# Boundary output feedback stabilization for 2-D and 3-D parabolic equations

Hugo Lhachemi, Ionut Munteanu, Christophe Prieur

► To cite this version:

Hugo Lhachemi, Ionut Munteanu, Christophe Prieur. Boundary output feedback stabilization for 2-D and 3-D parabolic equations. Automatica, 2025, 176 (June), pp.112259. 10.1016/j.automatica.2025.112259. hal-05023673

HAL Id: hal-05023673

https://hal.science/hal-05023673v1

Submitted on 11 Apr 2025

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_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Boundary output feedback stabilization for 2-D and 3-D parabolic equations*

Hugo Lhachemi<sup>a</sup>, Ionut Munteanu<sup>b</sup>, Christophe Prieur<sup>c</sup>

## Abstract

The present paper addresses the topic of boundary output feedback stabilization of parabolic-type equations, governed by linear differential operators which can be diagonalized by the introduction of adequate weighting functions (by means of the Sturm-Liouville method), and which evolve in bounded spatial domains that are subsets of $\mathbb{R}^d$, $d = 2, 3$. Generalizing previous works for the boundary output feedback control of 1-D parabolic PDEs and for the state feedback control of multi-D parabolic PDEs, we report in this paper an output feedback boundary stabilizing control with internal Dirichlet measurements designed by means of a finite-dimensional observer. The reported control design procedure is shown to be systematic for 2-D and 3-D parabolic equations.

**Key words:** second order parabolic equations; exponential asymptotic stabilization; observer design; eigenvalues and eigenfunctions; spectral decomposition; proportional feedback control.

## 1 Introduction

There is now a large amount of literature dealing with boundary control of dynamical infinite-dimensional systems. In particular, various techniques have been developed for the design of control strategies for 1-D partial differential equations (PDEs). These include: Lyapunov methods [8], backstepping design [21], linear quadratic control methods [11], characteristic analysis [9], among other approaches. Even if some of these approaches have been generalized to multi-D PDEs, such as LQR methods (see e.g., [37]), constructive methods for multi-D PDEs are not so developed. Among the contributions,

one can find in [7] the design of simple proportional-type boundary stabilizing controllers for multi-D parabolic-type equations under a restrictive assumption concerning the linear independence of the traces of the normal derivatives of the eigenfunctions. Extensions of this approach while removing the aforementioned restrictive assumption have been reported in the recent textbook [28]. Boundary control and observation of heat equations on multi-dimensional domains has also been solved in [14] under the restrictive assumption that all the unstable modes are simple.

In this paper, we study the problem of output feedback stabilization of 2-D and 3-D parabolic PDEs using spectral reduction methods. Spectral decomposition techniques consist first of the projection of the PDE into a finite-dimensional unstable system plus a residual stable infinite-dimensional system. In this framework, the control strategy is designed on the unstable finite-dimensional part of the plant. Although simple in their basic concepts, spectral-based control design methods are challenging because one must ensure that the control strategy does not introduce any spillover effect, that is: the control strategy originally designed on a finite-dimensional approximation of the PDE plant may actually

<footnote><sup>a</sup> Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France</footnote>

<footnote><sup>b</sup> Faculty of Mathematics, Al. I. Cuza University, Bd. Carol I, 11, Iasi 700506, Romania and O. Mayer Institute of Mathematics, Romanian Academy, Bd. Carol I, 8, Iasi 700505, Romania</footnote>

<footnote><sup>c</sup> Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France</footnote>

<footnote>* The authors would like to thank the anonymous Referees for carefully reading this paper and for pertinent comments which helped a lot to improve the initial submission. The work of C. Prieur is partly supported by MIAI @ Grenoble Alpes (ANR-19-P3IA-0003). This paper was not presented at any IFAC meeting. Corresponding author: I. Munteanu, e-mail: ionut.munteanu@uaic.ro.</footnote>

<footnote>Email addresses: hugo.lhachemi@centralesupelec.fr (Hugo Lhachemi), ionut.munteanu@uaic.ro (Ionut Munteanu), Christophe.Prieur@gipsa-lab.fr (Christophe Prieur).</footnote>
ally fail to stabilize the infinite-dimensional system due to the interconnection of the controller with the infinite-dimensional residual dynamics (see, e.g., [5,4]). Spectral reduction-based methods are very attractive in practice because they allow the design of finite-dimensional control strategies for parabolic PDEs. Their use for control design for infinite-dimensional dynamical systems goes back to the 60s in the work of Russell [31] and to the 80s in the work [35]. See also the works [3] and [2] where finite-dimensional projections are used for the stabilization by means of state-feedback laws. These methods are particularly relevant because they allow the computation of reduced order models, making numerical computations and practical implementations much easier to handle compared to infinite-dimensional control and observation strategies [20] (they avoid the necessity to perform late lumping approximations that generally conduct to the loss of the stability guarantees derived based on the original infinite-dimensionality). For other related ideas developed in various directions, see e.g. [10,25,30] for the one-dimensional case and [6,28,36,38] for the multi-dimensional case. In the case of output-feedback, a pioneer paper on finite-dimensional observer-based stabilizing controllers is [32] for the case where the both observation and control operators are bounded. See also the recent paper [14] where a boundary finite-dimensional control is designed and an infinite-dimensional observer is suggested for a multi-D Laplace operator. Adopting spectral reduction-based representation of parabolic PDEs [10,31] and leveraging on pioneer works [35,32] augmented with Linear Matrix Inequalities (LMIs) procedures [19], stabilization problems for 1-D parabolic PDEs have been solved in a systematic manner for various boundary control and boundary output [23]. Also note that the capability to handle systems of 1-D parabolic PDEs was demonstrated in [17] using small gain methods. In this paper, we further develop and generalize these methods to the case of multi-D parabolic PDEs.

The aim of this paper is to report a constructive control design procedure for the output feedback boundary stabilization of multi-D parabolic PDEs that is systematic in the 2-D and 3-D cases. Extending procedures reported in [23] for 1-D PDEs and combining this latter work with analyses on the eigenspaces inspired by [28], we succeed to design a finite-dimensional observer-based output feedback control strategy for multi-D parabolic-type equations. More precisely, we consider in this work the following boundary-controlled parabolic-type equation evolving in $O$, an open, bounded and connected subset of $\mathbb{R}^d$ with $d \in \{2, 3\}$, with $C^3$-smooth boundary, denoted by $\partial O$. Then, the real-valued system is described by

$$ egin{cases} 
\partial_t z(x,t) + \sum_{i,j=1}^{d} a_{ij}(x) \partial_{ij} z(x,t) + \sum_{i=1}^{d} b_i(x) \partial_i z(x,t) \\ 
+ c(x) z(x,t) = 0, \quad t > 0, \quad x \in O, \\ 
z(x,t) = u(x,t) = \chi(x)v(x,t), \quad x \in \partial O, \quad t > 0, \\ 
z(x,0) = z_o(x), \quad x \in O. 
\end{cases} 
\tag{1} $$

The control input to-be-designed is the function $v$. The function $\chi \in C^2(\partial O)$ is a fixed boundary weight function, with values in $[0, 1]$, with support in a subset $\Gamma_1 \subseteq \partial O$, and equal to 1 in an open non-empty subset $\Gamma_0 \subseteq \Gamma_1$. The assumed regularity of $\chi$ will be used to ensure some regularity properties for the control input $u$.

The system output consists of $M \ge 1$ in-domain measurements:

$$ y(t) = (z(\xi_1, t), z(\xi_2, t), \dots, z(\xi_M, t)), \tag{2} $$

with $\xi_i \in O$ that are mutually distinct. As we shall see, the number of measurements $M$ to be selected will depend on the maximum multiplicity of the unstable eigenvalues of the plant to be stabilized by the control strategy. In this context, our only assumption in the present work is that the second order governing differential operator can be diagonalized in a suitable Riesz-basis (this will be described in details in Section 2 below). We emphasize the fact that the previous results in the literature cannot be applied to non-diagonal systems with eigenvalues that are not simple. Therefore, the present work is a first result into that direction.

The main contribution of this paper can be informally stated as follows (see Theorem 15 for a precise statement)

**Contribution:** In this work, our aim is to show that assuming that the governing linear operator of equation (1) is in divergence form (as written in (4) below), there exists an explicit output feedback controller $v$ (see (19) below with $U$ provided by (32)) which exponentially stabilizes the reaction-diffusion equation (1) based on the pointwise internal measurement (2).

We provide below an example that fits within the above framework, except that $O$ is a square in $\mathbb{R}^2$ hence the $C^3$-smoothness assumption for the boundary $\partial O$ is not satisfied. However, this is compatible with the developments reported in this paper because the $C^3$-smoothness assumption for $\partial O$ is only requested to guarantee the $H^3$-regularity of the eigenfunctions; see the discussion after Equation (5). It appears that for the below example, the eigenfunctions do present this $H^3$-regularity. Note that this choice for Example 1 is motivated by the fact that we know for a square domain the analytical expression
of the eigenfunctions, allowing to explicitly carry on the numerical simulations reported at the end of this paper.

**Example 1** Consider the plant (1) with $d = 2$, $O = (0, \pi) \times (0, \pi)$, $\Gamma_1 = \{x_1 \in (0, \pi), x_2 = 0\}$. Let $\chi(\cdot)$ be a smooth function with support included in $\Gamma_1 \subset \partial O$, as in the discussion following system (1). Consider the equation

$$ egin{cases} 
\partial_t z(x,t) - \Delta z(x,t) - \epsilon \nabla \cdot z(x,t) - az(x,t) = 0, & t > 0, x \in O, \\
z(x,t) = \chi(x)v(x,t), x \in \partial O, t > 0, \\
z(x,0) = z_o(x), x \in O. 
\end{cases} $$

for some $\epsilon > 0$ and $a \in \mathbb{R}$. Denote by $Af = -\Delta f - \epsilon \nabla \cdot f - af$. It can be seen that $A$ is non-diagonal. However, introducing $\mu(x) = e^{\epsilon x_1 + \epsilon x_2}$, which is positive and satisfies $1 \le \mu(x) \le e^{2\epsilon\pi}$, we can diagonalize $A$ in the sense that

$$ \mu Af = - \sum_{i=1}^{2} \partial_i (e^{\epsilon x_1 + \epsilon x_2} \partial_i f) - a e^{\epsilon x_1 + \epsilon x_2} f. $$

The associated eigenfunctions are described by $\varphi_{i,j}(x) = \frac{2}{\pi} e^{-\frac{\epsilon x_1 - \epsilon x_2}{2}} \sin(ix_1) \sin(jx_2)$, $i, j \in \mathbb{N} \setminus \{0\}$, $x \in O$, with the corresponding eigenvalues $\lambda_{i,j} = i^2 + j^2 - a + \epsilon^2/2$. Then, it is seen that the associated bi-orthogonal system is given by $\psi_{i,j}(x) = \frac{2}{\pi} e^{\frac{\epsilon x_1 + \epsilon x_2}{2}} \sin(ix_1) \sin(jx_2)$, $i, j \in \mathbb{N} \setminus \{0\}$, $x \in O$. Let us note that, at the boundary $\Gamma_1$, we have

$$ T_n(\varphi_{i,j})(x_1, 0) = -\frac{2}{\pi} j \sin(ix_1), \quad x_1 \in (0, \pi). $$

Consider the particular case $\epsilon = 0.1$ and $a = 5.1$. In this setting, the open-loop system is unstable with a total of three modes that are not exponentially stable: $-\lambda_{1,1} = 3.095$ and $-\lambda_{1,2} = -\lambda_{2,1} = 0.095$ with the corresponding three eigenfunctions $\frac{2}{\pi} e^{-\frac{0.1x_1-0.1x_2}{2}} \sin x_1 \sin x_2$, $\frac{2}{\pi} e^{-\frac{0.1x_1-0.1x_2}{2}} \sin 2x_1 \sin x_2$ and $\frac{2}{\pi} e^{-\frac{0.1x_1-0.1x_2}{2}} \sin x_1 \sin 2x_2$, respectively. Hence, the unstable part of the system is of dimension $N_0 = 3$ with one unstable mode of multiplicity two. This leads to consider the case of two pointwise measurements, that is (2) with $M = 2$. Choosing the points of observation $\xi_i = (\xi_{i1}, \xi_{i2}) \in (0, \pi)^2$, $i = 1, 2$, such that a certain observability condition holds true (namely, Equation (29) below), Theorem 15 below can be applied to this example to provide a boundary output stabilizing controller.

The outline of the paper is as follows. Various notations, assumptions, and preliminary properties are summarized in Section 2. Then, the proposed control strategy along with the main stability result, is reported in

Section 3. Numerical simulation for Example 1 is developed in Section 4. Concluding remarks are formulated in Section 5.

## 2 Notation and preliminary properties

### 2.1 Notation and basic definitions

We denote by $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$ the set of all positive integers. Spaces $\mathbb{R}^n$ are endowed with the Euclidean scalar product $\langle \cdot, \cdot \rangle_n$ and norm $\| \cdot \|_n$. The associated induced norms of matrices are denoted by $\| \cdot \|$. $L^2(O)$ stands for the space of square Lebesgue integrable functions on $O$ and is endowed with the inner product $\langle f, g \rangle = \int_O f(x)g(x)dx$ with associated norm denoted by $\| \cdot \|_{L^2}$. For $m \ge 1$, the $m$-order fractional Sobolev space is denoted by $H^m(O)$ and is endowed with its usual norm denoted by $\| \cdot \|_{H^m}$. We set $H_0^1(O)$ for the completion of the space of infinitely differentiable functions, which are nonzero only on a compact subset of $O$, with respect to the Sobolev norm $\| \cdot \|_{H^1}$. For a symmetric matrix $P \in \mathbb{R}^{n \times n}$, $P \succeq 0$ (resp. $P \succ 0$) means that $P$ is positive semi-definite (resp. positive definite). For a positive semi-definite matrix $P$, one can introduce its positive semi-definite square root matrix, that is a matrix, denoted by $P^{\frac{1}{2}}$, such that

$$ P^{\frac{1}{2}} P^{\frac{1}{2}} = P. $$

Let $\{\varphi_n\}$, $n \in \mathbb{N}^*$ be a sequence in a Hilbert space $(H, \langle \cdot, \cdot \rangle_H, \| \cdot \|_H)$. It is called a Riesz basis if: (i) $\text{span}\{\varphi_n\} = H$; and (ii) there exist constants $0 < c \le C < \infty$ such that

$$ c \sum_{n \ge 1} |\alpha_n|^2 \le \left\| \sum_{n \ge 1} \alpha_n \varphi_n \right\|_H^2 \le C \sum_{n \ge 1} |\alpha_n|^2, $$

for all sequences of scalars $\{\alpha_n\}_n$ so that $\sum_{n \ge 1} |\alpha_n|^2 < \infty$.

For any given positive function $\mu$ in $C(O)$, we introduce the weighted Lebesgue space

$$ L_{\mu}^{2}(O) = \left\{ f : O \to \mathbb{R} : \int_{O} \mu(x) f^{2}(x) dx < \infty \right\}. $$

If there exist constants $0 < \mu_m, \mu_M < \infty$ such that $0 < \mu_m \le \mu(x) \le \mu_M$ almost everywhere, then the two spaces $L^2(O)$ and $L^2_\mu(O)$ are both algebraically and topologically equivalent. This implies, in particular, that a Riesz basis in $L^2(O)$ is also a Riesz basis in $L^2_\mu(O)$ and vice versa.
## 2.2 Differential operator in divergence form

Let us denote by $\mathcal{A}: H^2(\mathcal{O}) \to L^2(\mathcal{O})$ the second order differential operator:

$$ \mathcal{A}f = \sum_{i,j=1}^{d} a_{ij}(x)\partial_{ij}f + \sum_{i=1}^{d} b_i(x)\partial_i f + c(x)f. \quad (3) $$

**Assumption 2** We assume that there exists a multiplier $\mu \in C^2(\bar{\mathcal{O}})$, with $0 < \mu_m \le \mu(x) \le \mu_M$ for all $x \in \mathcal{O}$ for some constants $0 < \mu_m, \mu_M < \infty$, such that $\mu\mathcal{A}$ can be rewritten in divergence form:

$$ \mu\mathcal{A}f = -\sum_{i=1}^{d} \partial_i(\tilde{a}_i(x)\partial_i f) + \tilde{c}(x)f \quad (4) $$

with $C^1(\bar{\mathcal{O}})$-smooth coefficients $\tilde{a}_i$, $\tilde{c}$, for which there exist constants $0 < \tilde{a}_m < \tilde{a}_M$ and $\tilde{c}_m < \tilde{c}_M$ such that $0 < \tilde{a}_m \le \tilde{a}_i(x) \le \tilde{a}_M$ and $\tilde{c}_m \le \tilde{c}(x) \le \tilde{c}_M$ for all $x \in \bar{\mathcal{O}}$ and all $1 \le i \le d$.

**Remark 3** We emphasize that the form (4) can be seen as a generalization of Sturm-Liouville operators from the one-dimensional case. For details, see e.g. [12]. Similarly with the one-dimensional case, the spectrum of the operator $\mu\mathcal{A}$ consists of a countable set of eigenvalues with a corresponding countable set of eigenfunctions that form a Riesz basis (see below for details). However, output feedback control design is challenging in the multi-dimensional case. Compared to the one-dimensional case, the multi-dimensional case requires to reshape the form of the output feedback controller. This is essentially due to the fact that, in the multi-dimensional case, we may encountered unstable eigenvalues with multiplicity greater than one (note however that, for the studied class of systems, the multiplicity of the eigenvalues is always finite). This also requires the demonstration of a unique continuation property for the eigenfunctions, the definition of suitable lifting operators to embrace the geometry of the boundary, and the introduction of an adequate observer, among other modifications.

Previous assumption implies that $-\mathcal{A}_0 := -\mathcal{A}|_{H^2(\mathcal{O}) \cap H_0^1(\mathcal{O})}$ is the generator of a $C_0$-analytic semigroup in $L^2(\mathcal{O})$. To show this, we apply the well-known Hille-Yosida theorem. First, $\mathcal{D}(\mathcal{A}_0) = H^2(\mathcal{O}) \cap H_0^1(\mathcal{O})$ is dense in $L^2(\mathcal{O})$. Then, for $\lambda > 0$, we consider the equation

$$ (\lambda + \mathcal{A}_0)f = g. $$

It follows, by scalarly multiplying this equation by $\mu f$ that

$$ \begin{aligned} \langle g, \mu f \rangle &= \langle (\lambda + \mathcal{A}_0)f, \mu f \rangle \\ &= \lambda \langle \mu f, f \rangle + \langle \mu \mathcal{A}_0 f, f \rangle \\ &\ge (\lambda \mu_m + \tilde{c}_m) \|f\|_{L^2(\mathcal{O})}^2 + \tilde{a}_m \|\nabla f\|_{L^2(\mathcal{O})}^2. \end{aligned} $$

Hence, for $\lambda$ large enough, we have $\lambda\mu_m + \tilde{c}_m > 0$ and

$$ (\lambda\mu_m + \tilde{c}_m)\|f\|_{L^2(\mathcal{O})}^2 \le \mu_M \|g\|_{L^2(\mathcal{O})} \|f\|_{L^2(\mathcal{O})} $$

or, equivalently,

$$ \|(\lambda + \mathcal{A}_0)^{-1}g\|_{L^2(\mathcal{O})} = \|f\|_{L^2(\mathcal{O})} \le \frac{\mu_M}{\lambda\mu_m + \tilde{c}_m} \|g\|_{L^2(\mathcal{O})}. $$

Thus, $-\mathcal{A}_0$ is the generator of a $C_0$-analytic semigroup $\{e^{-t\mathcal{A}_0}: t \ge 0\}$ in $L^2(\mathcal{O})$.

Besides this, the above inequalities, together with the compact embedding of $H_0^1(\mathcal{O})$ in $L^2(\mathcal{O})$, implies the compactness of the resolvent of $-\mathcal{A}_0$. Therefore, $-\mathcal{A}_0$ has a countable set of eigenvalues, which accumulates at infinity, and for which the corresponding eigenfunctions form a Riesz basis in $L^2(\mathcal{O})$. More exactly, in view of the divergence form (4), we consider the following weighted eigenvalue problem ($\lambda \in \mathbb{C}$):

$$ -\sum_{i=1}^{d} \partial_i(\tilde{a}_i \partial_i \varphi) + \tilde{c}\varphi = \mu\lambda\varphi, \quad x \in \mathcal{O}; \quad \varphi = 0 \text{ on } \partial\mathcal{O}. $$

Owing to classical theory on spectral properties of elliptic self-adjoint operators (see, e.g. [16, Chapter 8]) we know that the above problem has a countable set of solutions formed by an increasing sequence of real eigenvalues $\{\lambda_n\}_{n \in \mathbb{N}^*}$ which accumulate at infinity and with corresponding eigenfunctions $\{\varphi_n\}_{n \in \mathbb{N}^*}$ that form an orthonormal basis in $L^2_\mu(\mathcal{O})$. By the above discussions, we know that $\{\varphi_n\}_{n \in \mathbb{N}^*}$ forms a Riesz basis in $L^2(\mathcal{O})$ as well (but is not necessarily orthonormal). Let us define $\psi_n := \mu\varphi_n$, $n \ge 1$. Then $\{\varphi_n\}_{n \in \mathbb{N}^*}$ and $\{\psi_n\}_{n \in \mathbb{N}^*}$ are bi-orthonormal in $L^2(\mathcal{O})$, i.e.,

$$ \langle \varphi_i, \psi_j \rangle = \delta_{i,j}, \quad i, j \ge 1, $$

where $\delta_{i,j}$ is the Kronecker symbol. In particular, there exist constants $c_1, c_2 > 0$ such that

$$ c_1 \sum_{n \ge 1} \langle f, \psi_n \rangle^2 \le \|f\|_{L^2(\mathcal{O})}^2 \le c_2 \sum_{n \ge 1} \langle f, \psi_n \rangle^2, \quad (5) $$

for all $f \in L^2(\mathcal{O})$. Besides this, since the coefficients $\tilde{a}_i, \tilde{c}$ are $C^1$-smooth and the boundary is $C^3$-smooth, it follows by [16, Theorem 8.13] that $\varphi_n \in H^3(\mathcal{O})$ for all $n$. Invoking the general Sobolev embeddings, we deduce that $\varphi_n$ belongs to the Hölder space $C^{0,\alpha}(\mathcal{O})$, for some $\alpha \in (0, 1)$. Therefore, it makes sense to consider point values of $\varphi_n$.

The following lemma, which is the key ingredient for the introduction of the Lyapunov functional in the proof of our main result stated in Theorem 15, provides a direct relation between the $H_0^1$-norm and the coefficients of projection onto the Riesz basis $\{\phi_n\}_n$.
**Lemma 4** Let us fix $\nu \ge 0$ so that $\tilde{c}_m + \nu\mu_m > 0$. Then, there exist constants $c_3, c_4 > 0$ such that

$$c_3 \|f\|_{H_0^1(\mathcal{O})}^2 \le \sum_{n \ge 1} (\lambda_n + \nu) \langle f, \psi_n \rangle^2 \le c_4 \|f\|_{H_0^1(\mathcal{O})}^2, \quad (6)$$

for all $f \in H^2(\mathcal{O}) \cap H_0^1(\mathcal{O})$.

**Proof.** Recall that, thanks to the Poincaré inequality, we have that $\|\nabla \cdot \mathbf{L}^2(\mathcal{O})\|$ is an equivalent norm in $H_0^1(\mathcal{O})$. In particular, there exists a constant $C > 0$ such that

$$\|\nabla f\|_{\mathbf{L}^2(\mathcal{O})}^2 \ge C \|f\|_{H_0^1(\mathcal{O})}^2, \quad \forall f \in H_0^1(\mathcal{O}).$$

Next, we observe for any $f \in H^2(\mathcal{O}) \cap H_0^1(\mathcal{O})$ the following:

$$\begin{aligned} \langle (\mathcal{A}_0 + \nu) f, f \rangle_{L^2_{\mu}(\mathcal{O})} &= \int_{\mathcal{O}} \mu \{ (\mathcal{A}_0 f) f + \nu f^2 \} dx \\ &= -\sum_{i=1}^{d} \int_{\mathcal{O}} \partial_i (\tilde{a}_i \partial_i f) f dx + \int_{\mathcal{O}} (\tilde{c} + \nu\mu) f^2 dx \\ &= \sum_{i=1}^{d} \int_{\mathcal{O}} \tilde{a}_i (\partial_i f)^2 dx + \int_{\mathcal{O}} (\tilde{c} + \nu\mu) f^2 dx \end{aligned}$$

where we have applied integration by parts. This shows that

$$\langle (\mathcal{A}_0 + \nu) f, f \rangle_{L^2_{\mu}(\mathcal{O})} \ge (\tilde{c}_m + \nu\mu_m) \|f\|_{L^2(\mathcal{O})}^2,$$

and so $\lambda_n > -\nu$ for all $n \ge 1$; and that

$$C \tilde{a}_m \|f\|_{H_0^1(\mathcal{O})}^2 \le \|(\mathcal{A}_0 + \nu)^{1/2} f\|_{L^2_{\mu}(\mathcal{O})}^2 \le C_1 \|f\|_{H_0^1(\mathcal{O})}^2.$$

Where, $C_1 = \max(\tilde{a}_M, \tilde{c}_M + \nu\mu_M)$. Then it can be seen from (5) that

$$\begin{aligned} \sum_{n \ge 1} (\lambda_n + \nu) \langle f, \psi_n \rangle^2 &\le \frac{1}{c_1} \|(\mathcal{A}_0 + \nu)^{1/2} f\|_{L^2(\mathcal{O})}^2 \\ &\le \frac{1}{c_1 \mu_m} \|(\mathcal{A}_0 + \nu)^{1/2} f\|_{L^2_{\mu}(\mathcal{O})}^2 \\ &\le \frac{C_1}{c_1 \mu_m} \|f\|_{H_0^1(\mathcal{O})}^2 \end{aligned}$$

while

$$\begin{aligned} \sum_{n \ge 1} (\lambda_n + \nu) \langle f, \psi_n \rangle^2 &\ge \frac{1}{c_2} \|(\mathcal{A}_0 + \nu)^{1/2} f\|_{L^2(\mathcal{O})}^2 \\ &\ge \frac{1}{c_2 \mu_M} \|(\mathcal{A}_0 + \nu)^{1/2} f\|_{L^2_{\mu}(\mathcal{O})}^2 \\ &\ge \frac{C \tilde{a}_m}{c_2 \mu_M} \|f\|_{H_0^1(\mathcal{O})}^2. \end{aligned}$$

This concludes the proof of Lemma 4. □

Due to [1], it is known that the set of eigenvalues accumulates only at infinity. This allows to fix $N_0 \in \mathbb{N}^*$ such that $\lambda_j \ge \lambda_{N_0+1} > 0$ for all $j \ge N_0 + 1$. We also set the integer $N \in \mathbb{N}$ large enough such that $N \ge N_0$ and $\lambda_n \ge 1$, $\forall n \ge N + 1$. A key element in the application of the control strategy reported in this paper relies on the asymptotic behavior of the eigenfunctions $\varphi_n$ evaluated at the measurement locations $\xi_i$. This asymptotic behavior is assessed through the following lemma.

**Lemma 5** The following holds:

$$\sum_{n \ge N+1} \frac{\varphi_n^2(\xi_i)}{\lambda_n^2} < \infty, \quad \forall i \in \{1, 2, \dots, M\}. \quad (7)$$

**Proof.** Considering $\mathcal{O}$ as a manifold with the Riemannian metric $\mu dx$, it follows that $\mathcal{A}_0$ is a self-adjoint operator in the Lebesgue $L^2$ space associated to this manifold. Hence, one may argue as in [18] (or as in [33] for the particular case of the Laplace operator) to deduce the existence of a constant $C > 0$ such that, for any $\lambda \ge 1$, we have

$$\sum_{\sqrt{\lambda_j} \in [\lambda, \lambda+1)} |\varphi_j(\xi)|^2 \le C \lambda^{d-1}, \quad \forall \xi \in \bar{\mathcal{O}}. \quad (8)$$

Let $0 < \frac{1}{5-d} < \beta < 1$, where we recall that $d \in \{2, 3\}$. We note that $[1, \infty) = \cup_{m \ge 1}[m^\beta, m^\beta + 1)$. Therefore, taking $\lambda = m^\beta$ in (8), we infer for any $\xi \in \bar{\mathcal{O}}$ that

$$\begin{aligned} \sum_{n \ge N+1} \frac{\varphi_n^2(\xi)}{\lambda_n^2} &\le \sum_{m=1}^{\infty} \left( \sum_{j \ge N+1, \sqrt{\lambda_j} \in [m^\beta, m^\beta+1)} \frac{\varphi_j^2(\xi)}{\lambda_j^2} \right) \\ &\le \sum_{m=1}^{\infty} \frac{1}{m^{4\beta}} \left( \sum_{j \ge N+1, \sqrt{\lambda_j} \in [m^\beta, m^\beta+1)} \varphi_j^2(\xi) \right) \\ &\le C \sum_{m=1}^{\infty} \frac{1}{m^{4\beta}} m^{(d-1)\beta} = C \sum_{m=1}^{\infty} \frac{1}{m^{(5-d)\beta}} < \infty, \end{aligned}$$

since $\beta(5-d) > 1$. □

**Remark 6** It is worth being noted that for dimensions $d \ge 4$ the quantities $\sum_{n \ge N+1} \frac{\varphi_n^2(\xi_i)}{\lambda_n^2}$, $i \in \{1, 2, \dots, M\}$, might not be finite, in general. This is the key point that restricts the application of the proposed control strategy to multi-D equations with $d \ge 4$. More exactly, invoking the Weyl's law (see [34, Chapter 11]), we know that $\lambda_n$ is of order $n^2$. Hence, $\sum_{n \ge N+1} \frac{1}{\lambda_n^2}$ is comparable with $\sum_{n \ge N+1} \frac{1}{n^d}$ which diverges for $d \ge 4$.
Let us denote by $\mathbf{n} = (n_1, \dots, n_d)$ the unit outward normal to the boundary of $\mathcal{O}$. For later purpose, we need to show a unique continuation property of the eigenfunctions of the operator $\mathcal{A}_0$, as stated in the following lemma.

**Lemma 7** Let any $\varphi \neq 0$ satisfying

$$-\sum_{i=1}^{d} \partial_i (\tilde{a}_i(x) \partial_i \varphi) + \tilde{c}(x) \varphi - \mu(x) \lambda \varphi = 0 \text{ in } \mathcal{O}$$

with $\varphi = 0$ on $\partial\mathcal{O}$. Then, for each open non-empty subset $\Gamma$ of $\partial\mathcal{O}$, $\sum_{i=1}^d n_i(\cdot)\tilde{a}_i(\cdot)\partial_i\varphi(\cdot)$ is not identically zero on $\Gamma$.

**Proof.** This property holds true because the principal part of the differential operator is uniformly elliptic (which, usually, is called the elliptic continuation principle from the boundary).

Let us assume by contradiction that $\sum_{i=1}^d n_i(\cdot)\tilde{a}_i(\cdot)\partial_i\varphi(\cdot) = 0$ on $\Gamma$. This implies first that $\partial_n\varphi := \sum_{i=1}^d n_i\partial_i\varphi = 0$ on $\Gamma$. Indeed, since $\varphi = 0$ on $\partial\mathcal{O}$, we have that $\nabla\varphi$ is normal to $\partial\mathcal{O}$, that is so $\nabla\varphi = (\partial_n\varphi)\mathbf{n}$. Hence,

$$0 = \sum_{i=1}^{d} n_i \tilde{a}_i \partial_i \varphi = \sum_{i=1}^{d} n_i^2 \tilde{a}_i \partial_n \varphi,$$

which, together with the fact that $\sum_{i=1}^d n_i^2 \tilde{a}_i > \tilde{a}_m > 0$, gives $\partial_n\varphi := \sum_{i=1}^d n_i\partial_i\varphi = 0$ on $\Gamma$.

Choose now some $x_0 \in \Gamma$. Pick coordinates $x = (x', x_d)$ so that $x_0 = 0$ and, for some $r > 0$,

$$\mathcal{O} \cap B(0, r) = \{x \in B(0, r); x_d > g(x')\},$$

where $g: \mathbb{R}^{d-1} \to \mathbb{R}$ is a $C^\infty$-function. We extend the domain near $x_0$ by choosing $\psi \in C_c^\infty(\mathbb{R}^{d-1})$ with $\psi = 0$ for $\|x'\|_{d-1} \ge r/2$ and $\psi = 1$ for $\|x'\|_{d-1} \le r/4$, and by defining

$$\mathcal{O}^* = \mathcal{O} \cup \{x \in B(0, r); x_d > g(x') - \varepsilon\psi(x')\}.$$

Here $\varepsilon > 0$ is chosen sufficiently small so that

$$\{(x', x_d); \|x'\|_{d-1} \le r/2, x_d = g(x') - \varepsilon\psi(x')\}$$

is contained in $B(0, r)$. Then, $\mathcal{O}^*$ is a connected open set with smooth boundary.

Define the functions

$$\varphi^*(x) = \begin{cases} \varphi(x) & \text{if } x \in \mathcal{O}, \\ 0 & \text{if } x \in \mathcal{O}^* \setminus \mathcal{O}, \end{cases}$$

$$\begin{align*} \tilde{a}_i^*(x) &= \begin{cases} \tilde{a}_i(x) & \text{if } x \in \mathcal{O}, \\ 0 & \text{if } x \in \mathcal{O}^* \setminus \mathcal{O}, \end{cases} \\ (\tilde{c} - \mu\lambda)^*(x) &= \begin{cases} (\tilde{c} - \mu(x)\lambda)(x) & \text{if } x \in \mathcal{O}, \\ 0 & \text{if } x \in \mathcal{O}^* \setminus \mathcal{O}. \end{cases} \end{align*}$$

Since $\varphi^*$ is of regularity $H^2$ in both regions $\mathcal{O}$ and $\mathcal{O}^* \setminus \mathcal{O}$, and the trace on $\Gamma$ of $\varphi^*$ and the normal derivative $\partial_n\varphi^*$ are both equal to zero, it yields via Green's formula that:

$$\begin{align*} \int_{\mathcal{O}^*} \Delta \varphi^* \psi dx &= \int_{\mathcal{O}} \Delta \varphi^* \psi dx + \int_{\mathcal{O}^* \setminus \mathcal{O}} \Delta \varphi^* \psi dx \\ &= \int_{\mathcal{O}} \varphi^* \Delta \psi dx + \int_{\mathcal{O}^* \setminus \mathcal{O}} \varphi^* \Delta \psi dx \\ &= \int_{\mathcal{O}^*} \varphi^* \Delta \psi dx, \end{align*}$$

for all $\psi \in C_0^\infty(\mathcal{O}^*)$. Therefore, $\varphi^* \in H^2(\mathcal{O}^*)$ and is solution to

$$-\sum_{i=1}^{d} \partial_i (\tilde{a}_i^*(x) \partial_i \varphi^*(x)) + (\tilde{c} - \mu\lambda)(x) \varphi^*(x) = 0 \text{ a.e. in } \mathcal{O}^*,$$

with $\varphi^* \equiv 0$ in some open ball contained in $\mathcal{O}^* \setminus \mathcal{O}$. Invoking the result in [15], we immediately get that $\varphi \equiv 0$ in $\mathcal{O}$, which is in contradiction with the hypothesis. It concludes the proof of Lemma 7. $\square$

We shall denote by

$$T_{\mathbf{n}}(\cdot) := \sum_{i=1}^{d} n_i \tilde{a}_i \partial_i (\cdot)$$

the trace operator of generalized normal derivative. Let us note that, since $\varphi_j \in H^3(\mathcal{O})$, it follows that $T_{\mathbf{n}}\varphi_j \in H^{\frac{3}{2}}(\partial\mathcal{O})$, for all $j \in \mathbb{N}^*$. Then, with similar arguments as in [13, Lemma 5.3] one can show that the truncation $\chi T_{\mathbf{n}}\varphi_j$ remains in $H^{\frac{3}{2}}(\partial\mathcal{O})$.

With all these elements in hands, we are now in position to introduce the output feedback control strategy that exponentially stabilizes the reaction-diffusion equation (1) based solely on the internal measurement (2). This is discussed in the next section.

### 3 Design of the control strategy

Let us denote $A_0 := (\langle -\mathcal{A}\varphi_i, \psi_j \rangle)_{1 \le i,j \le N_0}$. It is seen that $A_0 = -\text{diag}(\lambda_1, \lambda_2, \dots, \lambda_{N_0})$. For dimension $d \ge 2$, it is highly possible to have multiple eigenvalues. Note however that the first eigenvalue is always simple. Therefore, the number of scalar measurements $M$ from the
system output (2), as well as the structure of the control strategy, needs to be slightly adapted in function of the multiplicity of the unstable eigenvalues. To fix the ideas and to keep the presentation as concise as possible, we make the following choice for the spectrum structure (other possible choices can be treated in a similar manner as below without any supplementary effort): the second eigenvalue is of multiplicity equal two, while the rest of the first $N_0$ eigenvalues are simple, i.e., we have

$$ \lambda_1 < \lambda_2 = \lambda_3 < \lambda_4 < \dots < \lambda_{N_0}. $$

This configuration leads us to consider $M = 2$ scalar outputs, i.e.,

$$ y(t) = (z(\xi_1, t), z(\xi_2, t))^{\top}. \quad (9) $$

**Remark 8** Note that the other configurations, in terms of multiplicity of the different eigenvalues, can be handled in a similar way by setting the number of scalar measurements $M$ as the maximum of multiplicity for the eigenvalues $\lambda_1, \dots, \lambda_{N_0}$. This is possible because, invoking the results in [1], each eigenvalue has a finite multiplicity. It is worth being noted that the sequence of multiplicities might be unbounded in certain cases. However, since for control design we are only interested in the multiplicities of the first $N_0$ eigenvalues, the number $M$ of pointwise measurements is always finite. Note also that, due to [1], the set of eigenvalues accumulates only at infinity.

### 3.1 Preliminary control design

Assuming $\gamma > 0$ large enough, we have that, for each $v \in H^{\frac{3}{2}}(\partial O)$, there exists a unique solution, $D \in H^2(O)$, to the equation

$$ \begin{aligned} & \mathcal{A}D - 2 \sum_{i=1}^{N_0} \lambda_i \langle D, \psi_i \rangle \varphi_i + \gamma D = 0 \quad \text{in } O; \\ & D = \chi v \quad \text{on } \partial O. \end{aligned} \quad (10) $$

This is indeed so, since, likewise in [13, Lemma 5.3], one can show that, for $v \in H^{\frac{3}{2}}(\partial O)$, we have that the truncation $\chi v$ remains in $H^{\frac{3}{2}}(\partial O)$. Then, with similar arguments as in [26, p.188] (see also [27, p.226]), one can easily arrive to the wanted conclusion.

This allows us to introduce the following continuous linear operator:

**Definition 9** Let $D_\gamma : H^{\frac{3}{2}}(\partial O) \to H^2(O)$ be defined by $D_\gamma v := D$ where, for any given $v \in H^{\frac{3}{2}}(\partial O)$, $D \in H^2(O)$ is the unique solution to (10).

**Lemma 10** We have

$$ \begin{pmatrix} \langle D_\gamma v, \psi_1 \rangle \\ \langle D_\gamma v, \psi_2 \rangle \\ \vdots \\ \langle D_\gamma v, \psi_{N_0} \rangle \end{pmatrix} = -\Lambda_\gamma \begin{pmatrix} \langle \chi v, T_{\mathbf{n}} \varphi_1 \rangle_{L^2(\partial O)} \\ \langle \chi v, T_{\mathbf{n}} \varphi_2 \rangle_{L^2(\partial O)} \\ \vdots \\ \langle \chi v, T_{\mathbf{n}} \varphi_{N_0} \rangle_{L^2(\partial O)} \end{pmatrix}, \quad (11) $$

where

$$ \Lambda_\gamma = \operatorname{diag} \left( \frac{1}{\gamma - \lambda_1}, \frac{1}{\gamma - \lambda_2}, \dots, \frac{1}{\gamma - \lambda_{N_0}} \right), \quad (12) $$

while

$$ \langle D_\gamma v, \psi_k \rangle = - \frac{1}{\gamma + \lambda_k} \langle \chi v, T_{\mathbf{n}} \varphi_k \rangle_{L^2(\partial O)}, \quad k \ge N_0 + 1. \quad (13) $$

**Proof.** Scalarly multiplying equation (10) by $\psi_k = \mu\varphi_k$, $k = 1, 2, \dots, N_0$, while taking advantage of the bi-orthogonality of the sequences $\{\varphi_i\}_{i \ge 1}$ and $\{\psi_i\}_{i \ge 1}$, we get

$$ \langle \mu \mathcal{A} D_\gamma v, \varphi_k \rangle - 2 \lambda_k \langle D_\gamma v, \psi_k \rangle + \gamma \langle D_\gamma v, \psi_k \rangle = 0. \quad (14) $$

Let us compute $\langle \mu \mathcal{A} D_\gamma v, \varphi_k \rangle$. Using the integration by parts formula and invoking the boundary conditions of $\varphi_j$ and $D_\gamma v$, we obtain that

$$ \begin{aligned} \langle \mu \mathcal{A} D_\gamma v, \varphi_k \rangle &= \left\langle - \sum_{i=1}^d \partial_i (\tilde{a}_i \partial_i D_\gamma v) + \tilde{c} D_\gamma v, \varphi_k \right\rangle \\ &= \sum_{i=1}^d \langle D_\gamma v, n_i \tilde{a}_i \partial_i \varphi_k \rangle_{L^2(\partial O)} \\ &\quad + \left\langle D_\gamma v, - \sum_{i=1}^d \partial_i (\tilde{a}_i \partial_i \varphi_k) + \tilde{c} \varphi_k \right\rangle \end{aligned} $$

Consequently,

$$ \langle \mu \mathcal{A} D_\gamma v, \varphi_k \rangle = \langle \chi v, T_{\mathbf{n}} \varphi_k \rangle_{L^2(\partial O)} + \lambda_k \langle D_\gamma v, \psi_k \rangle. $$

Substituting the above identity into equation (14), we infer that

$$ (-\lambda_k + \gamma) \langle D_\gamma v, \psi_k \rangle = - \langle \chi v, T_{\mathbf{n}} \varphi_k \rangle_{L^2(\partial O)}, $$

$k \in \{1, \dots, N_0\}$. This gives (11). Proceeding similarly, we infer that (13) holds. $\square$

Next, we fix $N_0$ positive constants $0 < \gamma_1 < \gamma_2 < \dots < \gamma_{N_0}$, selected sufficiently large such that, for each $k \in \{1, 2, \dots, N_0\}$, equation (10) is well-posed for $\gamma = \gamma_k$. Following Definition 9, we denote for $k \in \{1, 2, \dots, N_0\}$ by $D_{\gamma_k}$ the corresponding operators as defined by (10).
We now introduce the positive semi-definite Gram matrix **B**, defined by:

$$ \mathbf{B} := \left( \langle \chi T_{\mathbf{n}} \varphi_k, T_{\mathbf{n}} \varphi_l \rangle_{L^2(\partial \mathcal{O})} \right)_{1 \le k,l \le N_0} \quad (15) $$

and we set

$$ B_k := \Lambda_{\gamma_k} \mathbf{B} \Lambda_{\gamma_k}, \quad k \in \{1, 2, \dots, N_0\}. \quad (16) $$

Note that, for all $k = 1, 2, \dots, N_0$, $B_k$ is positive semi-definite. We also define

$$ \mathcal{L}(x) = (T_{\mathbf{n}}\varphi_1(x), \dots, T_{\mathbf{n}}\varphi_{N_0}(x))^{\top} \quad (17) $$

for all $x \in \partial \mathcal{O}$. In the given hypotheses, $B_1 + B_2 + \dots + B_{N_0}$ is invertible (see the Lemma A in the Appendix for details), and we denote by **A** its inverse. Clearly seen, **A** is a positive definite matrix.

Next, introducing an auxiliary control input $U: [0, \infty) \to \mathbb{R}^{N_0}$, which will be specified later in Subsection 3.3, we define

$$ u_k(x,t) := \langle \Lambda_{\gamma_k} AU(t), \mathcal{L}(x) \rangle_{N_0}, \quad x \in \partial \mathcal{O}, t \ge 0, \quad (18) $$

for all $k = 1, 2, \dots, N_0$, and

$$ \begin{aligned} v(x,t) &:= u_1(x,t) + u_2(x,t) + \dots + u_{N_0}(x,t) \\ &= \sum_{k=1}^{N_0} \langle \Lambda_{\gamma_k} AU(t), \mathcal{L}(x) \rangle_{N_0}, \quad x \in \partial \mathcal{O}, t \ge 0. \end{aligned} \quad (19) $$

Then, we define the boundary control $u$ appearing in the plant (1) as

$$ u(x,t) = \chi(x)v(x,t) = \chi(x)[u_1(x,t) + \dots + u_{N_0}(x,t)]. $$

As it can be easily seen, $u$ is supported in $\Gamma_1$.

**Remark 11** Note that $u_k(\cdot, \cdot)$, $k = 1, \dots, N_0$, is of separable variables form. More precisely, it can be written as

$$ u_k(x,t) = \sum_{i=1}^{N_0} u_i^k(t) T_{\mathbf{n}} \varphi_i(x). $$

Therefore, applying $D_\gamma$ (defined by Definition 9), we have

$$ D_\gamma u_k(t) = \sum_{i=1}^{N_0} u_i^k(t) D_\gamma T_{\mathbf{n}} \varphi_i. $$

As we already saw before, $T_{\mathbf{n}}\varphi_i \in H^{\frac{3}{2}}(\partial \mathcal{O})$, thus, invoking the results in [26], $D_\gamma u_k(t)$ is well-defined for each $t \ge 0$ and belongs to $H^2(\mathcal{O})$. It follows by the general Sobolev

embeddings that $D_\gamma u_k(t)$ belongs to the Hölder space $C^{0,\alpha}(\mathcal{O})$, for some $\alpha \in (0,1)$. Hence, it makes sense to consider pointwise evaluations of $D_\gamma u_k(t)$ as made below, for all $t \ge 0$. Moreover, since $u \in C(0, \infty; H^{\frac{3}{2}}(\partial \mathcal{O}))$, we deduce that once we plug it into (1), the corresponding solution of (1) with $z_0 \in H^2(\mathcal{O})$ satisfies $z(\cdot, t) \in H^2(\mathcal{O})$ for all $t \ge 0$. Hence, it makes sense to consider point values of $z(\cdot, t)$ for all $t \ge 0$. (For details see [27, Theorem 4] combined with the arguments in [27, p226].)

**Lemma 12** It holds, for all $k = 1, \dots, N_0$,

$$ \left( \begin{array}{c} \langle D_{\gamma_k} u_k, \psi_1 \rangle \\ \langle D_{\gamma_k} u_k, \psi_2 \rangle \\ \vdots \\ \langle D_{\gamma_k} u_k, \psi_{N_0} \rangle \end{array} \right) = -B_k A U, \quad (20) $$

**Proof.** Based on (11), we infer that

$$ \left( \langle D_{\gamma_k} u_k, \psi_1 \rangle \langle D_{\gamma_k} u_k, \psi_2 \rangle \dots \langle D_{\gamma_k} u_k, \psi_{N_0} \rangle \right)^{\top} = -\Lambda_{\gamma_k} \begin{pmatrix} \langle \chi u_k, T_{\mathbf{n}} \varphi_1 \rangle_{L^2(\partial \mathcal{O})} \\ \langle \chi u_k, T_{\mathbf{n}} \varphi_2 \rangle_{L^2(\partial \mathcal{O})} \\ \vdots \\ \langle \chi u_k, T_{\mathbf{n}} \varphi_{N_0} \rangle_{L^2(\partial \mathcal{O})} \end{pmatrix}. $$

Taking into account (15) and (18), this yields

$$ \left( \begin{array}{c} \langle D_{\gamma_k} u_k, \psi_1 \rangle \\ \langle D_{\gamma_k} u_k, \psi_2 \rangle \\ \vdots \\ \langle D_{\gamma_k} u_k, \psi_{N_0} \rangle \end{array} \right) = -\Lambda_{\gamma_k} \mathbf{B} \Lambda_{\gamma_k} A U $$

and so, owing to the definition of $B_k$ given by (16), the claimed identity (20) is proved. □

### 3.2 Spectral reduction

Our objective is now to specify the auxiliary control input $U$ that appears in (18). To do so, we first need to carry on a spectral reduction of the system formed by the plant (1) along with the preliminary control input (19). This is done in this subsection.

**Lemma 13** Consider the following change of variable :

$$ w := z - \sum_{k=1}^{N_0} D_{\gamma_k} u_k \quad (21) $$
and define the coefficients of projection $w_n(t) = \langle w(t, \cdot), \psi_n \rangle$ and $z_n(t) = \langle z(t, \cdot), \psi_n \rangle$. Then, we have

$$ \frac{d}{dt}z_n = -\lambda_n z_n + \sum_{k=1}^{N_0} (\lambda_n + \gamma_k) \langle D_{\gamma_k} u_k, \psi_n \rangle - 2 \sum_{k,i=1}^{N_0} \lambda_i \langle D_{\gamma_k} u_k, \psi_i \rangle \delta_{i,n} \quad (22) $$

for all $n \ge 1$. Moreover, we have

$$ \frac{d}{dt}w_n = -\lambda_n w_n + \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle - \sum_{k=1}^{N_0} \left\langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \right\rangle \quad (23) $$

for all $n \ge N_0 + 1$.

**Proof.** We first equivalently rewrite (1) as an internal-type control problem. More precisely, invoking the change of variable (21), we have

$$ \begin{aligned} \frac{d}{dt}w &= \frac{d}{dt}z - \frac{d}{dt}\sum_{k=1}^{N_0} D_{\gamma_k} u_k \\ &\stackrel{(1)}{=} -Az - \frac{d}{dt}\sum_{k=1}^{N_0} D_{\gamma_k} u_k \\ &= -A_0 w - \sum_{k=1}^{N_0} AD_{\gamma_k} u_k - \frac{d}{dt}\sum_{k=1}^{N_0} D_{\gamma_k} u_k \\ &\stackrel{(10)}{=} -A_0 w - 2 \sum_{k,i=1}^{N_0} \lambda_i \langle D_{\gamma_k} u_k, \psi_i \rangle \varphi_i \\ &\quad + \sum_{k=1}^{N_0} \gamma_k D_{\gamma_k} u_k - \frac{d}{dt}\sum_{k=1}^{N_0} D_{\gamma_k} u_k, \quad t > 0. \end{aligned} \quad (24) $$

Then, recalling that $w_n(t) = \langle w(t, \cdot), \psi_n \rangle$ and $z_n(t) = \langle z(t, \cdot), \psi_n \rangle$, the projection of (24) gives

$$ \frac{d}{dt}w_n = -\lambda_n w_n - \sum_{k=1}^{N_0} \left\langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \right\rangle - 2 \sum_{k,i=1}^{N_0} \lambda_i \langle D_{\gamma_k} u_k, \psi_i \rangle \delta_{i,n} + \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle \quad (25) $$

for all $n \ge 1$. This gives (23) because $\delta_{i,n} = 0$ for all $1 \le i \le N_0$ and $n \ge N_0 + 1$. Now, in view of the

change of variable formula (21), we have

$$ w_n = z_n - \sum_{k=1}^{N_0} \langle D_{\gamma_k} u_k, \psi_n \rangle. \quad (26) $$

Thus, combining (25)-(26), we infer that (22) holds. $\square$

Our objective is now to write an ODE describing the dynamics of a finite number of modes of the system composed of the plant (1) and the control input (19). To do so, we define

$$ Z^{N_0} := (\langle z, \psi_1 \rangle \langle z, \psi_2 \rangle \dots \langle z, \psi_{N_0} \rangle)^\top. $$

In view of equation (22) and taking into account the relation (20), we deduce that

$$ \begin{aligned} \frac{d}{dt}Z^{N_0}(t) &= A_0 Z^{N_0}(t) + \sum_{k=1}^{N_0} [-A_0 + \gamma_k I] \begin{pmatrix} \langle D_{\gamma_k} u_k, \psi_1 \rangle \\ \langle D_{\gamma_k} u_k, \psi_2 \rangle \\ \vdots \\ \langle D_{\gamma_k} u_k, \psi_{N_0} \rangle \end{pmatrix} \\ &\quad + \sum_{k=1}^{N_0} 2A_0 \begin{pmatrix} \langle D_{\gamma_k} u_k, \psi_1 \rangle \\ \langle D_{\gamma_k} u_k, \psi_2 \rangle \\ \vdots \\ \langle D_{\gamma_k} u_k, \psi_{N_0} \rangle \end{pmatrix} \\ &= A_0 Z^{N_0}(t) - \sum_{k=1}^{N_0} [A_0 + \gamma_k I] B_k A U(t) \\ &= A_0 Z^{N_0}(t) + \left( -A_0 - \sum_{k=1}^{N_0} \gamma_k B_k A \right) U(t). \end{aligned} \quad (27) $$

where we recall that $A$ is the inverse of the sum $B_1 + \dots + B_{N_0}$ and $A_0 = -\text{diag}(\lambda_1, \dots, \lambda_{N_0})$. Introducing now a second integer $N \ge N_0 + 1$ to be specified later, we define

$$ Z^{N-N_0} := (\langle z, \psi_{N_0+1} \rangle \langle z, \psi_{N_0+2} \rangle \dots \langle z, \psi_N \rangle)^\top $$

along with $A_1 = -\text{diag}(\lambda_{N_0+1}, \dots, \lambda_N)$ and $H^{N-N_0} : \mathbb{R}^{N_0} \to \mathbb{R}^{N-N_0}$ defined by

$$ H^{N-N_0} := \left( \langle T_n \varphi_i, \chi T_n \varphi_j \rangle_{L^2(\partial \mathcal{O})} \right)_{\substack{N_0+1 \le i \le N \\ 1 \le j \le N_0}} \sum_{k=1}^{N_0} \Lambda_{\gamma_k} A. $$

Then, using again (22), we deduce that

$$ \frac{d}{dt} Z^{N-N_0}(t) = A_1 Z^{N-N_0}(t) + H^{N-N_0} U(t), \quad t > 0. \quad (28) $$
**Remark 14** In the case of a state-feedback, the proposed control strategy reduces to the application of the input (19) while setting $U = Z^{N_0}$. In that case (27) becomes $\frac{d}{dt}Z^{N_0}(t) = -\sum_{k=1}^{N_0} \gamma_k B_k A Z^{N_0}(t)$. It can be shown (see the proof of Theorem 15 below) that $\gamma_1 < \gamma_2 < \dots < \gamma_{N_0}$ can be selected large enough so that the matrix $-\sum_{k=1}^{N_0} \gamma_k B_k A$ is Hurwitz. From there, one can conclude that the proposed state-feedback control strategy achieves the exponential stabilization of the plant (1). The case of an output feedback using the internal measurement (2) requires the introduction of an observer. This is the objective of the next subsection.

### 3.3 Observer design and definition of the auxiliary control input U

We are now in position to properly define in this subsection the auxiliary control input $U$ that appears in (18). We select the measurement locations $\xi_1, \xi_2 \in O$ from (9) such that

$$ \begin{gathered} |\varphi_i(\xi_1)| + |\varphi_i(\xi_2)| \neq 0, \quad \forall i \in \{1, 4, 5, \dots, N_0\} \\ \text{and det} \left| \begin{array}{c} \varphi_2(\xi_1) \quad \varphi_3(\xi_1) \\ \varphi_2(\xi_2) \quad \varphi_3(\xi_2) \end{array} \right| \neq 0. \end{gathered} \quad (29) $$

Note that such a selection is always possible due to the fact that $\varphi_i, i \in \{1, 2, \dots, N_0\}$, cannot vanish on any open ball included in $O$ and the fact that $\varphi_2, \varphi_3$ correspond to the same eigenvalue and are linearly independent.

Second, introducing $\tilde{y}(t) = (w(\xi_1, t), w(\xi_2, t))^{\top}$, we deduce from the change of variable formula (21) that

$$ \begin{aligned} y(t) &= \tilde{y}(t) + \sum_{k=1}^{N_0} \begin{pmatrix} D_{\gamma_k} u_k(\xi_1, t) \\ D_{\gamma_k} u_k(\xi_2, t) \end{pmatrix} \\ &= \sum_{i \ge 1} \begin{pmatrix} \varphi_i(\xi_1) \\ \varphi_i(\xi_2) \end{pmatrix} w_i(t) + \sum_{k=1}^{N_0} \begin{pmatrix} D_{\gamma_k} u_k(\xi_1, t) \\ D_{\gamma_k} u_k(\xi_2, t) \end{pmatrix}. \end{aligned} \quad (30) $$

Defining $C_0 = \begin{pmatrix} \varphi_1(\xi_1) & \varphi_2(\xi_1) & \dots & \varphi_{N_0}(\xi_1) \\ \varphi_1(\xi_2) & \varphi_2(\xi_2) & \dots & \varphi_{N_0}(\xi_2) \end{pmatrix}$ and $C_1 = \begin{pmatrix} \varphi_{N_0+1}(\xi_1) & \varphi_{N_0+2}(\xi_1) & \dots & \varphi_N(\xi_1) \\ \varphi_{N_0+1}(\xi_2) & \varphi_{N_0+2}(\xi_2) & \dots & \varphi_N(\xi_2) \end{pmatrix}$, it immediately follows from (29) that the pair $(A_0, C_0)$ satisfies the Kalman condition. Hence, we can fix $L \in M_{N_0 \times 2}(\mathbb{R})$ such that $A_0 - LC_0$ is Hurwitz with arbitrary spectral abscissa.

In view of (26), (27), and (28), we now define the follow-

ing observer dynamics:

$$ \left\{ \begin{aligned} \hat{w}_n &:= \hat{z}_n - \sum_{k=1}^{N_0} \langle D_{\gamma_k} u_k, \psi_n \rangle, & 1 \le n \le N \\ \hat{Z}^{N_0} &:= (\hat{z}_1 \quad \dots \quad \hat{z}_{N_0})^{\top} \\ \hat{Z}^{N-N_0} &:= (\hat{z}_{N_0+1} \quad \dots \quad \hat{z}_N)^{\top} \\ \frac{d}{dt} \hat{Z}^{N_0}(t) &:= A_0 \hat{Z}^{N_0}(t) \\ & \quad + \left( -A_0 - \sum_{k=1}^{N_0} \gamma_k B_k A \right) U(t) \\ & \quad - L \left\{ \sum_{i=1}^{N} \begin{pmatrix} \varphi_i(\xi_1) \\ \varphi_i(\xi_2) \end{pmatrix} \hat{w}_i(t) \right. \\ & \quad \left. + \sum_{k=1}^{N_0} \begin{pmatrix} (D_{\gamma_k} u_k)(\xi_1, t) \\ (D_{\gamma_k} u_k)(\xi_2, t) \end{pmatrix} - y(t) \right\} \\ \frac{d}{dt} \hat{Z}^{N-N_0}(t) &:= A_1 \hat{Z}^{N-N_0}(t) + H^{N-N_0} U(t) \end{aligned} \right. \quad (31) $$

for $t > 0$. Recall here that $y(t)$ is the measurement provided by (9). We can now complete the definition of the control strategy by setting the auxiliary control input $U$ as<fnref n="1" />:

$$ U = \hat{Z}^{N_0}. \quad (32) $$

Overall, the control strategy is composed of (19), (31), and (32).

### 3.4 Main stabilization result

We can now state the main result of this work.

**Theorem 15** Assume that Assumption 2 holds. Let $\delta > 0$ and $N_0 \ge 1$ be such that $\lambda_n > \delta$ for all $n \ge N_0 + 1$. Assume that the first $N_0$ eigenvalues of the operator $A$ are simple except for the second and the third one which are equal. With corresponding measurement (9), pick $\xi_1, \xi_2 \in O$ so that (29) holds true. Let $L \in \mathbb{R}^{N_0}$ be such that $A_0 - LC_0$ is Hurwitz with eigenvalues that have a real part strictly less than $-\delta$. Then, $\gamma_1 < \gamma_2 < \dots < \gamma_{N_0}$ can be selected large enough such that the matrix $-\sum_{k=1}^{N_0} \gamma_k B_k A$ is Hurwitz with eigenvalues that have a real part strictly less than $-\delta$. Furthermore, for $N \ge N_0 + 1$ selected to be large enough, there exists a constant $C > 0$ such that, for any initial condition $z_0 \in H^2(O)$, the trajectory of the closed-loop system composed of the plant (1), the internal measurement (2) and the

<footnote n="1">¹ This is consistent with Remark 14: since we cannot apply the state-feedback $U = Z^{N_0}$, we replace the state of the $N_0$ first modes of the plant by their estimates computed by the observer (31).</footnote>
controller (19) with U given by (32), satisfies

$$
\begin{equation}
\begin{aligned}
& \|z(\cdot, t)\|_{H^1(\mathcal{O})} + \sum_{k=1}^{N} |\hat{z}_k(t)| \\
& \le Ce^{-\delta t} \left( \|z(0)\|_{H^1(\mathcal{O})} + \sum_{k=1}^{N} |\hat{z}_k(0)| \right), \quad t \ge 0.
\end{aligned}
\tag{33}
\end{equation}
$$

**Proof.** We first need to rewrite the dynamics of the closed-loop system formed by (19), (31), and (32) in a suitable format for the upcoming stability analysis. To do so, we first define the errors of observation as

$$
E^{N_0} := Z^{N_0} - \hat{Z}^{N_0}, \quad \tilde{E}^{N-N_0} := \Lambda^{N-N_0}(Z^{N-N_0} - \hat{Z}^{N-N_0})
$$

where $\Lambda^{N-N_0} = \text{diag}(\lambda_{N_0+1}, \dots, \lambda_N)$ while

$$
\zeta := \sum_{n \ge N+1} \begin{pmatrix} \varphi_n(\xi_1) \\ \varphi_n(\xi_2) \end{pmatrix} w_n, \quad \tilde{C}_1 := C_1 (\Lambda^{N-N_0})^{-1}.
$$

Hence, we obtain from (19), (31), and (32) that

$$
\begin{equation}
\begin{aligned}
\frac{d}{dt} \hat{Z}^{N_0} &= \left( -\sum_{k=1}^{N_0} \gamma_k B_k A \right) \hat{Z}^{N_0} + LC_0 E^{N_0} \\
&\quad + L \tilde{C}_1 \tilde{E}^{N-N_0} + L \zeta,
\end{aligned}
\tag{34a}
\end{equation}
$$

$$
\frac{d}{dt} E^{N_0} = (A_0 - LC_0)E^{N_0} - L\tilde{C}_1\tilde{E}^{N-N_0} - L\zeta, \quad (34\text{b})
$$

$$
\frac{d}{dt} \hat{Z}^{N-N_0} = A_1 \hat{Z}^{N-N_0} + H^{N-N_0} \hat{Z}^{N_0}, \quad (34c)
$$

$$
\frac{d}{dt} \tilde{E}^{N-N_0} = A_1 \tilde{E}^{N-N_0}. \qquad (34\text{d})
$$

Introducing the finite-dimensional state vector $X := (\hat{Z}^{N_0} E^{N_0} \tilde{E}^{N-N_0})^\top$ along with the matrices

$$
F := \begin{pmatrix}
-\sum_{k=1}^{N_0} \gamma_k B_k A & LC_0 & L\tilde{C}_1 \\
0 & A_0 - LC_0 & -L\tilde{C}_1 \\
0 & 0 & A_1
\end{pmatrix}, \quad (35)
$$

$$
\mathbb{L} := (L^{\top} - L^{\top} \ 0)^{\top},
$$

we infer that the closed-loop system dynamics is de-

scribed by

$$
\frac{d}{dt} X = F X + \mathbb{L} \zeta \quad (36a)
$$

$$
\frac{d}{dt} \hat{Z}^{N-N_0} = A_1 \hat{Z}^{N-N_0} + H^{N-N_0} \hat{Z}^{N_0} \quad (36\text{b})
$$

$$
\begin{equation}
\begin{split}
\frac{d}{dt} w_n = -\lambda_n w_n &+ \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle \\
& - \sum_{k=1}^{N_0} \left\langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \right\rangle, \quad n \ge N+1
\end{split}
\tag{36c}
\end{equation}
$$

Let us now show that we can fix the real numbers $0 < \gamma_1 < \gamma_2 < \dots < \gamma_{N_0}$, large enough, such that the matrix $-\sum_{k=1}^{N_0} \gamma_k B_k A$ is Hurwitz with eigenvalues that have a real part strictly less than $-\delta < 0$. To do so, let $\lambda \in \mathbb{C}$ and a non-zero vector $Z \in \mathbb{C}^{N_0}$ be such that

$$
- \sum_{k=1}^{N_0} \gamma_k B_k A Z = \lambda Z.
$$

Or, equivalently, by recalling that $A$ is the inverse of the sum of $B_k$s, we have

$$
\lambda Z = - \sum_{k=1}^{N_0} \gamma_k B_k A Z = -\gamma_1 Z + \sum_{k=2}^{N_0} (\gamma_1 - \gamma_k) B_k A Z.
$$

Next, recalling that $A$ is positive definite and multiplying
the above equality by $Az$, we get

$$
\begin{align*}
& \Re \lambda \| A^{\frac{1}{2}} Z \|_{N_0}^2 \\
& = -\gamma_1 \| A^{\frac{1}{2}} Z \|_{N_0}^2 + \sum_{k=2}^{N} (\gamma_1 - \gamma_k) \langle B_k A Z, A Z \rangle_{N_0}.
\end{align*}
$$

This yields $\Re\lambda \le -\gamma_1$, because $B_k$ is positive semi-
definite and $\gamma_k \ge \gamma_1$. Here, $A^{1/2}$ is the square root matrix
of the positive definite matrix $A$. The conclusion follows
immediately by taking $\gamma_1$ large enough. This, in particu-
lar, implies that the matrix $F$ defined by (35) is Hurwitz
with eigenvalues that have a real part strictly less than
$-\delta$.

We now carry out a Lyapunov stability analysis. In view
of (6), let us introduce the Lyapunov function, for all
$(X, w) \in \mathbb{R}^{N+N_0} \times H^1(\mathcal{O})$

$$
V(X, w) = X^\top P X + \sum_{n \ge N+1} (\lambda_n + \nu) w_n^2. \quad (37)
$$

The computation of the time derivative of V along the
system trajectories (36) gives

$$
\begin{aligned}
\dot{V} = {}& 2X^\top P(FX + L\zeta) \\
& + 2 \sum_{n \ge N+1} (\lambda_n + \nu) \left\{ -\lambda_n w_n + \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle \right. \\
& \left. - \left\{ \sum_{k=1}^{N_0} \langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \rangle \right\} w_n \right\}
\end{aligned}
$$

Therefore

$$
\begin{aligned}
\dot{V} = \tilde{X}^{\top} \begin{pmatrix} F^{\top} P + PF & PL \\ L^{\top} P & 0 \end{pmatrix} \tilde{X} - 2 \sum_{n \ge N+1} \lambda_n (\lambda_n + \nu) w_n^2 \\
+ 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle w_n \\
- 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \left\langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \right\rangle w_n
\end{aligned}
$$

where $\tilde{X} := \text{col}(X, \zeta)$. Let us estimate the penultimate term. To do so, let us note that

$$
\begin{aligned}
\langle D_{\gamma_k} u_k, \psi_n \rangle &\stackrel{(18)}{=} - \langle D_{\gamma_k} \langle \Lambda_{\gamma_k} AU, \mathcal{L} \rangle_{N_0}, \psi_n \rangle \\
&= - \sum_{l=1}^{N_0} \frac{1}{\gamma_k - \lambda_l} A_{L_l} U \langle D_{\gamma_k} T_{\mathbf{n}} \phi_l, \psi_n \rangle
\end{aligned}
$$

where $A_{L_l}$ is the $l$th line of the matrix $A$. Therefore, using Young's inequality, we deduce for any $\epsilon > 0$ that

$$
\begin{aligned}
& 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle w_n \\
&= -2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k,l=1}^{N_0} \frac{\gamma_k}{\gamma_k - \lambda_l} A_{L_l} U \langle D_{\gamma_k} T_{\mathbf{n}} \phi_l, \psi_n \rangle w_n \\
&\le 2 \sum_{k,l=1}^{N_0} \sum_{n \ge N+1} \left| \frac{\gamma_k}{\gamma_k - \lambda_l} \right| \|A_{L_l}\|_{N_0} \langle D_{\gamma_k} T_{\mathbf{n}} \phi_l, \psi_n \rangle \|U\|_{N_0} \\
&\qquad \times (\lambda_n + \nu) |w_n|
\end{aligned}
$$

Hence

$$
\begin{aligned}
& 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle w_n \\
&\le \epsilon \sum_{k,l=1}^{N_0} \left( \frac{\gamma_k}{\gamma_k - \lambda_l} \right)^2 \|A_{L_l}\|_{N_0}^2 \sum_{n \ge N+1} \langle D_{\gamma_k} T_{\mathbf{n}} \phi_l, \psi_n \rangle^2 \|U\|_{N_0}^2 \\
&\quad + \frac{1}{\epsilon} \sum_{k,l=1}^{N_0} \sum_{n \ge N+1} (\lambda_n + \nu)^2 w_n^2
\end{aligned}
$$

Invoking (5), we get

$$
\begin{aligned}
& 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \gamma_k \langle D_{\gamma_k} u_k, \psi_n \rangle w_n \\
&\le \epsilon S_{1,N} \|U\|_{N_0}^2 + \frac{N_0^2}{\epsilon} \sum_{n \ge N+1} (\lambda_n + \nu)^2 w_n^2
\end{aligned}
$$

where

$$
S_{1,N} := c_1 \sum_{k,l=1}^{N_0} \left( \frac{\gamma_k}{\gamma_k - \lambda_l} \right)^2 \|A_{L_l}\|_{N_0}^2 \|\mathcal{R}_N D_{\gamma_k} T_{\mathbf{n}} \phi_l\|^2
$$

and $\mathcal{R}_N f := \sum_{n \ge N+1} \langle f, \psi_n \rangle \varphi_n$ and

$U = \hat{Z}^{N_0} = E_1 X$ with $E_1 = [I \ 0 \ 0]$. Similarly, we have

$$
\begin{aligned}
& 2 \sum_{n \ge N+1} (\lambda_n + \nu) \sum_{k=1}^{N_0} \left\langle D_{\gamma_k} \frac{d}{dt} u_k, \psi_n \right\rangle w_n \\
&\stackrel{(5)}{\le} \epsilon S_{2,N} \left\| \frac{d}{dt} U \right\|_{N_0}^2 + \frac{N_0^2}{\epsilon} \sum_{n \ge N+1} (\lambda_n + \nu)^2 w_n^2
\end{aligned}
$$

where

$$
S_{2,N} := \epsilon c_1 \sum_{k,l=1}^{N_0} \left( \frac{1}{\gamma_k - \lambda_l} \right)^2 \|A_{L_l}\|_{N_0}^2 \|\mathcal{R}_N D_{\gamma_k} T_{\mathbf{n}} \phi_l\|^2
$$

and $\frac{d}{dt} U = \frac{d}{dt} \hat{Z}^{N_0} = E_2 \tilde{X}$ with

$$
E_2 = \left[ - \sum_{k=1}^{N_0} \gamma_k B_k A \, LC_0 \, L \tilde{C}_1 \, L \right].
$$

Finally, we have by Cauchy-Schwarz inequality that

$$
\begin{aligned}
\|\zeta\|^2 &= \left( \sum_{n \ge N+1} \varphi_n(\xi_1) w_n \right)^2 + \left( \sum_{n \ge N+1} \varphi_n(\xi_2) w_n \right)^2 \\
&\le \underbrace{\sum_{n \ge N+1} \frac{\varphi_n(\xi_1)^2 + \varphi_n(\xi_2)^2}{(\lambda_n + \nu)^2}}_{:=S_{\varphi,N}} \times \sum_{n \ge N+1} (\lambda_n + \nu)^2 w_n^2.
\end{aligned}
$$

Gathering all the above estimates, we deduce that

$$
\dot{V} + 2\delta V \le \tilde{X}^{\top} \Theta_1 \tilde{X} + \sum_{n \ge N+1} (\lambda_n + \nu) \Psi_n w_n^2 \quad (38)
$$
for $\delta > 0$ fixed such that $F + \delta I$ is Hurwitz and where

$$
\begin{align*}
\Theta_1 &= \begin{pmatrix} F^\top P + PF + 2\delta P + \epsilon S_{1,N} E_1^\top E_1 & P\mathbb{L} \\ \mathbb{L}^\top P & -\eta I \end{pmatrix} \\
&\quad + \epsilon S_{2,N} E_2^\top E_2, \\
\Psi_n &= \begin{bmatrix} -2 \left(1 - \frac{N_0^2}{\epsilon}\right) + \eta S_{\varphi,N} \\ -2\delta, \quad n \ge N+1 \end{bmatrix} \lambda_n + \begin{bmatrix} \frac{2N_0^2}{\epsilon} + \eta S_{\varphi,N} \\ \nu \end{bmatrix}
\end{align*}
$$

for an arbitrary $\eta > 0$.

Assume for the moment that $\Theta_1 \preceq 0$ and $\Psi_n \le 0$ for all $n \ge N+1$. Then, in view of (38), we get that $\dot{V} + 2\delta V \le 0$. Combining this estimate with the definition (37) of the Lyapunov function $V$, the direct integration of the dynamics $\dot{Z}^{N-N_0}$ from (36), the use of the estimates (6), and invoking the change of variable formula (21), we directly infer the existence of a constant $C > 0$, independent of the initial condition, such that the claimed stability estimate (33) holds.

To conclude the proof, it thus remains to show that $N$ can be selected so that $\Theta_1 \preceq 0$ and $\Psi_n \le 0$ for all $n \ge N+1$. To do so, let us set $\epsilon = 2N_0^2$ and $\eta = 1/\sqrt{S_{\varphi,N}}$ if $S_{\varphi,N} \ne 0$, $\eta = N$ otherwise. This implies that $\eta \to +\infty$ while $\eta S_{\varphi,N} \to 0$ as $N \to +\infty$. Hence, for $N$ large enough we have $\eta S_{\varphi,N} \le 1/2$ and $\lambda_n \ge \lambda_{N+1} > 0$ for all $n \ge N+1$, which implies that

$$
\Psi_n \le \Theta_2 := -\frac{1}{2}\lambda_{N+1} + \frac{3}{2}\nu + 2\delta, \quad n \ge N+1
$$

with $\Theta_2 \to -\infty$ as $N \to +\infty$. Now, since $F$ defined by (35) is Hurwitz, we define $P > 0$ as the unique solution to the Lyapunov equation $F^\top P + PF + 2\delta P = -I$. Owing to Lemma 5, it can be seen that $\|\tilde{C}_1\| = O(1)$, hence $\|L\tilde{C}_1\| = O(1)$, as $N \to +\infty$. Therefore, a result similar to [23, Lemma in Appendix] shows that $\|P\| = O(1)$ as $N \to +\infty$. Therefore, we have

$$
\Theta_1 = \underbrace{\begin{pmatrix} -I + \epsilon S_{1,N} E_1^\top E_1 & P\mathbb{L} \\ \mathbb{L}^\top P & -\eta I \end{pmatrix}}_{:=\Theta_{1,p}} + \epsilon S_{2,N} E_2^\top E_2. \quad (39)
$$

Using the Schur complement for $N$ sufficiently large so that $\eta > 1/2$, we see that $\Theta_{1,p} \preceq -\frac{1}{2}I$ if and only if $-\frac{1}{2}I + \epsilon S_{1,N}E_1^\top E_1 + \frac{1}{\eta-\frac{1}{2}}P\mathbb{L}\mathbb{L}^\top P \preceq 0$. Noting that $\|P\| = O(1)$ and $S_{2,N} \to 0$ as $N \to +\infty$ while $\|E_1\|$ and $\|\mathbb{L}\|$ are constants independent of $N$, we deduce that $\Theta_{1,p} \preceq -\frac{1}{2}I$ for all $N$ selected to be large enough. In that case, $\Theta_1 \preceq -\frac{1}{2}I + \epsilon S_{2,N}E_2^\top E_2$ for all $N$ selected to be large enough. Since $\|E_2\| = O(1)$ and $S_{2,N} \to 0$ as $N \to +\infty$, we deduce that $\Theta_1 \preceq 0$ for $N$ large enough. $\square$

**Remark 16** The proof of Theorem 15 provides sufficient conditions regarding the selection of the order $N$ of the observer. More precisely, for a fixed order $N \ge N_0 + 1$, if there exist decision variables $P > 0$, $\eta, \epsilon > 0$ for which the following LMI conditions hold<fnref n="2" />:

$$
\left(
\begin{array}{cc}
F^\top P + PF + \epsilon S_{1,N} E_1^\top E_1 & P\mathbb{L} \\
\mathbb{L}^\top P & -\eta I
\end{array}
\right) + \epsilon S_{2,N} E_2^\top E_2 \prec 0,
$$
$$
\left(
\begin{array}{cc}
-2\lambda_{N+1} + \eta S_{\varphi,N}(\lambda_{N+1} + \nu) & \sqrt{2N_0^2(\lambda_{N+1} + \mu)} \\
\sqrt{2N_0^2(\lambda_{N+1} + \mu)} & -\epsilon
\end{array}
\right) \prec 0,
$$

then the closed-loop system is exponentially stable in the sense that there exists $\delta > 0$ such that (33) holds. Following the proof of Theorem 15, these LMI conditions are always feasible for $N$ selected to be large enough.

4 Numerical example

We provide in this section numerical simulations for Ex-
ample 1 in the case $\epsilon = 0.1$ and $a = 5.1$. We recall that,
in this setting, the open-loop system is unstable with a
total of three modes that are unstable: $-\lambda_{1,1} = 3.095$
and $-\lambda_{1,2} = -\lambda_{2,1} = 0.095$. Hence, we fix $N_0 = 3$. We
select the two measurement locations as $\xi_1 = (1,0.5)$
and $\xi_2 = (3,1)$ for which the observability condition (29)
is satisfied. We select the observer gain $L \in \mathbb{R}^{3 \times 2}$ such
that the eigenvalues of the matrix $A_0 - LC_0$ are placed
at $-4, -5$, and $-6$. We choose

$$
\chi(x) = \begin{cases} \chi_p(x/h), & \text{for } x \in [0, h], \\ 1, & \text{for } x \in [h, 1-h], \\ \chi_p((1-x)/h), & \text{for } x \in [1-h, 1], \end{cases}
$$

where $\chi_p(x) = x^3(10 - 15x + 6x^2)$ and $h = 0.1$. Then, we
select $\gamma_1 = 2$, $\gamma_2 = 3.5$, $\gamma_3 = 5$ for which we check that
the matrix $-\sum_{k=1}^{N_0} \gamma_k B_k A$ is Hurwitz. Theorem 15 en-
sures that the plant (1) with internal measurement (2) is
exponentially stabilized (in the sense of inequality (33))
by the controller (19) with $U$ given by (32) provided the
order $N$ of the observer is selected to be large enough.

We perform a finite-dimensional modal approximation of
the PDE plant (1) based on its 300 first modes. We check
on numerical simulations that the closed-loop system is
unstable for an order of the observer up to $N = 69$ but
is stabilized for $N = 70$. In that case, the first mode that
is not estimated by the observer is $-\lambda_{10,1} = -95.905$.
Setting $N = 70$, Figure 1 shows the time-evolution of
the $H^1$-norm of the closed-loop system trajectory for
100 initial conditions. The initial conditions are of the

<footnote n="2">² The second condition is derived from the application of Schur’s complement.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Norm $||z(\cdot, t)||_{H^1(\mathcal{O})} + \sum_{k=1}^N |\hat{z}_k(t)|$ of the closed-loop system trajectories for 100 initial conditions of the form $z_o(x_1, x_2) = \sum_{0 \le i+j \le 3} a_{i,j} x_1^i x_2^j$ with $a_{i,j} \in [-1, 1]$ randomly selected</figcaption></figure>

form $z_o(x_1, x_2) = \sum_{0 \le i+j \le 3} a_{i,j} x_1^i x_2^j$ with $a_{i,j} \in [-1, 1]$ randomly selected. It is worth pointing out that the sufficient LMI conditions of Remark 16 are found to be unfeasible up to $N = 100$ using Matlab LMI solver. This seems to indicate the conservatism of the reported sufficient conditions.

# 5 Conclusion

This paper discussed the design of an observer-based feedback stabilizing controller for multi-dimensional parabolic type equations governed by diagonalizable second order differential operators. To fix the ideas and to ease the presentation, we focused the developments on the case of three unstable eigenvalues: one of multiplicity one and one of multiplicity two. However, the approach reported in this paper easily extends to any other case with a finite number of unstable modes with arbitrary finite multiplicity. We note that, due to the computations done in Example 1 in Section 1, the approach requires precise knowledge of eigenfunctions and eigenvalues of the leading linear operator. This may be an issue in case of complex geometries of the domain $\mathcal{O}$, where only approximations of the spectrum are available. This may lead to errors in the eigenfunctions and eigenvalues sets. Therefore, as a future research subject, we propose to address the robustness problem associated to the observer-based controller we designed.

To conclude, it is worth to mention that, based on the technique presented in this work, a natural perspective is the study of non-linear multi-dimensional parabolic equations combining the present design method with [24], and the study of multi-dimensional parabolic equations with delays as done in e.g., [25,22].

# References

[1] J.H. Albert. Genericity of simple eigenvalues for elliptic PDEs. *Proceedings of the American Mathematical Society*, 48(2):413–418, 1975.

[2] M. Badra and T Takahashi. Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: applications to the Navier-Stokes system. *SIAM Journal of Control and Optimization*, (49):420–463, 2011.

[3] M. Badra and T Takahashi. On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems. *ESAIM Control Optimization and Calculus of Variations*, (20):924–956, 2014.

[4] M. J. Balas. Feedback control of flexible systems. *IEEE Transactions on Automatic Control*, 23(4):673–679, 1978.

[5] M. J. Balas. Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters. *Journal of Mathematical Analysis and Applications*, 133(2):283–296, 1988.

[6] V. Barbu. *Stabilization of Navier-Stokes flows*. Springer, 2011.

[7] V. Barbu. Boundary stabilization of equilibrium solutions to parabolic equations. *IEEE Transactions on Automatic Control*, 58(9):2416–2420, 2013.

[8] G. Bastin and J.-M. Coron. *Stability and boundary stabilization of 1-D hyperbolic systems*, volume 88 of *Progress in Nonlinear Differential Equations and Their Applications*. Springer, 2016.

[9] A. Bressan. *Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem*, volume 20. Oxford University Press, USA, 2000.

[10] J.-M. Coron and E. Trélat. Global steady-state controllability of one-dimensional semilinear heat equations. *SIAM Journal on Control and Optimization*, 43(2):549–569, 2004.

[11] R. Curtain and H. Zwart. *Introduction to infinite-dimensional systems theory: a state-space approach*, volume 71. Springer Nature, 2020.

[12] C. Delattre, D. Dochain, and J. Winkin. Sturm-Liouville systems are Riesz-spectral systems. *International Journal of Applied Mathematics and Computer Science*, 4(13):481–484, 2003.

[13] G. Palatucci E. Di Nezza and E. Valdinoci. Hitchhiker's guide to the fractional Sobolev spaces. *Bulletin des Sciences Mathematiques*, 136(4):521–573, 2012.

[14] H. Feng, P.-H. Lang, and J. Liu. Boundary stabilization and observation of a weak unstable heat equation in a general multi-dimensional domain. *Automatica*, 138:110152, 2022.

[15] N. Garofalo and F.-H. Lin. Unique continuation for elliptic operators: a geometric-variational approach. *Communications on Pure and Applied Mathematics*, 40(3):347–366, 1987.

[16] D. Gilbarg and N. S. Trudinger. *Elliptic partial differential equations of second order*, volume 224. Springer, 1977.

[17] L. Grüne and T. Meurer. Finite-dimensional output stabilization for a class of linear distributed parameter systems—a small-gain approach. *Systems & Control Letters*, 164:105237, 2022.

[18] L. Hörmander. The spectral function of an elliptic operator. *Acta Mathematica*, 121:193–218, 1968.

[19] R. Katz and E. Fridman. Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs. *Automatica*, 122:109285, 2020.
[20] S. Koga and M. Krstic. Materials phase change PDE control & estimation: from additive manufacturing to polar ice. Springer Nature, 2020.

[21] M. Krstic and A. Smyshlyaev. Boundary control of PDEs: A course on backstepping designs. SIAM, 2008.

[22] H. Lhachemi and Ch. Prieur. Boundary output feedback stabilisation of a class of reaction–diffusion PDEs with delayed boundary measurement. International Journal of Control, 96(9):1–11, 2022.

[23] H. Lhachemi and Ch. Prieur. Finite-dimensional observer-based boundary stabilization of reaction–diffusion equations with either a Dirichlet or Neumann boundary measurement. Automatica, 135:109955, 2022.

[24] H. Lhachemi and Ch. Prieur. Global Output Feedback Stabilization of Semilinear Reaction-Diffusion PDEs. 4th IFAC Workshop on Control of Systems Governed by Partial Differential Equations (CPDE’22). Kiel, Germany, 2022.

[25] H. Lhachemi and R. Shorten. Boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and state-delay. Automatica, 116:108931, 2020.

[26] J.L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, I. Lecture Notes in Mathematics 181, Springer, Berlin, 1972.

[27] V.P. Mikhailov. Partial differential equations. Translated from the Russian by P.C. Sinha. Revised from the 1976 Russian ed., Moscow: Mir Publishers, 1978.

[28] I. Munteanu. Boundary stabilization of parabolic equations. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser-Springer, 2019.

[29] I. Munteanu. A simple pole-shifting gain matrix K which avoids solving Lyapunov equations. arXiv:2402.10489, 2024.

[30] Y. V. Orlov. Discontinuous unit feedback control of uncertain infinite-dimensional systems. IEEE Transactions on Automatic Control, 45(5):834–843, 2000.

[31] D. L. Russell. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Review, 20(4):639–739, 1978.

[32] Y. Sakawa. Feedback stabilization of linear diffusion systems. SIAM Journal on Control and Optimization, 21(5):667–676, 1983.

[33] C. D. Sogge. Eigenfunction and Bochner Riesz estimates on manifolds with boundary. Mathematical Research Letter, 9:205–216, 2002.

[34] W.A. Strauss. Partial Differential Equations. John Wiley & Sons, 2008.

[35] R. Triggiani. Boundary feedback stabilizability of parabolic equations. Applied Mathematics and Optimization, 6(1):201–220, 1980.

[36] R. Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete and Continuous Dynamical Systems-B, 8(2):279–314, 2007.

[37] B. Van Keulen. H∞-Control for distributed parameter systems: A state-space approach. Springer Science & Business Media, 1993.

[38] C. Xu, E. Schuster, R. Vazquez, and M. Krstic. Stabilization of linearized 2-D magnetohydrodynamic channel flow by backstepping boundary control. Systems & Control Letters, 57(10):805–812, 2008.

# Appendix

**Lemma A** The sum $B_1+B_2+\cdots+B_{N_0}$ is an invertible matrix.

**Proof.** The proof is given in the general assumption of semi-simple eigenvalues framework. More precisely, assume that we have $p$ distinct eigenvalues, $\{\mu_1, \mu_2, \dots, \mu_p\}$ of multiplicity $n_1, n_2, \dots, n_p$, respectively, such that $n_1 + n_2 + \cdots + n_p = N_0$. Invoking iteratively the unique continuation property of the eigenfunctions of the operator $\mathcal{A}$, stated by Lemma 7, we observe that $\mathcal{L}$, defined in (17), has non-zero entries for all $x$ in the non-empty open $\Gamma_0$. It implies that if $\varphi_{j_1}, \varphi_{j_1+1}, \dots, \varphi_{j_M}$ are eigenfunctions corresponding to the same eigenvalue, then $T_n\varphi_{j_1}, T_n\varphi_{j_2}, \dots, T_n\varphi_{j_M}$ are linearly independent in $L^2(\partial\mathcal{O})$. This implies that $\chi^{\frac{1}{2}}T_n\varphi_{j_1}, \chi^{\frac{1}{2}}T_n\varphi_{j_2}, \dots, \chi^{\frac{1}{2}}T_n\varphi_{j_M}$ are linearly independent in $L^2(\partial\mathcal{O})$, since, otherwise, recalling that $\chi \equiv 1$ on $\Gamma_0$, we would have a linear combination of the traces of the eigenfunctions corresponding to the same eigenvalue equal zero in $\Gamma_0$. But this, via Lemma 7, contradicts the linear independence of the eigenfunctions in $\mathcal{O}$. The fact that $\chi^{\frac{1}{2}}T_n\varphi_{j_1}, \chi^{\frac{1}{2}}T_n\varphi_{j_2}, \dots, \chi^{\frac{1}{2}}T_n\varphi_{j_M}$ are linearly independent in $L^2(\partial\mathcal{O})$ is equivalent to saying that $T_n\varphi_{j_1}, T_n\varphi_{j_2}, \dots, T_n\varphi_{j_M}$ are linearly independent in $L^2_\chi(\partial\mathcal{O})$. Consequently, the Gram matrix, associated to the set $\{T_n\varphi_{j_1}, T_n\varphi_{j_1+1}, \dots, T_n\varphi_{j_M}\}$ in $L^2_\chi(\partial\mathcal{O})$, is nonsingular. Here, $L^2_\chi(\partial\mathcal{O})$ is the weighted Lebesgue space of weight $\chi$ (see Subsection 2.1).

Clearly, the Gram matrix $\mathbf{B}$ can be written as

$$ \mathbf{B} = \begin{pmatrix} G_1 & * & \cdots & * \\ * & G_2 & \cdots & * \\ \vdots & \vdots & \ddots & \vdots \\ * & * & \cdots & G_p \end{pmatrix}, $$

where $G_k$ are the Gram matrices, in $L^2_\chi(\partial\mathcal{O})$, corresponding to the traces of the generalized normal derivatives of the eigenfunctions of the same eigenvalue. Therefore, $G_k$ is invertible for all $k = 1, 2, \dots, p$.

Matrix $A_0 \in \mathbb{R}^{N_0 \times N_0}$ can be written as

$$ A_0 = \operatorname{diag}(J_1, J_2, \dots, J_p), $$

where $J_j := \operatorname{diag}(-\mu_j, -\mu_j, \dots, -\mu_j) \in \mathbb{R}^{n_j \times n_j}$, $j = 1, 2, \dots, p$. Then, the Kalman matrix

$$ [\mathbf{B} \quad A_0\mathbf{B} \quad A_0^2\mathbf{B} \quad \dots \quad A_0^{N_0-1}\mathbf{B}] $$

has full rank. This is indeed so, since otherwise, we have, by the Hautus test, that there exists $V \in \mathbb{R}^{N_0}$, $V \neq 0$
and $k \in \{1, 2, \dots, p\}$ such that

$$A_0V = \mu_k V \text{ and } \mathbf{B}^\top V = 0.$$

We set $V = (V_1 V_2 \dots V_p)^\top$ where $V_j = (v_1^j v_2^j \dots v_{n_j}^j)^\top$, $j = 1, 2, \dots, p$. It follows that

$$A_0V - \mu_k V = ((J_1 - \mu_k)V_1 \dots (J_k - \mu_k)V_k \dots (J_p - \mu_k)V_p)^\top = 0.$$

Hence, $V_i = 0$ for all $i \neq k$. To conclude, we note that $\mathbf{B}^\top V = 0$ together with the fact that $V_i = 0$ for all $i \neq k$, implies that $G_k V_k = 0$, and so $V_k = 0$ as well, since $G_k$ is invertible. Consequently, $V = 0$, which contradicts the assumption $V \neq 0$.

Note that, the matrix

$$[\mathbf{B}^{\frac{1}{2}} \quad A_0 \mathbf{B}^{\frac{1}{2}} \quad \dots \quad A_0^{N_0-1} \mathbf{B}^{\frac{1}{2}}]$$

has full rank as well. Indeed, applying the Hautus test, we need to study the vector $V$ such that

$$A_0V = \mu V \text{ and } \mathbf{B}^{\frac{1}{2}}V = 0.$$

Since $\mathbf{B}^{\frac{1}{2}}V$ implies $BV = 0$, we get

$$A_0V = \mu V \text{ and } BV = 0,$$

which, as seen before, yields that $V = 0$.

Now, assume by contradiction that the sum of the $B_k$s is not invertible. Hence there exists $V \in \mathbb{R}^{N_0}, V \neq 0$ such that $\sum_{k=1}^{N_0} B_k V = 0$. Or, equivalently

$$\sum_{k=1}^{N_0} (A_0 + \gamma_k I)^{-1} \mathbf{B} (A_0 + \gamma_k I)^{-1} V = 0.$$

Scalarly multiplying in $\mathbb{R}^{N_0}$ the above relation by $V$, it follows that

$$\sum_{k=1}^{N_0} \| \mathbf{B}^{\frac{1}{2}} (A_0 + \gamma_k I)^{-1} V \|_{N_0}^2 = 0$$

or, equivalently,

$$V^\top \mathbf{B}^{\frac{1}{2}} (A_0 + \gamma_k I)^{-1} = 0, \text{ for all } k = 1, 2, \dots, N_0.$$

This means that the matrix

$$[\mathbf{B}^{\frac{1}{2}} (A_0 + \gamma_1 I)^{-1} \quad \dots \quad \mathbf{B}^{\frac{1}{2}} (A_0 + \gamma_{N_0} I)^{-1}]$$

does not have full rank, since there exists a non-trivial linear combination of its lines which is zero. Or, by taking the transpose, the matrix

$$[(A_0 + \gamma_1 I)^{-1} \mathbf{B}^{\frac{1}{2}} \quad \dots \quad (A_0 + \gamma_{N_0} I)^{-1} \mathbf{B}^{\frac{1}{2}}]$$

does not have full rank.

It is well-known that if we multiply a matrix by an invertible square matrix, either on the left or on the right, the resulting matrix has the same rank. That is, the elementary row operations and the corresponding elementary column operations on a matrix preserve the rank of a matrix. Performing such elementary operations, we get that

$$\operatorname{rank}[(A_0 + \gamma_1 I)^{-1} \mathbf{B}^{\frac{1}{2}} \quad \dots \quad (A_0 + \gamma_{N_0} I)^{-1} \mathbf{B}^{\frac{1}{2}}] = \operatorname{rank}[\mathbf{B}^{\frac{1}{2}} \quad A_0 \mathbf{B}^{\frac{1}{2}} \quad \dots \quad A_0^{N_0-1} \mathbf{B}^{\frac{1}{2}}].$$

(For more details see [29, Lemma 3].) This is in contradiction with the above results. □