# Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation

Hugo Lhachemi, Christophe Prieur

► To cite this version:

Hugo Lhachemi, Christophe Prieur. Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation. IEEE Transactions on Automatic Control, 2022, 67 (11), pp.6143-6150. 10.1109/TAC.2021.3130874 . hal-03472730

HAL Id: hal-03472730

https://hal.science/hal-03472730v1

Submitted on 25 Feb 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
# Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation

Hugo Lhachemi and Christophe Prieur

*Abstract*—This paper investigates the output regulation control of a reaction-diffusion equation by means of boundary control. The considered reaction-diffusion plant may be open-loop unstable. The proposed control strategy consists of the coupling of a finite-dimensional observer and a PI controller in order to achieve the boundary regulation control of various system outputs such as the Dirichlet and Neumann traces. In this context, it is shown that the order of the finite-dimensional observer can always be selected large enough, with explicit criterion, to achieve both the stabilization of the plant and the regulation of the system output.

*Index Terms*—Reaction-diffusion equation, finite-dimensional observer, output feedback, PI control, boundary regulation control, boundary measurement.

## I. INTRODUCTION

The problem of controlling the output of a system so as to achieve asymptotic tracking of prescribed trajectories is one of the most fundamental problems in control theory. In the general context of finite-dimensional linear time-invariant (LTI) control systems, the problem of setpoint regulation control is very classical and has been widely investigated. One possible way to solve this problem is based on the augmentation of the state-space representation of the plant with an integral component of the tracking error and the use of the separation principle by exploiting separately a Luenberger observer (which allows the estimation of the state based on the measure only) and a stabilizing full-state feedback (see, e.g., [1], [9]). Even if this approach has reached a very high level of maturity for finite-dimensional control systems, its possible extension to infinite-dimensional systems, as those considered in this paper, is still an open problem.

Infinite-dimensional systems emerge in many practical applications due to the occurrence of delays, reaction-diffusion dynamics, or even flexible behavior (see, e.g., [11], [14], [15] for introductory textbooks on dedicated control theory for infinite-dimensional systems). While many efficient control design methods have been reported for the stabilization of distributed parameter systems, very few have been extended to the problem of output regulation. The main reason is that all techniques that have been developed for finite-dimensional LTI cannot be easily generalized to infinite-dimensional plants. For instance, the frequency domain approach has been generalized to the infinite-dimensional setting, but it requires to deal with an infinite number of poles, yielding an infinite-dimensional pole allocation problem (which is one rationale behind the frequency domain approach for regulation of finite-dimensional LTI systems). The state-space approach is followed in this work.

We propose in this paper, for the first time, an output feedback control design procedure to achieve the setpoint regulation control of a reaction-diffusion system by means of a finite-dimensional observer coupled with a PI controller. The considered reaction-diffusion plant,

Hugo Lhachemi is with L2S, CentraleSupélec, 91192 Gif-sur-Yvette, France (e-mail: hugo.lhachemi@centralesupelec.fr).

Christophe Prieur is with Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France (e-mail: christophe.prieur@gipsa-lab.fr). This work has been partially supported by MIAI@Grenoble Alpes (ANR- 19-P3IA-0003)

which might be unstable, is modeled by a Sturm-Liouville operator as those classical introduced in the context of parabolic partial differential equation. Note that the case of PI regulation of this system by means of a state feedback was reported in [13] (see also [3], [4], [7], [13], [18]–[22] for various approaches about PI control design for different types of PDEs). Here we go beyond by proposing an output feedback PI control strategy. Even if the proposed procedure also applies to bounded control inputs and bounded observations, we focus the presentation on boundary controls and boundary measurements. This is because these configurations are the most interesting in terms of practical applications. They are also the most challenging since they require to deal with unbounded control and observation operators (see e.g., [6] for further explanations). Several cases for the input-to-output map are investigated. This includes in particular Dirichlet control inputs (which can easily be extended to Neumann control inputs as discussed in conclusion) along with Dirichlet and/or Neumann to-be-regulated outputs and measured outputs. We also show that our procedure can be used to regulate a system output that is distinct of the measured one. Therefore, our approach gives a complete framework to study every associated input-to-output maps. The proposed control design strategy is based on the introduction of an adequate integral component for setpoint regulation purposes and an appropriate coupling of a state-feedback with a finite-dimensional observer.

It is worth noting that the design of finite-dimensional observer-based controllers for distributed parameter plants is a challenging task due to the fact that the separation principle, that is classically used for finite-dimensional systems, does not apply for infinite-dimensional systems [2], [5], [8]. A constructive method for solving this stabilization problem for a reaction-diffusion system was reported in [10] in the case where the either control or observation operator is bounded. This approach was extended in [12] to the case where both control and observation operators are unbounded, including both Dirichlet and Neumann settings. The present work, taking advantage of [12], goes beyond the simple problem of closed-loop stabilization by embracing the issue of output setpoint regulation control. Our approach is based on Lyapunov direct methods. Let us emphasize that when Dirichlet/Neumann boundary conditions are considered for either the control input or the to-be-regulated output, the solutions need to be sufficiently regular. However, our conditions do not need further regularity than the ones required by the existence results of classical solutions. Our main results take the form of sets of explicit sufficient conditions ensuring the both stability and regulation control of the closed-loop plant. We assess that these sufficient conditions are always feasible when selecting the order of the finite-dimensional observer large enough. Therefore, we show in a constructive manner that the setpoint regulation control of a reaction-diffusion plant can always be achieved by the coupling of an integral component, a finite-dimensional observer, and a state-feedback.

The paper is organized by considering successively different input-output maps for the reaction-diffusion equation depending on the selected boundary measured output, the to-be-regulated output, and the control input. After recalling classical notations and properties for the Sturm-Liouville operators in Section II, the case of a Dirichlet
observation and a Dirichlet control input is considered in Section III. Then the case of a Neumann measurement and a Dirichlet control input is considered in Section IV. While the to-be-regulated output and the measured output are the same in the two latter sections, a crossed configuration is considered in Section V. The regulation problem is solved for a Dirichlet measured output, a Neumann to-be-regulated output, and a Dirichlet control input. This final result completes the picture and gives a full study of the different cases for the input-to-output map of the considered class of distributed parameter systems. Some numerical simulations are given in Section VI for this final result. Section VII collects some concluding remarks.

## II. NOTATION AND PROPERTIES

Spaces $\mathbb{R}^n$ are endowed with the Euclidean norm denoted by $\| \cdot \|$. The associated induced norms of matrices are also denoted by $\| \cdot \|$. Given two vectors $X$ and $Y$, $\text{col}(X, Y)$ denotes the vector $[X^\top, Y^\top]^\top$. $L^2(0, 1)$ stands for the space of square integrable functions on $(0, 1)$ and is endowed with the inner product $\langle f, g \rangle = \int_0^1 f(x)g(x) dx$ with associated norm denoted by $\| \cdot \|_{L^2}$. For an integer $m \ge 1$, the $m$-order Sobolev space is denoted by $H^m(0, 1)$ and is endowed with its usual norm denoted by $\| \cdot \|_{H^m}$. For a symmetric matrix $P \in \mathbb{R}^{n \times n}$, $P \succeq 0$ (resp. $P > 0$) means that $P$ is positive semi-definite (resp. positive definite) while $\lambda_M(P)$ (resp. $\lambda_m(P)$) denotes its maximal (resp. minimal) eigenvalue.

Let $p \in C^1([0, 1])$ and $q \in C^0([0, 1])$ with $p > 0$ and $q \ge 0$. Let the Sturm-Liouville operator $\mathcal{A}: D(\mathcal{A}) \subset L^2(0, 1) \to L^2(0, 1)$ be defined by $\mathcal{A}f = -(pf)' + qf$ on the domain $D(\mathcal{A}) \subset L^2(0, 1)$ given by either $D(\mathcal{A}) = \{f \in H^2(0, 1) : f(0) = f(1) = 0\}$ or $D(\mathcal{A}) = \{f \in H^2(0, 1) : f'(0) = f(1) = 0\}$. The eigenvalues $\lambda_n$, $n \ge 1$, of $\mathcal{A}$ are simple, non negative, and form an increasing sequence with $\lambda_n \to +\infty$ as $n \to +\infty$. Moreover, the associated unit eigenvectors $\phi_n \in L^2(0, 1)$ form a Hilbert basis. We also have $D(\mathcal{A}) = \{f \in L^2(0, 1) : \sum_{n \ge 1} |\lambda_n|^2 |\langle f, \phi_n \rangle|^2\}$ and $\mathcal{A}f = \sum_{n \ge 1} \lambda_n \langle f, \phi_n \rangle \phi_n$.

Let $p_*, \bar{p}^*, q^* \in \mathbb{R}$ be such that $0 < p_* \le p(x) \le p^*$ and $0 \le q(x) \le q^*$ for all $x \in [0, 1]$, then it holds [16]:

$$0 \le \pi^2(n-1)^2 p_* \le \lambda_n \le \pi^2 n^2 p^* + q^* \quad (1)$$

for all $n \ge 1$. Assuming further than $p \in C^2([0, 1])$, we have for any $x \in \{0, 1\}$ that $\phi_n(x) = O(1)$ and $\phi'_n(x) = O(\sqrt{\lambda_n})$ as $n \to +\infty$ [16]. Finally, one can check that, for all $f \in D(\mathcal{A})$,

$$\sum_{n \ge 1} \lambda_n \langle f, \phi_n \rangle^2 = \langle \mathcal{A}f, f \rangle = \int_0^1 p(f')^2 + qf^2 dx. \quad (2)$$

This implies that, for any $f \in D(\mathcal{A})$, the series expansion $f = \sum_{n \ge 1} \langle f, \phi_n \rangle \phi_n$ holds in $H^1(0, 1)$ norm. Then, using the definition of $\mathcal{A}$ and the fact that it is a Riesz-spectral operator, we obtain that the latter series expansion holds in $H^2(0, 1)$ norm. Due to the continuous embedding $H^1(0, 1) \subset L^\infty(0, 1)$, we obtain that $f(x) = \sum_{n \ge 1} \langle f, \phi_n \rangle \phi_n(x)$ and $f'(x) = \sum_{n \ge 1} \langle f, \phi_n \rangle \phi'_n(x)$ for all $x \in [0, 1]$.

## III. DIRICHLET MEASUREMENT AND REGULATION CONTROL

We consider the reaction-diffusion system with Dirichlet boundary observation described for $t > 0$ and $x \in (0, 1)$ by

$$z_t(t, x) = (p(x)z_x(t, x))_x + (q_c - q(x))z(t, x) \quad (3a)$$

$$z_x(t, 0) = 0, \quad z(t, 1) = u(t) \quad (3b)$$

$$z(0, x) = z_0(x) \quad (3c)$$

$$y(t) = z(t, 0) \quad (3d)$$

in the case $p \in C^2([0, 1])$. Here $q_c \in \mathbb{R}$ is a constant, $u(t) \in \mathbb{R}$ is the command input, $y(t) \in \mathbb{R}$ with $c \in L^2(0, 1)$ is the measurement, $z_0 \in L^2(0, 1)$ is the initial condition, and $z(t, \cdot) \in L^2(0, 1)$ is the state. Our control design objective is to design a finite-dimensional observer and an estimated state feedback to achieve the setpoint regulation of $y(t)$ to some prescribed reference signal $r(t)$.

### A. Spectral reduction

We introduce the change of variable

$$w(t, x) = z(t, x) - x^2 u(t). \quad (4)$$

Then we have

$$w_t(t, x) = (p(x)w_x(t, x))_x + (q_c - q(x))w(t, x) + a(x)u(t) + b(x)\dot{u}(t) \quad (5a)$$

$$w_x(t, 0) = 0, \quad w(t, 1) = 0 \quad (5b)$$

$$w(0, x) = w_0(x) \quad (5c)$$

$$\bar{y}(t) = w(t, 0) \quad (5d)$$

with $a, b \in L^2(0, 1)$ defined by $a(x) = 2p(x) + 2xp'(x) + (q_c - q(x))x^2$ and $b(x) = -x^2$, respectively, $\bar{y}(t) = y(t)$, and $w_0(x) = z_0(x) - x^2u(0)$. Introducing the auxiliary command input $v(t) = \dot{u}(t)$, we infer that

$$\dot{u}(t) = v(t) \quad (6a)$$

$$\frac{dw}{dt}(t, \cdot) = -Aw(t, \cdot) + q_c w(t, \cdot) + au(t) + bv(t) \quad (6b)$$

with $D(\mathcal{A}) = \{f \in H^2(0, 1) : f'(0) = f(1) = 0\}$. We introduce the coefficients of projection $w_n(t) = \langle w(t, \cdot), \phi_n \rangle$, $a_n = \langle a, \phi_n \rangle$, and $b_n = \langle b, \phi_n \rangle$. Considering classical solutions associated with any $z_0 \in H^2(0, 1)$ and any $u(0) \in \mathbb{R}$ such that $z'_0(0) = 0$ and $z_0(1) = u(0)$ (their existence for the upcoming closed-loop dynamics is an immediate consequence of [17, Chap. 6, Thm. 1.7]), we have $w(t, \cdot) \in D(\mathcal{A})$ for all $t \ge 0$ and we infer that

$$\dot{u}(t) = v(t) \quad (7a)$$

$$\dot{w}_n(t) = (-\lambda_n + q_c)w_n(t) + a_n u(t) + b_n v(t), \quad n \ge 1 \quad (7b)$$

$$\bar{y}(t) = \sum_{i \ge 1} \phi_i(0) w_i(t) \quad (7c)$$

### B. Control design

Let $N_0 \ge 1$ and $\delta > 0$ be given such that $-\lambda_n + q_c < -\delta < 0$ for all $n \ge N_0 + 1$. Let $N \ge N_0 + 1$ be arbitrary. Inspired by [10], we design an observer to estimate the $N$ first modes of the plant while the state-feedback is performed on the $N_0$ first modes of the plant. Specifically, introducing $W^{N_0}(t) = [w_1(t) \quad \dots \quad w_{N_0}(t)]^\top$, $A_0 = \text{diag}(-\lambda_1 + q_c, \dots, -\lambda_{N_0} + q_c)$, $B_{0,a} = [a_1 \quad \dots \quad a_{N_0}]^\top$, and $B_{0,b} = [b_1 \quad \dots \quad b_{N_0}]^\top$, we have

$$\dot{W}^{N_0}(t) = A_0 W^{N_0}(t) + B_{0,a} u(t) + B_{0,b} v(t). \quad (8)$$

Our objective is to introduce an integral component to achieve the setpoint regulation control of the system output $y(t)$. To do so, proceeding as in [13], we consider first the case of the following classical integral component: $\dot{z}_i = y(t) - r(t) = \sum_{n \ge 1} \phi_n(0)w_n(t) - r(t)$. Then, introducing $\xi_p(t) = z_i(t) - \sum_{n \ge N_0+1} \frac{\phi_n(0)}{-\lambda_n + q_c}w_n(t)$, we obtain that $\dot{\xi}_p(t) = \sum_{n=1}^{N_0} \phi_n(0)w_n(t) + \alpha_0 u(t) + \beta_0 v(t) - r(t)$ with

$$\alpha_0 = - \sum_{n \ge N_0+1} \frac{a_n \phi_n(0)}{-\lambda_n + q_c}, \quad \beta_0 = - \sum_{n \ge N_0+1} \frac{b_n \phi_n(0)}{-\lambda_n + q_c}. \quad (9)$$
Since $w_n(t)$ are not measured, we replace them by their estimated version $\hat{w}_n(t)$ which will be described below. Hence, the employed integral component takes the form:

$$ \dot{\xi}(t) = \sum_{n=1}^{N_0} \phi_n(0)\hat{w}_n(t) + \alpha_0 u(t) + \beta_0 v(t) - r(t). \quad (10) $$

We now define for $1 \le n \le N$ the observer dynamics:

$$ \dot{\hat{w}}_n(t) = (-\lambda_n + q_c)\hat{w}_n(t) + a_n u(t) + b_n v(t) - l_n \left( \sum_{i=1}^{N} \phi_i(0)\hat{w}_i(t) - \alpha_1 u(t) - \tilde{y}(t) \right) \quad (11) $$

with

$$ \alpha_1 = \sum_{n \ge N+1} \frac{a_n \phi_n(0)}{-\lambda_n + q_c} \quad (12) $$

and where $l_n \in \mathbb{R}$ are the observer gains. We impose $l_n = 0$ for $N_0+1 \le n \le N$ and the initial condition of the observer as $\hat{w}_n(0) = 0$ for all $1 \le n \le N$. We define for $1 \le n \le N$ the observation error as $e_n(t) = w_n(t) - \hat{w}_n(t)$. Hence we have

$$ \dot{\hat{w}}_n(t) = (-\lambda_n + q_c)\hat{w}_n(t) + a_n u(t) + b_n v(t) + l_n \sum_{i=1}^{N_0} \phi_i(0)e_i(t) + l_n \sum_{i=N_0+1}^{N} \frac{\phi_i(0)}{\sqrt{\lambda_i}} \tilde{e}_i(t) + l_n \alpha_1 u(t) + l_n \zeta(t) \quad (13) $$

with $\tilde{e}_n(t) = \sqrt{\lambda_n} e_n(t)$ and $\zeta(t) = \sum_{n \ge N+1} \phi_n(0) w_n(t)$. Hence, introducing $\hat{W}^{N_0}(t) = [\hat{w}_1(t) \dots \hat{w}_{N_0}(t)]^\top$, $E^{N_0}(t) = [e_1(t) \dots e_{N_0}(t)]^\top$, $\tilde{E}^{N-N_0}(t) = [\tilde{e}_{N_0+1} \dots \tilde{e}_N]^\top$, $C_0 = [\phi_1(0) \dots \phi_{N_0}(0)]$ and $C_1 = [\frac{\phi_{N_0+1}(0)}{\sqrt{\lambda_{N_0+1}}} \dots \frac{\phi_{N}(0)}{\sqrt{\lambda_N}}]$, and $L = [l_1 \dots l_{N_0}]^\top$, we obtain that

$$ \dot{\hat{W}}^{N_0}(t) = A_0 \hat{W}^{N_0}(t) + B_{0,a} u(t) + B_{0,b} v(t) + LC_0 E^{N_0}(t) + LC_1 \tilde{E}^{N-N_0}(t) + \alpha_1 Lu(t) + L\zeta(t). \quad (14) $$

With

$$ \hat{W}_a^{N_0}(t) = \operatorname{col}(u(t), \hat{W}^{N_0}(t), \xi(t)), \quad (15) $$

$\tilde{L} = \operatorname{col}(0, L, 0)$, and defining

$$ A_1 = \begin{bmatrix} 0 & 0 & 0 \\ B_{0,a} & A_0 & 0 \\ \alpha_0 & C_0 & 0 \end{bmatrix}, \quad B_1 = \begin{bmatrix} 1 \\ B_{0,b} \\ \beta_0 \end{bmatrix}, \quad B_r = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad (16) $$

we deduce that

$$ \dot{\hat{W}}_a^{N_0}(t) = A_1 \hat{W}_a^{N_0}(t) + B_1 v(t) - B_r r(t) + \tilde{L} C_0 E^{N_0}(t) + \tilde{L} C_1 \tilde{E}^{N-N_0}(t) + \alpha_1 \tilde{L} u(t) + \tilde{L} \zeta(t). \quad (17) $$

Setting the auxiliary command input as

$$ v(t) = K \hat{W}_a^{N_0}(t), \quad (18) $$

and defining

$$ A_{\mathrm{cl}}(\alpha_1) = A_1 + B_1 K + \alpha_1 \tilde{L} \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}, \quad (19) $$

we obtain that

$$ \dot{\hat{W}}_a^{N_0}(t) = A_{\mathrm{cl}}(\alpha_1) \hat{W}_a^{N_0}(t) - B_r r(t) + \tilde{L} C_0 E^{N_0}(t) + \tilde{L} C_1 \tilde{E}^{N-N_0}(t) + \tilde{L} \zeta(t) \quad (20) $$

and, from (8) and (14),

$$ \dot{E}^{N_0}(t) = (A_0 - LC_0)E^{N_0}(t) - LC_1 \tilde{E}^{N-N_0}(t) - \alpha_1 L \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \hat{W}_a^{N_0} - L\zeta(t). \quad (21) $$

We now define $\hat{W}^{N-N_0}(t) = [\hat{w}_{N_0+1}(t) \dots \hat{w}_N(t)]^\top$, $A_2 = \operatorname{diag}(-\lambda_{N_0+1} + q_c, \dots, -\lambda_N + q_c)$, $B_{2,a} = [a_{N_0+1} \dots a_N]^\top$, $B_{2,b} = [b_{N_0+1} \dots b_N]^\top$. We obtain from (11) with $l_n = 0$ for $N_0+1 \le n \le N$ that

$$ \begin{aligned} \dot{\hat{W}}^{N-N_0}(t) &= A_2 \hat{W}^{N-N_0}(t) + B_{2,a} u(t) + B_{2,b} v(t) \\ &= A_2 \hat{W}^{N-N_0}(t) + (B_{2,b} K + [B_{2,a} \quad 0 \quad 0]) \hat{W}_a^{N_0}(t) \end{aligned} \quad (22) $$

and, using (7b) and (11),

$$ \dot{\tilde{E}}^{N-N_0}(t) = A_2 \tilde{E}^{N-N_0}(t). \quad (23) $$

Putting now together (20-23) while introducing

$$ X(t) = \operatorname{col} (\hat{W}_a^{N_0}(t), E^{N_0}(t), \hat{W}^{N-N_0}(t), \tilde{E}^{N-N_0}(t)), \quad (24) $$

we obtain that

$$ \dot{X}(t) = FX(t) + \mathcal{L}\zeta(t) - \mathcal{L}_r r(t) \quad (25) $$

where

$$ F = \begin{bmatrix} A_{\mathrm{cl}}(\alpha_1) & \tilde{L}C_0 & 0 & \tilde{L}C_1 \\ -\alpha_1 L \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} & A_0 - LC_0 & 0 & -LC_1 \\ B_{2,b} K + \begin{bmatrix} B_{2,a} & 0 & 0 \end{bmatrix} & 0 & A_2 & 0 \\ 0 & 0 & 0 & A_2 \end{bmatrix} \quad , $$

$$ \mathcal{L} = \operatorname{col}(\tilde{L}, -L, 0, 0), \quad \mathcal{L}_r = \operatorname{col}(B_r, 0, 0, 0). $$

Defining $E = [1 \ 0 \ \dots \ 0]$ and $\tilde{K} = [K \ 0 \ 0 \ 0]$, we obtain from (15), (18), and (24) that

$$ u(t) = EX(t), \quad v(t) = \tilde{K}X(t) \quad (26) $$

and we can introduce

$$ G = \|a\|_{L^2}^2 E^\top E + \|b\|_{L^2}^2 \tilde{K}^\top \tilde{K} \le gI \quad (27) $$

with $g = \|a\|_{L^2}^2 + \|b\|_{L^2}^2 \|K\|^2$ a constant independent of $N$.

**Lemma 1:** $(A_1, B_1)$ is controllable and $(A_0, C_0)$ is observable.

**Proof.** From [13, Lem. 2], $(A_1, B_1)$ is controllable if and only if

$$ \left( \begin{bmatrix} 0 & 0 \\ B_{0,a} & A_0 \end{bmatrix}, \begin{bmatrix} 1 \\ B_{0,b} \end{bmatrix} \right) \quad (28) $$

satisfies the Kalman condition and the matrix

$$ T = \begin{bmatrix} 0 & 0 & 1 \\ B_{0,a} & A_0 & B_{0,b} \\ \alpha_0 & C_0 & \beta_0 \end{bmatrix} \quad (29) $$

is invertible. The former condition was assessed in [12]. Hence we focus on the latter one. Let $[u_e \ w_{1,e} \ \dots \ w_{N_0,e} \ v_e]^\top \in \ker(T)$. We obtain that

$$ v_e = 0, \quad (29a) $$

$$ a_n u_e + (-\lambda_n + q_c) w_{n,e} = 0, \quad 1 \le n \le N_0, \quad (29b) $$

$$ \alpha_0 u_e + \sum_{n=1}^{N_0} \phi_n(0) w_{n,e} = 0. \quad (29c) $$

Defining for $n \ge N_0 + 1$ the quantity $w_{n,e} = - \frac{a_n}{-\lambda_n + q_c} u_e$, we have $(-\lambda_n + q_c)w_{n,e} + a_n u_e = 0$ for all $n \ge 1$. Hence $(w_{n,e})_{n \ge 1}, (\lambda_n w_{n,e})_{n \ge 1} \in l^2(\mathbb{N})$ ensuring that $w_e \triangleq \sum_{n \ge 1} w_{n,e} \phi_n \in D(\mathcal{A})$ and $\mathcal{A}w_e = \sum_{n \ge 1} \lambda_n w_{n,e} \phi_n \in D(\mathcal{A})$. This shows that $-\mathcal{A}w_e + q_c w_e + au_e = 0$. Moreover, from (29c) and using (9), we infer that $w_e(0) = 0$. From the two last identities, we have that $(pw_e')' + (q_c - q)w_e + au_e = 0$, $w_e(0) = w_e'(0) = 0$, and $w_e(1) = 0$. Introducing the change of variable $z_e(x) = w_e(x) + x^2 u_e$, we deduce that $(pz_e')' + (q_c - q)z_e = 0$, $z_e(0) = z_e'(0) = 0$, and $z_e(1) = u_e$. By Cauchy uniqueness, we infer that $z_e = 0$ hence $u_e = z_e(1) = 0$. Thus we have $w_e = z_e - x^2 u_e = 0$ hence $w_{n,e} = 0$
for all $n \ge 1$. We deduce that $\ker(T) = \{0\}$. Overall, we have shown that $(A_1, B_1)$ is controllable. Finally, the pair $(A_0, C_0)$ is observable because 1) $A_0$ is diagonal with simple eigenvalues, 2) by Cauchy uniqueness, $\phi_n(0) \ne 0$ for all $n \ge 1$.

Hence we can select in the sequel $K \in \mathbb{R}^{1 \times (N_0+2)}$ and $L \in \mathbb{R}^{N_0}$ such that $A_1 + B_1 K$ and $A_0 - LC_0$ are Hurwitz.

### C. Equilibrium condition and dynamics of deviations

We aim at characterizing the equilibrium condition of the closed-loop system composed of the reaction-diffusion system (3), the auxiliary command input dynamics (6a), the integral action (10), the observer dynamics (11), and the state-feedback (18). To do so let $r(t) = r_e \in \mathbb{R}$ be arbitrary. We must solve the system of equations:

$$0 = (-\lambda_n + q_c)w_{n,e} + a_n u_e + b_n v_e = 0, \quad n \ge 1, \quad (30a)$$

$$0 = v_e = K \hat{W}_{a,e}^{N_0}, \quad (30b)$$

$$0 = \sum_{n=1}^{N_0} \phi_n(0) \hat{w}_{n,e} + \alpha_0 u_e + \beta_0 v_e - r_e, \quad (30c)$$

$$0 = (-\lambda_n + q_c) \hat{w}_{n,e} + a_n u_e + b_n v_e \quad (30d)$$

$$-l_n \left\{ \sum_{i=1}^{N} \phi_i(0) \hat{w}_{i,e} - \alpha_1 u_e - \tilde{y}_e \right\}, \quad 1 \le n \le N_0,$$

$$0 = (-\lambda_n + q_c) \hat{w}_{n,e} + a_n u_e + b_n v_e, \quad N_0 + 1 \le n \le N, \quad (30e)$$

$$\tilde{y}_e = \sum_{n \ge 1} \phi_n(0) w_{n,e}. \quad (30f)$$

We first note from (30b) that $v_e = 0$. Then, from (30a) we have $w_{n,e} = -\frac{a_n}{-\lambda_n + q_c} u_e$ for all $n \ge N_0+1$. In particular, from (30e), we have $\hat{w}_{n,e} = w_{n,e} = -\frac{a_n}{-\lambda_n + q_c} u_e$ for all $N_0+1 \le n \le N$. Defining $e_{n,e} = w_{n,e} - \hat{w}_{n,e}$ and $\zeta_e = \sum_{n \ge N+1} \phi_n(0)w_{n,e}$, we obtain that $e_{n,e} = 0$ for all $N_0+1 \le n \le N$. Hence, from (30d), we infer that $0 = (-\lambda_n + q_c) \hat{w}_{n,e} + a_n u_e + l_n \sum_{i=1}^{N_0} \phi_i(0) e_{i,e} + l_n \alpha_1 u_e + l_n \zeta_e$ for all $1 \le n \le N_0$. Combining this latter identity with (30a), we obtain that $(A_0 - LC_0)E_e^{N_0} - L\alpha_1 u_e - L\zeta_e = 0$. Invoking (12), we note that $\alpha_1 u_e = -\sum_{n \ge N+1} \phi_n(0)w_{n,e} = -\zeta_e$, implying that $(A_0 - LC_0)E_e^{N_0} = 0$. Since $A_0 - LC_0$ is Hurwitz, we infer that $e_{n,e} = 0$ for all $1 \le n \le N_0$. In particular, $\hat{w}_{n,e} = w_{n,e}$ for all $1 \le n \le N$. From (30b-30d) we deduce that $0 = A_{cl}(\alpha_1) \hat{W}_{a,e}^{N_0} - B_r r_e + \tilde{L} \zeta_e$. Recalling that $\zeta_e = -\alpha_1 u_e$ and $A_{cl}(\alpha_1)$ is defined by (19), we obtain that $(A_1 + B_1 K) \hat{W}_{a,e}^{N_0} = B_r r_e$. Since $A_1 + B_1 K$ is Hurwitz, we infer that $\hat{W}_{a,e}^{N_0} = [u_e \ \hat{w}_{1,e} \ \dots \ \hat{w}_{N_0,e} \ \xi_e]^\top = (A_1 + B_1 K)^{-1} B_r r_e$. This is in particular compatible with (30b) since, based on (16), we indeed obtain that $K \hat{W}_{a,e}^{N_0} = 0$. We note that $(w_{n,e})_{n \ge 1}, (\lambda_n w_{n,e})_{n \ge 1} \in l^2(\mathbb{N})$ ensuring that $w_e \triangleq \sum_{n \ge 1} w_{n,e} \phi_n \in D(A)$ and $Aw_e = \sum_{n \ge 1} \lambda_n w_{n,e} \phi_n \in D(A)$. Using (30a), we obtain that $-Aw_e + q_c w_e + au_e + bv_e = 0$. Introducing the change of variable $z_e = w_e + x^2 u_e$, $z_e$ is a static solution of (3a-3b) associated with the constant control input $u(t) = u_e$. Denoting by $y_e \triangleq z_e(0) = w_e(0) = \tilde{y}_e$, we infer from (30c) while invoking (9) that

$$r_e = \sum_{n=1}^{N_0} \phi_n(0) \hat{w}_{n,e} + \alpha_0 u_e = \sum_{n \ge 1} \phi_n(0) w_{n,e} = y_e.$$

Hence, for an arbitrarily given constant reference signal $r(t) = r_e \in \mathbb{R}$, the equilibrium condition of the closed-loop system is unique, fully characterized by $r_e$, and is such that $y_e = r_e$.

We can now introduce the dynamics of deviation of the different quantities w.r.t the equilibrium condition characterized by $r_e \in \mathbb{R}$. In particular:

$$\Delta w(t, x) = \Delta z(t, x) - x^2 \Delta u(t), \quad (31a)$$

$$\Delta \dot{X}(t) = F \Delta X(t) + \mathcal{L} \Delta \zeta(t) - \mathcal{L}_r \Delta r(t), \quad (31b)$$

$$\Delta \zeta(t) = \sum_{n \ge N+1} \phi_n(0) \Delta w_n(t), \quad (31c)$$

$$\Delta \hat{w}_n(t) = (-\lambda_n + q_c) \Delta w_n(t) + a_n \Delta u(t) + b_n \Delta v(t), \quad (31d)$$

$$\Delta v(t) = K \Delta \hat{W}_a^{N_0}(t), \quad (31e)$$

$$\Delta \tilde{y}(t) = \Delta y(t) = \sum_{n \ge 1} \phi_n(0) \Delta w_n(t) \quad (31f)$$

with $\Delta w_n(t) = \langle \Delta w(t, \cdot), e_n \rangle$.

### D. Stability analysis and regulation assessment

We define the constant $M_{1,\phi} = \sum_{n \ge 2} \frac{\phi_n(0)^2}{\lambda_n}$, which is finite when $p \in C^2([0, 1])$ because we recall that $\phi_n(0) = O(1)$ as $n \to +\infty$ and (1) hold.

**Theorem 1:** Let $p \in C^2([0, 1])$ with $p > 0$, $q \in C^0([0, 1])$ with $q \ge 0$, and $q_c \in \mathbb{R}$. Consider the reaction-diffusion system described by (3). Let $N_0 \ge 1$ and $\delta > 0$ be given such that $-\lambda_n + q_c < -\delta < 0$ for all $n \ge N_0 + 1$. Let $K \in \mathbb{R}^{1 \times (N_0+2)}$ and $L \in \mathbb{R}^{N_0}$ be such that $A_1 + B_1 K$ and $A_0 - LC_0$ are Hurwitz with eigenvalues that have a real part strictly less than $-\delta < 0$. Assume that there exist $N \ge N_0 + 1$, $P > 0$, and $\alpha, \beta, \gamma > 0$ such that

$$\Theta = \begin{bmatrix} F^\top P + PF + 2\delta P + \alpha \gamma G & P\mathcal{L} \\ \mathcal{L}^\top P^\top & -\beta \end{bmatrix} < 0, \quad (32)$$

$$\Gamma_n = -\lambda_n + q_c + \delta + \frac{\lambda_n}{\alpha} + \frac{\beta M_{1,\phi}}{2\gamma} \le 0$$

for all $n \ge N+1$. Then, for any $\eta \in [0, 1]$, there exists $M > 0$ such that, for any $z_0 \in H^2(0, 1)$ and any $u(0) \in \mathbb{R}$ such that $z_0'(0) = 0$ and $z_0(1) = u(0)$, the classical solution of the closed-loop system composed of the plant (3), the integral actions (6a) and (10), the observer dynamics (11) with null initial condition, and the state feedback (18) satisfies

$$\begin{aligned} & \Delta u(t)^2 + \Delta \xi(t)^2 + \sum_{n=1}^{N} \Delta \hat{w}_n(t)^2 + \|\Delta z(t)\|_{H^1}^2 \\ & \le Me^{-2\delta t} (\Delta u(0)^2 + \Delta \xi(0)^2 + \|\Delta z_0\|_{H^1}^2) \\ & + M \sup_{\tau \in [0,t]} e^{-2\eta\delta(t-\tau)} \Delta r(\tau)^2 \end{aligned} \quad (33)$$

for all $t \ge 0$. Moreover, the above constraints are always feasible for $N$ large enough.

**Proof.** Let $P > 0$ and $\gamma > 0$ and consider the Lyapunov function candidate defined by

$$V(\Delta X, \Delta w) = \Delta X^\top P \Delta X + \gamma \sum_{n \ge N+1} \lambda_n \langle \Delta w, \phi_n \rangle^2. \quad (34)$$

with $\Delta X \in \mathbb{R}^{2N+2}$ and $\Delta w \in D(A)$. Proceeding exactly as in [12] but taking into account the extra contribution of the reference signal appearing in (25), we obtain for $t \ge 0$ that

$$\begin{aligned} \dot{V}(t) + 2\delta V(t) &\le \begin{bmatrix} \Delta X(t) \\ \Delta \zeta(t) \end{bmatrix}^\top \Theta \begin{bmatrix} \Delta X(t) \\ \Delta \zeta(t) \end{bmatrix} \\ &\quad - 2\Delta X(t)^\top P \mathcal{L}_r \Delta r(t) + 2\gamma \sum_{n \ge N+1} \lambda_n \Gamma_n \Delta w_n(t)^2 \end{aligned} \quad (35)$$

with $\alpha, \beta > 0$ arbitrary and where, with a slight abuse of notation, $\dot{V}(t)$ denotes the time derivative of $V(X(t), w(t))$ along the system trajectories (31). From (32), there exists $\epsilon > 0$ such that $\theta < -\epsilon I$. Hence the assumptions imply that $\dot{V}(t) + 2\delta V(t) \le -\epsilon \|\Delta X(t)\|^2 - 2\Delta X(t)^\top P \mathcal{L}_r \Delta r(t) \le \frac{\|P \mathcal{L}_r\|^2}{\epsilon} \Delta r(t)^2$ where Young's inequality has been used to derive the latter estimate. After integration, we obtain for any $\eta \in [0, 1)$ the existence of a constant $M_1 > 0$ such that $V(t) \le e^{-2\delta t} V(0) + M_1 \sup_{\tau \in [0,t]} e^{-2\eta\delta(t-\tau)} \Delta r(\tau)^2$ for all
$t \ge 0$. The claimed estimate (33) easily follows from the definition (34) of the Lyapunov function, the use of (2), Poincaré's inequality, and the change of variable (31a).

We now show that we can always select $N \ge N_0 + 1$, $P > 0$ and $\alpha, \beta, \gamma > 0$ such that $\Theta < 0$ and $\Gamma_n \le 0$ for all $n \ge N + 1$. By the Schur complement, $\Theta < 0$ is equivalent to $F^\top P + PF + 2\delta P + \alpha\gamma G + \frac{1}{\beta} P L L^\top P^\top < 0$. We define $F = F_1 + F_2$ where

$$F_1 = \begin{bmatrix} A_1 + B_1 K & \tilde{L}C_0 & 0 & \tilde{L}C_1 \\ 0 & A_0 - LC_0 & 0 & -LC_1 \\ B_{2,b}K + [B_{2,a} & 0 & 0] & A_2 & 0 \\ 0 & 0 & 0 & A_2 \end{bmatrix}, \quad (36a)$$

$$F_2 = \begin{bmatrix} \alpha_1 \tilde{L} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} & 0 & 0 & 0 \\ -\alpha_1 L \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \quad (36b)$$

with $\|F_2\| \to 0$, because $\alpha_1 \to 0$, when $N \to +\infty$. In order to apply the Lemma reported in the Appendix, we note that $A_1 + B_1 K + \delta I$ and $A_0 - LC_0 + \delta I$ are Hurwitz while $\|e^{(A_2+\delta I)t}\| \le e^{-\kappa_0 t}$ with $\kappa_0 = \lambda_{N_0+1} - q_c - \delta > 0$. Moreover, $\|\tilde{L}C_1\| \le \|L\|\|C_1\|$, $\|LC_1\| \le \|L\|\|C_1\|$, with $\|C_1\| = O(1)$ as $N \to +\infty$ while $\|B_{2,b}K + [B_{2,a} & 0 & 0]\| \le \|b\|_{L^2} \|K\| + \|a\|_{L^2}$ where the right-hand side is a constant independent of $N$. Hence, the application of the Lemma reported in the Appendix to $F_1 + \delta I$ yields the existence of $P > 0$ such that $F_1^\top P + PF_1 + 2\delta P = -I$ and $\|P\| = O(1)$ as $N \to +\infty$. Therefore, we have $F^\top P + PF + 2\delta P + \alpha\gamma G + \frac{1}{\beta} P L L^\top P^\top = -I + F_2^\top P + PF_2 + \alpha\gamma G + \frac{1}{\beta} P L L^\top P^\top$ where $G$ satisfies (27) and $\|L\| = \sqrt{2}\|L\|$, which is independent of $N$. Hence, setting $\alpha = \beta = \sqrt{N}$ and $\gamma = N^{-1}$, we infer the existence of $N \ge N_0 + 1$ such that $\Theta < 0$ and $\Gamma_n \le 0$ for all $n \ge N + 1$. $\square$ We can now assess the setpoint regulation control of the left Dirichlet trace.

**Theorem 2:** Under both assumptions and conclusions of Theorem 1, for any $\eta \in [0, 1)$, there exists $M_r > 0$ such that

$$|y(t) - r(t)| \le M_r e^{-\delta t} (|\Delta u(0)| + |\Delta \xi(0)| + \|\Delta z_0\|_{H^1}) \quad (37) \\ + M_r \sup_{\tau \in [0,t]} e^{-\eta \delta(t-\tau)} |\Delta r(\tau)|$$

for all $t \ge 0$.

**Proof.** Recalling that $y_e = r_e$, one has $|y(t) - r(t)| \le |\Delta y(t)| + |\Delta r(t)|$. From (31f) and Cauchy-Schwarz inequality, we infer that $|\Delta y(t)| \le \sqrt{\sum_{n \ge 1} \frac{\phi_n(0)^2}{\lambda_n}} \sqrt{\sum_{n \ge 1} \lambda_n \Delta w_n(t)^2}$. Using now (2) we infer the existence of a constant $M_2 > 0$ such that $|\Delta y(t)| \le M_2 \|\Delta w(t)\|_{H^1}$. The proof is completed by invoking the change of variable (31a) and the stability result (33). $\square$

## IV. NEUMANN MEASUREMENT AND REGULATION CONTROL

We now consider the reaction-diffusion system with Neumann boundary observation described for $t > 0$ and $x \in (0, 1)$ by

$$z_t(t, x) = (p(x)z_x(t, x))_x + (q_c - q(x))z(t, x) \quad (38a)$$

$$z(t, 0) = 0, \quad z(t, 1) = u(t) \quad (38b)$$

$$z(0, x) = z_0(x) \quad (38c)$$

$$y(t) = z_x(t, 0) \quad (38d)$$

in the case $p \in C^2([0, 1])$.

### A. Control design

Introducing the change of variable

$$w(t, x) = z(t, x) - xu(t) \quad (39)$$

we obtain

$$w_t(t, x) = (p(x)w_x(t, x))_x + (q_c - q(x))w(t, x) \quad (40a) \\ + a(x)u(t) + b(x)\dot{u}(t)$$

$$w(t, 0) = 0, \quad w(t, 1) = 0 \quad (40b)$$

$$w(0, x) = w_0(x) \quad (40c)$$

$$\tilde{y}(t) = w_x(t, 0) \quad (40d)$$

with $a, b \in L^2(0, 1)$ defined by $a(x) = p'(x) + (q_c - q(x))x$ and $b(x) = -x$, respectively, $\tilde{y}(t) = y(t) - u(t)$, and $w_0(x) = z_0(x) - xu(0)$. Introducing the auxiliary command input $v(t) = \dot{u}(t)$, we infer that (6) still holds but the domain of $\mathcal{A}$ is now replaced by $D(\mathcal{A}) = \{f \in H^2(0, 1) : f(0) = f(1) = 0\}$. Then, considering classical solutions associated with any $z_0 \in H^2(0, 1)$ and any $u(0) \in \mathbb{R}$ such that $z_0(0) = 0$ and $z_0(1) = u(0)$ (their existence for the upcoming closed-loop dynamics is an immediate consequence of [17, Chap. 6, Thm. 1.7]), (7a-7b) is still valid while (7c) is replaced by

$$\tilde{y}(t) = \sum_{i \ge 1} \phi_i'(0) w_i(t). \quad (41)$$

Based on similar motivations than the ones reported in Section III, we consider the integral component

$$\dot{\xi}(t) = \sum_{n=1}^{N_0} \phi_n'(0) \hat{w}_n(t) + \alpha_0 u(t) + \beta_0 v(t) - r(t). \quad (42)$$

with

$$\alpha_0 = 1 - \sum_{n \ge N_0+1} \frac{a_n \phi_n'(0)}{-\lambda_n + q_c}, \quad (43a)$$

$$\beta_0 = - \sum_{n \ge N_0+1} \frac{b_n \phi_n'(0)}{-\lambda_n + q_c} \quad (43b)$$

and where the observation dynamics, for $1 \le n \le N$, take the form:

$$\dot{\hat{w}}_n(t) = (-\lambda_n + q_c)\hat{w}_n(t) + a_n u(t) + b_n v(t) \quad (44) \\ - l_n \left( \sum_{i=1}^{N} \phi_i'(0)\hat{w}_i(t) - \alpha_1 u(t) - \tilde{y}(t) \right)$$

with

$$\alpha_1 = \sum_{n \ge N+1} \frac{a_n \phi_n'(0)}{-\lambda_n + q_c} \quad (45)$$

and where $l_n \in \mathbb{R}$ are the observer gains. We impose $l_n = 0$ for $N_0 + 1 \le n \le N$ and the initial condition of the observer as $\hat{w}_n(0) = 0$ for all $1 \le n \le N$. Proceeding now as in Section III but with the updated versions of the matrices $C_0$ and $C_1$ now given by $C_0 = [\phi_1'(0) \dots \phi_{N_0}'(0)]$ and $C_1 = \left[ \frac{\phi_{N_0+1}'(0)}{\lambda_{N_0+1}} \dots \frac{\phi_N'(0)}{\lambda_N} \right]$ while redefining $\tilde{e}_n(t)$ and $\zeta(t)$ as $\tilde{e}_n(t) = \lambda_n e_n(t)$ and $\zeta(t) = \sum_{n \ge N+1} \phi_n'(0) w_n(t)$, we infer that (25) holds.

**Lemma 2:** $(A_1, B_1)$ is controllable and $(A_0, C_0)$ is observable.

The proof of this Lemma is analogous to the one of Lemma 1 and is thus omitted. We select in the sequel $K \in \mathbb{R}^{1 \times (N_0+2)}$ and $L \in \mathbb{R}^{N_0}$ such that $A_1 + B_1 K$ and $A_0 - LC_0$ are Hurwitz.

### B. Equilibrium condition and dynamics of deviations

Proceeding similarly to Section III, we can characterize the equilibrium condition of the closed-loop system composed of the reaction-diffusion system (38), the auxiliary command input dynamics (6a), the integral action (42), the observer dynamics (44), and the state-feedback (18). In particular, setting $r(t) = r_e \in \mathbb{R}$, it can be shown that there exists a unique solution to:

$$0 = (-\lambda_n + q_c)w_{n,e} + a_n u_e + b_n v_e = 0, \quad n \ge 1, \quad (46a)$$
$$0 = v_e = K \hat{W}_{a,e}^{N_0}, \quad (46b)$$

$$0 = \sum_{n=1}^{N_0} \phi'_{n}(0) \hat{w}_{n,e} + \alpha_0 u_e + \beta_0 v_e - r_e, \quad (46c)$$

$$0 = (-\lambda_n + q_c)\hat{w}_{n,e} + a_n u_e + b_n v_e \\ - l_n \left\{ \sum_{i=1}^{N} \phi'_{i}(0)\hat{w}_{i,e} - \alpha_1 u_e - \tilde{y}_e \right\}, \quad 1 \le n \le N_0, \quad (46d)$$

$$0 = (-\lambda_n + q_c)\hat{w}_{n,e} + a_n u_e + b_n v_e, \quad N_0 + 1 \le n \le N, \quad (46e)$$

$$\tilde{y}_e = \sum_{n \ge 1} \phi'_{n}(0) w_{n,e}. \quad (46f)$$

Moreover we can define $w_e \triangleq \sum_{n \ge 1} w_{n,e} \phi_n \in D(\mathcal{A})$. Introducing the change of variable $z_e = w_e + xu_e$, $z_e$ is a static solution of (38a-38b) associated with the constant control input $u(t) = u_e$. Denoting by $y_e \triangleq z'_e(0)$, we also infer that $y_e = r_e$, achieving the desired reference tracking. This allows the introduction of the dynamics of deviation of the different quantities w.r.t the equilibrium condition characterized by $r_e \in \mathbb{R}$. We have:

$$\Delta w(t, x) = \Delta z(t, x) - x \Delta u(t), \quad (47a)$$

$$\Delta \dot{X}(t) = F \Delta X(t) + \mathcal{L} \Delta \zeta(t) - \mathcal{L}_r \Delta r(t), \quad (47b)$$

$$\Delta \zeta(t) = \sum_{n \ge N+1} \phi'_{n}(0) \Delta w_{n}(t), \quad (47c)$$

$$\Delta \dot{w}_{n}(t) = (-\lambda_{n} + q_{c})\Delta w_{n}(t) + a_{n}\Delta u(t) + b_{n}\Delta v(t), \quad (47d)$$

$$\Delta v(t) = K \Delta \hat{W}_{a}^{N_0}(0), \quad (47e)$$

$$\Delta \bar{y}(t) = \Delta y(t) - \Delta u(t) = \sum_{n \ge 1} \phi'_{n}(0) \Delta w_{n}(t). \quad (47f)$$

### C. Stability analysis and regulation assessment

We define, for any $\epsilon \in (0, 1/2]$, the constant $M_{2,\phi}(\epsilon) = \sum_{n \ge 2} \frac{\phi'_{n}(0)^2}{\lambda_n^{3/2+\epsilon}}$, which is finite when $p \in C^2([0, 1])$ because we recall that $\phi'_{n}(0) = O(\sqrt{\lambda_n})$ as $n \to +\infty$ and (1) hold.

**Theorem 3:** Let $p \in C^2([0, 1])$ with $p > 0$, $q \in C^0([0, 1])$ with $q \ge 0$, and $q_c \in \mathbb{R}$. Consider the reaction-diffusion system described by (38). Let $N_0 \ge 1$ and $\delta > 0$ be given such that $-\lambda_n + q_c < -\delta < 0$ for all $n \ge N_0 + 1$. Let $K \in \mathbb{R}^{1 \times (N_0+2)}$ and $L \in \mathbb{R}^{N_0}$ be such that $A_1 + B_1 K$ and $A_0 - LC_0$ are Hurwitz with eigenvalues that have a real part strictly less than $-\delta < 0$. Assume that there exist $N \ge N_0 + 1$, $P > 0$, $\epsilon \in (0, 1/2]$, and $\alpha, \beta, \gamma > 0$ such that $\Theta < 0$

where $\Theta$ is defined by (32), and

$$\Gamma_n = -\lambda_n + q_c + \delta + \frac{\lambda_n}{\alpha} + \frac{\beta M_{2,\phi}(\epsilon)}{2\gamma} \lambda_n^{1/2+\epsilon} \le 0$$

for all $n \ge N + 1$. Then, for any $\eta \in [0, 1)$, there exists $M > 0$ such that, for any $z_0 \in H^2(0, 1)$ and any $u(0) \in \mathbb{R}$ such that $z_0(0) = 0$ and $z_0(1) = u(0)$, the classical solution of the closed-loop system composed of the plant (38), the integral actions (6a) and (42), the observer dynamics (44) with null initial condition, and the state feedback (18) satisfies (33) for all $t \ge 0$. Moreover, the above constraints are always feasible for $N$ large enough.

**Proof.** Let $P > 0$ and $\gamma > 0$ and consider the Lyapunov function candidate defined by (34). Then, proceeding as in [12] but taking into account the extra contribution of the reference signal appearing in (25), we obtain that (35) holds for all $t \ge 0$. Now the proof of the stability estimate (33) is analogous to the one reported in the proof of Theorem 1.

It remains to show that we can always select $N \ge N_0 + 1$, $P > 0$, $\epsilon \in (0, 1/2]$, and $\alpha, \beta, \gamma > 0$ such that $\Theta < 0$ and $\Gamma_n \le 0$ for

all $n \ge N + 1$. To handle the constraint $\Theta < 0$, we proceed as in the last part of the proof of Theorem 1. This is allowed because $\|C_1\| = O(1)$ as $N \to +\infty$. Noting now that, because $\epsilon \in (0, 1/2]$, $\lambda_n^{1/2+\epsilon} = \lambda_n/\lambda_n^{1/2-\epsilon} \le \lambda_n/\lambda_{N+1}^{1/2-\epsilon}$ for all $n \ge N + 1$, we infer that $\Gamma_n \le -\left(1 - \frac{1}{\alpha} - \frac{\beta M_{2,\phi}(\epsilon)}{2\gamma\lambda_{N+1}^{1/2-\epsilon}}\right)\lambda_n + q_c + \delta$ for all $n \ge N + 1$. Setting $\epsilon = 1/8$, $\alpha = \beta = N^{1/8}$, and $\gamma = N^{-3/16}$, we deduce the existence of an integer $N \ge N_0 + 1$ large enough such that $\Theta < 0$ and $\Gamma_n \le 0$ for all $n \ge N + 1$. $\square$

We are now in position to assess the setpoint regulation control of the left Dirichlet trace.

**Theorem 4:** Under both assumptions and conclusions of Theorem 1, for any $\eta \in [0, 1)$, there exists $M_r > 0$ such that

$$|y(t) - r(t)| \le M_r e^{-\delta t} (|\Delta u(0)| + |\Delta \zeta(0)| + \|\Delta z_0\|_{H^1} + \|\mathcal{A}\Delta w_0\|_{L^2}) + M_r \sup_{\tau \in [0,t]} e^{-\eta \delta(t-\tau)} |\Delta r(\tau)| \quad (48)$$

for all $t \ge 0$ where $\Delta w_0 = \Delta z_0 - x \Delta u(0)$.

**Proof.** Recalling that $y_e = r_e$, one has $|y(t) - r(t)| \le |\Delta y(t)| + |\Delta r(t)|$. We infer from (47f) and Cauchy-Schwarz inequality that $|\Delta y(t)| \le \sqrt{\sum_{n \ge 1} \frac{\phi'_{n}(0)^2}{\lambda_n^2}} \sqrt{\sum_{n \ge 1} \lambda_n^2 \Delta w_n(t)^2} + |\Delta u(t)|$. In view of the stability estimate (33) provided by Theorem 3, we only need to study the term $\sum_{n \ge 1} \lambda_n^2 \Delta w_n(t)^2$. This can be done as in [13, Proof of Theorem 2], yielding the claimed estimate (48). $\square$

### V. DIRICHLET MEASUREMENT AND NEUMANN REGULATION CONTROL

We now consider the reaction-diffusion system described by (3a-3c), still in the case $p \in C^2([0, 1])$, but this time with the boundary measurement $y_m(t)$ and the (unmeasured) output to-be-regulated $y_r(t)$ which are distinct, described by:

$$y_m(t) = z(t, 0), \quad y_r(t) = z_x(t, 1). \quad (49)$$

#### A. Control design

Using the change of variable (4), we obtain (5a-5c) still hold while (5d) is replaced by

$$\tilde{y}_m(t) = w(t, 0) = z(t, 0) = y_m(t), \quad (50a)$$

$$\tilde{y}_r(t) = w_x(t, 1) = z_x(t, 1) - 2u(t) = y_r(r) - 2u(t). \quad (50b)$$

Then, considering classical solutions, (7a-7b) is still valid while (7c) is replaced by

$$\tilde{y}_m(t) = \sum_{i \ge 1} \phi_i(0) w_i(t), \quad \tilde{y}_r(t) = \sum_{i \ge 1} \phi'_i(1) w_i(t). \quad (51)$$

Based on similar motivations that the ones reported in Section III, we consider the integral component

$$\dot{\xi}(t) = \sum_{n=1}^{N_0} \phi'_{n}(1) \hat{w}_{n}(t) + \alpha_{0} u(t) + \beta_{0} v(t) - r(t). \quad (52)$$

with

$$\alpha_0 = 2 - \sum_{n \ge N_0+1} \frac{a_n \phi'_{n}(1)}{-\lambda_n + q_c}, \quad (53a)$$

$$\beta_0 = - \sum_{n \ge N_0+1} \frac{b_n \phi'_{n}(1)}{-\lambda_n + q_c} \quad (53b)$$

and where the observation dynamics, for $1 \le n \le N$, takes the form:

$$\dot{w}_n(t) = (-\lambda_n + q_c)w_n(t) + a_n u(t) + b_n v(t)$$
$$ -l_n \left( \sum_{i=1}^{N} \phi_i(0) \hat{w}_i(t) - \alpha_1 u(t) - \tilde{y}_m(t) \right) \quad (54) $$

with

$$ \alpha_1 = \sum_{n \ge N+1} \frac{a_n \phi_n(0)}{-\lambda_n + q_c} \quad (55) $$

and where $l_n \in \mathbb{R}$ are the observer gains. We impose $l_n = 0$ for $N_0+1 \le n \le N$ and the initial condition of the observer as $\hat{w}_n(0) = 0$ for all $1 \le n \le N$. Adopting now the same definitions as the ones used in Section III except that the matrix $A_1$, originally defined by (16), is now replaced by

$$ A_1 = \begin{bmatrix} 0 & 0 & 0 \\ B_{0,a} & A_0 & 0 \\ \alpha_0 & C_0^* & 0 \end{bmatrix} $$

where $C_0^* = [\phi'_1(1) \dots \phi'_{N_0}(1)]$, we infer that (25) holds.

**Lemma 3:** The pair $(A_0, C_0)$ is observable. Moreover, if the unique solution of $(pf')' + (q_c - q)f = 0$ with $f(1) = 1$ and $f'(1) = 0$ is such that $f'(0) \ne 0$, then the pair $(A_1, B_1)$ is controllable.

**Proof.** The observability of $(A_0, C_0)$ was assessed in Lemma 1. From [13, Lem. 2], and because the pair (28) is controllable, then $(A_1, B_1)$ is controllable if and only if the matrix

$$ T = \begin{bmatrix} 0 & 0 & 1 \\ B_{0,a} & A_0 & B_{0,b} \\ \alpha_0 & C_0^* & \beta_0 \end{bmatrix} $$

is invertible. Let $[u_e \ w_1, e \ \dots \ w_{N_0, e} \ v_e]^\top \in \ker(T)$. We obtain that $v_e = 0$, $a_n u_e + (-\lambda_n + q_c)w_{n,e} = 0$ for all $1 \le n \le N_0$, and $\alpha_0 u_e + \sum_{n=1}^{N_0} \phi'_n(1)w_{n,e} = 0$. Defining for $n \ge N_0 + 1$ the quantity $w_{n,e} = -\frac{a_n}{-\lambda_n + q_c}u_e$, we have $(-\lambda_n + q_c)w_{n,e} + a_n u_e = 0$ for all $n \ge 1$. Hence $(w_{n,e})_{n \ge 1}$, $(\lambda_n w_{n,e})_{n \ge 1} \in l^2(\mathbb{N})$ ensuring that $w_e \triangleq \sum_{n \ge 1} w_{n,e} \phi_n \in D(\mathcal{A})$ and $\mathcal{A}w_e = \sum_{n \ge 1} \lambda_n w_{n,e} \phi_n \in D(\mathcal{A})$. This shows that $-\mathcal{A}w_e + q_c w_e + au_e = 0$. Moreover, using (53), we also have $0 = \alpha_0 u_e + \sum_{n=1}^{N_0} \phi'_n(1)w_{n,e} = 2u_e + w'_e(1)$. From the two latter identities, we infer that $(pw'_e)' + (q_c - q)w_e + au_e = 0$, $w'_e(0) = w_e(1) = 0$, and $w'_e(1) + 2u_e = 0$. Introducing the change of variable $z_e(x) = w_e(x) + x^2 u_e$, we deduce that $(pz'_e)' + (q_c - q)z_e = 0$, $z'_e(0) = z'_e(1) = 0$, and $z_e(1) = u_e$. From our assumption, we infer that then $z_e = 0$ hence $u_e = z_e(1) = 0$. Thus we have $w_e = z_e - x^2 u_e = 0$ hence $w_{n,e} = 0$ for all $n \ge 1$. We deduce that $\ker(T) = \{0\}$. Overall, we have shown that $(A_1, B_1)$ is controllable. □

We now select $K \in \mathbb{R}^{1 \times (N_0+2)}$ and $L \in \mathbb{R}^{N_0}$ such that $A_1+B_1 K$ and $A_0 - LC_0$ are Hurwitz.

## B. Equilibrium condition and dynamics of deviations

Proceeding as in Section III, we can characterize the equilibrium condition of the closed-loop system composed of the reaction-diffusion system (3a-3c) with (49), the auxiliary command input dynamics (6a), the integral action (52), the observer dynamics (54), and the state-feedback (18). In particular, setting $r(t) = r_e \in \mathbb{R}$, it can be shown that there exists a unique solution to:

$$ 0 = (-\lambda_n + q_c)w_{n,e} + a_n u_e + b_n v_e = 0, \quad n \ge 1, \quad (56a) $$

$$ 0 = v_e = K \hat{W}_{a,e}^{N_0}, \quad (56b) $$

$$ 0 = \sum_{n=1}^{N_0} \phi'_n(1) \hat{w}_{n,e} + \alpha_0 u_e + \beta_0 v_e - r_e, \quad (56c) $$

$$ 0 = (-\lambda_n + q_c)\hat{w}_{n,e} + a_n u_e + b_n v_e \quad (56d) $$

$$ -l_n \left\{ \sum_{i=1}^{N} \phi_i(0) \hat{w}_{i,e} - \alpha_1 u_e - \tilde{y}_e \right\}, \quad 1 \le n \le N_0, $$

$$ 0 = (-\lambda_n + q_c)\hat{w}_{n,e} + a_n u_e + b_n v_e, \quad N_0 + 1 \le n \le N, \quad (56e) $$

$$ \tilde{y}_{m,e} = \sum_{n \ge 1} \phi_n(0)w_{n,e}, \quad (56f) $$

$$ \tilde{y}_{r,e} = \sum_{n \ge 1} \phi'_n(1)w_{n,e}. \quad (56g) $$

Moreover we can define $w_e \triangleq \sum_{n \ge 1} w_{n,e} \phi_n \in D(\mathcal{A})$. Introducing the change of variable $z_e = w_e + x^2 u_e$, $z_e$ is a static solution of (38a-38b) associated with the constant control input $u(t) = u_e$. Denoting by $y_{r,e} \triangleq z'_e(1)$, we also infer that $y_{r,e} = r_e$, achieving the desired reference tracking. Consequently, we obtain the following dynamics of deviations:

$$ \Delta w(t, x) = \Delta z(t, x) - x^2 \Delta u(t), \quad (57a) $$

$$ \Delta \dot{X}(t) = F \Delta X(t) + \mathcal{L} \Delta \zeta(t) - \mathcal{L}_r \Delta r(t), \quad (57b) $$

$$ \Delta \zeta(t) = \sum_{n \ge N+1} \phi_n(0) \Delta w_n(t), \quad (57c) $$

$$ \Delta \hat{w}_n(t) = (-\lambda_n + q_c) \Delta w_n(t) + a_n \Delta u(t) + b_n \Delta v(t), \quad (57d) $$

$$ \Delta v(t) = K \Delta \hat{W}_a^{N_0}(t), \quad (57e) $$

$$ \Delta \tilde{y}_m(t) = \Delta y_m(t) = \sum_{n \ge 1} \phi_n(0) \Delta w_n(t), \quad (57f) $$

$$ \Delta \tilde{y}_r(t) = \Delta y_r(t) - 2\Delta u(t) = \sum_{n \ge 1} \phi'_n(1) \Delta w_n(t). \quad (57g) $$

## C. Stability analysis and regulation assessment

The proof of the following Theorem directly follows from the proofs reported in the previous sections.

**Theorem 5:** Under the assumption of Lemma 3, the stability result stated by Theorem 1 also applies to the closed-loop system composed of the plant (3a-3c) with (49), the integral actions (6a) and (52), the observer dynamics (54) with null initial condition, and the state feedback (18). Moreover, for any $\eta \in [0, 1)$, there exists $M_r > 0$ such that

$$ \begin{aligned} & |y_r(t) - r(t)| \\ & \le M_r e^{-\delta t} (|\Delta u(0)| + |\Delta \xi(0)| + \|\Delta z_0\|_{H^1} + \|\mathcal{A} \Delta w_0\|_{L^2}) \\ & + M_r \sup_{\tau \in [0,t]} e^{-\eta \delta(t-\tau)} |\Delta r(\tau)| \end{aligned} \quad (58) $$

for all $t \ge 0$ where $\Delta w_0 = \Delta z_0 - x^2 \Delta u(0)$.

## VI. NUMERICAL ILLUSTRATION

We illustrate the result of Section V corresponding to Dirichlet measurement and Neumann regulation using a modal approximation that captures the 50 dominant modes of the reaction-diffusion plant. We set $p = 1$, $q = 0$, and $q_c = 3$ for which the open-loop plant is unstable. Selecting $\delta = 0.5$, we obtain $N_0 = 1$, the feedback gain $K = [-10.4134 \quad -11.3747 \quad 2.3100]$, and the observer gain $L = 1.4373$. The conditions of Theorem 5 are found feasible for $N = 3$. The time-domain evolution of the closed-loop system trajectories are depicted in fig. 1, confirming the theoretical predictions.

## VII. CONCLUSION

We proposed the design of a finite-dimensional observer-based PI controller in order to achieve the output stabilization of a reaction-diffusion equation, as well as the regulation control of various system outputs such as Dirichlet and Neumann traces. Even if presented in the case of a Dirichlet boundary control input, the presented results easily extend to Neumann boundary control (the only significant modification consists of the modification of the change of
<figure><img src="image_2.png" /><figcaption>Fig. 1. Time evolution in closed-loop with Dirichlet boundary measurement $y_m(t) = z(t, 0)$ and Neumann boundary regulation $y_r(t) = z_x(t, 1)$ for the reaction-diffusion system (3a-3c)</figcaption></figure>

variable formula to obtain an homogeneous PDE, yielding different $a, b \in L^2(0, 1)$). Moreover, the case of in-domain measurements and/or regulated outputs can also be handled with the same approach provided the satisfaction of adequate observability conditions.

# APPENDIX
## TECHNICAL LEMMA

The following Lemma is an immediate generalization of the result presented in [10].

**Lemma 4:** Let $n, m, N \ge 1$, $M_{11} \in \mathbb{R}^{n \times n}$ and $M_{22} \in \mathbb{R}^{m \times m}$ Hurwitz, $M_{12} \in \mathbb{R}^{n \times m}$, $M_{14}^N \in \mathbb{R}^{n \times N}$, $M_{24}^N \in \mathbb{R}^{m \times N}$, $M_{31}^N \in \mathbb{R}^{N \times n}$, $M_{33}^N$, $M_{44}^N \in \mathbb{R}^{N \times N}$, and

$$F^N = \begin{bmatrix} M_{11} & M_{12} & 0 & M_{14}^N \\ 0 & M_{22} & 0 & M_{24}^N \\ M_{31}^N & 0 & M_{33}^N & 0 \\ 0 & 0 & 0 & M_{44}^N \end{bmatrix}.$$

We assume that there exist constants $C_0, \kappa_0 > 0$ such that $\|e^{M_{33}^N t}\| \le C_0 e^{-\kappa_0 t}$ and $\|e^{M_{44}^N t}\| \le C_0 e^{-\kappa_0 t}$ for all $t \ge 0$ and all $N \ge 1$.

Moreover, we assume that there exists a constant $C_1 > 0$ such that $\|M_{14}^N\| \le C_1$, $\|M_{24}^N\| \le C_1$, and $\|M_{31}^N\| \le C_1$ for all $N \ge 1$. Then there exists a constant $C_2 > 0$ such that, for any $N \ge 1$, there exists a symmetric matrix $P^N \in \mathbb{R}^{n+m+2N}$ with $P^N > 0$ such that $P^N F^N + (F^N)^\top P^N = -I$ and $\|P^N\| \le C_2$.

# REFERENCES

[1] P. J. Antsaklis and A. N. Michel, Linear systems. Springer Science & Business Media, 2006.

[2] M. J. Balas, "Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters," *Journal of Mathematical Analysis and Applications*, vol. 133, no. 2, pp. 283–296, 1988.

[3] M. Barreau, F. Gouaisbaut, and A. Seuret, "Practical stability analysis of a drilling pipe under friction with a PI-controller," *IEEE Transactions on Control Systems Technology*, 2019.

[4] J.-M. Coron and A. Hayat, "PI controllers for 1-D nonlinear transport equation," *IEEE Transactions on Automatic Control*, vol. 64, no. 11, pp. 4570–4582, 2019.

[5] R. Curtain, "Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input," *IEEE Transactions on Automatic Control*, vol. 27, no. 1, pp. 98–104, 1982.

[6] R. F. Curtain and H. Zwart, *An introduction to infinite-dimensional linear systems theory: A State-Space Approach*. Springer, 2020, vol. 71.

[7] V. Dos Santos, G. Bastin, J.-M. Coron, and B. d'Andréa Novel, "Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments," *Automatica*, vol. 44, no. 5, pp. 1310–1318, 2008.

[8] C. Harkort and J. Deutscher, "Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers," *International journal of control*, vol. 84, no. 1, pp. 107–122, 2011.

[9] J. P. Hespanha, *Linear systems theory*. Princeton university press, 2018.

[10] R. Katz and E. Fridman, "Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs," *Automatica*, vol. 122, p. 109285, 2020.

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

[12] H. Lhachemi and C. Prieur, "Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with a either Dirichlet or Neumann boundary measurement," *submitted to Automatica*. [Online]. Available: http://www.gipsa-lab.grenoble-inp.fr/%7Echristophe.prieur/Papers/submitted.pdf

[13] H. Lhachemi, C. Prieur, and E. Trélat, "PI regulation of a reaction-diffusion equation with delayed boundary control," *IEEE Transactions on Automatic Control*, vol. DOI: 10.1109/TAC.2020.2996598, 2021.

[14] T. Meurer, *Control of higher-dimensional PDEs: Flatness and backstepping designs*. Springer Science & Business Media, 2012.

[15] K. A. Morris, *Controller Design for Distributed Parameter Systems*. Springer, 2020.

[16] Y. Orlov, "On general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on "ISS with respect to boundary disturbances for 1-D parabolic PDEs"," *IEEE Transactions on Automatic Control*, vol. 62, no. 11, pp. 5970–5973, 2017.

[17] A. Pazy, *Semigroups of linear operators and applications to partial differential equations*. Springer Science & Business Media, 2012, vol. 44.

[18] S. Pohjolainen, "Robust multivariable PI-controller for infinite dimensional systems," *IEEE Transactions on Automatic Control*, vol. 27, no. 1, pp. 17–30, 1982.

[19] A. Terrand-Jeanne, V. Andrieu, V. D. S. Martins, and C. Xu, "Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems," *IEEE Transactions on Automatic Control*, vol. 65, no. 11, pp. 4481–4492, 2020.

[20] A. Terrand-Jeanne, V. Andrieu, M. Tayakout-Fayolle, and V. D. S. Martins, "Regulation of inhomogeneous drilling model with a PI controller," *IEEE Transactions on Automatic Control*, vol. 65, no. 1, pp. 58–71, 2020.

[21] N.-T. Trinh, V. Andrieu, and C.-Z. Xu, "Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations," *IEEE Transactions on Automatic Control*, vol. 62, no. 9, pp. 4527–4536, 2017.

[22] C.-Z. Xu and H. Jerbi, "A robust PI-controller for infinite-dimensional systems," *International Journal of Control*, vol. 61, no. 1, pp. 33–45, 1995.