ \in \mathcal{L}(\mathbb{U}_\nu^+, \mathbb{H}_\nu^+), \quad B_\nu^- := (\mathrm{Id} - P_\nu^+) B i_\nu^- \in \mathcal{L}(\mathbb{U}_\nu^-, \mathbb{H}_\nu^-).$$

It is proved in [1] (see also [2] and [18]) that

$$P_\nu^+ B = B_\nu^+ p_\nu^+, \quad (\mathrm{Id} - P_\nu^+) B = B_\nu^- p_\nu^-.$$

We introduce also the orthogonal projection $Q_\nu^+$ from $\mathbb{Y}$ into $\mathbb{Y}_\nu^+ = C\mathbb{H}_\nu^+$ and define

$$C_\nu^+ = C i_\nu^+,$$

where $i_\nu^+$ designates the injection operator from $\mathbb{H}_\nu^+$ to $\mathbb{H}$.

Consider now $\sigma^* > \sigma > 0$. We take $\nu = \sigma$ or $\nu = \sigma^*$ in the maps and spaces defined previously. Since A is

self-adjoint with compact resolvent, we deduce the existence of $\varepsilon > 0$ such that for all $t \ge 0$

$$\|e^{A_\sigma^- t}\|_{\mathcal{L}(\mathbb{H}_\sigma^-)} \le e^{-(\sigma+\varepsilon)t}, \qquad (3.2)$$

and

$$\|e^{A_{\sigma^*} t}\|_{\mathcal{L}(\mathbb{H}_{\sigma^*}^-)} \le e^{-\sigma^* t}. \qquad (3.3)$$

The system (1.3) splits into

$$(z_{\sigma^*}^+)'(t) = A_{\sigma^*}^+ z_{\sigma^*}^+(t) + B_{\sigma^*}^+ p_{\sigma^*} v(t), \quad z_{\sigma^*}^+(0) = P_{\sigma^*}^+ z^0, \qquad (3.4)$$

$$(z_{\sigma^*}^-)'(t) = A_{\sigma^*}^- z_{\sigma^*}^-(t) + B_{\sigma^*}^- p_{\sigma^*} v(t), \quad z_{\sigma^*}^-(0) = (\mathrm{Id} - P_{\sigma^*}^+) z^0. \qquad (3.5)$$
In other words, the control can also be written in the form

$$v(t) = \sum_{i=1}^{N_{\sigma}} \langle \hat{z}, \zeta_i \rangle_{\mathbb{H}} w_i.$$

In the special case where there is only one unstable simple eigenvalue (with an eigenspace spanned by an eigen-function $\varepsilon_1 \in \mathbb{H}$), the above relations take simpler forms. Indeed, we have then

$$w_1 = B^* \varepsilon_1,$$

and for all $\varphi \in \mathbb{H}_{\sigma}^{+} = \text{Span}\{\varepsilon_1\}$:

$$K_1(\varphi) = -\langle B^* \varepsilon_1, B^* \Pi P_{\sigma}^{+} \varphi \rangle_{\mathbb{H}} = \langle \zeta_1, \varphi \rangle_{(\mathbb{H}_{\sigma}^{+})^*, \mathbb{H}_{\sigma}^{+}}, \quad \zeta_1 = -(P_{\sigma}^{+})^* \Pi B B^* \varepsilon_1.$$

# 3 Finite dimensional observer

## 3.1 Spectral decomposition of the system

In this section, we assume hypotheses (H2.A), (H2.B), (H2.C) and (H2.D) to hold true.

Consider $\nu > 0$ and let us introduce the projection operators $P_{\nu}^{+}$ as in (1.5) where in this case $\Gamma_{\nu}^{+}$ is a circle enclosing $\Sigma_{\nu}^{+}$ but no other point of the spectrum of A and oriented counterclockwise (see [12, V.5, p.272]). Since A is a self-adjoint operator, then $P_{\nu}^{+}$ is well defined. Moreover from the expression of the projections, it follows that

$$(P_{\nu}^{+})^{*} = P_{\nu}^{+}.$$

Thus, $P_{\nu}^{+}$ is orthogonal projection of norm equal to 1. We set

$$\mathbb{H}_{\nu}^{+} = P_{\nu}^{+} \mathbb{H}, \quad \mathbb{H}_{\nu}^{-} = (\text{Id} - P_{\nu}^{+}) \mathbb{H}, \quad \mathbb{H} = \mathbb{H}_{\nu}^{+} \oplus \mathbb{H}_{\nu}^{-},$$

and

$$A_{\nu}^{+} := A_{|\mathbb{H}_{\nu}^{+}} : \mathbb{H}_{\nu}^{+} \to \mathbb{H}_{\nu}^{+}, \quad A_{\nu}^{-} := A_{|\mathbb{H}_{\nu}^{-}} : \mathcal{D}(A) \cap \mathbb{H}_{\nu}^{-} \to \mathbb{H}_{\nu}^{-}.$$

We also define as before

$$\mathbb{U}_{\nu}^{+} := B^{*} \mathbb{H}_{\nu}^{+}, \quad \mathbb{U}_{\nu}^{-} := B^{*} (\mathcal{D}(A) \cap \mathbb{H}_{\nu}^{-}),$$

and

$$p_{\nu}^{+} : \mathbb{U} \to \mathbb{U}_{\nu}^{+}, \quad p_{\nu}^{-} : \mathbb{U} \to \mathbb{U}_{\nu}^{-}, \quad i_{\nu}^{+} : \mathbb{U}_{\nu}^{+} \to \mathbb{U}, \quad i_{\nu}^{-} : \mathbb{U}_{\nu}^{-} \to \mathbb{U},$$

the orthogonal projections and the inclusion maps. Note that we have the following relations for the above maps:

$$i_{\nu}^{+} = (p_{\nu}^{+})^{*}, \quad i_{\nu}^{-} = (p_{\nu}^{-})^{*}. \qquad (3.1)$$

We can thus define

$$B_{\nu}^{+} := P_{\nu}^{+} B i_{\nu}^{+} \in \mathcal{L}(\mathbb{U}_{\nu}^{+}, \mathbb{H}_{\nu}^{+}), \quad B_{\nu}^{-} := (\text{Id} - P_{\nu}^{+}) B i_{\nu}^{-} \in \mathcal{L}(\mathbb{U}_{\nu}^{-}, \mathbb{H}_{\nu}^{-}).$$

It is proved in [2] (see also [3] and [20]) that

$$P_{\nu}^{+} B = B_{\nu}^{+} p_{\nu}^{+}, \quad (\text{Id} - P_{\nu}^{+}) B = B_{\nu}^{-} p_{\nu}^{-}.$$

We introduce also the orthogonal projection $Q_{\nu}^{+}$ from $\mathbb{Y}$ into $\mathbb{Y}_{\nu}^{+} = C\mathbb{H}_{\nu}^{+}$ and define

$$C_{\nu}^{+} = C i_{\nu}^{+},$$

where $i_{\nu}^{+}$ designates the injection operator from $\mathbb{H}_{\nu}^{+}$ to $\mathbb{H}$.
## 3.2 Finite dimensional observer based control and stability of the closed-loop system

We are now in position to prove Theorem 1.2. The matrices $K_\sigma^+$ and $L_\sigma^+$ being respectively given by (2.11) and (2.15), we define the finite dimensional observer-based feedback control by

$$v(t) = K_{\sigma}^{+} \hat{z}_{\star}(t) \in B^{*} P_{\sigma}^{+} \mathbb{H} \subset B^{*} P_{\sigma^{*}}^{+} \mathbb{H}, \quad (3.6)$$

where the finite dimensional observer $\hat{z}_\star \in P_{\sigma^*}^+ \mathbb{H}$ solves

$$\left\{ \begin{array}{l} \hat{z}_{\star}(t) = A_{\sigma^{*}}^{+} \hat{z}_{\star}(t) + B_{\sigma^{*}}^{+} v(t) + L_{\sigma}^{+} Q_{\sigma}^{+} (C_{\sigma^{*}}^{+} \hat{z}_{\star}(t) - y(t)), \\ \hat{z}_{\star}(0) = 0. \end{array} \right. \qquad (3.7)$$

Our goal is to prove that the coupled system (1.3), (3.6) and (3.7) is exponentially stable. We define the error $e = z_{\sigma^*}^+ - \hat{z}_\star$ which satisfies the following system

$$\left\{ \begin{array}{l} e'(t) = A_{\sigma^*}^+ e(t) + L_{\sigma}^+ Q_{\sigma}^+ (C_{\sigma^*}^+ e(t) + C z_{\sigma^*}^{-}(t)), \\ e(0) = z_{\sigma^*}^+(0). \end{array} \right. \qquad (3.8)$$

We prove that $\hat{z}$ and $e$ are exponentially stable with decay rate $-\sigma$. Let us set

$$X = \begin{pmatrix} \hat{z}_{\star} \\ e \end{pmatrix}, \qquad X^0 = \begin{pmatrix} 0 \\ z_{\sigma^{*}}^{+}(0) \end{pmatrix}.$$

Using (3.7) and (3.8), $X$ satisfies the system

$$\left\{ \begin{array}{l} X'(t) = AX(t) + L(z_{\sigma^*}^{-}(t)), \\ X(0) = X^0, \end{array} \right. \qquad (3.9)$$

where

$$A = \begin{pmatrix} A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+ & -L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma^*}^+ \\ 0 & A_{\sigma^*}^+ + L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma^*}^+ \end{pmatrix}, \quad L(z_{\sigma^*}^{-}) = \begin{pmatrix} -L_{\sigma}^+ Q_{\sigma}^+ C z_{\sigma^*}^{-} \\ L_{\sigma}^+ Q_{\sigma}^+ C z_{\sigma^*}^{-} \end{pmatrix}. \qquad (3.10)$$

Let us first prove that $A$ is stable matrix. In the sequel, the constant $M$ is a generic constant that can change from a line to another but need to be independent of $\sigma^*$.

**Lemma 3.1.** The matrices $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+$ and $A_{\sigma^*}^+ + L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma^*}^+$ are exponentially stable with a decay rate less than $-\sigma$.

*Proof.* Let $\xi^0 \in \mathbb{H}_{\sigma^*}^+$ be given. To prove that $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+$ is exponentially stable, we only need to show that the solution $\xi(t)$ of the finite dimensional system

$$\left\{ \begin{array}{l} \xi'(t) = (A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+) \xi(t), \\ \xi(0) = \xi^0, \end{array} \right. \qquad (3.11)$$

is exponentially decaying. Consider then the infinite dimensional system

$$\left\{ \begin{array}{l} x'(t) = (A + BK_{\sigma}^{+})x(t), \\ x(0) = \xi^{0}. \end{array} \right. \qquad (3.12)$$

From Section 2.2, we see that the system (3.12) is exponentially stable of decay rate $-\sigma - \varepsilon'$. It implies that

$$\|x(t)\|_{\mathbb{H}} \leq M e^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}. \qquad (3.13)$$
Consider now $\sigma^* > \sigma > 0$. We take $\nu = \sigma$ or $\nu = \sigma^*$ in the maps and spaces defined previously. Since $A$ is self-adjoint with compact resolvent, we deduce the existence of $\varepsilon > 0$ such that for all $t \ge 0$

$$ \|e^{A_{\bar{\sigma}}^* t}\|_{\mathcal{L}(\mathbb{H}_{\bar{\sigma}})} \le e^{-(\sigma+\varepsilon)t}, \qquad (3.2) $$

and

$$ \|e^{A_{\sigma^*} t}\|_{\mathcal{L}(\mathbb{H}_{\sigma^*})} \le e^{-\sigma^* t}. \qquad (3.3) $$

The system (1.3) splits into

$$ (z_{\sigma^*}^+)'(t) = A_{\sigma^*}^+ z_{\sigma^*}^+(t) + B_{\sigma^*}^+ p_{\sigma^*}^+ v(t), \quad z_{\sigma^*}^+(0) = P_{\sigma^*}^+ z^0, \qquad (3.4) $$

$$ (z_{\sigma^*}^-)'(t) = A_{\sigma^*}^- z_{\sigma^*}^- (t) + B_{\sigma^*}^- p_{\sigma^*}^- v(t), \quad z_{\sigma^*}^-(0) = (\mathrm{Id} - P_{\sigma^*}^+) z^0. \qquad (3.5) $$

## 3.2 Finite dimensional observer-based control and stability of the closed-loop system

We are now in position to prove Theorem 1.2. The matrices $K_\sigma^+$ and $L_\sigma^+$ being respectively given by (2.11) and (2.15), we define the finite dimensional observer-based feedback control by

$$ v(t) = K_{\sigma}^{+} \hat{z}_{\star}(t) \in B^{*} P_{\sigma}^{+} \mathbb{H} \subset B^{*} P_{\sigma^{*}}^{+} \mathbb{H}, \qquad (3.6) $$

where the finite dimensional observer $\hat{z}_\star \in P_{\sigma^*}^+ \mathbb{H}$ solves

$$ \begin{cases} \hat{z}'_\star(t) = A_{\sigma^*}^+ \hat{z}_\star(t) + B_{\sigma^*}^+ v(t) + L_\sigma^+ Q_\sigma^+ (C_{\sigma^*}^+ \hat{z}_\star(t) - y(t)), \\ \hat{z}_\star(0) = 0. \end{cases} \qquad (3.7) $$

Our goal is to prove that the coupled system (1.3), (3.6) and (3.7) is exponentially stable. We define the error $e = z_{\sigma^*}^+ - \hat{z}_\star$ which satisfies the following system

$$ \begin{cases} e'(t) = A_{\sigma}^{+} e(t) + L_{\sigma}^{+} Q_{\sigma}^{+} (C_{\sigma}^{+} e(t) + C z_{\sigma}^{-}(t)), \\ e(0) = z_{\sigma}^{+}(0). \end{cases} \qquad (3.8) $$

We prove that $\hat{z}$ and $e$ are exponentially stable with decay rate $-\sigma$. Let us set

$$ \mathbb{X} = \begin{pmatrix} \hat{z}_{\star} \\ e \end{pmatrix}, \quad \mathbb{X}^{0} = \begin{pmatrix} 0 \\ z_{\sigma^{*}}^{+}(0) \end{pmatrix}. $$

Using (3.7) and (3.8), $\mathbb{X}$ satisfies the system

$$ \begin{cases} \mathbb{X}'(t) = A\mathbb{X}(t) + \mathbb{L}(z_{\sigma^*}^{-}(t)), \\ \mathbb{X}(0) = \mathbb{X}^{0}, \end{cases} \qquad (3.9) $$

where

$$ A = \begin{pmatrix} A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+ & -L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma^*}^+ \\ 0 & A_{\sigma^*}^+ + L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma^*}^+ \end{pmatrix}, \quad \mathbb{L}(z_{\sigma^*}^{-}) = \begin{pmatrix} -L_{\sigma}^+ Q_{\sigma}^+ C z_{\sigma^*}^{-} \\ L_{\sigma}^+ Q_{\sigma}^+ C z_{\sigma^*}^{-} \end{pmatrix}. \qquad (3.10) $$

Let us first prove that $A$ is stable matrix. In the sequel, the constant $M$ is a generic constant that can change from a line to another but need to be independent of $\sigma^*$.

**Lemma 3.1.** The matrices $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_\sigma^+$ and $A_{\sigma^*}^+ + L_\sigma^+ Q_\sigma^+ C_{\sigma^*}^+$ are exponentially stable with a decay rate less than $-\sigma$.
$$ \left\{ \begin{array}{l} (x_{\sigma^*}^+)'(t) = A_{\sigma^*}^+ x_{\sigma^*}^+(t) + B_{\sigma^*}^+ K_{\sigma^*}^+ x(t) = (A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+) x_{\sigma^*}(t), \\ x_{\sigma^*}^+(0) = \xi^0. \end{array} \right. \qquad (3.14) $$

This shows that $x_{\sigma^*}^+(t)$ is the unique solution $\xi(t)$ of (3.11), and we get from (3.13), that

$$ \|\xi(t)\|_{\mathbb{H}} = \|x_{\sigma^*}^+(t)\|_{\mathbb{H}} = \|P_{\sigma^*}^+ x(t)\|_{\mathbb{H}} \leq M e^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}, $$

for all $\xi^0 \in \mathbb{H}_{\sigma^*}^+$. Hence

$$ \left\| e^{t(A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+)} \xi^0 \right\|_{\mathbb{H}} \leq M e^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}, $$

$$ \left\{ \begin{array}{l} \mathbf{X}'(t) = \mathbf{A}\mathbf{X}(t) + \mathcal{L}(z_{\sigma^*}^{-}(t)), \\ (z_{\sigma^*}^{-})'(t) = A_{\sigma^*}^{-}z_{\sigma^*}^{-}(t) + B_{\sigma^*}^{-}p_{\sigma^*}^{-}\mathbb{K}_{\sigma}^{+}\mathbf{X}(t), \end{array} \right. \qquad (3.15) $$

where $\mathbb{K}_{\sigma}^{+} = (K_{\sigma}^{+}, 0)$ and with the initial conditions

$$ \left\{ \begin{array}{l} \mathbf{X}(0) = \mathbf{X}^0, \\ z_{\sigma^*}^{-}(0) = (\mathrm{Id} - P_{\sigma^*}^{+})z^0. \end{array} \right. $$

From Duhamel's formula, the two first equations in (3.15) also read

$$ \left\{ \begin{array}{l} \mathbf{X}(s) = e^{s\mathbb{A}}\mathbf{X}(0) + \int_{0}^{s} e^{(s-\tau)\mathbb{A}}\mathcal{L}(z_{\sigma^*}^{-}(\tau))d\tau, \\ z_{\sigma^*}^{-}(t) = e^{tA_{\sigma^*}^{-}}z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}^{-}}(B_{\sigma^*}^{-}p_{\sigma^*}^{-}\mathbb{K}_{\sigma}^{+}\mathbf{X}(s)) ds. \end{array} \right. $$

$$ z_{\sigma^*}^{-}(t) = e^{tA_{\sigma^*}^{-}} z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}^{-}} B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} \left( e^{s\mathbb{A}} \mathbf{X}(0) + \int_{0}^{s} e^{(s-\tau)\mathbb{A}} \mathcal{L}(z_{\sigma^*}^{-}(\tau)) d\tau \right) ds. \quad (3.16) $$

Setting

$$ Z^0 := e^{tA_{\sigma^*}^{-}} z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}^{-}} (B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} (e^{s\mathbb{A}} \mathbf{X}(0))) ds, $$

$$ z_{\sigma^*}^{-}(t) = \int_{0}^{t} \int_{0}^{s} e^{(t-s)A_{\sigma^*}^{-}} B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} \left( e^{(s-\tau)\mathbb{A}} \mathcal{L}(z_{\sigma^*}^{-}(\tau)) d\tau ds \right) + Z^0. \quad (3.17) $$

To prove the stability of $z_{\sigma^*}^{-}$, we prove the existence of a unique solution to (3.17) in a weighted space by using a fixed point argument. More precisely, let us define the following map

$$ \Phi : L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-}) \longrightarrow L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-}) \\ g \longmapsto \Phi(g) := \int_{0}^{t} \int_{0}^{s} e^{(t-s)A_{\sigma^*}^{-}} B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} e^{(s-\tau)\mathbb{A}} \mathcal{L}(g(\tau)) d\tau ds + Z^0. $$

<footnote n="1">1 On the other hand, applying $P_{\sigma^*}^+$ to (3.12) and recalling that $K_{\sigma}^+ x(t)$ acts only on the projected part of $x(t)$ on</footnote>

<footnote n="2">2 $\mathbb{H}_{\sigma}^{+}$, we obtain that</footnote>

<footnote n="3">3 and the matrix $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma}^+$ is exponentially stable with a decay rate less than $-\sigma$. We use the same argument</footnote>

<footnote n="4">4 for $A_{\sigma^*}^+ + L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma}^+$ by considering its adjoint $(A_{\sigma^*}^+)^* + (C_{\sigma^*}^+)^* (L_{\sigma}^+)^*$ that has exactly the same form as the one</footnote>

<footnote n="5">5 previously studied. $\square$</footnote>

<footnote n="6">6 Since $\mathbb{A}$ is a triangular matrix, using Lemma 3.1 and Duhamel's formula, we obtain that $\mathbb{A}$ is stable with</footnote>

<footnote n="7">7 exponential rate strictly less than $-\sigma$.</footnote>

<footnote n="8">8 We can now prove the exponential stability of the full closed-loop system (3.7) and (1.3):</footnote>

<footnote n="9">9 Substituting the first equation above into the second one yields</footnote>

<footnote n="10">10 relation (3.16) can be written</footnote>
Proof. Let $\xi^0 \in \mathbb{H}_{\sigma^*}^+$ be given. To prove that $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+$ is exponentially stable, we only need to show that the solution $\xi(t)$ of the finite dimensional system

$$ \left\{ \begin{array}{l} \xi'(t) = (A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+) \xi(t), \\ \xi(0) = \xi^0, \end{array} \right. \qquad (3.11) $$

is exponentially decaying. Consider then the infinite dimensional system

$$ \left\{ \begin{array}{l} x'(t) = (A + BK_{\sigma}^{+})x(t), \\ x(0) = \xi^{0}. \end{array} \right. \qquad (3.12) $$

From Section 2.2, we see that the system (3.12) is exponentially stable of decay rate $-\sigma - \varepsilon'$. It implies that

$$ \|x(t)\|_{\mathbb{H}} \leq Me^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}. \qquad (3.13) $$

On the other hand, applying $P_{\sigma^*}^+$ to (3.12) and recalling that $K_{\sigma^*}^+ x(t)$ acts only on the projected part of $x(t)$ on $\mathbb{H}_{\sigma^*}^+$, we obtain that

$$ \left\{ \begin{array}{l} (x_{\sigma^*}^+)'(t) = A_{\sigma^*}^+ x_{\sigma^*}^+(t) + B_{\sigma^*}^+ K_{\sigma^*}^+ x(t) = (A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+) x_{\sigma^*}^+(t), \\ x_{\sigma^*}^+(0) = \xi^0. \end{array} \right. \qquad (3.14) $$

This shows that $x_{\sigma^*}^+(t)$ is the unique solution $\xi(t)$ of (3.11), and we get from (3.13), that

$$ \|\xi(t)\|_{\mathbb{H}} = \|x_{\sigma^*}^+(t)\|_{\mathbb{H}} = \|P_{\sigma^*}^+ x(t)\|_{\mathbb{H}} \leq Me^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}, $$

for all $\xi^0 \in \mathbb{H}_{\sigma^*}^+$. Hence

$$ \left\| e^{t(A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+)} \xi^0 \right\|_{\mathbb{H}} \leq Me^{-t(\sigma+\varepsilon')} \|\xi^0\|_{\mathbb{H}}, $$

and the matrix $A_{\sigma^*}^+ + B_{\sigma^*}^+ K_{\sigma^*}^+$ is exponentially stable with a decay rate less than $-\sigma$. We use the same argument for $A_{\sigma^*}^+ + L_{\sigma}^+ Q_{\sigma}^+ C_{\sigma}^+$. by considering its adjoint $(A_{\sigma^*}^+)^* + (C_{\sigma^*}^+)^* (L_{\sigma}^+)^*$ that has exactly the same form as the one previously studied. $\square$

Since $\mathbb{A}$ is a triangular matrix, using Lemma 3.1 and Duhamel's formula, we obtain that $\mathbb{A}$ is stable with exponential rate strictly less than $-\sigma$.

We can now prove the exponential stability of the full closed-loop system (3.7) and (1.3):

$$ \left\{ \begin{array}{l} \mathbb{X}'(t) = \mathbb{A}\mathbb{X}(t) + \mathbb{L}(z_{\sigma^*}^{-}(t)), \\ (z_{\sigma^*}^{-})'(t) = A_{\sigma^*}^{-}z_{\sigma^*}^{-}(t) + B_{\sigma^*}^{-}p_{\sigma^*}^{-}\mathbb{K}_{\sigma}^{+}\mathbb{X}(t), \end{array} \right. \qquad (3.15) $$

where $\mathbb{K}_{\sigma}^{+} = (K_{\sigma}^{+}, 0)$ and with the initial conditions

$$ \left\{ \begin{array}{l} \mathbb{X}(0) = \mathbb{X}^0, \\ z_{\sigma^*}^{-}(0) = (\mathrm{Id} - P_{\sigma^*}^{+})z^0. \end{array} \right. $$

From Duhamel's formula, the two first equations in (3.15) also read

$$ \left\{ \begin{array}{l} \mathbb{X}(s) = e^{s\mathbb{A}}\mathbb{X}(0) + \int_{0}^{s} e^{(s-\tau)\mathbb{A}}\mathbb{L}(z_{\sigma^*}^{-}(\tau))d\tau, \\ z_{\sigma^*}^{-}(t) = e^{tA_{\sigma^*}^{-}}z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}^{-}}(B_{\sigma^*}^{-}p_{\sigma^*}^{-}\mathbb{K}_{\sigma}^{+}\mathbb{X}(s)) ds. \end{array} \right. $$
Then, equation (3.16) simply reads

$$ \Phi(z_{\sigma^*}^{-}) = z_{\sigma^*}^{-}. $$

First, we prove that the function $\Phi$ is well defined. Given $g \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$. Since $C$ is bounded, $\mathbb{L}$ is also bounded. Hence using the last relation, (3.3) and (H2.B), we obtain that there exists $\varepsilon'' > 0$ such that

$$ \begin{aligned} \|\Phi(g)(t)\|_{\mathbb{H}} &\le M \int_0^t \int_0^s e^{-(t-s)\sigma^*} e^{-(s-\tau)(\sigma+\varepsilon'')} e^{-\tau\sigma} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau ds \\ &\quad + M \|z^0\|_{\mathbb{H}} \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} ds + e^{-t\sigma^*} \|z^0\|_{\mathbb{H}}. \end{aligned} $$

$$ \text{1 Consequently} \qquad \|\Phi(g)(t)\|_{\mathbb{H}} \le I_1 + I_2 + e^{-t\sigma^*} \|z^0\|_{\mathbb{H}}, \qquad (3.18) $$

where we have set

$$ \begin{aligned} I_1 &:= M \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} \left( \int_0^s e^{-(s-\tau)\varepsilon''} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau \right) ds, \\ I_2 &:= M \|z^0\|_{\mathbb{H}} \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} ds. \end{aligned} $$

Noticing that

$$ \int_0^s e^{-(s-\tau)\varepsilon''} d\tau = \frac{1}{\varepsilon''} [1 - e^{-s\varepsilon''}] \le \frac{1}{\varepsilon''}, $$

we get that

$$ \int_0^s e^{-(s-\tau)\varepsilon''} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau \le \frac{1}{\varepsilon''} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})}, $$

and hence

$$ I_1 \le M \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} \frac{e^{-t\sigma}}{\varepsilon''} \int_0^t e^{(t-s)(\sigma-\sigma^*)} ds. $$

Since

$$ \int_0^t e^{(t-s)(\sigma-\sigma^*)} ds = \frac{1 - e^{-t(\sigma^*-\sigma)}}{(\sigma^*-\sigma)} \le \frac{1}{(\sigma^*-\sigma)}, $$

we have

$$ I_1 \le \frac{Me^{-\sigma t}}{\varepsilon''(\sigma^* - \sigma)} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} $$

and similarly

$$ I_2 \le Me^{-\sigma t} \left( \int_0^t e^{(t-s)(\sigma-\sigma^*)} ds \right) \|z^0\|_{\mathbb{H}} \le \frac{Me^{-\sigma t}}{(\sigma^* - \sigma)} \|z^0\|_{\mathbb{H}}. $$

Using the above estimates in (3.18), we get that

$$ \|\Phi(g)(t)\|_{\mathbb{H}} \le Me^{-\sigma t} \left( \frac{1}{\varepsilon''(\sigma^* - \sigma)} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} + \|z^0\|_{\mathbb{H}} + \frac{1}{\sigma^* - \sigma} \|z^0\|_{\mathbb{H}} \right), $$

2 and hence $\Phi(g) \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$.

It remains to show that $\Phi$ is a contraction mapping. Given $g_1, g_2 \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$, the same calculations as above show that

$$ \|\Phi(g_1) - \Phi(g_2)\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} \le \frac{M}{\varepsilon''(\sigma^* - \sigma)} \|g_1 - g_2\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})}. $$
Substituting the first equation above into the second one yields

$$z_{\sigma^*}^{-}(t) = e^{tA_{\sigma^*}} z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}} B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} \left( e^{sA} \mathbb{X}(0) + \int_{0}^{s} e^{(s-\tau)A} \mathbb{L}(z_{\sigma^*}^{-}(\tau)) d\tau \right) ds. \quad (3.16)$$

Setting

$$Z^0 := e^{tA_{\sigma^*}} z_{\sigma^*}^{-}(0) + \int_{0}^{t} e^{(t-s)A_{\sigma^*}} (B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} (e^{sA} \mathbb{X}(0))) ds,$$

relation (3.16) can be written

$$z_{\sigma^*}^{-}(t) = \int_{0}^{t} \int_{0}^{s} e^{(t-s)A_{\sigma^*}^{-}} B_{\sigma^*}^{-} p_{\sigma^*}^{-} \mathbb{K}_{\sigma}^{+} \left( e^{(s-\tau)A} \mathbb{L}(z_{\sigma^*}^{-}(\tau)) d\tau ds \right) + Z^0. \quad (3.17)$$

To prove the stability of $z_{\sigma^*}^{-}$, we prove the existence of a unique solution to (3.17) in a weighted space by using a fixed point argument. More precisely, let us define the following map

$$\Phi : L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-}) \to L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-}) \\ g \mapsto \Phi(g) := \int_{0}^{t} \int_{0}^{s} e^{(t-s)A_{\sigma^{*}}^{-}} B_{\sigma^{*}}^{-} p_{\sigma^{*}}^{-} \mathbb{K}_{\sigma}^{+} e^{(s-\tau)A} \mathbb{L}(g(\tau)) d\tau ds + Z^{0}.$$

Then, equation (3.16) simply reads

$$\Phi(z_{\sigma^*}^{-}) = z_{\sigma^*}^{-}.$$

First, we prove that the function $\Phi$ is well defined. Given $g \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-})$. Since $C$ is bounded, $\mathbb{L}$ is also bounded. Hence, using the definition of $\Phi$, (H2.B) (boundedness of $B$), (3.3) (exponential decay of $A_{\sigma^*}$) and the exponential decay of $\mathbb{A}$, we obtain that there exists $\varepsilon'' > 0$ such that

$$\|\Phi(g)(t)\|_{\mathbb{H}} \le M \int_0^t \int_0^s e^{-(t-s)\sigma^*} e^{-(s-\tau)(\sigma+\varepsilon'')} e^{-\tau\sigma} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau ds \\ + M \|z^0\|_{\mathbb{H}} \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} ds + e^{-t\sigma^*} \|z^0\|_{\mathbb{H}}.$$

Consequently

$$\|\Phi(g)(t)\|_{\mathbb{H}} \le I_1 + I_2 + e^{-t\sigma^*} \|z^0\|_{\mathbb{H}}, \quad (3.18)$$

where we have set

$$I_1 := M \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} \left( \int_0^s e^{-(s-\tau)\varepsilon''} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau \right) ds, \\ I_2 := M \|z^0\|_{\mathbb{H}} \int_0^t e^{-(t-s)\sigma^*} e^{-s\sigma} ds.$$

Noticing that

$$\int_0^s e^{-(s-\tau)\varepsilon''} d\tau = \frac{1}{\varepsilon''} [1 - e^{-s\varepsilon''}] \le \frac{1}{\varepsilon''},$$

we get that

$$\int_0^s e^{-(s-\tau)\varepsilon''} \|e^{\tau\sigma}g(\tau)\|_{\mathbb{H}} d\tau \le \frac{1}{\varepsilon''} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-})},$$

and hence

$$I_1 \le M \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^{*}}^{-})} \frac{e^{-t\sigma}}{\varepsilon''} \int_0^t e^{(t-s)(\sigma-\sigma^{*})} ds.$$

Since

$$\int_0^t e^{(t-s)(\sigma-\sigma^*)} ds = \frac{1 - e^{-t(\sigma^*-\sigma)}}{(\sigma^*-\sigma)} \le \frac{1}{(\sigma^*-\sigma)},$$
The application $\Phi$ is thus a contraction provided that $\sigma^*$ is chosen large enough to ensure that

$$ \frac{M}{\varepsilon''(\sigma^* - \sigma)} < 1. $$

Then, applying the fixed point theorem we get that there exists a unique $z_{\sigma^*} \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*})$ such that
$\Phi(z_{\sigma^*}) = z_{\sigma^*}$ and

$$ \|z_{\sigma^*}^{-}(t)\|_{\mathbb{H}} \leq M e^{-\sigma t} \|z^{0}\|_{\mathbb{H}}. $$

Moreover, going back to the first equation in (3.15), and using Duhamel's formula again, we easily obtain that

$$ \|X(t)\|_{\mathbb{H} \times \mathbb{H}} \leq M e^{-\sigma t} \|z^0\|_{\mathbb{H}}. $$

1. This completes the proof of Theorem 1.2.

## 4 Stabilization of the reaction-diffusion equation

3. Let $\Omega \subset \mathbb{R}^N$ ($N \ge 1$) be a bounded domain of class $C^{1,1}$. In this section, we apply Theorem 1.1 and Theorem 1.2 for the stabilization of the heat equation. Let us consider $\Gamma$ a non-empty open subset of $\partial\Omega$ and the control problem:

4.
5.

$$ \left\{ \begin{array}{ll} \partial_t z(t,x) - \nabla \cdot (b \nabla z(t,x)) - cz(t,x) = 0 & \text{in } (0, \infty) \times \Omega, \\ z(t,x) = v(t,x) & \text{on } (0, \infty) \times \Gamma, \\ z(t,x) = 0 & \text{on } (0, \infty) \times (\partial\Omega \setminus \Gamma), \\ z(0, \cdot) = z^0 & \text{in } \Omega, \\ y(t,x) = 1_O z(t,x) & \text{in } (0, \infty) \times \Omega, \end{array} \right. \qquad (4.1) $$

where $b, c \in L^\infty(\Omega)$ and $\mathcal{O}$ an open subset of $\mathbb{R}^N$ with $\bar{\mathcal{O}} \subset \Omega$. In order to write (4.1) under the form (1.3), we introduce the following functional setting:

$$ \mathbb{H} = L^2(\Omega), \quad \mathbb{U} = L^2(\Gamma), $$

$$ Az = \nabla \cdot (b \nabla z) + cz, \quad \mathcal{D}(A) = H^2(\Omega) \cap H_0^1(\Omega), $$

The operator $(A, \mathcal{D}(A))$ generates an analytical semigroup, this is a direct consequence of [3, Theorem 2.12, p.115]. Thus (H1.A) is satisfied. Moreover, the operator $A$ is self-adjoint, then in particular, we see that (H2.A) holds true. To define the control operator $B$, we use a standard method (see, for instance [21, pp.341-343] or [17]): we first consider the lifting operator $D_0 \in \mathcal{L}(L^2(\partial\Omega); L^2(\Omega))$ such that for any $v \in L^2(\partial\Omega)$, $w = D_0v$ is the unique solution of the following system

$$ \left\{ \begin{array}{ll} \mu_0 w - \nabla \cdot (b \nabla w) - cw = 0 & \text{in } \Omega, \\ w = v & \text{on } \partial \Omega, \end{array} \right. $$

where $\mu_0 \in \rho(A)$. Then, we set

$$ B = (\mu_0 \mathrm{Id} - A)D_0 : \mathbb{U} \longrightarrow (\mathcal{D}(A^*))', $$

where we have extended the operator $A$ as an operator from $L^2(\Omega)$ into $(\mathcal{D}(A^*))'$ and where we see $\mathbb{U}$ as a closed subspace of $L^2(\partial\Omega)$ (by extending by zero in $\partial\Omega \setminus \Gamma$ any $v \in \mathbb{U}$). Using standard results on elliptic equations, we have that $B$ satisfies (H1.B) for any $\gamma > 3/4$ (see for instance [16, Theorem 2.6]). We set also

$$ \mathbb{Y} = L^2(\mathcal{O}), \quad C = 1_O, $$

we see that $C \in \mathcal{L}(\mathbb{H}, \mathbb{Y})$. Let us recall how we can reduce (4.1) to an evolution problem (1.3). We set $\tilde{z} = z - w$, with $w = D_0v$. Then $\tilde{z}$ satisfies the system

$$ \left\{ \begin{array}{ll} \partial_t \tilde{z} - \nabla \cdot (b \nabla \tilde{z}) - c \tilde{z} = - \partial_t w + \mu_0 w & \text{in } (0, \infty) \times \Omega, \\ \tilde{z} = 0 & \text{on } (0, \infty) \times \partial \Omega, \\ \tilde{z}(0, \cdot) = \tilde{z}^0 := z^0 - w(0, \cdot) & \text{in } \Omega. \end{array} \right. $$
we have

$$I_1 \le \frac{Me^{-\sigma t}}{\varepsilon''(\sigma^* - \sigma)} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})},$$

and similarly

$$I_2 \le Me^{-\sigma t} \left( \int_0^t e^{(t-s)(\sigma-\sigma^*)} ds \right) \|z^0\|_{\mathbb{H}} \le \frac{Me^{-\sigma t}}{(\sigma^* - \sigma)} \|z^0\|_{\mathbb{H}}.$$

Using the above estimates in (3.18), we get that

$$\|\Phi(g)(t)\|_{\mathbb{H}} \le Me^{-\sigma t} \left( \frac{1}{\varepsilon''(\sigma^* - \sigma)} \|g\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} + \|z^0\|_{\mathbb{H}} + \frac{1}{\sigma^* - \sigma} \|z^0\|_{\mathbb{H}} \right),$$

and hence $\Phi(g) \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$.

It remains to show that $\Phi$ is a contraction mapping. Given $g_1, g_2 \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$, the same calculations as above show that

$$\|\Phi(g_1) - \Phi(g_2)\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})} \le \frac{M}{\varepsilon''(\sigma^* - \sigma)} \|g_1 - g_2\|_{L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})}.$$

The application $\Phi$ is thus a contraction provided that $\sigma^*$ is chosen large enough to ensure that

$$\frac{M}{\varepsilon''(\sigma^* - \sigma)} < 1. \qquad (3.19)$$

Then, applying the fixed point theorem we get that there exists a unique $z_{\sigma^*}^{-} \in L_{\sigma}^{\infty}(0, \infty; \mathbb{H}_{\sigma^*}^{-})$ such that $\Phi(z_{\sigma^*}^{-}) = z_{\sigma^*}^{-}$ and

$$\|z_{\sigma^*}^{-}(t)\|_{\mathbb{H}} \le Me^{-\sigma t} \|z^0\|_{\mathbb{H}}.$$

Moreover, going back to the first equation in (3.15), and using Duhamel's formula again, we easily obtain that

$$\|X(t)\|_{\mathbb{H} \times \mathbb{H}} \le Me^{-\sigma t} \|z^0\|_{\mathbb{H}}.$$

This completes the proof of Theorem 1.2.

# 4 Stabilization of the reaction-diffusion equation by an infinite dimensional observer

In this section, we apply Theorem 1.1 for the stabilization of the heat equation. Let $\Omega \subset \mathbb{R}^N$ ($N \ge 1$) be a bounded domain with smooth boundary. Let us consider $\Gamma$ a non-empty open subset of $\partial\Omega$ and the control problem:

$$\left\{
\begin{array}{ll}
\partial_t z(t, x) - \Delta z(t, x) - cz(t, x) = 0 & \text{in } (0, \infty) \times \Omega, \\
z(t, x) = v(t, x) & \text{on } (0, \infty) \times \Gamma, \\
z(t, x) = 0 & \text{on } (0, \infty) \times (\partial\Omega \setminus \Gamma), \\
z(0, \cdot) = z^0 & \text{in } \Omega, \\
y(t, x) = \mathbb{1}_{\mathcal{O}} z(t, x) & \text{in } (0, \infty) \times \Omega,
\end{array}
\right.
\qquad (4.1)$$

where $c \in L^{\infty}(\Omega)$, $\mathcal{O}$ a non empty open subset of $\mathbb{R}^N$, with $\bar{\mathcal{O}} \subset \Omega$. We assume that $z^0 \in L^2(\Omega)$. In order to write (4.1) under the form (1.3), we introduce the following functional setting:

$$\mathbb{H} = L^2(\Omega), \quad \mathbb{U} = L^2(\Gamma), \\
Az = \Delta z + cz, \quad \mathcal{D}(A) = H^2(\Omega) \cap H_0^1(\Omega), \\
\mathbb{Y} = L^2(\mathcal{O}), \quad C = \mathbb{1}_{\mathcal{O}}, \quad C \in \mathcal{L}(\mathbb{H}, \mathbb{Y}).$$
Using the Duhamel formula, we have

$$ \tilde{z}(t) = e^{tA} \tilde{z}^0 + \int_0^t e^{(t-s)A} (-\partial_t w(s) + \mu_0 w(s)) ds. $$

By integrating by parts, we obtain

$$ z(t) = e^{tA} z^0 + \int_0^t e^{(t-s)A} (\mu_0 \mathrm{Id} - A) w(s) ds, $$

that is

$$ \begin{cases} z'(t) = Az(t) + Bv(t), & z(0) = z^0, \\ y(t) = Cz(t). & \end{cases} $$

To apply Theorem 1.1, we only need to check (H2.D). We recall that

$$ \mathcal{D}(A^*) = H^2(\Omega) \cap H_0^1(\Omega), \quad A^* = A. $$

Moreover, by classical results (see [21, Proposition 10.6.7]), we see that

$$ D_0^* := -\frac{\partial}{\partial \nu} (\mu_0 \mathrm{Id} - A^*)^{-1} = -\frac{\partial}{\partial \nu} (\mu_0 \mathrm{Id} - A)^{-1}, $$

and thus

$$ B^*\varepsilon := -\frac{\partial\varepsilon}{\partial\nu}|_{\Gamma}. $$

Thus if $\varepsilon$ satisfies $A^*\varepsilon = \lambda\varepsilon$ and $B^*\varepsilon = 0$, then

$$ \left\{ \begin{array}{ll} \lambda\varepsilon - \nabla \cdot (b\nabla\varepsilon) - c\varepsilon = 0 & \text{in } \Omega, \\ \varepsilon = 0 & \text{on } \partial\Omega, \\ \frac{\partial\varepsilon}{\partial\nu} = 0 & \text{on } \Gamma. \end{array} \right. $$

From standard results on the unique continuation of the Laplace operator (see for instance [9, Theorem 5.3.1, p.125]), we deduce that $\varepsilon = 0$. In the other hand, if $\varepsilon$ satisfies $A\varepsilon = \lambda\varepsilon$ and $C\varepsilon = 0$, then

$$ \left\{ \begin{array}{ll} \lambda\varepsilon - \nabla \cdot (b\nabla\varepsilon) - c\varepsilon = 0 & \text{in } \Omega, \\ \varepsilon = 0 & \text{on } \partial\Omega, \\ \varepsilon = 0 & \text{in } \mathcal{O}. \end{array} \right. $$

$$ \left\{ \begin{array}{ll} \partial_t \hat{z} - \nabla \cdot (b \nabla \hat{z}) - c \hat{z} & = \displaystyle\sum_{i=1}^{N_\sigma} \langle 1_O(\hat{z} - z), w_i^* \rangle_Y \chi_i \quad \text{in } (0, \infty) \times \Omega, \\ \hat{z} & = \displaystyle\sum_{i=1}^{N_\sigma} \langle \hat{z}, \zeta_i \rangle_{\mathbb{H}} w_i \quad \text{on } (0, \infty) \times \Gamma, \\ \hat{z} & = 0 \quad \text{on } (0, \infty) \times (\partial \Omega \setminus \Gamma), \\ \hat{z}(0, \cdot) & = 0 \quad \text{in } \Omega, \end{array} \right. \quad (4.2) $$

<footnote n="1">1 We obtain again from standard results on the unique continuation of the Laplace operator (see for instance [9,</footnote>

<footnote n="2">2 Theorem 5.3.1, p.125]), we get also that $\varepsilon = 0$. Thus (H2.D) holds for any $\sigma$.</footnote>

<footnote n="3">3 Now, we define the observer $\hat{z}$ using Remark 2.3. Let us define $N_\sigma$ by (2.10) and $\hat{z}$ the solution of the closed</footnote>

<footnote n="4">4 loop system</footnote>

<footnote n="5">5 where $(w_i^*)_{1 \le i \le N_\sigma} \subset \mathbb{Y}$, $(\chi_i)_{1 \le i \le N_\sigma} \subset \mathbb{H}$, $(\zeta_i)_{1 \le i \le N_\sigma} \subset \mathcal{D}(A^*)$ and $(w_i)_{1 \le i \le N_\sigma} \subset \mathbb{U}$ (see equations (2.15)-(2.16)</footnote>

<footnote n="6">6 and Remark 2.3). We deduce the following result by applying Theorem 1.1:</footnote>
The operator $(A, \mathcal{D}(A))$ generates an analytical semigroup with compact resolvent, this is a direct consequence of [4, Theorem 2.12, p.115]. Thus (H1.A) is satisfied.

To define the control operator $B$, we use a standard method (see, for instance [24, pp.341-343] or [19]): we first consider the lifting operator $D_0 \in \mathcal{L}(L^2(\partial\Omega); L^2(\Omega))$ such that for any $v \in L^2(\partial\Omega)$, $w = D_0v$ is the unique solution of the following system

$$ \left\{ \begin{array}{ll} \mu_0 w - \Delta w - cw = 0 & \text{in } \Omega, \\ w = v & \text{on } \partial\Omega, \end{array} \right. $$

where $\mu_0 \in \rho(A)$. Let us recall how we can reduce (4.1) to an evolution problem (1.3). We set $\tilde{z} = z - w$, with $w = D_0v$. Then $\tilde{z}$ is the weak solution of the system

$$ \left\{ \begin{array}{ll} \partial_t \tilde{z} - \Delta \tilde{z} - c \tilde{z} = - \partial_t w + \mu_0 w & \text{in } (0, \infty) \times \Omega, \\ \tilde{z} = 0 & \text{on } (0, \infty) \times \partial \Omega, \\ \tilde{z}(0, \cdot) = \tilde{z}^0 := z^0 - w(0, \cdot) & \text{in } \Omega. \end{array} \right. $$

Using Duhamel's formula, we have

$$ \tilde{z}(t) = e^{tA} \tilde{z}^0 + \int_0^t e^{(t-s)A} (-\partial_t w(s) + \mu_0 w(s)) ds. $$

By integrating by parts, we obtain

$$ z(t) = e^{tA} z^0 + \int_0^t e^{(t-s)A} (\mu_0 \mathrm{Id} - A) w(s) ds. \quad (4.2) $$

**Remark 4.1.** The above computations are valid only for smooth solutions. It is worth noticing that formula (4.2) makes sense for $v \in L^2(\partial\Omega)$. In fact, to handle the general case, we refer to [24, pp.342] and the definition of weak solutions therein.

We get finally that the problem (4.1) can be rewritten as

$$ \left\{ \begin{array}{ll} z'(t) = Az(t) + Bv(t), & t > 0, \quad z(0) = z^0, \\ y(t) = Cz(t), & \end{array} \right. $$

with

$$ B = (\mu_0 \mathrm{Id} - A)D_0 : \mathbb{U} \to (\mathcal{D}(A^*))', $$

where we have extended the operator $A$ as an operator from $L^2(\Omega)$ into $(D(A^*))'$ and where we see $\mathbb{U}$ as a closed subspace of $L^2(\partial\Omega)$ (by extending by zero in $\partial\Omega \setminus \Gamma$ any $v \in \mathbb{U}$). Using standard results on elliptic equations, we have that $B$ satisfies (H1.B) for any $\gamma > 3/4$ (see for instance [18, Theorem 2.6]).

To apply Theorem 1.1, we only need to check (H1.D). We recall that

$$ \mathcal{D}(A^*) = H^2(\Omega) \cap H_0^1(\Omega), \quad A^* = A. $$

Moreover, by classical results (see [24, Proposition 10.6.7]), we see that

$$ D_0^* := -\frac{\partial}{\partial \nu} (\mu_0 \mathrm{Id} - A^*)^{-1} = -\frac{\partial}{\partial \nu} (\mu_0 \mathrm{Id} - A)^{-1}, $$

and thus

$$ B^* \varepsilon := - \frac{\partial \varepsilon}{\partial \nu} |_{\Gamma}. $$

Thus if $\varepsilon$ satisfies $A^*\varepsilon = \lambda\varepsilon$ and $B^*\varepsilon = 0$, then

$$ \left\{ \begin{array}{ll} \lambda\varepsilon - \Delta\varepsilon - c\varepsilon = 0 & \text{in } \Omega, \\ \varepsilon = 0 & \text{on } \partial\Omega, \\ \frac{\partial\varepsilon}{\partial\nu} = 0 & \text{on } \Gamma. \end{array} \right. $$
1 Theorem 4.1. Assume $\sigma > 0$. There exists a control

$$v(t) = \sum_{i=1}^{N_{\sigma}} \left( \int_{\Omega} \hat{z}(t) \zeta_i \, dx \right) v_i, \quad (4.3)$$

2 with $\zeta_k \in H^2(\Omega) \cap H_0^1(\Omega)$, $v_k \in H^{1/2}(\Gamma)$, $k = 1, \dots, N_\sigma$ such that the coupled system (4.1) and (4.2) is

3 exponentially stable that satisfies for $z^0 \in L^2(\Omega)$ the estimate

$$\|z(t)\|_{L^2(\Omega)} \le C e^{-\sigma t} \|z^0\|_{L^2(\Omega)}. \quad (4.4)$$

4 Acknowledgments.

5 The authors were supported by the ANR research project ANR ODISSE (ANR-19-CE48-0004-01).

6 References

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The unique continuation property is a consequence of Holmgren's uniqueness theorem (see [11, Theorem 3.5.1, p.125]), by using an extension of the domain. On the other hand, if $\varepsilon$ satisfies $A\varepsilon = \lambda\varepsilon$ and $C\varepsilon = 0$, then

$$ \left\{ \begin{array}{ll} \lambda\varepsilon - \Delta\varepsilon - c\varepsilon = 0 & \text{in } \Omega, \\ \varepsilon = 0 & \text{on } \partial\Omega, \\ \varepsilon = 0 & \text{in } \mathcal{O}. \end{array} \right. $$

Applying Holmgren's theorem again, we obtain $\varepsilon = 0$. Thus (H1.D) holds for any $\sigma$.

Now, we define the observer $\tilde{z}$ using Remark 2.3. Let us define $N_\sigma$ by (2.10) and $\tilde{z}$ the solution of the closed loop system

$$ \left\{ \begin{array}{ll} \partial_t \tilde{z} - \Delta \tilde{z} - c \tilde{z} = & \displaystyle \sum_{i=1}^{N_\sigma} \langle 1_{\mathcal{O}}(\tilde{z} - z), w_i^* \rangle_{\mathbb{Y}} \chi_i \quad \text{in } (0, \infty) \times \Omega, \\ \\ \tilde{z} = & \displaystyle \sum_{i=1}^{N_\sigma} \langle \tilde{z}, \zeta_i \rangle_{\mathbb{H}} w_i \quad \text{on } (0, \infty) \times \Gamma, \\ \\ \tilde{z} = 0 & \displaystyle \quad \text{on } (0, \infty) \times (\partial \Omega \setminus \Gamma), \\ \\ \tilde{z}(0, \cdot) = 0 & \displaystyle \quad \text{in } \Omega, \end{array} \right. \quad (4.3) $$

where $(w_i^*)_{1\le i \le N_\sigma} \subset \mathbb{Y}$, $(\chi_i)_{1\le i \le N_\sigma} \subset \mathbb{H}$, $(\zeta_i)_{1\le i \le N_\sigma} \subset \mathcal{D}(A^*)$ and $(w_i)_{1\le i \le N_\sigma} \subset \mathbb{U}$ (see equations (2.15)-(2.16) and Remark 2.3). We deduce the following result by applying Theorem 1.1:

**Theorem 4.2.** Assume $\sigma > 0$. There exists a control

$$ v(t) = \sum_{i=1}^{N_\sigma} \left( \int_{\Omega} \tilde{z}(t) \zeta_i \, dx \right) w_i, \quad (4.4) $$

with $\zeta_i \in H^2(\Omega) \cap H_0^1(\Omega)$, $w_i \in H^{1/2}(\Gamma)$, $i = 1, \dots, N_\sigma$ such that the coupled system (4.1) and (4.3) is exponentially stable that satisfies for $z^0 \in L^2(\Omega)$ the estimate

$$ \|z(t)\|_{L^2(\Omega)} \le C e^{-\sigma t} \|z^0\|_{L^2(\Omega)}. \quad (4.5) $$

**Remark 4.3.** In dimension 2, we can obtain the same result for the operator

$$ Az = \nabla \cdot (b\nabla z) + cz, \quad \mathcal{D}(A) = H^2(\Omega) \cap H_0^1(\Omega), $$

with b a positive-definite symmetric matrix of variable coefficients in $L^\infty(\Omega)$, using the unique continuation property from [1, Theorem 3.2].

# 5 Stabilization of the reaction-diffusion equation by a finite dimensional observer

Let $\Omega$ be a domain of $\mathbb{R}^N$ ($N \ge 1$) with smooth boundary. In this section, we apply Theorem 1.2 to the following initial boundary value problem

$$ \left\{ \begin{array}{ll} \partial_t z(t,x) - \Delta z(t,x) - cz(t,x) & = \mathbb{1}_{\mathcal{O}'} v(t,x) \quad \text{in } (0, \infty) \times \Omega, \\ z(t,x) & = 0 \quad \text{on } (0, \infty) \times \partial \Omega, \\ z(0, \cdot) & = z^0 \quad \text{in } \Omega, \\ y(t,x) & = \mathbb{1}_{\mathcal{O}} z(t,x) \quad \text{in } (0, \infty) \times \Omega, \end{array} \right. \quad (5.1) $$

where $\mathcal{O}$ and $\mathcal{O}'$ are two non empty open sets such that $\bar{\mathcal{O}} \subset \Omega$ and $\bar{\mathcal{O}}' \subset \Omega$. In this case, we have

$$ \mathbb{H} = L^2(\Omega), \quad \mathbb{U} = L^2(\mathcal{O}'), $$
1

[14] K. RAMDANI, M. TUCSNAK, AND J. VALEIN, *Detectability and state estimation for linear age-structured population diffusion models*, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 1731–1761.

2

3

[15] K. RAMDANI, J. VALEIN, AND J.-C. VIVALDA, *Adaptive observer for age-structured population with spatial diffusion*, North-Western European Journal of Mathematics, 4 (2018), pp. 39–58.

4

5

[16] J.-P. RAYMOND, *Feedback boundary stabilization of the two-dimensional Navier-Stokes equations*, SIAM J. Control Optim., 45 (2006), pp. 790–828.

6

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[17] ——, *Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition*, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 1537–1564.

8

10

[18] J.-P. RAYMOND AND L. THEVENET, *Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers*, Discrete Contin. Dyn. Syst., 27 (2010), pp. 1159–1187.

11

[19] L. THEVENET, *Lois de feedback pour le contrôle d'écoulements*, PhD thesis, Université de Toulouse, 2009.

12

[20] L. THEVENET, J.-M. BUCHOT, AND J.-P. RAYMOND, *Nonlinear feedback stabilization of a two-dimensional Burgers equation*, ESAIM, Control Optim. Calc. Var., 16 (2010), pp. 929–955.

13

14

[21] M. TUCSNAK AND G. WEISS, *Observation and Control for Operator Semigroups*, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2009.

15
$$Az = \Delta z + cz, \quad \mathcal{D}(A) = H^2(\Omega) \cap H_0^1(\Omega).$$

The operator A is self-adjoint with compact resolvent, then in particular, we see that (H2.A) holds true. Moreover,

$$B: \mathbb{U} \to \mathbb{H}, \quad B = \mathbb{1}_{\mathcal{O}'},$$

and

$$\mathbb{Y} = L^2(\mathcal{O}), \quad C: \mathbb{H} \to \mathbb{Y}, \quad C = \mathbb{1}_{\mathcal{O}}.$$

We can check as the previous section that (H2.B), (H2.C) and (H2.D) are verified. Let fix $\sigma > 0$ and $\sigma^* > \sigma$. The hypothesis (H2.A) implies in particular that A is diagonalizable. Hence

$$\mathbb{H}_{\sigma}^{+} = \bigoplus_{\lambda \in \Sigma_{\sigma}^{+}} \ker(A - \lambda \mathrm{Id}), \quad \mathbb{H}_{\sigma^{*}}^{+} = \bigoplus_{\lambda \in \Sigma_{\sigma^{*}}^{+}} \ker(A - \lambda \mathrm{Id}).$$

The map $P_{\sigma}^{+}$ (resp. $P_{\sigma^{*}}^{+}$) is then the orthogonal projection on $\mathbb{H}_{\sigma}^{+}$ (resp. $\mathbb{H}_{\sigma^{*}}^{+}$) spanned by the eigenvectors associated to $\Sigma_{\sigma}^{+}$ (resp. $\Sigma_{\sigma^{*}}^{+}$).

Thus $A_{\sigma}^{+}$ (resp. $A_{\sigma^{*}}^{+}$) can be viewed as the endomorphism on $\mathbb{H}_{\sigma}^{+}$ (resp. $\mathbb{H}_{\sigma^{*}}^{+}$) associated to the diagonal matrix of eigenvalues $\Sigma_{\sigma}^{+}$ (resp. $\Sigma_{\sigma^{*}}^{+}$).

We define the finite dimensional observer

$$\begin{cases} \hat{z}_{\star}(t) = A_{\sigma^{*}}^{+} \hat{z}_{\star}(t) + \sum_{i=1}^{N_{\sigma}} \langle \hat{z}_{\star}, \zeta_{i} \rangle_{\mathbb{H}} B_{\sigma^{*}}^{+} w_{i} + \sum_{i=1}^{N_{\sigma}} \langle (C_{\sigma^{*}}^{+} \hat{z}_{\star} - y), w_{i}^{*} \rangle_{\mathbb{Y}} \chi_{i}, \\ \hat{z}_{\star}(0) = 0, \end{cases} \quad (5.2)$$

where $(w_i^*)_1 \le i \le N_\sigma \subset \mathbb{Y}$, $(\chi_i)_1 \le i \le N_\sigma \subset \mathbb{H}$, $(\zeta_i)_1 \le i \le N_\sigma \subset \mathcal{D}(A^*)$ and $(w_i)_1 \le i \le N_\sigma \subset \mathbb{U}$ are chosen as the previous section (see equations (2.15)-(2.16) and Remark 2.3).

Using Theorem 1.2, we deduce the following result:

**Theorem 5.1.** Let $\sigma > 0$ and $\sigma^* > \sigma$. There exists a control $v$ based on $\hat{z}_\star$ solution of the system (5.2) where

$$v(t) = \sum_{i=1}^{N_{\sigma}} \left( \int_{\Omega} \hat{z}_{\star}(t) \zeta_{i} \, dx \right) w_{i}, \quad (5.3)$$

with $\zeta_i \in H^2(\Omega) \cap H_0^1(\Omega)$, $w_i \in L^2(\mathcal{O}')$, $i = 1, \dots, N_\sigma$ such that for $\sigma^*$ large enough, the coupled system (5.1) and (5.2) is exponentially stable that satisfies for $z^0 \in L^2(\Omega)$ the estimate

$$\|z(t)\|_{L^2(\Omega)} \le C e^{-\sigma t} \|z^0\|_{L^2(\Omega)}. \quad (5.4)$$

# 6 Conclusion

In this paper, we proposed a constructive method to stabilize a class of linear parabolic evolution systems $z' = Az + Bv$ by means of a finite dimensional control $v$, under the observation $y = Cz$. The control is based on the design of a Luenberger observer which can be infinite dimensional or finite dimensional of dimension large enough (under stronger assumptions). In both cases, we show that if $(A, B)$ and $(A, C)$ verify the Fattorini-Hautus Criterion, then we can construct an observer-based control $v$ of finite dimension such that the closed-loop system is exponentially stable. The results are applied to study the stabilization of the diffusion system with Dirichlet boundary control and an internal observation. Possible future work is to implement this method numerically for the example studied here, or for more involved examples related to population dynamics, in the spirit of [16].

# Acknowledgments.

The authors were supported by the ANR research project ANR ODISSE (ANR-19-CE48-0004-01).
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