# Underactuated boundary control of a linearized Saint-Venant equation

Jean Auriol, Adam Braun, Mohand Ouidir Amirat, Mathieu Bajodek

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Jean Auriol, Adam Braun, Mohand Ouidir Amirat, Mathieu Bajodek. Underactuated boundary control of a linearized Saint-Venant equation. IMA Journal of Mathematical Control and Information, 2025, 10.1093/imamci/dnnxxx. hal-05033220

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# Underactuated boundary control of a linearized Saint-Venant equation

JEAN AURIOL¹, ADAM BRAUN¹, MOHAND OUIDIR AMIRAT², MATHIEU BAJODEK²

¹Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France.

²Laboratoire d'Automatique, de Génie des Procédés et de Génie Pharmaceutique (LAGEPP), UMR CNRS 5007, Université Claude Bernard Lyon 1, Lyon France

*Corresponding author e-mail: jean.auriol@centralesupelec.fr*

This paper deals with the stabilization of one-dimensional linearized, shock-free Saint-Venant equations, which are widely used in fluid modeling. The approach focuses on four under-actuated hyperbolic partial differential equations, combining Volterra transformations inspired by backstepping techniques. The system is reduced to an integral difference equation, allowing for stability analysis the use of spectral methods. A distributed state feedback controller is designed to eliminate integral terms by solving a Fredholm equation. This result effectively addresses challenges related to under-actuation and lays the foundation for extending stabilization techniques to nonlinear scenarios.

**Keywords:** Hyperbolic systems, Under-actuation, Stabilization, Integral Difference Equations, Backstepping method.

## 1. Introduction

Hyperbolic equations are widely recognized for their ability to model complex physical phenomena, such as advection, wave propagation, gas dynamics (as seen in the Euler equations), traffic flows, and shallow water flows (described by the Saint-Venant equations) (Xu and Sallet 2002, Yu and Krstić 2023). Our focus in this paper is on the one-dimensional conservative Saint-Venant equations, which are essential for modeling navigable rivers, irrigation canals, and urban drainage networks. These equations form a nonlinear 2 × 2 hyperbolic system and are often studied for their effectiveness in accurately representing flow phenomena. One of the key challenges in hydraulic engineering is the control and stabilization of this class of systems (Bastin and Coron 2016).

Early results focused on boundary control and observer design for the linearized equations, relying on energy-based Lyapunov methods and characteristics analysis (Coron, Andréa-Novel and Bastin 1999). Nonlinear control approaches have also been explored, particularly in the context of feedback linearization (Bastin and Coron 2016), Lyapunov-based techniques (Hayat and Shang 2019) and proportional-integral controllers (Hayat 2022). The challenge of controlling cascaded hyperbolic systems has led to new methodologies, including predictor-based and integral transformation approaches as in (Auriol and Di Meglio 2016, Diagne, Diagne, Tang and Krstic 2017). However, the stabilization of weakly hyperbolic formulations and underactuated configurations remains an open problem, requiring innovative control strategies such as those proposed in this work.

In this paper, we address the question of the stabilization of underactuated Saint-Venant equations in a linearized and shock-free configuration, i.e. four hyperbolic Partial Differential Equations (PDEs) with only two control inputs. Although this particular configuration is not directly applicable in practical scenarios, it establishes an initial theoretical foundation that can be expanded to include more general cases involving nonlinearities and shock waves. The methodology uses Volterra backstepping transformations (Krstic and Smyshlyaev 2008b) to rewrite the initial PDE system as a time-delay system (Auriol 2024a). More precisely, we will show that the linearized PDE system has stability properties
equivalent to those of an Integral Difference Equation (IDE). Due to the under-actuation, the control input will appear through pointwise and distributed delay terms in the IDE, requiring specific and novel stabilization techniques. We will adjust the approach proposed in (Auriol 2024b) to design an appropriate stabilizing control law by solving an appropriate set of Fredholm equations.

Among existing stabilization techniques for hyperbolic PDEs, the backstepping approach, introduced by (Krstic and Smyshlyaev 2008a), has proven to be a powerful tool. This method employs suitable integral transformations to map the original system into a target system that possesses desirable stability properties (Krstic 2008, Redaud, Auriol and Le Gorrec 2024). The backstepping technique has been applied to Saint-Venant-type systems, successfully stabilizing linearized models with arbitrary decay rates, (Diagne, Shang and Wang 2015) or general heterodirectional systems (Auriol and Di Meglio 2016). The present paper leverages the backstepping framework to transition from a fully interconnected model to a cascaded system within the context of underactuated boundary control. In this context, the target system can be reformulated as an IDE with pointwise and distributed delay terms. Interestingly, despite being under-actuated in PDE framework, the system may be over-actuated using the time-delay representation.

When designing stabilizing controllers for systems with both pointwise and distributed delays in the state and actuator, several challenges arise. These delays can often lead to instability in the closed-loop system or result in suboptimal performance of the proposed output-feedback control laws. Additionally, the presence of a distributed delayed term in the control input means that the system's stabilizability is not guaranteed. Therefore, researchers have derived necessary and sufficient conditions for stabilizability and controllability in the literature, using complex analysis (Hale and Verduyn Lunel 1993, Bellman and Cooke 1963, Mounier 1998). The analysis and control of these delayed systems have been rigorously examined in works such as (Michiels and Niculescu 2007), employing advanced spectral techniques. A constructive approach has recently been introduced in the literature (Redaud, Auriol and Niculescu 2022, Redaud, Auriol and Niculescu 2021) to stabilize a specific class of IDEs. The proposed methodology involved employing a PDE system as a comparison system (Niculescu 2001). Subsequently, this PDE system is then stabilized using the backstepping method, incorporating a Fredholm transformation (Yoshida 1960), whose invertibility must be verified. Notably, the invertibility of such a transformation, usually not guaranteed except under specific structural assumptions (Coron, Hu and Olive 2017, Bribiesca-Argomedo and Krstic 2015), was related to a spectral controllability condition. Recently, this approach was generalized and refined in (Auriol 2024b) by proposing a direct spectral analysis of the IDE.

In Section II, we present the framework for the problem that leads to the stabilization of a specific class of linear shock-free Saint-Venant equations. In Section III, we propose a transformation that converts a system of four PDEs into an IDE. In Section IV, we propose a control law ensuring exponential stability of the closed-loop system.

*Notation:* Let $m, n$ two positive integers. Throughout this paper, $\text{Id}_{n \times n}$ and $0_{n \times m}$ denote the identity matrix of size $n$ and the zero matrix of size $n \times m$, respectively. Furthermore, for any matrix $M \in \mathbb{R}^{n \times n}$, $M^\top \in \mathbb{R}^{m \times n}$ stands for its transpose. We denote $L^2([0, 1], \mathbb{R}^n) = (L^2(0, 1))^n$ the space of real-valued square-integrable functions defined on $[0, 1]$ with the standard $L^2$ norm, i.e., for any $f \in L^2([0, 1], \mathbb{R}^n)$, we have $||f||_{L^2} = \sqrt{\int_0^1 f^\top(x)f(x)dx}$. For any $\tau > 0$, we denote $(L_\tau^2)^n = L^2([-\tau, 0], \mathbb{R}^n)$ the Banach space of $L^2$ functions mapping the interval $[-\tau, 0]$ into $\mathbb{R}^n$. For any function $\phi : [-\tau, \infty) \to \mathbb{R}^n$, we define its partial trajectory $\phi_{[t]}$ by $\phi_{[t]}(\theta) = \phi(t + \theta)$, $-\tau \le \theta \le 0$. The associated $L^2$ norm is given by $||\phi_{[t]}|| = (\int_{-\tau}^0 \phi^T(t+s)\phi(t+s)ds)^{\frac{1}{2}}$. Finally, the variable $s$ denotes the Laplace variable. Provided it is
well defined, the Laplace transform of a time-varying function $y(t)$ will be denoted $y(s)$.

## 2. Problem under consideration

### 2.1 Linearized shock-free Saint-Venant equation

We focus on the boundary state feedback stabilization of the one-dimensional Saint-Venant equation subject to hydraulic jumps (Bastin, Coron, Hayat and Shang 2019). The Saint-Venant model is often used to describe hydraulic flows and is given by

$$ \partial_t H(t, x) + \partial_x Q(t, x) = 0, \qquad (2.1) $$

$$ \partial_t Q(t, x) + \partial_x \left( \frac{Q^2}{H} + \frac{1}{2} g H^2 \right) (t, x) = 0, \qquad (2.2) $$

for a.e. $(t, x) \in [0, +\infty) \times [0, 1]$ where $L > 0$ is the channel's length, $H(t, x)$ denotes the water depth and $Q(t, x)$ the momentum. The nonlinear Saint-Venant problem (2.1)-(2.2) is a quasi-linear hyperbolic system and can be seen as a Cauchy problem in Riemann coordinates (Bastin and Coron 2016, Chapter 1.1, Appendix B). At each boundary point, the incoming information is determined by the outgoing information having the relation. The well-posedness of this Cauchy problem may require that the initial condition be compatible with the boundary condition, which gives a condition on the relation $H$ that depends on the functional space to which the solutions belong. We will clearly state the well-posedness of the linearized system in Section 2.2

A hydraulic jump is a phenomenon that frequently occurs in open channel flows, such as rivers and spillways. It describes a transition between a torrential regime and a fluvial regime, i.e. a transition between a fast and a slow flow with a higher height. Discontinuous shock solutions of the Saint-Venant equation above can represent hydraulic jumps. In (Bastin et al. 2019), the authors considered solutions $(H, Q)^{\top}$ that may have a jump discontinuity at some point $x_s \in (0, L)$ and are classical otherwise. The jump location dynamic is derived by using the well-known Rankine-Hugoniot relationships yielding the ordinary differential equation (ODE)

$$ \dot{x}_s(t) = \frac{[Q]}{[H]}, \qquad (2.3) $$

and an additional boundary condition

$$ [Q]^2 = [H] \left[ \frac{Q^2}{H} + \frac{1}{2} g H^2 \right], \qquad (2.4) $$

where for any bounded function $g$ defined in the neighborhood of $x_s$, we have denoted:

$$ [g] = g(x_s^+(t)) - g(x_s^-(t)). $$

One can see that the shock location $x_s$ depends on time in general. System (2.1), (2.2), (2.3) and (2.4) is then equivalent to a 4 × 4 hyperbolic PDE coupled with an ODE:

$$ \mathbf{w}_t(t, x) = (A(x_s) + B(\mathbf{w}, x_s))\mathbf{w}_x(t, x) + C(\mathbf{w}, x_s)\mathbf{w}_x^*(x), \qquad (2.5) $$

$$ \dot{x}_s(t) = \frac{v_1(t, 0) + v_2(t, 0) - v_3(t, 0) - u(t, 0)}{\frac{v_1(t, 0)}{\mu_1} + \frac{v_2(t, 0)}{\mu_2} + \frac{v_3(t, 0)}{\mu_3} - \frac{u(t, 0)}{\lambda} + (H_1^* - H_2^*)}, \qquad (2.6) $$

$$ \frac{2Q_s^*}{H_2^*} \left( -\frac{v_3(t, 0)}{\mu_3} + \frac{u(t, 0)}{\lambda} \right) + \left( \frac{Q_s^*}{H_2^*} \right)^2 (v_3(t, 0) + u(t, 0)) = \frac{2Q_s^*}{H_1^*} \left( \frac{v_1(t, 0)}{\mu_1} + \frac{v_2(t, 0)}{\mu_2} \right) + \left( \frac{Q_s^*}{H_1^*} \right)^2 (v_1(t, 0) + v_2(t, 0)), \qquad (2.7) $$
defined for a.e. $x \in (0,1)$ where $A(x_s) = \text{diag}(\mu_1(x_s), \mu_2(x_s), \mu_3(x_s), -\lambda(x_s))$, $(B, C)$ are two matrices of size $4 \times 4$ with $B(0,0) = 0$ and $C(0,0) = 0$, $x_s$ denotes the shock-location error around its equilibrium $x_s^*$ and $w = (v_1, v_2, v_3, u)$ denotes the shock-free PDE deviation state around the steady state before and after the shock. A change of coordinates using Riemann invariants (see for instance (Bastin and Coron 2013)) and a common change of the space coordinate on the fixed domain $[0,1]$ (see for instance (Diagne et al. 2015, de Andrade, Vazquez and Pagano 2020, Li and Yu 1985)) have been performed to obtain this diagonal representation and avoid time-varying domains. We encourage the reader to refer to (Bastin et al. 2019) for further details.

In the present work, we assume that the steady-state flow is in the torrential regime before the jump and in the fluvial regime after the jump, meaning that we have $\mu_1 > 0, \mu_2 > 0, \mu_3 > 0, \lambda > 0$. We also assume that before the jump, the system is weakly hyperbolic, meaning that the eigenvalues are equal $\mu_1 = \mu_2$ and that after the jump, the eigenvalue is larger, i.e. $\mu_3 > \mu_1$. In addition, we can act at the start of the channel, meaning that $v_1(t, 1) = U_2(t) - U_1(t)$ and $v_2(t, 1) = U_1(t)$. We also consider that the top and bottom of the channel are connected, which gives continuity of flow $v_1(t, 1) + v_2(t, 1) = v_3(t, 1) + u(t, 1)$. Finally, we are mostly interested in a linearized version of system (2.5)-(2.7) in the case where the ODE part is already at equilibrium that can be expressed as:

$$ w(t,x) = \Lambda w_x(t,x) + \Sigma(x)w(t,x), \quad (2.8) $$

$$ u(t,0) = Qv(t,0), \quad (2.9) $$

$$ v(t,1) = Ru(t,1) + BU(t), \quad (2.10) $$

where $\Lambda = A(0) = \text{diag}(\mu_1(0), \mu_2(0), \mu_3(0), -\lambda(0))$ is a diagonal matrix, $\Sigma(x)$ is a full $4 \times 4$ matrix which comes from the linearization of the term $C(w,0)w_x^*(x)$, $Q = (Q_{11} \ Q_{12} \ Q_{13})$ is in $\mathbb{R}^{1 \times 3}$,

$$ B = \begin{pmatrix} -1 & 1 \\ 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad R = \begin{pmatrix} R_{11} \\ R_{21} \\ R_{31} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}, \quad (2.11) $$

and the control input $U(t) = (U_1(t), U_2(t)) \in \mathbb{R}^2$ simultaneously acts on the last and first equations.

## 2.2 A class of linear hetero-directional hyperbolic system

In this paper, we focus on the stabilization of system (2.8)-(2.10), i.e. the pre-stabilization of system (2.5)-(2.7), that can be seen as an underactuated $1 \times 3$ linear hetero-directional hyperbolic system

$$ u_t(t,x) + \lambda u_x(t,x) = \Sigma^{+-}(x)v(t,x), \quad (2.12) $$

$$ v_t(t,x) - \Lambda^{-} v_x(t,x) = \Sigma^{-+}(x)u(t,x) + \Sigma^{--}(x)v(t,x), \quad (2.13) $$

$$ u(t,0) = Qv(t,0), \quad (2.14) $$

$$ v(t,1) = Ru(t,1) + BU(t), \quad (2.15) $$

for a.e. $(t,x) \in [0, +\infty) \times [0, 1]$. System (2.12)-(2.15) is more general than (2.8)-(2.10) so that we can state more general theoretical results on underactuated systems. The state of the system is $(u(t, \cdot), v(t, \cdot))$ in $L^2([0, 1]; \mathbb{R})^{1+3}$. The matrix $\Lambda^{-} = \text{diag}(\mu_1, \mu_2, \mu_3)$ is diagonal with $0 < \mu_1 = \mu_2 < \mu_3$ and represents (counter-convecting) transport velocities. It can be seen that the two states $v_1$ and $v_2$ have identical transport speeds. This situation is referred to as an isotachic case (Hu, Di Meglio, Vazquez and Krstic 2016). The spatially-varying coupling matrices $\Sigma^{--}$ are regular matrices (we assume here that each coefficient of the matrix is a continuous function). Without loss of generality, we can assume that
the diagonal entries of $\Sigma^{++}$ are equal to zero (Coron, Vazquez, Krstic and Bastin 2013). This also explains why there are no $u$-terms on the right-hand side of equation (2.12). The initial condition is taken as $(u_0, v_0) \in H^1([0, 1]; \mathbb{R})^{1+3}$ and satisfies the appropriate zero-order compatibility conditions $(u_0(0) = Qv_0(0)$ and $v_0(1) = Ru_0(1)$, as stated in (Bastin and Coron 2016)), so that the open-loop system is well-posed in the sense of (Bastin and Coron 2016, Theorem A.1, page 224). The system is **underactuated** as the matrix $B$ is not invertible. The number of inputs (two) is less than the number of counter-convecting states $v$ (three).

Our objective is to stabilize the system (2.12)-(2.15) in the sense of the spatial $L^2([0, 1], \mathbb{R})^4$-norm. More precisely we want to guarantee that, in closed loop with an appropriate state feedback control law $U$, there exist two constants $C_0 > 0$ and $v > 0$ such that for any initial condition $(u_0, v_0)$ and all $t > 0$, we have

$$ \|(u(t, \cdot), v(t, \cdot))\|_{L^2} \le C_0 e^{-vt} \|(u_0, v_0)\|_{L^2}. \qquad (2.16) $$

The problem of stabilizing heterodirectional hyperbolic systems has been solved in (Auriol and Di Meglio 2016, Coron et al. 2017). However, these contributions assumed that the right boundary of the PDE was fully actuated, which means that $B = \mathrm{Id}_{3 \times 3}$. This is not the case here, as the system is underactuated, which makes its stabilization more challenging.

**Remark 1.** Since the control operator is admissible, the system remains well-posed for any continuous feedback such that $v_0(1) = Ru_0(1) + U(0)$. Although the control law we design in this paper will be continuous, it may not satisfy such a compatibility condition. A solution to this issue has been proposed in (Coron et al. 2013), showing that the control law can be modified in such a way that the compatibility conditions remain satisfied while preserving its stabilizing properties. This modification involves adding a dynamic extension to the controller. We refer the reader to (Coron et al. 2013) for more details.

## 2.3 Methodology and hypotheses

The design strategy is based on rewriting the original Partial Differential Equations (PDEs) as an Integral Difference Equation (IDE). We begin by applying an invertible Volterra backstepping transformation, which converts the interconnected and unstructured system of PDEs into a cascaded, self-looped structure (see Figure 1). This transformation allows us to express the transport PDE system as a delay system. Specifically, the resulting system exhibits stability properties equivalent to those of an integral difference equation with three discrete delays and three integral terms. The control inputs act on this IDE through pointwise and distributed terms. To achieve stabilization, we adopt an auto-regressive structure for the controller (Ammari, Boussaada, Niculescu and Tliba 2024). In particular, the control input gains are derived from the solutions of Fredholm equations. To design the proposed stabilizing control law, we make the following assumption.

**Assumption 1.** The boundary coupling terms verify $|QR_{11} + QR_{21}| + |QR_{31}| < 1$, $R_{31} - R_{21} - R_{11} \neq 0$, and $Q_{12} \neq Q_{11}$.

The first condition in this assumption can be related to the delay-robustness condition stated in (Auriol and Di Meglio 2019). More precisely, when $\frac{1}{\lambda} + \frac{1}{\mu_1}$ and $\frac{1}{\lambda} + \frac{1}{\mu_3}$ are rationally independent, it can be shown that the condition $|QR_{11} + QR_{21}| + |QR_{31}| < 1$ is necessary and sufficient to ensure that the open-loop system (2.12)-(2.15) does not have an infinite number of unstable roots. The presence of an infinite number of unstable roots would result in vanishing delay-robustness margins (Logemann, Rebarber and
Weiss 1996, Auriol, Bribiesca Argomedo and Di Meglio 2023). Consequently, introducing any arbitrarily small delay in the control input would render the closed-loop system unstable. This would prevent any practical implementation, thereby justifying the necessity of the proposed condition. Note that when the delays $\frac{1}{\lambda} + \frac{1}{\mu_1}$ and $\frac{1}{\lambda} + \frac{1}{\mu_3}$ are not rationally independent, this condition is no longer necessary (Hale and Verduyn Lunel 1993) Finally, we underline that this condition implies $|(R_{31} - R_{21} - R_{11})Q_{11}| < 1$, which will be used later in the paper.

The two other conditions are conservative but are necessary to simplify the design of the stabilizing control law in Section 4. In particular, these conditions ensure that the operator $\mathcal{T}$ introduced in equation (4.7) is a Fredholm operator. If $R_{31} = R_{21} + R_{11}$ or $Q_{12} = Q_{11}$, the approach developed in Section 4 no longer applies. Nevertheless, as demonstrated in (Auriol 2024b), the proposed control strategy could then be slightly modified, provided that suitable regularity conditions are met. In addition to Assumption 1, a spectral condition (linked to the controllability of the PDE system (2.12)-(2.15)) will also be required when designing the control law in Section 4.

<figure><img src="image_2.png" /><figcaption>FIG. 1. Diagram of the backstepping transformation</figcaption></figure>

### 3. Backstepping transformation and time-delay formulation

In this section, we will introduce a backstepping transformation to simplify the PDE system (2.12)-(2.15) before rewriting it as an IDE system, which will have equivalent stability properties. This approach greatly simplifies the design of the stabilizing controller and opens ways of managing the underactuated case.

#### 3.1 Isotachic blocks

The concept of “isotachic” blocks arises when multiple states share the same transport speed. This scenario was first mentioned in (Hu et al. 2016), where the authors highlighted that applying the backstepping method directly, in this case, would result in singularities within the kernel equations. To address this issue, they proposed a diagonalizing transformation to decouple the isotachic states before applying the backstepping transformation. This idea was further developed and explored in (Chen, Vazquez, Qiao and Krstic 2023) in the context of multilayer Timoshenko beams. More precisely, define the new state $\bar{v}$ for all $t > 0$ and all $x \in [0, 1]$ as

$$ \bar{v}(t,x) = \begin{pmatrix} A(x) & 0_{2,1} \\ 0_{1,2} & 1 \end{pmatrix} v(t,x), \quad (3.1) $$
where $A(x)$ is computed from the initial value problem

$$ \frac{d}{dx} A(x) = \frac{1}{\mu_1} A(x) \begin{pmatrix} \Sigma_{1,1}^{--}(x) & \Sigma_{1,2}^{--}(x) \\ \Sigma_{2,1}^{--}(x) & \Sigma_{2,2}^{--}(x) \end{pmatrix}, \quad A(0) = \mathrm{Id}_{2 \times 2} \qquad (3.2) $$

The transformation (3.1) is invertible (Hu et al. 2016) and the matrix $A^{-1}$ is obtained from the initial value problem

$$ \frac{d}{dx} A^{-1}(x) = -\frac{1}{\mu_1} \begin{pmatrix} \Sigma_{1,1}^{--}(x) & \Sigma_{1,2}^{--}(x) \\ \Sigma_{2,1}^{--}(x) & \Sigma_{2,2}^{--}(x) \end{pmatrix} A^{-1}(x), \quad A^{-1}(0) = \mathrm{Id}_{2 \times 2}. \qquad (3.3) $$

Applying the transformation (3.1) to the PDE system (2.12)-(2.15), we obtain

$$ u_t(t,x) + \lambda u_x(t,x) = \bar{\Sigma}^{+-}(x) \bar{v}(t,x), \qquad (3.4) $$

$$ \bar{v}_t(t,x) - \Lambda^{-} \bar{v}_x(t,x) = \bar{\Sigma}^{-+}(x) u(t,x) + \bar{\Sigma}^{--}(x) \bar{v}(t,x), \qquad (3.5) $$

$$ u(t,0) = Qv(t,0), \qquad (3.6) $$

$$ v(t,1) = Ru(t,1) + BU(t), \qquad (3.7) $$

where for all $x \in [0, 1]$, we have

$$ \bar{\Sigma}^{+-}(x) = \Sigma^{+-}(x) \begin{pmatrix} A^{-1}(x) & 0_{2,1} \\ 0_{1,2} & 1 \end{pmatrix}, \quad \bar{\Sigma}^{-+}(x) = \begin{pmatrix} A(x) & 0_{2,1} \\ 0_{1,2} & 1 \end{pmatrix} \Sigma^{+-}(x), $$

$$ \bar{\Sigma}^{--}(x) = \begin{pmatrix} 0_{2 \times 1} & 0_{2 \times 1} & A(x) \begin{pmatrix} \Sigma_{1,3}^{--}(x) \\ \Sigma_{2,3}^{--}(x) \\ 0 \end{pmatrix} \\ \Sigma_{3,1}^{--}(x) & \Sigma_{3,2}^{--}(x) & 0 \end{pmatrix}. $$

To avoid overloading the notations, we will remove the $\bar{\cdot}$, assuming that $\Sigma_{1,2}^{--}(x) = \Sigma_{2,1}^{--}(x) = 0$ (which is the case once the transformation (3.1) is performed).

## 3.2 Backstepping transformation

Inspired by (Auriol and Di Meglio 2016) and (Hu et al. 2016), we consider the integral transformation defined for all $x \in [0, 1]$ by

$$ \alpha(t,x) = u(t,x) + \int_{0}^{x} K^{uu}(x,y)u(t,y) + K^{uv}(x,y)v(t,y)dy, \qquad (3.8) $$

$$ \beta(t,x) = v(t,x) + \int_{0}^{x} K^{vu}(x,y)u(t,y) + K^{vv}(x,y)v(t,y)dy, \qquad (3.9) $$

where the kernels $K^{--}$ are piecewise continuous functions defined on $\{(x,y) \in [0,1]^2 \mid y \le x\}$ (their regularity is determined by the regularity of the matrix functions $\Sigma$) (Di Meglio, Bribiesca-Argomedo, Hu and Krstic 2018). They verify the following set of PDEs

$$ \lambda \partial_x K^{uu}(x,y) + \lambda \partial_y K^{uu}(x,y) = -K^{uv}(x,y)\Sigma^{-+}(y), \qquad (3.10) $$

$$ \lambda \partial_x K^{uv}(x,y) + \partial_y K^{uv}(x,y) \Lambda^{-} = -K^{uu}(x,y) \Sigma^{+-}(y) - K^{uv}(x,y) \Sigma^{--}(y), \qquad (3.11) $$

$$ -\Lambda^{-} \partial_x K^{vu}(x,y) + \lambda \partial_y K^{vu}(x,y) = -K^{vv}(x,y) \Sigma^{-+}(y) - \Omega(x) K^{vu}(x,y), \qquad (3.12) $$

$$ \Lambda^{-} \partial_x K^{vv}(x,y) + \partial_y \Lambda^{-} K^{vu}(x,y) = K^{vu}(x,y) \Sigma^{+-}(y) + K^{vv}(x,y) \Sigma^{--}(y) - \Omega(x) K^{vu}(x,y), \qquad (3.13) $$
with the boundary conditions

$$ \lambda K^{uv}(x,x) + K^{uv}(x,x)\Lambda^{-} = -\Sigma^{+-}(x), \quad \Lambda^{-}K^{vu}(x,x) + \lambda K^{vu}(x,x) = \Sigma^{-+}(x), \quad (3.14) $$

$$ K^{uu}(x,0) = 0, \quad K^{vv}(x,0) = K^{vu}(x,0)\lambda Q(\Lambda^{-})^{-1}, \quad (3.15) $$

$$ (\mu_i - \mu_j) K_{i,j}^{vv}(x,x) = \Sigma_{ij}^{--}(x), \text{ if } i > j, \quad (3.16) $$

where $\Omega$ is the strictly upper triangular matrix define for all $x \in [0, 1]$ and all $i < j$ by

$$ \Omega_{ij}(x) = (\Sigma^{--}(x) + K^{vv}(x,x)\Lambda^{-} - \Lambda^{-}K^{vv}(x,x))_{ij}. \quad (3.17) $$

One can notice, that in our isotachic case, we have for all $x \in [0, 1]$, $\Omega_{12}(x) = 0$, since $\Sigma_{12}^{--} = 0$, meaning that

$$ \Omega(x) = \begin{pmatrix} 0 & 0 & \Omega_{13}(x) \\ 0 & 0 & \Omega_{23}(x) \\ 0 & 0 & 0 \end{pmatrix}. $$

The kernel equations (3.10)-(3.17) admit a unique piecewise continuous solution (Auriol and Di Meglio 2016). The transformation (3.8)-(3.9) is a Volterra transformation and is consequently boundedly invertible (Krstic and Smyshlyaev 2008b). The inverse transformation is defined for all $x \in [0, 1]$ by

$$ u(t,x) = \alpha(t,x) + \int_{0}^{x} L^{\alpha\alpha}(x,y)\alpha(t,y) + L^{\alpha\beta}(x,y)\beta(t,y)dy, \quad (3.18) $$

$$ v(t,x) = \beta(t,x) + \int_{0}^{x} L^{\beta\alpha}(x,y)\alpha(t,y) + L^{\beta\beta}(x,y)\beta(t,y)dy, \quad (3.19) $$

where the kernels $L^{--}$ are also piecewise continuous functions defined on $\{(x,y) \in [0,1]^2 | y \le x\}$. The transformation (3.8)-(3.9) maps the original system (2.12)-(2.15) to the target system

$$ \alpha_t(t,x) + \lambda \alpha_x(t,x) = p(x)\beta(t,0), \quad (3.20) $$

$$ \beta_t(t,x) - \Lambda^{-} \beta_x(t,x) = \Omega(x) \beta(t,x), \quad (3.21) $$

$$ \alpha(t,0) = Q\beta(t,0), \quad (3.22) $$

$$ \beta_1(t, 1) = r\alpha(t, 1) + V_2(t) - V_1(t) + \int_0^1 N^\alpha(y)\alpha(t,y) + N^\beta(y)\beta(t,y)dy, \quad (3.23) $$

$$ \beta_2(t, 1) = V_1(t), \quad (3.24) $$

$$ \beta_3(t, 1) = V_2(t), \quad (3.25) $$

where

$$ r = R_{11} + R_{21} - R_{31}, \quad (3.26) $$

$$ p(x) = \lambda K^{uu}(x,0)Q - K^{uv}(x,0)\Lambda^{-}, \quad (3.27) $$

$$ N^\alpha(y) = (R_{11} + R_{21} - R_{31})L^{\alpha\alpha}(1,y) - L_{11}^{\beta\alpha}(1,y) - L_{21}^{\beta\alpha}(1,y) + L_{31}^{\beta\alpha}(1,y), \quad (3.28) $$

$$ N^\beta(y) = \begin{pmatrix} L_{31}^{\beta\beta}(1,y) - R_{31}L_{11}^{\alpha\beta}(1,y) - L_{21}^{\beta\beta}(1,y) + R_{21}L_{11}^{\alpha\beta}(1,y) - L_{11}^{\beta\beta}(1,y) \\ L_{32}^{\beta\beta}(1,y) - R_{31}L_{12}^{\alpha\beta}(1,y) - L_{22}^{\beta\beta}(1,y) + R_{21}L_{12}^{\alpha\beta}(1,y) - L_{12}^{\beta\beta}(1,y) \\ L_{33}^{\beta\beta}(1,y) - R_{31}L_{13}^{\alpha\beta}(1,y) - L_{23}^{\beta\beta}(1,y) + R_{21}L_{13}^{\alpha\beta}(1,y) - L_{13}^{\beta\beta}(1,y) \end{pmatrix}^\top, \quad (3.29) $$
and where we have defined the control inputs $U_1$ and $U_2$ as

$$ 
\begin{aligned}
U_1(t) &= V_1(t) - R_{21}u(t, 1) - \left( \int_0^1 K^{vu}(1,y)u(t,y) + K^{vv}(1,y)v(t,y) \right) dy \Big|_{21}, \\
&= V_1(t) - R_{21}\alpha(t, 1) + \left( \int_0^1 (L^{\beta\alpha}(1,y) - RL^{\alpha\alpha}(1,y))\alpha(t,y)dy \right)_{21} \\
&\quad + \left( \int_0^1 (L^{\beta\beta}(1,y) - RL^{\alpha\beta}(1,y))\beta(t,y)dy \right)_{21}
\end{aligned} 
\quad (3.30) $$

$$ 
\begin{aligned}
U_2(t) &= V_2(t) - R_{31}u(t, 1) - \left( \int_0^1 K^{vu}(1,y)u(t,y) + K^{vv}(1,y)v(t,y) \right) dy \Big|_{31}, \\
&= V_2(t) - R_{31}\alpha(t, 1) + \left( \int_0^1 (L^{\beta\alpha}(1,y) - RL^{\alpha\alpha}(1,y))\alpha(t,y)dy \right)_{31} \\
&\quad + \left( \int_0^1 (L^{\beta\beta}(1,y) - RL^{\alpha\beta}(1,y))\beta(t,y)dy \right)_{31}
\end{aligned} 
\quad (3.31) $$

the variables $V_1$ and $V_2$ being the new control inputs that have to be designed.

This new representation of our system is depicted on the right in Figure 1. It represents four cascaded PDEs looped by the boundary of a single PDE. This structure makes it possible to use the method of characteristics to switch to the formalism of systems with delays, as explained in the next subsection.

### 3.3 Time domain formulation

In what follows, we denote $\tau = \tau_1 = \frac{1}{\lambda} + \frac{1}{\mu_1} = \frac{1}{\lambda} + \frac{1}{\mu_2}$ and $\tau_3 = \frac{1}{\lambda} + \frac{1}{\mu_3}$

Applying the method of characteristics, we have for all $t > \frac{1}{\mu_1} + \frac{1}{\lambda}$ and all $x \in [0, 1]$,

$$ \beta_3(t,x) = \beta_3(t - \frac{1-x}{\mu_3}, 1) = V_2(t - \frac{1-x}{\mu_3}), $$

$$ \beta_2(t,x) = V_1(t - \frac{1-x}{\mu_2}) + \int_0^\tau G_2(x, \eta)V_2(t - \eta)d\eta, $$

$$ \beta_1(t,x) = \beta_1(t - \frac{1-x}{\mu_1}, 1) + \int_0^\tau G_1(x, \eta)V_2(t - \eta, 1)d\eta, $$

$$ 
\begin{aligned}
\alpha(t,x) = & Q_{11}\beta_1(t - \frac{1}{\mu_1} - \frac{x}{\lambda}, 1) + Q_{12}V_1(t - \frac{1}{\mu_2} - \frac{x}{\lambda}) + Q_{13}V_2(t - \frac{1}{\mu_3} - \frac{x}{\lambda}) \\
& + \int_0^\tau H_1(x, \eta)V_1(t - \eta) + H_2(x, \eta)V_2(t - \eta) + F(x, \eta)\beta_1(t - \eta, 1)d\eta,
\end{aligned}
$$
where

$$G_2(x, \eta) = \Omega_{23} \left( \frac{\mu_2 \mu_3}{\mu_3 - \mu_2} \left( \eta - \frac{x}{\mu_2} - \frac{1}{\mu_3} \right) \right) \mathbf{1}_{\left[ \frac{1-x}{\mu_3}, \frac{1-x}{\mu_2} \right]}(\eta),$$

$$G_1(x, \eta) = \Omega_{13} \left( \frac{\mu_1 \mu_3}{\mu_3 - \mu_1} \left( \eta - \frac{x}{\mu_1} - \frac{1}{\mu_3} \right) \right) \mathbf{1}_{\left[ \frac{1-x}{\mu_3}, \frac{1-x}{\mu_1} \right]}(\eta),$$

$$H_1(x, \eta) = p_2(x - \lambda(\eta - \frac{1}{\mu_2})) \mathbf{1}_{\left[ \frac{1}{\mu_2}, \frac{x}{\lambda} + \frac{1}{\mu_2} \right]}(\eta),$$

$$H_2(x, \eta) = Q_{11} G_1\left(0, \eta - \frac{x}{\lambda}\right) \mathbf{1}_{\left[\frac{x}{\lambda}, \frac{x}{\lambda} + \frac{1}{\mu_1}\right]}(\eta) + Q_{12} G_2\left(0, \eta - \frac{x}{\lambda}\right) \mathbf{1}_{\left[\frac{x}{\lambda}, \frac{x}{\lambda} + \frac{1}{\mu_1}\right]}(\eta) \\ + p_3\left(x - \lambda\left(\eta - \frac{1}{\mu_3}\right)\right) \mathbf{1}_{\left[\frac{1}{\mu_3}, \frac{x}{\lambda} + \frac{1}{\mu_3}\right]}(\eta) + \int_0^\tau p_2\left(x - \lambda s\right) G_2\left(0, \eta - s\right) \mathbf{1}_{\left[s, \frac{1}{\tau} + s\right]}(\eta) \mathbf{1}_{\left[0, \frac{x}{\lambda}\right]}(s) ds \\ + \int_0^\tau p_1\left(x - \lambda s\right) G_1\left(0, \eta - s\right) \mathbf{1}_{\left[s, \frac{1}{\tau} + s\right]}(\eta) \mathbf{1}_{\left[0, \frac{x}{\lambda}\right]}(s) ds,$$

$$F(x, \eta) = p_1(x - \lambda(\eta - \frac{1}{\mu_1})) \mathbf{1}_{\left[\frac{1}{\mu_1}, \frac{x}{\lambda} + \frac{1}{\mu_1}\right]}(\eta),$$

and where $\mathbf{1}$ is the indicator function.

Injecting this expression into equation (3.23) and denoting $z(t) = \beta_1(t, 1)$, we obtain

$$z(t) = V_2(t) - V_1(t) + rQ_{11}z(t - \tau) + rQ_{12}V_1(t - \tau) + rQ_{13}V_2(t - \tau_3) + \int_0^\tau N(\eta)z(t - \eta)d\eta \\ + \int_0^\tau M_1(\eta)V_1(t - \eta) + M_2(\eta)V_2(t - \eta)d\eta, \quad (3.32)$$

where

$$N(\eta) = rF(1, \eta) + N^{\alpha}(\lambda\eta - \frac{\lambda}{\mu_1})Q_{11}\mathbf{1}_{\left[\frac{1}{\mu_1}, \tau_1\right]}(\eta) + N_1^{\beta}(1 - \mu_1\eta)\mathbf{1}_{\left[0, \frac{1}{\mu_1}\right]}(\eta) + \int_0^1 N^{\alpha}(y)F(y, \eta)dy,$$

$$M_1(\eta) = rH_1(1, \eta) + N^{\alpha}(\lambda\eta - \frac{\lambda}{\mu_1})Q_{12}\mathbf{1}_{\left[\frac{1}{\mu_1}, \tau_1\right]}(\eta) + N_2^{\beta}(1 - \mu_2\eta)\mathbf{1}_{\left[0, \frac{1}{\mu_2}\right]}(\eta) + \int_0^1 N^{\alpha}(y)H_1(y, \eta)dy,$$

$$M_2(\eta) = rH_2(1, \eta) + N^{\alpha}(\lambda\eta - \frac{\lambda}{\mu_3})Q_{13}\mathbf{1}_{\left[\frac{1}{\mu_3}, \tau_3\right]}(\eta) + N_3^{\beta}(1 - \mu_3\eta)\mathbf{1}_{\left[0, \frac{1}{\mu_3}\right]}(\eta) + \int_0^1 N^{\alpha}(y)H_2(y, \eta)dy \\ + \int_0^1 N_2^{\beta}(y)G_2(y, \eta) + N_1^{\beta}(y)G_1(y, \eta)dy.$$

Equation (3.32) is an integral difference equation (IDE). Its initial condition can be seen as a function of $L^2([-\tau, 0], \mathbb{R})$. Due to Assumption 1, it only has a finite number of unstable roots (Hale and Verduyn Lunel 1993) since its principal part is exponentially stable. Due to the distributed delay term $(\int_0^{\tau_0} N(v)x(t - v)dv)$, the open-loop system may still be unstable. The IDE system (3.32) is not strictly equivalent to the PDE system (3.20)-(3.25). Instead, it serves as a comparison system since the IDE (3.32) and the PDE system (2.12)-(2.15) share equivalent stability properties. More precisely, we have the following lemma.

**Lemma 1.** The exponential stability of the IDE (3.32) in the sense of the $L^2([-\tau, 0], \mathbb{R})$-norm implies the exponential stability of the PDE (2.12)-(2.15) in the sense of the $L^2([0, 1], \mathbb{R}^4)$-norm.
*Proof.* The proof can be found in (Auriol 2024a, Theorem 6.3.1) and is a consequence of the invertibility and boundedness of the different backstepping transformations. □

Therefore, we now aim to stabilize the IDE (3.32), i.e., we want to design the controllers $V_1$ and $V_2$ so that there exists $C_1$ and $v_1$ such that for any $z_0 \in L^2([-\tau, 0], \mathbb{R})$ and all $t > 0$, we have $||z_{[t]}||_{L^2([-\tau, 0])} \le C_1 e^{-v_1 t} ||z_0||_{L^2([-\tau, 0])}$. Additionally, we also aim to guarantee the exponential stability of the control inputs $V_1$ and $V_2$. The actuation term in equation (3.32) involves both pointwise and distributed delay terms. This aspect has been rarely addressed in the literature (Auriol 2024b, Bekiaris-Liberis and Krstic 2016, Ponomarev 2015) and represents a significant distinction from existing findings. The challenges in stabilizing equation (3.32) stem from the simultaneous presence of both distributed-delay terms in the actuation and the state. A stabilizing controller was proposed in (Redaud et al. 2021, Redaud et al. 2022) for an equation similar to (3.32) using a PDE comparison system, which required complex computations and the application of Fredholm integral transformations. In the sequel, our approach will be inspired by the one developed in (Auriol, Kong and Bresch-Pietri 2022, Auriol 2024b), the main difference being the multiple pointwise actions of the control inputs.

# 4. Design of the stabilizing control law through Fredholm equations

In this section, we will choose $V_2 = 0$ for simplicity reasons. Although we are restricting the range of controllable systems, this significantly simplifies the analysis and a stabilizing control law based on Fredholm transformation kernels can be designed under not so strong controllability assumptions.

## 4.1 Candidate control law and characteristic equation

In this subsection, we present our methodology to stabilize the system (3.32). We will show that the stabilization problem boils down to the existence of a solution to simple integral equations with a convolution structure. We propose to look for the desired control law under the form

$$V_1(t) = \int_0^\tau f(v)z(t-v)d\nu + \int_0^\tau g(v)V_1(t-v)d\nu, \quad (4.1)$$

where $f$ and $g$ are piecewise continuously differentiable functions to be defined.

The stability analysis will be conducted in the Laplace domain and taking the Laplace transform of (3.32) with $V_2 \equiv 0$ yields

$$F_0(s)z(s) + F_1(s)V_1(s) = 0,$$

with $F_0$ and $F_1$ the two holomorphic functions defined for all $s \in \mathbb{C}$ by

$$\begin{aligned} F_0(s) &= 1 - rQ_{11}e^{-\tau s} - \int_0^\tau N(\eta)e^{-\eta s}d\eta, \\ F_1(s) &= 1 - rQ_{12}e^{-\tau s} - \int_0^\tau M_1(\eta)e^{-\eta s}d\eta. \end{aligned} \quad (4.2)$$

Combining the Laplace transform of equations (4.1) and (3.32) (with $V_2 \equiv 0$), we obtain:

$$\left( \int_0^\tau f(v)e^{-\nu s} d\nu \quad \int_0^\tau g(v)e^{-\nu s} \right) \left( \begin{array}{c} F_0(s) \\ F_1(s) \\ V_1(s) \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right)$$
The associated characteristic equation is defined for all $s \in \mathbb{C}$ by

$$ \det \left( \int_{0}^{\tau} f(v)e^{-vs}dv \quad \int_{0}^{\tau} g(v)e^{-vs}dv \right) = 0. \quad (4.3) $$

Our objective is to design the function $f$ and $g$ such that all the solutions of the characteristic equation have a real value smaller than a constant $-\eta_0$ ($\eta_0$ being positive), thus guaranteeing the exponential stability of the closed-loop system (3.32) (Henry 1974, Hale and Verduyn Lunel 1993)[Theorem 3.5]. Developing the characteristic equation (4.3), and using Fubini's theorem to deal with the double integral terms, we obtain

$$ \begin{aligned} 0 = 1 - rQ_{11}e^{-\tau s} - \int_{0}^{\tau} N(\eta)e^{-vs}d\nu - \int_{0}^{\tau} g(\nu)e^{-vs}d\nu + \int_{\tau}^{2\tau} rQ_{11}g(\nu - \tau)e^{-vs}d\nu \\ + \int_{0}^{\tau} e^{-vs}\int_{0}^{\nu} g(\eta)N(\nu - \eta)d\eta d\nu + \int_{\tau}^{2\tau} e^{-vs}\int_{\nu-\tau}^{\tau} g(\eta)N(\nu - \eta)d\eta d\nu \\ + \int_{0}^{\tau} f(\nu)e^{-vs}d\nu - \int_{\tau}^{2\tau} rQ_{12}f(\nu - \tau)e^{-vs}d\nu \\ - \int_{0}^{\tau} e^{-vs}\int_{0}^{\nu} f(\eta)M_{1}(\nu - \eta)d\eta d\nu - \int_{\tau}^{2\tau} e^{-vs}\int_{\nu-\tau}^{\tau} f(\eta)M_{1}(\nu - \eta)d\eta d\nu. \end{aligned} $$

The previous equation can be rewritten as

$$ 0 = 1 - rQ_{11}e^{-\tau s} - \int_{0}^{\tau} (I_{1}^{g,f}(\nu) + N(\nu))e^{-vs}d\nu - \int_{\tau}^{2\tau} I_{2}^{g,f}(\nu - \tau)e^{-vs}d\nu, \quad (4.4) $$

where for $\nu \in [0, \tau]$:

$$ I_{1}^{g,f}(\nu) = g(\nu) - f(\nu) + \int_{0}^{\nu} f(\eta)M_{1}(\nu - \eta)d\eta - \int_{0}^{\nu} g(\eta)N(\nu - \eta)d\eta, \quad (4.5) $$

$$ I_{2}^{g,f}(\nu) = -rQ_{11}g(\nu) + rQ_{12}f(\nu) + \int_{\nu}^{\tau} f(\eta)M(\nu + \tau - \eta)d\eta - \int_{\nu}^{\tau} g(\eta)N(\nu + \tau - \eta)d\eta. \quad (4.6) $$

Provided that we can choose $f$ and $g$ such that $I_1^{g,f} \equiv -N$, and $I_2^{g,f} \equiv 0$ on their respective domain of definition, we would obtain the characteristic equation $1 = rQ_{11}e^{-\tau_0 s}$, which implies the exponential stability of the state $(z, V_1)$ $|rQ_{11}| < 1$ due to Assumption 1.

## 4.2 Definition of the function $f$ and $g$

In this subsection, we prove that there exist $f$ and $g$ piecewise continuously differentiable functions such that the functions $I_1^{g,f} = -N$, and $I_2^{g,f} = 0$. Hence, we want to find $g$ and $f$ such that $\mathcal{T}(g, f) = (-N, 0)$ where $\mathcal{T}$ is defined on $X := L^2(0, \tau) \times L^2(0, \tau)$ by

$$ \begin{gathered} \mathcal{T}: X \to X \tag{4.7} \\ (g, f)^T \mapsto (I_1^{g,f}, I_2^{g,f})^T \end{gathered} $$

Under the conditions $r \neq 0$ and $Q_{12} \neq Q_{11}$ due to Assumption 1, the operator $\mathcal{T}$ is a Fredholm operator (Brezis 2010). To prove that the operator $\mathcal{T}$ is invertible, and therefore that the equation $\mathcal{T}(g, f) = (-N, 0)$ admits a unique solution, we will apply the following lemma.
**Lemma 2.** Consider two linear operators $\mathcal{A}, \mathcal{B}$, such that $D(\mathcal{A}) = D(\mathcal{B}) \subset X$. Consider an operator $\mathcal{T}: X \to X$ of the form $\mathcal{T} = D+K$ with $D$ an invertible operator on $X$ and $K$ a compact operator on $X$. Assume that

1. $\ker(\mathcal{T}) \subset D(\mathcal{A})$,

2. $\ker(\mathcal{T}) \subset \ker(\mathcal{B})$,

3. $\forall h \in \ker(\mathcal{T}), \mathcal{T}Ah = 0$,

4. $\forall s \in \mathbb{C}, \ker(sId - \mathcal{A}) \cap \ker(\mathcal{B}) = \{0\}$.

Then, the operator $\mathcal{T}$ is invertible.

*Proof.* The proof of this lemma can be found in (Coron, Hu and Olive 2016, Lemma 2.2). □

We can now prove the existence of adequate functions $f$ and $g$.

**Theorem 1.** Let us assume that Assumption 1 holds and that for all $s \in \mathbb{C}$, $rank(F_0(s), F_1(s)) = 1$ with $(F_0, F_1)$ defined in (4.2). Then, the Fredholm operator $\mathcal{T}$ defined in equation (4.7) is invertible.

*Proof.* Let us consider the two operators $\mathcal{A}$ and $\mathcal{B}$ defined by

$$ \mathcal{A}: D(\mathcal{A}) \to X \qquad \begin{pmatrix} \psi \\ \phi \end{pmatrix} \mapsto \begin{pmatrix} \psi' + \phi(0)M_1 \\ \phi' + \phi(0)N \end{pmatrix} $$

$$ \mathcal{B}: D(\mathcal{A}) \to X \qquad (\psi, \phi)^T \mapsto \psi(0) - \phi(0) $$

with $D(\mathcal{A}) = \{(\psi, \phi) \in H^1(0, \tau) \times H^1(0, \tau) : \phi(\tau) = rQ_{11}\phi(0), \psi(\tau) = rQ_{12}\phi(0)\}$. We will show that these operators satisfy the requirements of Lemma 2. First of all, the operator $\mathcal{T}$ defined in equation (4.7) is of the form $\mathcal{T} = D+K$ with

$$ D \begin{pmatrix} g \\ f \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -rQ_{11} & rQ_{12} \end{pmatrix} \begin{pmatrix} g \\ f \end{pmatrix}, $$

which is invertible from $X$ to $X$ due to Assumption 1 (and in particular, the conditions $r \neq 0$ and $Q_{11} \neq Q_{12}$), and

$$ K \begin{pmatrix} g \\ f \end{pmatrix} = \begin{pmatrix} \int_0^\nu f(\eta) M_1 (\nu - \eta) d\eta - \int_0^\nu g(\eta) N(\nu - \eta) d\eta \\ \int_\nu^\tau f(\eta) M(\nu + \tau - \eta) d\eta - \int_\nu^\tau g(\eta) N(\nu + \tau - \eta) d\eta, \end{pmatrix} $$

which is compact from $X$ to $X$ because an integral operator with an $L^2$ kernel is compact (Brezis 2010, Chapter 6). Let $(g, f) \in \ker(\mathcal{T})$. We have $I_1^{g,f}(v) = 0 = I_2^{g,f}(v)$ for all $v \in [0, \tau]$. Since $I_1^{g,f}(0) = 0 \Leftrightarrow g(0) - f(0) = 0$, we have proved $\ker(\mathcal{T}) \subset \ker(\mathcal{B})$. Next, $I_1^{g,f}(\tau) = 0$ and $I_2^{g,f}(0) = 0$ give us $g(\tau) - f(\tau) = rQ_{11}f(0) + rQ_{12}f(0)$. Since we already have $I_2^{g,f}(\tau) = 0 \Leftrightarrow -rQ_{11}g(\tau) + rQ_{12}f(\tau) = 0$, we obtain $f(\tau) = rQ_{11}f(0)$ and $g(\tau) = rQ_{12}f(0)$. Hence $\ker(\mathcal{T}) \subset D(\mathcal{A})$. Let us now prove that $\mathcal{T}\mathcal{A}(g, f) = 0$. Due to the redundancy of the computations, we only consider the case where $N$ and $M_1$ are continuously differentiable functions. The case in which both functions have a finite number of
discontinuities can be solved using an analogous approach. Performing integration by parts on each line for $v \in [0, \tau]$, we find (using $(g, f) \in D(\mathcal{A})$):

$$ \mathcal{T}\mathcal{A}\left(\frac{g}{f}\right)(v) = \left(\frac{dv}{d\nu} I_1^{g,f}(v)\right) $$

By differentiating the equations $I_1^{g,f} \equiv 0$ and $I_2^{g,f} \equiv 0$, we prove: $\forall h \in \ker(\mathcal{T})$, $\mathcal{T}\mathcal{A}h = 0$. Now, let us show that $\ker(s\mathrm{Id} - \mathcal{A}) \cap \ker(\mathcal{B}) = \{0\}$ for all $s \in \mathbb{C}$. Taking $(g, f) \in \ker(s\mathrm{Id} - \mathcal{A}) \cap \ker(\mathcal{B})$ implies that, for all $v \in [0, \tau]$:

$$ \begin{aligned} sg(v) &= g'(v) + f(0)M_1(v), \\ sf(v) &= f'(v) + f(0)N(v), \\ g(0) &= f(0). \end{aligned} $$

Hence, by the Duhamel formula, we deduce

$$ g(v) = e^{sv}f(0) - \int_{0}^{v} e^{s(v-\eta)} M_1(\eta) d\eta f(0) \quad (4.8) $$

$$ f(v) = e^{sv}f(0) - \int_{0}^{v} e^{s(v-\eta)} N(\eta) d\eta f(0) \quad (4.9) $$

We only need to prove $f(0) = 0$ to show $(g, f) = (0, 0)$. We evaluate equation (4.8) (respectively equation (4.9)) in $v = \tau$ multiply it by $e^{-s\tau}$. Then, using $g(\tau) = rQ_{12}f(0)$ (respectively $f(\tau) = rQ_{11}f(0)$), we obtain: $F_1(s)f(0) = 0$ and $F_0(s)f(0) = 0$. This implies that $f(0) = 0$ which concludes the proof. $\square$

The assumption $\mathrm{rank}(F_0(s) F_1(s)) = 1$ in the statement of Theorem 1 can be interpreted as a spectral controllability condition (Mounier 1998, Pandolfi 1976). By adjusting the method of moments (Russell 1978), it should be possible to show that this requirement is actually equivalent to the exact controllability of the IDE system (3.32) when $V_2 \equiv 0$, as suggested in (Coron et al. 2016). Clearly, setting $V_2 \equiv 0$ introduces additional conservatism. Ideally, in future work, we aim to consider the weaker spectral controllability assumption

$$ \mathrm{rank}(F_0(s) F_1(s) F_2(s)) = 1, $$

where

$$ F_2(s) = -1 - rQ_{13}e^{-\tau_3 s} - \int_{0}^{\tau} M_2(\eta)e^{-\eta s} d\eta, $$

and relate this condition to the controllability of the original underactuated PDE system (2.12)-(2.15). Then, we aim to stabilize the IDE system (3.32) using both $V_1$ and $V_2$ actuators. However, this requires adapting the proposed proof to the case of IDEs with multiple inputs. While we believe this extension is feasible, it poses several technical challenges related to the spectral analysis of the associated Fredholm operator. Finally, we emphasize that even the condition rank $(F_0(s) F_1(s) F_2(s)) = 1$ would not be a necessary condition to stabilize the system (3.32). However, it significantly simplifies our analysis by ensuring the invertibility of the Fredholm integral operator $\mathcal{T}$. In principle, stabilization could still be achieved under a **spectral stabilizability** condition, as “for all $s \in \mathbb{C}$ such that $\mathcal{R}(s) \ge 0$, $\mathrm{rank}(F_0(s) F_1(s) F_2(s)) = 1$” (or $\mathrm{rank}(F_0(s) F_1(s)) = 1$ in our case). Under this weaker assumption, we may not be able to fully eliminate the integral term $\int_0^{\tau_0} N(v)x(t-v)dv$, but rather only
a portion of it. However, complete elimination of this term is not strictly required for stabilization—as long as the remaining component is sufficiently small. Overall, employing a stabilizability condition rather than a controllability condition would necessitate a more intricate spectral analysis to design an appropriate stabilizing control law, beyond the scope of our current paper.

Regarding practical implementations, the controllability condition is verified using a zero-location algorithm and the functions $f$ and $g$ can be numerically obtained by solving the Fredholm integral equations. When the integral part of the Fredholm operator has a spectral radius smaller than one (Kress, Mazya and Kozlov 1989, Theorem 10.13), this can be done using a successive approximations approach (method of iterations). Among other possible numerical approaches, we can cite projection methods, collocation methods, Galerkin methods, or quadrature methods (Kress et al. 1989).

## 4.3 Main result

In this last subsection, we state our main theorem that guarantees exponential closed-loop stability.

**Theorem 2.** Consider the functions $I_1^{g,f}$, and $I_2^{g,f}$ defined in (4.5)-(4.6) and let $f$ and $g$ be the unique piecewise continuously differentiable functions that lead to $I_1^{g,f}(v) = -N(v)$ for all $v \in [0, \tau]$, and $I_2^{g,f}(v) = 0$ for $v \in [\tau, 2\tau]$ (as stated in Theorem 1). Then the closed-loop system consisting of the plant (3.32) and the control law (4.1) is exponentially stable. Consequently, the closed-loop system (2.12)-(2.15) with the control law (3.30)-(3.31) is exponentially stable.

*Proof.* The existence of $f$ and $g$ is implied by Theorem 1. For this choice of $f$ and $g$, the closed-loop characteristic equation rewrites $1 = rQ_{11}e^{-\tau s}$ which implies the exponential stability of the state and the control law since $|rQ_{11}| < 1$. The exponential stability of (2.12)-(2.15) is a consequence of Lemma 1. □

Although the controller designed in Theorem 2 fulfills the control objective and stabilizes the system (2.12)-(2.15), it requires the cancellation of some of the boundary reflection terms. This may lead to vanishing robustness margins, as shown in (Auriol, Aarsnes, Martin and Di Meglio 2018). However, taking the Laplace transform of equation (4.1), we obtain $V_1(s) = \frac{\int_0^\tau f(v)e^{-vs}dv}{1-\int_0^\tau g(v)e^{-vs}dv}$, which defines a strictly proper transfer function due to Riemann Lebesgues' lemma. Therefore, it may be possible to apply the robustification procedure presented in (Auriol et al. 2023) to guarantee the existence of robustness margins.

## 5. Conclusions

In this paper, we have developed a stabilizing control law for one-dimensional linearized shock-free Saint-Venant equations. Initially, we reformulated the system as an underactuated hyperbolic PDEs system, highlighting the challenges associated with the weakly hyperbolic nature of the equations and the presence of limited actuation. Subsequently, we employed the backstepping approach to transform the system into an IDE. This reformulation allowed us to define a suitable stabilizing controller characterized by an auto-regressive structure. The controller gains were derived by solving a set of Fredholm equations.

Despite its effectiveness, the proposed control design necessitates conservative controllability conditions, which could be weakened by utilizing the effect of the actuator $V_2$ in equation (3.32). However, this adjustment would introduce additional complexities, as it necessitates the stabilization of an IDE
featuring multiple pointwise delays that vary in both the state and the actuation. The analysis and control of such systems remain an open challenge, requiring advanced delay compensation techniques and possibly new integral transformation methods. Future research will focus on addressing these challenges, aiming to extend the applicability of the proposed approach to broader classes of hyperbolic PDE systems and delay equations. We also aim to investigate configurations where actuation is applied along the spatial domain rather than at the boundaries. In this case, the proposed approach can still be adapted; however, the operator $\mathcal{T}$ is no longer a Fredholm operator, which invalidates the proof presented in Section 4. Some alternative strategies to address this issue have been proposed in (Auriol 2024b) under conservative conditions.

## Acknowledgments

The work of the two first authors has received funding from the Agence Nationale de la Recherche (ANR) via the grant PANOPLY ANR-23-CE48-0001-01.

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