# Event-triggered gain scheduling of 2 x 2 linear hyperbolic PDEs with time and space varying coupling coefficients

Nicolás Espitia, Jean Auriol

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Nicolás Espitia, Jean Auriol. Event-triggered gain scheduling of 2 x 2 linear hyperbolic PDEs with time and space varying coupling coefficients. Automatica, 2025. hal-05082681v2

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# Event-triggered gain scheduling of 2 × 2 linear hyperbolic PDEs with time and space varying coupling coefficients

Nicolas Espitia,ª Jean Auriolᵇ

## Abstract

In this paper, we address the problem of exponential stabilization of 2 × 2 hyperbolic PDEs systems with time- and space-varying in-domain coupling coefficients using event-triggered gain scheduling. More precisely, at each triggering time, we apply the control input as the classical static backstepping control law while scheduling the gains of the boundary controller according to the triggering mechanism and by solely considering the spatial variation of the coefficients. We present two different event-triggered gain scheduling strategies. The first strategy relies on a Lyapunov-based event-triggering condition, and the $L^2$ closed-loop exponential stability is shown using a Lyapunov analysis. The second one is a small-gain based event-triggering condition and builds on the $L^\infty$-norm. The $L^\infty$ closed-loop exponential stability is shown by combining ISS estimates with small-gain arguments. In both cases, we prove that we avoid the Zeno phenomenon, provided that the coupling coefficients are slowly time-varying. Unlike existing results in the literature, the proposed approaches do not require solving time-varying backstepping kernel equations in real-time resulting in reduced computational burden and broader applicability.

**Key words:** hyperbolic equations, backstepping control design, event-triggered gain scheduling, Lyapunov analysis, small-gain analysis.

## 1 Introduction

Hyperbolic Partial Differential Equations (PDEs) play a fundamental role in the mathematical modeling of complex systems involving transport phenomena with finite propagation speeds, such as the transport of matter, sound waves, and information. These equations are extensively used to model hydraulic systems [10, Chapter 8], road traffic [48], and gas flow pipelines or oil drilling processes [1].

The boundary control of systems governed by PDEs is a critical area of research. In this context, the *backstepping approach* has emerged as a powerful tool for designing stabilizing controllers. This method enables the design of controllers that ensure exponential or even finite-time convergence for various systems. Additionally, it provides explicit control laws and allows for parameter-adaptive implementations with real-time parameter estimation. Originally developed to stabilize 1D reaction-diffusion parabolic PDEs in [12], this approach relies on integral transformations to map the original PDE system into a *target* system with de-

sired stability properties, facilitating subsequent Lyapunov stability analysis. The backstepping methodology has played a crucial role in the stabilization of coupled hyperbolic PDEs [15,17,24]. These results were then extended to design robust controllers [7], finite-time output regulators [18], or adaptive controllers [3] for such hyperbolic systems. Lately, several contributions focused on complex interconnected dynamics involving couplings with ordinary differential equations or other PDE systems [20,39], requiring the use of advanced tools such as Fredholm transformations [13,38]. The reader is referred to [43] for a complete overview of the recent developments of the backstepping approach for PDEs.

The aforementioned contributions primarily focus on time-invariant coupling parameters. Controlling and estimating hyperbolic PDE systems with time- and space-dependent coefficients presents a significantly greater challenge. Such systems are relevant in numerous applications, including physical models of balance laws linearized around time-varying trajectories, such as plug flow chemical reactor equations, Saint-Venant equations, and heat exchanger equations (see, e.g., [10]). There are relatively few contributions on backstepping-based control and estimation of hyperbolic systems with time- and space-varying coefficients. In [19], an observer was proposed for a hyperbolic Partial Integro-Differential Equation (PIDE). For boundary stabilization and tracking of hyperbolic PDEs, [2] and later [4]

<footnote>ªUniv. Lille, CNRS, Centrale Lille, UMR 9189 CRISTAL, F-59000 Lille, France.</footnote>

<footnote>ᵇUniversité Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France.</footnote>

<footnote>Email addresses: nicolas.espitia-hoyos@univ-lille.fr (Nicolas Espitia), jean.auriol@12s.centralesupelec.fr (Jean Auriol).</footnote>

<footnote n="1">¹ This project has received funding from the Agence Nationale de la Recherche (ANR) via grants PANOPLY ANR-23-CE48-0001-01, and PH-DIPSY ANR-24-CE48-1712.</footnote>
were the first to achieve finite-time stabilization using a time-varying backstepping transformation. By employing an appropriate time-dependent Volterra transformation, they converted the system into a fixed-time target system, ensuring stability through the bounded invertibility of the transformation. More recently, [16] addressed the problem of finite-time boundary stabilization for general hyperbolic systems of balance laws with both time- and space-varying coupling coefficients and transport speeds. Their approach relied on time-dependent Volterra integral transformations to convert the original system into suitable target systems: one in “control normal form” (as introduced in [14] and [40]), and another with a simple coupling structure exhibiting finite-time stability. The use of time-dependent Volterra transformations introduces additional complexity compared to the time-invariant case. Specifically, the resulting kernel equations involve a first-order derivative with respect to time. In [16], the authors rigorously analyzed the well-posedness of such kernel equations. However, implementing the associated controllers in real-world systems requires solving this set of time-varying kernel equations in real-time. This process can be computationally demanding, potentially limiting the practical applicability of the approach.

Instead of managing a time-varying kernel, we adopt in this paper a simpler approach that considers only the spatial variations of the coupling coefficients. This approach remains valid as long as the coupling coefficients are sampled over time, reducing the kernel PDEs to involve only space-dependent coefficients between successive sampling or triggering times. This fundamental idea was introduced in [25], to address the problem of exponential stabilization for reaction-diffusion PDEs with time- and space-dependent coefficients using event-triggered gain scheduling. The kernel computation was scheduled aperiodically, based on the current state information of the closed-loop system and the time- and space-varying reaction coefficient, which was treated as a distributed input disturbance. This resulted in the first event-triggered gain scheduling approach for PDEs.

It is worth mentioning that event-triggered gain scheduling draws inspiration from the framework of Event-Triggered Control (ETC) for PDEs, which has become an active research area. Numerous results and advanced schemes have been developed in recent years for different classes of PDEs, including hyperbolic PDEs, parabolic PDEs, the Schrödinger equation, the Korteweg–de Vries (KdV) equation, the wave equation, and coupled PDE-ODE systems, among others. Without providing an exhaustive literature review on ETC—since it is not the core focus of our paper—we refer to some key contributions on ETC (and the references therein), focusing specifically on 1D parabolic and hyperbolic PDEs. For instance, in the context of ETC for parabolic PDEs, notable contributions include [23, 27, 31, 33, 34, 37], with significant extensions and redesigns leading to more advanced strategies. These include performance-barrier-based event-triggered control [36], as well as Periodic Event-Triggered Control (PETC) and Self-Triggered Control (STC) [35]. The latter two approaches offer practical implementation advantages, as they eliminate the need for continuous monitoring of the triggering condi-

tion: PETC monitors the triggering condition periodically, while STC predicts the next triggering time based on the state information at the current sampling instant.

For hyperbolic PDEs, seminal studies have explored event-triggered control across different system classes. In particular, [22] addresses ETC for linear hyperbolic systems of conservation laws, while [42] extends the analysis to semilinear cases, and [21] investigates ETC for nonlinear hyperbolic PDEs. Moreover, ETC has been adapted to more complex frameworks, including coupled PDE-ODE and PDE-PDE systems, as demonstrated in [28, 45, 46]. Additionally, recent advancements have introduced PETC and STC strategies in this context [41], providing alternative approaches to triggering mechanisms and offering a key practical case study on the control of open channel of navigation modeled by the Saint-Venant equations.

Although in this work we do not deal with event-triggered control (the state value of the control is not updated on events) but gain scheduling, the main theoretical questions underlying the design of event-triggered mechanisms, such avoidance of the Zeno phenomena, well-posedness issues, and robustness to the sampling schedule while guaranteeing exponential stability, persist within this framework.

We propose two event-triggered gain scheduling schemes wherein we treat the time- and space-varying in-domain coupling coefficients as external (multiplicative) disturbances integrated into the control and event-triggered mechanism. As such, our proposed scheme is a feedforward controlled one (see, e.g., [11]) that compensate for the disturbances. At each triggering time, we recalculate the kernels of the Volterra backstepping transformation, thereby scheduling the gains of the controller according to the triggering mechanism while solely considering the spatial variation of the coefficients. Following the corresponding backstepping transformation, the resulting target system exhibits errors due to sampling. However, by ensuring these errors remain sufficiently small, we can still guarantee the well-posedness and the exponential stability of the closed-loop system. Under the first event-triggered gain scheduling scheme that we propose in this paper, we update the kernel gains according to a Lyapunov-based event-triggering condition. It accounts for the error when sampling, which is compared with the $L^2$-norm of the state, therefore ensuring that the error remain small enough. The $L^2$ closed-loop exponential stability is shown using a Lyapunov analysis. Under the second event-triggered gain scheduling scheme, we update the kernel gains according to a small-gain based event-triggering condition which accounts for the error terms resulting from the sampling procedure and compares with the $L^\infty$-norm of the state. The $L^\infty$ closed-loop exponential stability is shown by using suitable Input-to-State Stability (ISS) estimates and small-gain arguments. The main parameter of the triggering condition is selected according to a small-gain condition. For both strategies, we prove that we can avoid the Zeno phenomenon under the corresponding event-triggering policy, provided that the coupling coefficients are slowly time-varying.

**Notations:** For any integer $p > 0$, we denote $L^p([0, 1], \mathbb{R})$ the space of real-valued $p$-integrable functions defined on $[0, 1]$
with the standard $L^p$ norm, i.e., for any $f \in L^p([0,1], \mathbb{R})$,
we have $||f||_{L^p} := (\int_0^1 |f(x)|^p dx)^{\frac{1}{p}}$. The set $L^\infty([0,1], \mathbb{R})$ denotes the space of bounded real-valued functions defined on $[0,1]$. For any $f \in L^\infty([0,1], \mathbb{R})$, $||f||_\infty := \ess\sup_{x \in [0,1]} |f(x)|$.

# 2 Problem description and control design

## 2.1 System under consideration

Consider the following $2 \times 2$ linear hyperbolic system

$$ \partial_t u(t,x) + \lambda(x) \partial_x u(t,x) = \sigma^+(t,x) v(t,x), \quad (1) $$

$$ \partial_t v(t,x) - \mu(x) \partial_x v(t,x) = \sigma^-(t,x) u(t,x), \quad (2) $$

with the boundary conditions

$$ u(t,0) = qv(t,0), \quad v(t,1) = \rho u(t,1) + U(t), \quad (3) $$

where $(u(t,x), v(t,x))$ is the state of the system, the different arguments evolving in $\{(t,x) \text{ s.t. } t \ge t_0, x \in [0,1]\}$, where $t_0 \ge 0$ is the initial time. The function $U(t)$ corresponds to the boundary actuation. The velocities $\lambda(x) > 0$ and $\mu(x) > 0$ belong to $C^1([0,1])$ and are assumed to satisfy $-\mu(x) < 0 < \lambda(x)$ for all $x \in [0,1]$. The time- and space- varying coupling coefficients $\sigma^+$ and $\sigma^-$ belong to $C^0([t_0, \infty) \times [0,1]) \cap L^\infty([t_0, \infty) \times [0,1])$, while the boundary couplings $\rho$ (the proximal reflection) and $q$ (the distal reflection) are considered as constants. We assume that the time-varying coupling terms $\sigma^+$ and $\sigma^-$ are bounded, i.e., there exists a constant $M_\sigma$ s.t. for all $t \ge t_0$ and all $x \in [0,1]$

$$ |\sigma^-(t,x)|, |\sigma^+(t,x)| < M_\sigma. \quad (4) $$

We will refer to the well-posedness aspects in Section 2.5 as we will be considering different state spaces based on the gain scheduling mechanism. Furthermore, we have the following assumptions.

**Assumption 1.** The boundary coupling terms verify $|\rho q| < 1$. We also assume that $q \neq 0$.

The condition $|\rho q| < 1$ means that the boundary couplings are dissipative. Such a requirement is necessary to guarantee the delay-robustness of the closed-loop system as shown in [7,32]. Having $q \neq 0$ is not necessary but simplifies the backstepping transformation we use later in the paper.

**Assumption 2.** There exist a constant $l_\sigma > 0$ such that the following inequalities hold for all $x \in [0,1]$, and all $t, s \ge t_0$

$$ |\sigma^+(t,x) - \sigma^+(s,x)| \le l_\sigma |t-s|, \quad (5) $$

$$ |\sigma^-(t,x) - \sigma^-(s,x)| \le l_\sigma |t-s|. \quad (6) $$

This assumption means that the different time-varying coefficients are Lipschitz with respect to time. It is crucial in our analysis to ensure the avoidance of the Zeno phenomenon, as demonstrated in the proofs of Lemma 1 and Lemma 2. While it may be possible to relax this assumption and replace it with a Hölder regularity condition, removing it altogether could lead to fast-varying coefficients and the impossibility to guarantee a Zeno-free behavior.

**Remark 1.** Similarly to what has been proposed in [25], we will consider the system (1)-(3) not as a time-varying system but as a time-invariant system affected by the control input $U(t)$ and (multiplicative) disturbance inputs, i.e., the coupling coefficients $\sigma^-(t,x)$ and $\sigma^+(t,x)$. By regarding (1)-(3) as a time-invariant system, we can assume that the initial time is zero. As a result, the suggested control scheme can be viewed as a feedforward control that compensates for the impact of the disturbance inputs.

## 2.2 Control objective

For hyperbolic systems with time-independent coefficients, stabilizing boundary control laws were designed in [17,44] using the backstepping approach. The extension to time-dependent coefficients has been proposed in [16] using time-varying backstepping transformations but raises several computational challenges. In this paper, our objective is to stabilize the system (1)-(3) exponentially using a suitable backstepping-based boundary controller while avoiding solving time-varying kernel equations as those given in [16]. This is possible by scheduling the backstepping kernel gains at a specific increasing sequence of times. We will design two event-triggered gain scheduling schemes, (and the corresponding boundary feedback controllers). For a given sequence of time instants $\{t_j\}_{j \in \mathbb{N}}$, which will be characterized according to the even-triggered gain-scheduling policies, we will consider that all the time-varying coefficients are constant on the time interval $[t_j, t_{j+1})$, with their values set to those at time $t_j$. We then solve the corresponding kernel equations at $t = t_j$ to obtain the candidate control law.

## 2.3 Backstepping transformations

Let $\{t_j\}_{j \in \mathbb{N}}$ be an increasing sequence of times with $t_0 = 0$. For all $x \in [0,1]$, we denote

$$ \sigma_j^+(x) := \sigma^+(t_j, x), \quad \sigma_j^-(x) := \sigma^-(t_j, x), \quad (7) $$

which are the sampled versions of the different coupling coefficients that appear in equations (1)-(3). We also define the errors when sampling as

$$ \delta_{\sigma_j^+}(t,x) := \sigma^+(t,x) - \sigma^+(t_j, x), \quad (8) $$

$$ \delta_{\sigma_j^-}(t,x) := \sigma^-(t,x) - \sigma^-(t_j, x). \quad (9) $$

For all $t \in [t_j, t_{j+1})$, we can rewrite the system (1)-(3) as

$$ \partial_t u(t,x) + \lambda(x) \partial_x u(t,x) = \sigma_j^+(x) v(t,x) + \delta_{\sigma_j^+}(t,x) v(t,x), \quad (10) $$

$$ \partial_t v(t,x) - \mu(x) \partial_x v(t,x) = \sigma_j^-(x) u(t,x) + \delta_{\sigma_j^-}(t,x) u(t,x), \quad (11) $$

with the boundary conditions

$$ u(t,0) = qv(t,0), \quad v(t,1) = \rho u(t,1) + U(t). \quad (12) $$
Consider the following Volterra integral change of coordinates, defined for $j \ge 0$, for all $t \in [t_j, t_{j+1})$ and all $x \in [0, 1]$:

$$ \alpha_j(t,x) := u(t,x) + \int_0^x K_j^{uu}(x,y)u(t,y)dy \\ + \int_0^x K_j^{uv}(x,y)v(t,y)dy, \quad (13) $$

$$ \beta_j(t,x) := v(t,x) + \int_0^x K_j^{vu}(x,y)u(t,y)dy \\ + \int_0^x K_j^{vv}(x,y)v(t,y)dy, \quad (14) $$

which can be rewritten in the following compact form:

$$ (\alpha_j(t,x), \beta_j(t,x)) := (\bar{T}_j(u(t, \cdot), v(t, \cdot)))(x), \quad (15) $$

where the kernels $K_j^\cdot$ are continuous functions defined on the triangular domain $\mathcal{D} := \{(x, y) \in [0, 1]^2, 0 \le y \le x \le 1\}$. They satisfy [17,44]

$$ \Delta(x)\partial_x K_j(x, y) + \partial_y(K_j(x, y)\Delta(y)) = -K_j\Sigma_j(y), \quad (16) $$

with the boundary conditions

$$ \Delta(x)K_j(x,x) - K_j(x,x)\Delta(x) = -\Sigma_j(x) \quad (17) $$

$$ K_j^{uu}(x, 0) = \frac{\mu(0)}{\lambda(0)q} K_j^{uv}(x, 0), \quad (18) $$

$$ K_j^{vv}(x, 0) = \frac{\lambda(0)q}{\mu(0)} K_j^{vu}(x, 0), \quad (19) $$

$$ \text{where} \quad K_j(x, y) := \begin{pmatrix} K_j^{uu}(x, y) & K_j^{uv}(x, y) \\ K_j^{vu}(x, y) & K_j^{vv}(x, y) \end{pmatrix}, \\ \Delta(x) := \begin{pmatrix} \lambda(x) & 0 \\ 0 & -\mu(x) \end{pmatrix}, \quad \text{and} \quad \Sigma_j(x) := \begin{pmatrix} 0 & \sigma_j^+(x) \\ \sigma_j^-(x) & 0 \end{pmatrix}. $$

The backstepping boundary control design is performed by mapping the system (10)-(12) to a target system which will reflect the errors when sampling. The transformation (13)-(14) is a Volterra transformation and is invertible. Consequently, there exist bounded functions $L_j^\cdot$ defined on the triangular domain $\mathcal{D}$, such that for all $t \in [t_j, t_{j+1})$, all $x \in [0, 1]$, we have

$$ u(t,x) = \alpha_j(t,x) + \int_0^x L_j^{\alpha\alpha}(x,y)\alpha_j(t,y)dy \\ + \int_0^x L_j^{\alpha\beta}(x,y)\beta_j(t,y)dy, \quad (20) $$

$$ v(t,x) = \beta_j(t,x) + \int_0^x L_j^{\beta\alpha}(x,y)\alpha_j(t,y)dy \\ + \int_0^x L_j^{\beta\beta}(x,y)\beta_j(t,y)dy. \quad (21) $$

The kernels $L_j^\cdot$ verify PDE kernel equations that can be found in [44]. The inverse transformation is denoted $\bar{T}_j^{-1}$

such that for all $t \in [t_j, t_{j+1})$ we have

$$ (u(t,x), v(t,x)) := (\bar{T}_j^{-1}(\alpha_j(t, \cdot), \beta_j(t, \cdot))(x), \quad (22) $$

Again, we will use the concatenated notation where $L_j(x, y) := \begin{pmatrix} L_j^{\alpha\beta}(x, y) & L_j^{\alpha\beta}(x, y) \\ L_j^{\beta\alpha}(x, y) & L_j^{\beta\beta}(x, y) \end{pmatrix}$. We define the following matrix norms $||K_j||_\infty := \sup_{(x,y)\in\mathcal{D}}|K_j(x,y)|$, $||L_j||_\infty := \sup_{(x,y)\in\mathcal{D}}|L_j(x,y)|$. Applying [17, Prop. A.6], we have

$$ \max\{||K_j||_\infty, ||L_j||_\infty\} < \bar{\phi}e^{\bar{D}_j\kappa}, \quad (23) $$

where

$$ \kappa = \max_{x \in [0,1]} \left\{ \frac{1}{\lambda(x)}, \frac{1}{\mu(x)} \right\}, \quad \bar{q} = \max \left\{ \left| \frac{\lambda(0)q}{\mu(0)} \right|, \left| \frac{\mu(0)}{\lambda(0)q} \right| \right\}, $$

$$ \bar{D}_j = 16(1 + 2\bar{q}) \max \left\{ \max_{x \in [0,1]} |\lambda'(x)|, \max_{x \in [0,1]} |\mu'(x)|, \max_{x \in [0,1]} |\sigma_j^+(x)|, \max_{x \in [0,1]} |\sigma_j^-(x)| \right\}, $$

$$ \bar{\phi} = \bar{q} \max \left\{ \max_{x \in [0,1]} |\sigma_j^+(x)|, \max_{x \in [0,1]} |\sigma_j^-(x)| \right\}. $$

Consequently, due to the bound (4), we obtain that all the kernels are uniformly bounded by some constant that does not depend on $j$. More precisely, we have for all $j \ge 0$:

$$ \max\{||K_j||_\infty, ||L_j||_\infty\} < M_1, \quad (24) $$

where $M_1 := M_\sigma \bar{q} e^{\kappa \bar{D}} > 0$ with $\bar{D} = 16(1 + 2\bar{q}) \max\{\max_{x \in [0,1]} |\lambda'(x)|, \max_{x \in [0,1]} |\mu'(x)|, M_\sigma\}$. Let us now define $M_2$ and $M_\infty$ as

$$ M_2 := \sqrt{2 + 8M_1^2}, \quad M_\infty := (1 + 2M_1). \quad (25) $$

Using the Cauchy-Schwartz inequality on equations (15) and (22) we obtain the following estimates for $j \ge 0$:

$$ \begin{aligned} & ||\bar{T}_j^{-1}(\alpha_j, \beta_j)||_{L^2} \le M_2||(\alpha_j(t, \cdot), \beta_j(t, \cdot))||_{L^2}, \\ & ||\bar{T}_j(u, v)||_{L^2} \le M_2||(u(t, \cdot), v(t, \cdot))||_{L^2}, \end{aligned} \quad (26) $$

$$ \begin{aligned} & ||\bar{T}_j^{-1}(\alpha_j, \beta_j)||_\infty \le M_\infty||(\alpha_j(t, \cdot), \beta_j(t, \cdot))||_\infty, \\ & ||\bar{T}_j(u, v)||_\infty \le M_\infty||(u(t, \cdot), v(t, \cdot))||_\infty. \end{aligned} \quad (27) $$

## 2.4 Control law and target system

For all $t \in [t_j, t_{j+1})$, we define the boundary control as

$$ U(t) := \int_0^1 (\rho K_j^{uu}(1, y) - K_j^{vu}(1, y))u(t, y)dy \\ + \int_0^1 (\rho K_j^{uv}(1, y) - K_j^{vv}(1, y))v(t, y)dy. \quad (28) $$

Using the backstepping transformation (15), we map the original system (10)-(12) into the following target system,
defined for all $t \in [t_j, t_{j+1})$ by:

$$ \partial_t \alpha_j(t, x) + \lambda(x) \partial_x \alpha_j(t, x) = e_{j,1}(t, x), \quad (29) $$

$$ \partial_t \beta_j(t, x) - \mu(x) \partial_x \beta_j(t, x) = e_{j,2}(t, x), \quad (30) $$

with the boundary conditions

$$ \alpha_j(t, 0) = q\beta_j(t, 0), \quad \beta_j(t, 1) = \rho\alpha_j(t, 1), \quad (31) $$

where $e_{j,1}(t,x) := (\mathcal{T}_j(\delta_{\sigma_j^+}(t, \cdot)v(t, \cdot), \delta_{\sigma_j^-}(t, \cdot)u(t, \cdot)))_1(x)$ and $e_{j,2}(t,x) := (\mathcal{T}_j(\delta_{\sigma_j^+}(t, \cdot)v(t, \cdot), \delta_{\sigma_j^-}(t, \cdot)u(t, \cdot)))_2(x)$. More precisely, by the definition of $\mathcal{T}_j$ as in (15), we have

$$ \begin{aligned} e_{j,1}(t,x) &:= \delta_{\sigma_j^+}(t,x)v(t,x) \\ &\quad + \int_0^x K_j^{uu}(x,y)\delta_{\sigma_j^+}(t,y)v(t,y)dy \\ &\quad + \int_0^x K_j^{uv}(x,y)\delta_{\sigma_j^-}(t,y)u(t,y)dy, \end{aligned} \quad (32) $$

$$ \begin{aligned} e_{j,2}(t,x) &:= \delta_{\sigma_j^-}(t,x)u(t,x) \\ &\quad + \int_0^x K_j^{vu}(x,y)\delta_{\sigma_j^+}(t,y)v(t,y)dy \\ &\quad + \int_0^x K_j^{vv}(x,y)\delta_{\sigma_j^-}(t,y)u(t,y)dy. \end{aligned} \quad (33) $$

**Remark 2.** The target system (29)-(33) exhibits the errors when sampling after transformation. If the time-varying coefficients were equal to their sampled values, the control law (28) would exponentially stabilize the closed-loop system (1)-(3) in the sense of the $L^2$-norm [5, Lemma 2].

## 2.5 Well-posedness aspects

The boundary control input (28) can be rewritten as

$$ U(t) = \int_{0}^{1} F_{1}(t, y)u(t, y) + F_{2}(t, y)v(t, y)dy, \quad (34) $$

for all $t \in [0, \lim_{j \to \infty}(t_j))$ where $F_1 \in L^\infty((0, \lim_{j \to \infty}(t_j)) \times (0, 1))$ and $F_2 \in L^\infty((0, \lim_{j \to \infty}(t_j)) \times (0, 1))$ are piecewise constant functions (in time). The closed-loop system (1)-(3) with (28) is then a special case of the system considered in [16]. Therefore, we consider the concept of solution along the characteristics, use the explicit solutions of (1)-(3) via (22) applied to (A.1)-(A.2), (A.22)-(A.23) and apply [16] to conclude that for any initial condition $(u(0, \cdot), v(0, \cdot)) = (u_0, v_0) \in (L^2(0, 1))^2$, the closed-loop system (1)-(3), (28) admits a unique solution $(u, v) \in C^0([0, \lim_{j \to \infty}(t_j)); L^2(0, 1)^2)$. Similarly, if the initial condition belongs to $(L^\infty(0, 1))^2$, the closed-loop system admits a unique solution $(u, v) \in C^0([0, \lim_{j \to \infty}(t_j)); L^\infty(0, 1)^2)$. For the two event-triggered gain scheduling strategies we propose in this paper, it is essential to show that the triggering sequence verifies $\lim_{j \to \infty}(t_j) = +\infty$.

# 3 Lyapunov-based event-triggered gain scheduling strategy

In this section, we introduce our first event-triggering mechanism for gain scheduling. We assume the initial condition belongs to $(L^2(0, 1))^2$. The triggering condition determines the time instants at which the coefficients must be sampled and, thereby when the kernel computation and updating must be done. The triggering condition relies on the evolution of the errors when sampling $e_j(t, x) = (e_{j,1}(t, x), e_{j,2}(t, x))$ given in (32)-(33) and a function $V((\alpha_j(t, \cdot), \beta_j(t, \cdot)))$ (Lyapunov-like functional) defined by

$$ \begin{aligned} V(\gamma_j(t, \cdot)) &:= \int_0^1 \frac{e^{-\nu\phi_1(x)}}{\lambda(x)} \alpha_j^2(t, x) + \zeta \frac{e^{\nu\phi_2(x)}}{\mu(x)} \beta_j^2(t, x)dx, \\ &= \langle g(\cdot)\gamma_j(t, \cdot), \gamma_j(t, \cdot) \rangle. \end{aligned} \quad (35) $$

where the multiplication inside the scalar products should be understood in the Hadamard sense, $\gamma_j(t, x) := (\alpha_j(t, x), \beta_j(t, x))$ and

$$ \phi_1(x) := \int_0^x \frac{1}{\lambda(r)}dr, \quad \phi_2(x) := \int_0^x \frac{1}{\mu(r)}dr, \quad (36) $$

and $g$ is defined as

$$ g(x) := \left( \frac{e^{-\nu\phi_1(x)}}{\lambda(x)}, \zeta \frac{e^{\nu\phi_2(x)}}{\mu(x)} \right). \quad (37) $$

Notice that $V(t) = V(\gamma_j(t, \cdot)) = V(w(t, \cdot)) = \langle g(\cdot)\mathcal{T}_j(w(t, \cdot)), \mathcal{T}_j(w(t, \cdot)) \rangle$ via (15), where $w(t, x) := (u(t, x), v(t, x))$. The parameters $\nu$ and $\zeta$ are chosen such that

$$ q^2 - \zeta < 0, \quad \zeta \rho^2 e^{\nu(\phi_1(1) + \phi_2(1))} < 1. \quad (38) $$

These conditions are feasible since $|\rho q| < 1$ (see Assumption 1). It is sufficient to choose $|q|^2 < \zeta < \frac{1}{\rho^2}$ and $0 < \nu < -(\phi_1(1) + \phi_2(1))^{-1} \ln(\zeta \rho^2)$. The functional $V$ is equivalent to the $L^2$-norm of the state $(w(t, x))$, i.e., there exist $k_u$ and $k_\ell$ that do not depend on $j$ (due to estimates (26)) s.t.

$$ k_\ell \|w(t, \cdot)\|_{L^2}^2 \le V(t) \le k_u \|w(t, \cdot)\|_{L^2}^2. \quad (39) $$

More precisely, we can choose

$$ k_u := M_2^2 \max \left\{ \frac{\zeta e^{\nu\phi_2(1)}}{\mu_{\min}}, \frac{1}{\lambda_{\min}} \right\}, \quad (40) $$

$$ k_\ell := \frac{1}{M_2^2} \min \left\{ \frac{e^{-\nu\phi_1(1)}}{\lambda_{\max}}, \frac{\zeta}{\mu_{\max}} \right\}, \quad (41) $$

where $M_2$ is the bound characterized in (25). Finally, we define the following quantities:

$$ \lambda_{\min} := \min_{x \in [0,1]} \lambda(x), \quad \mu_{\min} := \min_{x \in [0,1]} \mu(x), \quad (42) $$

$$ \lambda_{\max} := \max_{x \in [0,1]} \lambda(x), \quad \mu_{\max} := \max_{x \in [0,1]} \mu(x). \quad (43) $$
### 3.1 Lyapunov-based Event-triggered gain scheduling with a static triggering condition

Let $R_L \in (0, 1)$ be a design parameter, $\nu$ be chosen such that (38) holds, and define the following set:

$$E_L(t_j) := \{t > t_j : 2\langle g(\cdot)T_j(w(t, \cdot)), e_j(t, \cdot) \rangle > \nu R_L V(t)\}. \quad (44)$$

The times of events $t_j \ge t_0$ with $t_0 = 0$ (see Remark 1) form a finite or countable set of times, which is determined by the following rules for any $j \ge 0$:

(1) if $E_L(t_j) = \emptyset$, then the set of the times of the events is $\{t_0, \dots, t_j\}$.

(2) if $E_L(t_j) \neq \emptyset$, then the next event time is given by

$$t_{j+1} := \inf E_L(t_j), \quad (45)$$

### 3.2 Avoidance of the Zeno phenomenon

To be able to implement our controller on digital platforms, we need to guarantee that there are no infinite triggering times that occur in a finite time interval (which would imply being able to sample infinitely fast). In other words, we need to guarantee there is no Zeno phenomenon.

**Lemma 1.** Under the event-triggered condition (44)-(45), there exists a minimal dwell time between two triggering times, i.e., there exists a constant $\tau_L > 0$ (independent of the initial condition of the state) s.t. $t_{j+1} - t_j \ge \tau_L$ for all $j \in \mathbb{N}$.

*Proof.* Assume that an event occurs at $t = t_{j+1}$. Then, from (44)-(45), we have

$$\nu R_L V(t_{j+1}) \le 2\langle g(\cdot)T_j(w(t_{j+1}, \cdot)), e_j(t_{j+1}, \cdot) \rangle.$$

Using the Cauchy-Schwartz inequality, we obtain

$$2\langle g(\cdot)T_j(w(t_{j+1}, \cdot)), e_j(t_{j+1}, \cdot)\rangle \\ \le 2\|g\|_{\infty}\|T_j(w(t_{j+1}, \cdot))\|_{L^2}\|e_j(t_{j+1}, \cdot)\|_{L^2}.$$

Using the definition of $e_j(t, x) = (e_{j,1}(t, x), e_{j,2}(t, x))$ given in (32)-(33) and inequalities in (26), we obtain

$$2\langle g(\cdot)T_j(w(t_{j+1}, \cdot)), e_j(t_{j+1}, \cdot)\rangle \le 2M_2\|g\|_{\infty}\|w(t_{j+1}, \cdot)\|_{L^2} \\ \times \|(\delta_{\sigma_j^+}(t_{j+1}, \cdot)v(t_{j+1}, \cdot), \delta_{\sigma_j^-}(t_{j+1}, \cdot)u(t_{j+1}, \cdot))\|_{L^2} \\ \le 2M_2\|g\|_{\infty}\|w(t_{j+1}, \cdot)\|_{L^2}^2 \\ \times \left( \max \left\{ \max_{x \in [0,1]} (\delta_{\sigma_j^+}^2(t_{j+1}, x)), \max_{x \in [0,1]} (\delta_{\sigma_j^-}^2(t_{j+1}, x)) \right\} \right)^{\frac{1}{2}}.$$

Finally, using the definition of $\delta_{\sigma_j^+}$ and $\delta_{\sigma_j^-}$ given in (8)-(9) and Assumption 2, we get

$$\nu R_L V(t_{j+1}) \le M_3 l_{\sigma} |t_{j+1} - t_j| \|w(t_{j+1}, \cdot)\|_{L^2}^2, \quad (46)$$

where $M_3 := 2\|g\|_{\infty} M_2$. Using (39), we obtain

$$0 \le (M_3 l_{\sigma} |t_{j+1} - t_j| - \nu R_L k_{\ell}) \|w(t_{j+1}, \cdot)\|_{L^2}^2,$$

which yields

$$|t_{j+1} - t_j| \ge \frac{\nu R_L k_{\ell}}{M_3 l_{\sigma}} =: \tau_L, \quad (47)$$

with $\tau_L > 0$ being the minimal dwell time, which is independent of the initial condition.

**Remark 3.** The minimal dwell time depends on the rate of change of the different coefficients. Larger values of $l_{\sigma}$ induce a smaller minimal dwell time, corresponding to a more frequent update of the kernel gains of the control law.

Lemma 1 implies that $\lim_{j \to \infty} (t_j) = +\infty$ and consequently the proposed control law rewrites as in equation (34), which implies the existence of solutions in $C^0([0, \infty); L^2(0, 1)^2)$ of the closed-loop system under the event-triggered gain scheduler (44)-(45).

### 3.3 Stability analysis

We now show the closed-loop stability under the proposed event-triggered gain scheduling algorithm.

**Theorem 1.** Let $\zeta$ and $\nu$ be selected such that (38) holds. Under Assumption 2, if the following condition is fulfilled,

$$l_{\sigma} < \frac{k_{\ell} \nu^2 (1 - R_L) R_L}{M_3 \ln \left( \frac{k_u}{k_{\ell}} \right)}, \quad (48)$$

where $k_{\ell}$, $k_u$ are defined in (40), $g$ is defined in (37), $M_3 = 2\|g\|_{\infty} M_2$ with $M_2$ given in (25), and $R_L \in (0, 1)$ is the parameter involved in (44)-(45); then the closed-loop system (1)-(3) with the control law (28) and the event-triggered gain scheduler (44)-(45) is globally exponentially stable; that is there exist $G_L > 0$ and $\nu_L > 0$ such that, for all $t > 0$

$$\| (u(t, \cdot), v(t, \cdot)) \|_{L^2}^2 \le G_L e^{-\nu_L t} \|(u_0, v_0)\|_{L^2}^2. \quad (49)$$

*Proof.* Taking the time derivative of (35) along the solutions (29)-(33), and integrating by parts, we obtain, for all $t \in (t_j, t_{j+1})$,

$$\begin{aligned} \dot{V}(t) = & -\nu V(t) + 2 \int_0^1 \frac{e^{-\nu \phi_1(x)}}{\lambda(x)} \alpha_j(t, x) e_{j,1}(t, x) dx \\ & + 2\zeta \int_0^1 \frac{e^{\nu \phi_2(x)}}{\mu(x)} \beta_j(t, x) e_{j,2}(t, x) dx + \beta_j^2(t, 0)(q^2 - \zeta) \\ & + (\zeta e^{\nu \phi_2(1)} \rho^2 - e^{-\nu \phi_1(1)}) \alpha_j^2(t, 1), \end{aligned}$$

which can be rewritten as follows:

$$\begin{aligned} \dot{V}(t) = & -\nu(1 - R_L)V(t) + 2\langle g(\cdot)T_j(w(t, \cdot)), e_j(t, \cdot) \rangle \\ & -\nu R_L V(t) + \beta_j^2(t, 0)(q^2 - \zeta) \\ & + (\zeta e^{\nu \phi_2(1)} \rho^2 - e^{-\nu \phi_1(1)}) \alpha_j^2(t, 1), \end{aligned} \quad (50)$$

with $R_L \in (0, 1)$. Under (38), we get the following estimate:

$$\begin{aligned} \dot{V}(t) \le & -\nu(1 - R_L)V(t) + 2\langle g(\cdot)T_j(w(t, \cdot)), e_j(t, \cdot) \rangle \\ & -\nu R_L V(t). \end{aligned} \quad (51)$$
Moreover, under (44)-(45), events are triggered to guarantee $2\langle g(\cdot)T_j(w(t, \cdot)), e_j(t, \cdot)\rangle - \nu R_L V(t) \le 0$ for all $t \in (t_j, t_{j+1})$. Therefore, we obtain for all $t \in [t_j, t_{j+1})$ $\dot{V}(t) \le -\nu_1 V(t)$ where $\nu_1 = \nu(1 - R_L) > 0$. Consequently, for all $t \in [t_j, t_{j+1})$, we have $V(t) \le e^{-\nu_1(t-t_j)}V(t_j)$, which implies, due to (39) and the norm equivalence (26), that for all $t \in [t_j, t_{j+1})$,

$$ \|(u(t, \cdot), v(t, \cdot))\|_{L^2}^2 \le \frac{k_u}{k_\ell} e^{-\nu_1(t-t_j)} \|(u(t_j, \cdot), v(t_j, \cdot))\|_{L^2}^2. \quad (52) $$

Since $(u, v) \in C^0([0, \infty); L^2(0, 1)^2)$, it follows that (52) holds for $t = t_{j+1}$ and we get

$$ \|(u(t_{j+1}, \cdot), v(t_{j+1}, \cdot))\|_{L^2}^2 \le \frac{k_u}{k_\ell} e^{-\nu_1(t_{j+1}-t_j)} \times \|(u(t_j, \cdot), v(t_j, \cdot))\|_{L^2}^2. $$

Therefore, for all $t \ge 0$, we can recursively obtain an estimate of $\|(u(t, \cdot), v(t, \cdot))\|_{L^2}^2$ in terms of $\|(u_0, v_0)\|_{L^2}^2$, since we know that there have been $j$ events and that $j\tau_L$ units of time have (at least) been passed until $t$ (where $\tau_L$ is defined by equation (47) in the proof of Lemma 1). Applying induction on $j$, we have, for all $j \in \mathbb{N}$

$$ \|(u(t_j, \cdot), v(t_j, \cdot))\|_{L^2}^2 \le \left(\frac{k_u}{k_\ell}\right)^j e^{-\nu_1 t_j} \|(u_0, v_0)\|_{L^2}^2, $$

and $t_j \ge j\tau_L$. For $j \ge 0$ (arbitrary) and $t \in [t_j, t_{j+1})$ (arbitrary), and since $j \le \frac{t}{\tau_L}$, we get

$$ \begin{aligned} \|(u(t, \cdot), v(t, \cdot))\|_{L^2}^2 &\le \left(\frac{k_u}{k_\ell}\right)^{j+1} e^{-\nu_1 t} \|(u_0, v_0)\|_{L^2}^2 \\ &\le \frac{k_u}{k_\ell} e^{-\left(\nu_1 - \ln\left(\frac{k_u}{k_\ell}\right)/\tau_L\right)t} \|(u_0, v_0)\|_{L^2}^2. \end{aligned} $$

Consequently, we obtain

$$ \|(u(t, \cdot), v(t, \cdot))\|_{L^2}^2 \le \frac{k_u}{k_\ell} e^{-\nu_L t} \|(u_0, v_0)\|_{L^2}^2, \quad (53) $$

with $\nu_L = \nu_1 - \frac{\ln(\frac{k_u}{k_\ell})}{\tau_L}$. Due to Lemma 1 with $\tau_L$ given in (47), and under condition (48), we guarantee $\nu_L > 0$. Therefore, we finally obtain the exponential stability of the closed-loop system (1)-(3), (28), (44)-(45). ■

**Remark 4.** We emphasize that Condition (48) of Theorem 1 is a sufficient condition and that closed-loop stability could be achieved even if this condition is not verified. The possible conservatism of this condition mostly comes from the coefficients $k_\ell$ and $k_u$ that depend on the uniform bound of the backstepping kernels $M_1$ defined in (24). Finally, the parameter $R_L$ can be tuned to alter the sampling speed, i.e., how often the kernels should be updated.

# 4 Small-Gain Based Event-triggered gain scheduling

In this section, we introduce our second event-triggering mechanism for gain scheduling. The triggering condition will now rely on a small-gain analysis based on the evolution of the supremum norm of the errors when sampling $e_j(t, x) = (e_{j,1}(t, x), e_{j,2}(t, x))$ given in (32)-(33). More precisely, the triggering condition requires computing the $L^\infty$-norm of the state, which implies having more regularity than in the $L^2$-setting we have considered so far. Therefore, we assume that the initial condition $(u_0, v_0)$ now belongs to $(L^\infty(0, 1))^2$.

## 4.1 Small-Gain Based Event-triggered gain scheduling

Let $R_s > 0$ be a positive design parameter. We define the following set

$$ E_s(t_j) := \{t > t_j : \begin{aligned}[t] & \|e_{j,1}(t, \cdot)\|_{\infty} + \|e_{j,2}(t, \cdot)\|_{\infty} \\ & > R_s (\|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty}) \end{aligned} \}. \quad (54) $$

The second event-triggered gain scheduler proposed in this paper is defined next. The times of events $t_j \ge t_0$ with $t_0 = 0$ form a finite or countable set of times, which is determined by the following rules for some $j \ge 0$:

(1) if $E_s(t_j) = \emptyset$, then the set of the times of the events is $\{t_0, \dots, t_j\}$.

(2) if $E_s(t_j) \ne \emptyset$, then the next event time is given by

$$ t_{j+1} := \inf E_s(t_j). \quad (55) $$

## 4.2 Avoidance of the Zeno phenomenon

**Lemma 2.** Under the event-triggered condition (54)-(55), there exists a minimal dwell time between two triggering times, i.e., there exists a constant $\tau_s > 0$ (independent of the initial condition of the state) such that $t_{j+1} - t_j \ge \tau_s$ for all $j \in \mathbb{N}$.

*Proof.* We proceed similarly as in the proof of Lemma 1. Assume that an event occurs at $t = t_{j+1}$. Then, from (55), we have

$$ \begin{aligned} & R_s (\|u(t_{j+1}, \cdot)\|_{\infty} + \|v(t_{j+1}, \cdot)\|_{\infty}) \\ & \le \|e_{j,1}(t_{j+1}, \cdot)\|_{\infty} + \|e_{j,2}(t_{j+1}, \cdot)\|_{\infty}. \end{aligned} \quad (56) $$

Using (32)-(33) along with (24) and Assumption 2, we obtain

$$ \begin{aligned} & R_s (\|u(t_{j+1}, \cdot)\|_{\infty} + \|v(t_{j+1}, \cdot)\|_{\infty}) \\ & \le 2M_{\infty}l_{\sigma}|t_{j+1} - t_j| (\|u(t_{j+1}, \cdot)\|_{\infty} + \|v(t_{j+1}, \cdot)\|_{\infty}), \end{aligned} \quad (57) $$

which yields, for all $j \ge 0$

$$ |t_{j+1} - t_j| \ge \frac{R_s}{2M_{\infty}l_{\sigma}} =: \tau_s, \quad (58) $$

with $\tau_s > 0$ being the minimal dwell time (independent of the initial condition). ■
## 4.3 Stability analysis

First, we will establish ISS estimates in the sup-norm (whose proof is given in Appendix A). The ISS estimates can be obtained by manipulating the explicit solutions of (29)-(33) via the method of characteristics, using Lyapunov analysis in $L_p$-norm, employing the superposition principle and taking $p \to +\infty$. The reasoning follows mainly [26].

**Lemma 3.** Consider the target system (29)-(33), for all $t \in [t_j, t_{j+1})$, with initial data $\alpha(t_j, x)$, $\beta(t_j, x)$ for all $x \in [0, 1]$. The following estimates hold, for some $\nu > 0$ and, for all $t \in [t_j, t_{j+1})$, $j \ge 0$:

$$ 
\begin{aligned}
\|\alpha_j(t, \cdot)\|_{\infty} &\le e^{-\nu(t-t_j)+\nu\phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} + \frac{e^{\left(1+\frac{\nu}{\lambda_{\min}}\right)}}{\lambda_{\min}} \\
&\quad \times \max_{\max\{t_j, t-\frac{1}{\lambda_{\min}}\} \le s \le t} \left( \|e_{j,1}(s, \cdot)\|_{\infty} e^{-\nu(t-s)} \right) + e^{\nu\phi_1(1)} |q| \\
&\quad \times \max_{\max\{t_j, t-\frac{1}{\lambda_{\min}}\} \le s \le t} \left( |\beta_j(s, 0)| e^{-\nu(t-s)} \right), \quad (59) \\
\|\beta_j(t, \cdot)\|_{\infty} &\le e^{-\nu(t-t_j)+\nu\phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} + e^{\nu\phi_2(1)} |\rho| \\
&\quad \times \max_{\max\{t_j, t-\frac{1}{\mu_{\min}}\} \le s \le t} \left( |\alpha_j(s, 1)| e^{-\nu(t-s)} \right) + \frac{e^{\left(1+\frac{\nu}{\mu_{\min}}\right)}}{\mu_{\min}} \\
&\quad \times \max_{\max\{t_j, t-\frac{1}{\mu_{\min}}\} \le s \le t} \left( \|e_{j,2}(s, \cdot)\|_{\infty} e^{-\nu(t-s)} \right), \quad (60)
\end{aligned}
$$

with $\phi_1(x)$, $\phi_2(x)$ defined in (36), and $\lambda_{\min}$, $\mu_{\min}$ defined in (42).

The second main result follows from small-gain arguments and proves exponential stability in the sup-norm. To that end, let us first state the following small-gain condition under which we can choose the parameter $R_s$ involved in the triggering condition (54)-(55):

$$ R_s M_{\infty} \left( \frac{e^1}{\lambda_{\min}} + \frac{e^1}{\mu_{\min}} \right) \left( |\rho| + |q| + 2 \right) + |\rho q| < 1, \quad (61) $$

where $M_{\infty}$ is as in (25). The choice of $R_s$ (small enough) is indeed possible thanks to Assumption 1 (which guarantees $|\rho q| < 1$). By continuity arguments, there exists $\nu > 0$ sufficiently small such that

$$ R_s M_{\infty} \Lambda(\nu) \left( e^{\nu\phi_1(1)} |\rho| + e^{\nu\phi_2(1)} |q| + 2 \right) + e^{\nu(\phi_1(1)+\phi_2(1))} |\rho q| < 1, \quad (62) $$

where $\phi_1(x)$, $\phi_2(x)$ are defined in (36), and

$$ \Lambda(\nu) := \left( \frac{1}{\lambda_{\min}} + \frac{1}{\mu_{\min}} \right) e^{1+\nu} \left( \frac{1}{\lambda_{\min}} + \frac{1}{\mu_{\min}} \right), \quad (63) $$

with $\lambda_{\min}$, $\mu_{\min}$ being defined in (42). Indeed, the existence of $\nu$ is guaranteed since the function $h(\nu) := R_s M_{\infty} \Lambda(\nu) (e^{\nu\phi_1(1)}|\rho| + e^{\nu\phi_2(1)}|q| + 2)$ +

$e^{\nu(\phi_1(1)+\phi_2(1))}|\rho q|$ is continuous at 0, and satisfies $h(0) < 1$. Moreover, it also holds $0 < R_s M_{\infty} \Lambda(\nu) < 1$. Let us define

$$ \Phi := (1 - \Lambda(\nu) R_s M_{\infty})^{-2} \left( e^{\nu\phi_1(1)} |q| + \Lambda(\nu) R_s M_{\infty} \right) \times \left( e^{\nu\phi_2(1)} |\rho| + \Lambda(\nu) R_s M_{\infty} \right). \quad (64) $$

Notice that (62) implies

$$ 
\begin{aligned}
& e^{\nu(\phi_1(1)+\phi_2(1))} |\rho q| + (R_s M_{\infty} \Lambda(\nu))^2 \\
& \quad + R_s M_{\infty} \Lambda(\nu) \left( e^{\nu\phi_1(1)} |\rho| + e^{\nu\phi_2(1)} |q| \right) \\
& < (1 - (R_s M_{\infty} \Lambda(\nu)))^2,
\end{aligned}
\quad (65) $$

Therefore, we obtain $(e^{\nu\phi_1(1)}|q| + \Lambda(\nu)R_sM_{\infty}) (e^{\nu\phi_2(1)}|\rho| + \Lambda(\nu)R_sM_{\infty}) < (1 - (R_sM_{\infty}\Lambda(\nu)))^2$. Hence, we can ensure that if (62) is verified, then $\Phi < 1$. The conditions $R_sM_{\infty}\Lambda(\nu) < 1$ and $\Phi < 1$ are instrumental in the proof of the main result. Finally, using definitions (63) and (64), we define the following quantity:

$$ G_s := 2e^{\nu \max\{\phi_1(1), \phi_2(1)\}} \max \left\{ \frac{1}{|q|}, \frac{1}{|\rho|} \right\} \times (1 - \Lambda(\nu) R_s M_{\infty})^{-1} (1 - \Phi)^{-1} M_{\infty}^2. \quad (66) $$

**Theorem 2.** Let $R_s$ be the parameter involved in (54)-(55) that is selected such that (61) holds. Let $\nu > 0$ be selected such that (62) holds. If, in addition, under Assumption 2 the following condition is fulfilled

$$ l_{\sigma} < \frac{\nu R_s}{M_{\infty} \ln(G_s)}, \quad (67) $$

where $G_s$ is given by (66); then the closed-loop system (1)-(3) with the control law (28) and the event-triggered gain scheduler (54)-(55) is globally exponentially stable; that is there exists $\nu_s > 0$ such that, for all $t > 0$

$$ \|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty} \le G_s e^{-\nu s t} (\|u_0\|_{\infty} + \|v_0\|_{\infty}). \quad (68) $$

*Proof.* From Lemma 3, the ISS estimates (59) and (60) imply, for all $t \in [t_j, t_{j+1})$, $j \ge 0$

$$ 
\begin{aligned}
\|\alpha_j(t, \cdot)\|_{\infty} &\le e^{-\nu(t-t_j)} e^{\nu\phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} \\
&\quad + e^{\nu\phi_1(1)} |q| \max_{t_j \le s \le t} \left( \|\beta_j(s, \cdot)\|_{\infty} e^{-\nu(t-s)} \right) \\
&\quad + \Lambda(\nu) \left[ \max_{t_j \le s \le t} \left( \|e_{j,1}(s, \cdot)\|_{\infty} e^{-\nu(t-s)} \right) \right. \\
&\quad \left. + \max_{t_j \le s \le t} \left( \|e_{j,2}(s, \cdot)\|_{\infty} e^{-\nu(t-s)} \right) \right],
\end{aligned}
\quad (69) $$
and

$$
\begin{align}
\|\beta_j(t, \cdot)\|_{\infty} \le & e^{-\nu(t-t_j)} e^{\nu\phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} \nonumber \\
& + e^{\nu\phi_2(1)} |\rho| \max_{t_j \le s \le t} (\|\alpha_j(s, \cdot)\|_{\infty} e^{-\nu(t-s)}) \nonumber \\
& + \Lambda(\nu) \left[ \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{\infty} e^{-\nu(t-s)}) \right. \nonumber \\
& \left. \qquad + \max_{t_j \le s \le t} (\|e_{j,2}(s, \cdot)\|_{\infty} e^{-\nu(t-s)}) \right], \tag{70}
\end{align}
$$

where $\Lambda(\nu)$ is defined in (63). We define the following quantities for all $t \in [t_j, t_{j+1})$:

$$
\|\alpha_j\|_{[t_j, t]} := \max_{t_j \le s \le t} (\|\alpha_j(s, \cdot)\|_{\infty} e^{\nu s}), \quad (71)
$$

$$
\|\beta_j\|_{[t_j, t]} := \max_{t_j \le s \le t} (\|\beta_j(s, \cdot)\|_{\infty} e^{\nu s}), \quad (72)
$$

$$
\|e_{j,1}\|_{[t_j, t]} := \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{\infty} e^{\nu s}), \quad (73)
$$

$$
\|e_{j,2}\|_{[t_j, t]} := \max_{t_j \le s \le t} (\|e_{j,2}(s, \cdot)\|_{\infty} e^{\nu s}). \quad (74)
$$

Using (69)-(70) and definitions (71)-(74), we get for all $t \in [t_j, t_{j+1})$

$$
\begin{align}
\|\alpha_j\|_{[t_j, t]} \le & e^{\nu t_j} e^{\nu \phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} + e^{\nu \phi_1(1)} |q| \|\beta_j\|_{[t_j, t]} \nonumber \\
& + \Lambda(\nu) (\|e_{j,1}\|_{[t_j, t]} + \|e_{j,2}\|_{[t_j, t]}), \tag{75}
\end{align}
$$

and

$$
\begin{align}
\|\beta_j\|_{[t_j, t]} \le & e^{\nu t_j} e^{\nu \phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} + e^{\nu \phi_2(1)} |\rho| \|\alpha_j\|_{[t_j, t]} \nonumber \\
& + \Lambda(\nu) (\|e_{j,1}\|_{[t_j, t]} + \|e_{j,2}\|_{[t_j, t]}), \tag{76}
\end{align}
$$

From the definition of the event-triggering condition (54)-(55), in conjunction with (27), events are triggered to guarantee, for all $t \in [t_j, t_{j+1})$,

$$
\|e_{j,1}(t, \cdot)\|_{\infty} + \|e_{j,2}(t, \cdot)\|_{\infty} \le \bar{R}_s (\|\alpha_j(t, \cdot)\|_{\infty} + \|\beta_j(t, \cdot)\|_{\infty}),
$$

with $\bar{R}_s = R_s M_\infty$. The previous inequality implies, for all $t \in [t_j, t_{j+1})$

$$
\|e_{j,1}\|_{[t_j, t]} + \|e_{j,2}\|_{[t_j, t]} \le \bar{R}_s \left( \|\alpha_j\|_{[t_j, t]} + \|\beta_j\|_{[t_j, t]} \right). \quad (77)
$$

Therefore using (77) along with (75)-(76), we get

$$
\begin{align}
\|\alpha_j\|_{[t_j, t]} \le & e^{\nu t_j} e^{\nu \phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} + e^{\nu \phi_1(1)} |q| \|\beta_j\|_{[t_j, t]} \nonumber \\
& + \Lambda(\nu) \bar{R}_s (\|\alpha_j\|_{[t_j, t]} + \|\beta_j\|_{[t_j, t]}), \tag{78}
\end{align}
$$

and

$$
\begin{equation}
\begin{split}
\|\beta_j\|_{[t_j, t]} \le & e^{\nu t_j} e^{\nu \phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} + e^{\nu \phi_2(1)} |\rho| \|\alpha_j\|_{[t_j, t]} \\
& + \Lambda(\nu) \bar{R}_s (\|\alpha_j\|_{[t_j, t]} + \|\beta_j\|_{[t_j, t]}).
\end{split}
\tag{79}
\end{equation}
$$

From (78)-(79), and since $\Lambda(\nu)\bar{R}_s < 1$ (by virtue of (61), and continuity arguments), we get, for all $t \in [t_j, t_{j+1})$

$$
\begin{align*}
\|\alpha_j\|_{[t_j, t]} \le {}& (1 - \Lambda(\nu) \bar{R}_s)^{-1} e^{\nu t_j} e^{\nu \phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} \\
& + (1 - \Lambda(\nu) \bar{R}_s)^{-1} (e^{\nu \phi_1(1)} |q| + \Lambda(\nu) \bar{R}_s) \|\beta_j\|_{[t_j, t]},
\end{align*}
$$

and

$$
\begin{align*}
\|\beta_j\|_{[t_j, t]} \le {}& (1 - \Lambda(\nu) \bar{R}_s)^{-1} e^{\nu t_j} e^{\nu \phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} \\
& + (1 - \Lambda(\nu) \bar{R}_s)^{-1} (e^{\nu \phi_2(1)} |\rho| + \Lambda(\nu) \bar{R}_s) \|\alpha_j\|_{[t_j, t]}.
\end{align*}
$$

Therefore, from the previous two estimates, we have

$$
\begin{equation}
\begin{aligned}
\|\alpha_j\|_{[t_j, t]} \le {}& (1 - \Lambda(\nu) \bar{R}_s)^{-1} e^{\nu t_j} e^{\nu \phi_1(1)} \|\alpha_j(t_j, \cdot)\|_{\infty} \\
& + (1 - \Lambda(\nu) \bar{R}_s)^{-2} (e^{\nu \phi_1(1)} |q| + \Lambda(\nu) \bar{R}_s) e^{\nu t_j} e^{\nu \phi_2(1)} \\
& \times \|\beta_j(t, \cdot)\|_{\infty} + \Phi \|\alpha_j\|_{[t_j, t]},
\end{aligned}
\tag{80}
\end{equation}
$$

and

$$
\begin{align}
\|\beta_j\|_{[t_j, t]} \le {}& (1 - \Lambda(\nu) \bar{R}_s)^{-1} e^{\nu t_j} e^{\nu \phi_2(1)} \|\beta_j(t_j, \cdot)\|_{\infty} \nonumber \\
& + (1 - \Lambda(\nu) \bar{R}_s)^{-2} (e^{\nu \phi_2(1)} |\rho| + \Lambda(\nu) \bar{R}_s) e^{\nu t_j} e^{\nu \phi_1(1)} \nonumber \\
& \times \|\alpha_j(t, \cdot)\|_{\infty} + \Phi \|\beta_j\|_{[t_j, t]}, \tag{81}
\end{align}
$$

where $\Phi$ is defined in (64). Combining (80) and (81), and since $\Phi < 1$; we get

$$
\begin{equation}
\begin{split}
\|\alpha_j\|_{[t_j, t]} + \|\beta_j\|_{[t_j, t]} \le & e^{\nu t_j} \bar{G}_0 \\
& \times (\|\alpha_j(t_j, \cdot)\|_{\infty} + \|\beta_j(t_j, \cdot)\|_{\infty}),
\end{split}
\tag{82}
\end{equation}
$$

and in turn,

$$
\|\alpha_j(t, \cdot)\|_{\infty} + \|\beta_j(t, \cdot)\|_{\infty} \le \bar{G}_0 e^{-\nu(t-t_j)} \\
\times (\|\alpha_j(t_j, \cdot)\|_{\infty} + \|\beta_j(t_j, \cdot)\|_{\infty}),
$$

where

$$
\begin{equation}
\begin{split}
\bar{G}_0 :={}& e^{\nu \max\{\phi_1(1), \phi_2(1)\}} \left(1 + \Phi \left(1 - \Lambda(\nu) \bar{R}_s\right) \right. \\
& \qquad \left. \times \max\left\{\frac{1}{(e^{\nu \phi_1(1)}|q| + \Lambda(\nu)\bar{R}_s)}, \frac{1}{(e^{\nu \phi_2(1)}|\rho| + \Lambda(\nu)\bar{R}_s)}\right\} \right) \\
& \qquad \times (1 - \Lambda(\nu)\bar{R}_s)^{-1} (1 - \Phi)^{-1}.
\end{split}
\tag{83}
\end{equation}
$$
Let us notice that $\tilde{G}_0$ can be upper bounded as follows (since $\Phi < 1$ and $0 < \Lambda(\nu)\tilde{R}_s < 1$):

$$ \tilde{G}_0 \le 2e^{\nu \max\{\phi_1(1), \phi_2(1)\}} \max\left\{\frac{1}{|q|}, \frac{1}{|\rho|}\right\} \quad (84) \\ \times (1 - \Lambda(\nu)\tilde{R}_s)^{-1} (1 - \Phi)^{-1} =: \tilde{G}_1. $$

Using the norm equivalence (27) we obtain

$$ \|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty} \le \tilde{G}_2 e^{-\nu(t-t_j)} \quad (85) \\ \times (\|u(t_j, \cdot)\|_{\infty} + \|v(t_j, \cdot)\|_{\infty}), $$

with $\tilde{G}_2 = M_{\infty}^2 \tilde{G}_1$. Since $(u,v) \in C^0([0, \infty); L^{\infty}(0, 1)^2)$, it follows that (85) holds for $t = t_{j+1}$ and we get

$$ \|u(t_{j+1}, \cdot)\|_{\infty} + \|v(t_{j+1}, \cdot)\|_{\infty} \le \tilde{G}_2 e^{-\nu(t_{j+1}-t_j)} \times (\|u(t_j, \cdot)\|_{\infty} + \|v(t_j, \cdot)\|_{\infty}). $$

As in the proof of Theorem 1, for all $t \ge 0$, we can recursively obtain an estimate of $\|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty}$ in terms of $\|u_0\|_{\infty}$ and $\|v_0\|_{\infty}$, by the fact that there have been $j$ events and that $j\tau_s$ units of time have (at least) been passed until $t$ (where $\tau_s$ is defined by (58) in the proof of Lemma 2). Thus, applying induction on $j$, we have, for all $j \in \mathbb{N}$,

$$ \|u(t_j, \cdot)\|_{\infty} + \|v(t_j, \cdot)\|_{\infty} \le (\tilde{G}_2)^j e^{-\nu t_j} (\|u_0\|_{\infty} + \|v_0\|_{\infty}). $$

and $t_j \ge j\tau_s$. For $j \ge 0$ (arbitrary) and $t \in [t_j, t_{j+1})$ (arbitrary), and since $j \le \frac{t}{\tau_s}$, we get

$$ \|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty} \le (\tilde{G}_2)^{j+1} e^{-\nu t} (\|u_0\|_{\infty} + \|v_0\|_{\infty}) \\ \le \tilde{G}_2 e^{-(\nu - \frac{\ln(\tilde{G}_2)}{\tau_s})t} (\|u_0\|_{\infty} + \|v_0\|_{\infty}). $$

Consequently, we obtain

$$ \|u(t, \cdot)\|_{\infty} + \|v(t, \cdot)\|_{\infty} \le \tilde{G}_2 e^{-\bar{\nu}t} (\|u_0\|_{\infty} + \|v_0\|_{\infty}), $$

with $\bar{\nu} = \nu - \frac{\ln(\tilde{G}_2)}{\tau_s}$. Due to Lemma 2 with $\tau_s$ given in (58), and under condition (67), we guarantee $\bar{\nu} > 0$. Therefore, we finally obtain (68) with $G_s := \tilde{G}_2$ and $\nu_s := \bar{\nu}$, which proves that the closed-loop system (1)-(3),(28),(54)-(55) is global exponentially stable. This concludes the proof. ■

## 5 Numerical simulations

In this section, we illustrate our theoretical results with numerical simulations. We consider the system (1)-(3) with the following coefficients:

$$ \lambda(x) = 1, \mu(x) = 2, q = 1.2, \rho = 0.3, $$

$$ \sigma^+(t, x) = 2 + 0.3xe^{2\sin^2(t)}, $$

$$ \sigma^-(t, x) = 1 + \frac{5}{\cosh^2((t-8)^2)} + 0.6 \cos(\pi t) + \frac{5}{\cosh^2(5x)}. $$

The time- and space-varying coupling coefficients are chosen so that the open-loop system is unstable and the

nominal backstepping controller with constant gains (computed at $t = 0$) is not robust to the time variations of the coefficients [8]. The time- and space-varying coefficient

<figure><img src="image_2.png" /><figcaption>Figure 1. Profile of the time- and space-varying coefficient $\sigma^-(t, x)$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. Sampled version of the profile of the time- and space-varying coefficient $\sigma^-(t_j, x)$ for all $\{t_j\}_{j \in \mathbb{N}}$ according to the Lyapunov-based event-triggered gain scheduler (44)-(45).</figcaption></figure>

$\sigma^-(t, x)$ is pictured in Figure 1. We simulate the evolution of the system using a classical finite volume method based on a Godunov scheme [30] with a space step of 0.025 and a Courant-Friedrichs-Lewy condition equal to 1. We run our simulations on a frame of 15s.

We close the loop using the boundary control law (28) whose kernel gains are updated according to I) the Lyapunov-based event-triggered gain scheduler (44)-(45) and II) the small-gain-based event-triggered gain scheduler (54)-(55). In both scenarios, the backstepping kernels are computed at each event using an iterative algorithm based on the method of successive approximations [6]. Solving the kernel equations at each event requires 0.1s in average, which can be done in real-time while simulating the system. Therefore, the proposed approaches do not require solving
time- and space-varying kernel equations, resulting in a simplified procedure.

In our simulations, the parameter $R_L$ of the Lyapunov-based event-triggered condition (44)-(45) is set as $\bar{R}_L = 0.5$. The parameters are chosen as $\zeta = 1.5$, $\nu = 1.2$. One can verify that condition (38) is satisfied. On the other hand, the parameter of the small-gain based event-triggering condition (54)-(55) is set as $R_s = 0.1$. The parameters $R_L$ and $R_s$ have been chosen such that the number of events generated by the two event-triggered gain schedulers are equal for this test case. We have also pictured in Figure 2 the sampled version of the profile of the time- and space-varying coefficient $\sigma^-(t_j, x)$ for all $\{t_j\}_{j \in \mathbb{N}}$ according to the event-triggered gain scheduler (44)-(45). Figure 3 illustrates the time evolution of the $L^2$-norm of the system for four different configurations: 1) open-loop system (U is set to zero), 2) closed-loop system with a control law using constant gains (i.e., the gains are computed based on the parameter values at $t=0$ and remain unchanged), 3) closed-loop system with a non-event-triggered strategy inspired by [16] (with time-varying kernels), 4) closed-loop system with (28) and the event-triggered gain scheduler (44)-(45). As observed in the figure, the nominal controller with constant gains lacks robustness against time variations in the system coefficients. In contrast, the two other controllers ensure closed-loop stability. Their performance is nearly identical; however, the controller with a Lyapunov-based event-triggered scheduler (44)-(45) requires significantly fewer computational resources. We now proceed to compare the Lyapunov-based and small-gain based event-triggered schedulers. Figure 4 shows the time-evolution of the $L^2$-norm and $L^\infty$-norm of the closed-loop system (1)-(3) using the Lyapunov-based event-triggered scheduler (44)-(45) and the small-gain event-triggered scheduler (54)-(55). The corresponding control efforts are pictured in Figure 5. As can be seen, the two controllers have analogous performance and guarantee closed-loop stability. For the considered test case, the number of events occurring within the given time period is 26 for the $L^2$ gain scheduler and 25 for the small-gain strategy. The numerical minimum inter-sampling time between two consecutive events are 0.048s and 0.1s, respectively.

## 5.1 Conservatism of the proposed event-triggered gain scheduling mechanisms

The stability condition (48) associated with the proposed Lyapunov-based event-triggered gain scheduling strategy (44)-(45) relies on a Lyapunov analysis, which, therefore, induces conservatism (e.g., on the choice of $\nu$ and $\zeta$). Moreover, the constant $k_\ell$ and $k_u$ are expressed as functions of the bound $M_1$, emphasizing the necessity of accurately estimating the upper bound of the backstepping kernels. In this context, the bound given in condition (23) is highly conservative. For instance, we have $\bar{D}_j > 500$ for the proposed test case, which gives a value of $\bar{M}_1$ considerably larger than the real one (around 50). Still, even for $\bar{M}_1 = 50$, the condition (48) can be restrictive, as in the considered example, choosing $R_L = 0.5$ leads to $\ell_\sigma < 10^{-9}$ which is unrealistic. This is primarily due to the fact that the quantity $\ell_\sigma$ is computed using the worst-case scenario corresponding

to the minimum dwell time computed in Lemma 1. Moreover, we could also reduce the conservatism of our approach by replacing the constant $M_1$ (that corresponds to the essential norm of the kernels) with the $L^2$-norm of the kernels (which would reduce the effect of large values of the kernels) and adequately adjusting our analysis. Similar considerations can be made for the small-gain-based event-triggered scheduling strategy (54)-(55). Indeed, the small-gain condition (61), although always feasible, may induce extremely small values of $R_s$ therefore reducing the applicability of condition (67). In this paper, our main goal was to demonstrate how to design event-triggered gain-scheduling mechanisms without the need for solving time-varying kernel equations. We emphasize that we purposely selected a critical test case example with large values of the coupling terms. This was done to highlight the potential limitations of our methodology and to demonstrate the conservative nature of the proposed estimates. Indeed, our gain scheduling algorithms can still guarantee closed-loop stability even if conditions (48) or (61) are not verified (as in [25]).

## 5.2 Comparison of the approaches and possible improvements

As can be seen through the simulations, the two event-triggered gain-scheduling strategies can lead to analogous performance while inducing the same number of events. However, they feature some specificities:

* The Lyapunov-based event-triggered gain scheduling approach directly relates to the $L^2$-norm of the system, and therefore, we can follow the strategy developed in [25] to introduce a dynamic variable that can be viewed as a filtered value of the static triggering condition (44)-(45). The expected output is a reduction of the updating times for the kernel scheduling and obtaining larger inter-execution times. Moreover, we could build on the ideas developed in the context of even-triggered control for PDEs [36] to propose a Performance-barrier based even-triggered gain scheduling under which the Lyapunov function of the closed-loop system is allowed to deviate from strict monotonic decrease, provided it remains below an acceptable performance barrier. Consequently, this would result in sparser kernel gains updates and larger inter-execution times.

* The small-gain-based event-triggered gain scheduling approach (54)-(55) requires more regularity for the system's initial condition, as we must be able to compute the $L^\infty$-norm of the system. Moreover, the conditions given in the statement of Theorem 2 might be more restrictive than the ones proposed in Theorem 1. However, the small-gain-based event-triggered gain scheduling approach may bring some benefits as soon as one wants to introduce event-triggered boundary control, as well; hence, keeping the control input constant between two events (see e.g., [23]). Indeed, combining event-triggered gain scheduling and control will also introduce boundary errors (deviation of control actuation), which may turn out to be very challenging to handled under ($L^2$) Lyapunov-based approach – specially when proving the avoidance of the Zeno phenomenon.
<figure><img src="image_4.png" /><figcaption>Figure 3. Time evolution of the L² norm of the system 1) in open-loop, 2) closed-loop with a nominal controller with constant backstepping gains, 3) closed-loop with a non-event-triggered strategy inspired by [16], 4) closed-loop with (28) and the event-triggered gain scheduler (44)-(45).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4. Time evolution of the L² (resp. L∞) norm of the closed-loop system with the Lyapunov-based event-triggered gain scheduler (44)-(45) (resp. with the small-gain event-triggered gain scheduler (54)-(55)).</figcaption></figure>

## 6 Concluding remarks and perspectives

In this paper, we solved the problem of achieving exponential stabilization of 2 × 2 hyperbolic PDE systems with time- and space-varying in-domain coupling coefficients. The control design is based on the backstepping method, and the gains are computed and updated based on two event-triggered gain scheduling schemes. Unlike existing results in the literature, our approach does not necessitate solving time-varying backstepping kernel equations in real-time, leading to reduced computational burden and enhanced applicability. We demonstrated that under the two proposed event-triggered gain scheduling schemes, Zeno behavior is avoided, enabling well-posedness and exponential stability of the closed-loop system. In future work, we aim to explore time-varying boundary couplings and velocities and propose a dynamic-triggering condition to minimize the number of events for the event-triggered gain scheduling strategies. We will explore the integration of the proposed gain scheduling strategies with an event-triggered control mechanism

<figure><img src="image_6.png" /><figcaption>Figure 5. Time-evolution of the boundary control input under the Lyapunov-based event-triggered gain scheduler (44)-(45) and under the small-gain event-triggered gain scheduler (54)-(55).</figcaption></figure>

to maintain constant control input between two triggering times. Additionally, our event-triggered gain scheduling algorithms can be enhanced with deep neural network approximation techniques (see e.g. [47,29]) to expedite the computation of backstepping kernels. Although, to the best of our knowledge, no existing results on neural operators have been developed for hyperbolic PDEs with time-varying coefficients, we can envision a possible strategy where instead of computing the kernels at each event using the successive approximation method [6]—as we have done—we could leverage kernel approximations derived from Neural Operators. This approach could potentially enhance the computation of the new feedback law at each event. However, in such a case, it would be crucial to carefully evaluate the impact of the approximation error on the triggering policy and overall system stability. The stability of the closed-loop system can then be demonstrated using the Input-to-State Stability properties of the system [9].

# Appendix

## A Proof of Lemma 3

By the method of characteristics, the explicit solution of (29) with boundary condition (31) and initial data $\alpha_j(t_j, x)$, $x \in [0, 1]$, is given as follows, for all $t \in [t_j, t_{j+1})$:

$$ \alpha_j(t, x) = \alpha_j(t_j, \phi_1^{-1}(\phi_1(x) - t + t_j)) + \int_{t_j}^{t} e_{j,1}(\tau, \phi_1^{-1}(\phi_1(x) - t + \tau)) d\tau, \quad (A.1) $$

for $t_j \le t \le t_j + \phi_1(x)$; and

$$ \alpha_j(t, x) = q\beta_j(t - \phi_1(x), 0) + \int_{t-\phi_1(x)}^{t} e_{j,1}(\tau, \phi_1^{-1}(\phi_1(x) - t + \tau)) d\tau, \quad (A.2) $$

for $t_j + \phi_1(x) \le t < t_{j+1}$. Recall that $\phi_1(x)$, which is defined in (36), is a monotonically increasing function of $x$, and thus invertible. From (A.1) and (A.2), the solution can be seen
as the sum of the following terms for all $t \in [t_j, t_{j+1})$:

$$ \alpha_j(t, x) = \bar{\alpha}_{j,1}(t, x) + \bar{\alpha}_{j,2}(t, x) + \bar{\alpha}_{j,3}(t, x) \quad (A.3) $$

where

$$ \bar{\alpha}_{j,1}(t, x) := \alpha_j(t_j, \phi_1^{-1}(\phi_1(x) - t + t_j)), \quad (A.4) $$

$$ \bar{\alpha}_{j,2}(t, x) := 0, \quad (A.5) $$

$$ \bar{\alpha}_{j,3}(t, x) := \int_{t_j}^{t} e_{j,1}(\tau, \phi_1^{-1}(\phi_1(x) - t + \tau)) d\tau, \quad (A.6) $$

for $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j \le t \le t_j + \phi_1(x)$; and

$$ \bar{\alpha}_{j,1}(t, x) := 0, \quad (A.7) $$

$$ \bar{\alpha}_{j,2}(t, x) := q\beta_j(t - \phi_1(x), 0), \quad (A.8) $$

$$ \bar{\alpha}_{j,3}(t, x) := \int_{t-\phi_1(x)}^{t} e_{j,1}(\tau, \phi_1^{-1}(\phi_1(x) - t + \tau)) d\tau, \quad (A.9) $$

for $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j + \phi_1(x) \le t \le t_{j+1}$.

We estimate the $L^p(0, 1)$ norm of $\bar{\alpha}_{j,1}$, with $p \in [1, \infty)$.
From (A.4), the following estimate holds for every $\nu > 0$:

$$ \begin{aligned} |\bar{\alpha}_{j,1}(t, x)|^p &= |\alpha_j(t_j, \phi_1^{-1}(\phi_1(x) - t + t_j))|^p \\ &\le e^{-p\nu(t-t_j)} e^{p\nu\phi_1(1)} |\alpha_j(t_j, \phi_1^{-1}(\phi_1(x) - t + t_j))|^p, \end{aligned} $$

for all $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j \le t \le t_j + \phi_1(x)$.
Therefore, we get

$$ \begin{aligned} \|\bar{\alpha}_{j,1}(t, \cdot)\|_{L^p} &\le e^{-\nu(t-t_j)} e^{\nu\phi_1(1)} \\ &\quad \times \left( \int_{\phi_1^{-1}(t-t_j)}^{1} |\alpha_j(t_j, \phi_1^{-1}(\phi_1(x) - t + t_j))|^p dx \right)^{1/p}. \end{aligned} $$

Using the change of variable $\xi = \phi_1^{-1}(\phi_1(x) - t + t_j)$ together with the fact that $\frac{\partial}{\partial x}(\phi_1^{-1}(\phi_1(x) - t + t_j)) = \frac{\lambda(\phi_1^{-1}(\phi_1(x) - t + t_j))}{\lambda(x)}$ (which follows from the inverse function theorem), and $\phi_1^{-1}(\phi_1(1) - (t - t_j)) \le 1$, we get

$$ \begin{aligned} \|\bar{\alpha}_{j,1}(t, \cdot)\|_{L^p} &\le e^{-\nu(t-t_j)} e^{\nu\phi_1(1)} \\ &\quad \times \left( \int_0^{\phi_1^{-1}(\phi_1(1)-t+t_j)} |\alpha_j(t_j, \xi)|^p \frac{\lambda(x)}{\lambda(\xi)} d\xi \right)^{1/p} \\ &\le e^{-\nu(t-t_j)} e^{\nu\phi_1(1)} \left( \int_0^1 |\alpha_j(t_j, \xi)|^p \frac{\lambda(x)}{\lambda(\xi)} d\xi \right)^{1/p}. \end{aligned} $$

Hence, we further get

$$ \|\bar{\alpha}_{j,1}(t, \cdot)\|_{L^p} \le e^{-\nu(t-t_j)} e^{\nu\phi_1(1)} \times \left(\frac{\lambda_{\max}}{\lambda_{\min}}\right)^{1/p} \|\alpha_j(t_j, \cdot)\|_{L^p}, \quad (A.10) $$

with $\lambda_{\min}$ and $\lambda_{\max}$ defined in (42) and (43), respectively. We estimate now the $L^p(0, 1)$ norm of $\bar{\alpha}_{j,2}(t, x)$. From (A.8),

for every $\nu > 0$, we get

$$ \begin{aligned} |\bar{\alpha}_{j,2}(t, x)|^p &= |q|^p |\beta_j(t - \phi_1(x), 0)|^p \\ &= e^{p\nu\phi_1(x)} e^{-p\nu(t-(t-\phi_1(x)))} |q|^p |\beta_j(t - \phi_1(x), 0)|^p \\ &\le e^{p\nu\phi_1(x)} |q|^p \max_{t-\phi_1(x) \le s \le t} \left( |\beta_j(s, 0)|^p e^{-p\nu(t-s)} \right), \end{aligned} \quad (A.11) $$

for all $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j + \phi_1(x) \le t \le t_{j+1}$.
Notice that

$$ \max\{t_j, t - \frac{x}{\lambda_{\min}}\} \le t - \phi_1(x) \le t. \quad (A.12) $$

Therefore from (A.11), we obtain the following estimate:

$$ \begin{aligned} |\bar{\alpha}_{j,2}(t, x)|^p &\le e^{p\nu\phi_1(1)} |q|^p \\ &\quad \times \max_{\max\{t_j, t - \frac{1}{\lambda_{\min}}\} \le s \le t} \left( |\beta_j(s, 0)|^p e^{-p\nu(t-s)} \right). \end{aligned} \quad (A.13) $$

Hence, we obtain for every $\nu > 0$, $p \in [1, \infty)$ and $t \ge t_j$

$$ \begin{aligned} \|\bar{\alpha}_{j,2}(t, \cdot)\|_{L^p} &\le e^{\nu\phi_1(1)} |q| \\ &\quad \times \max_{\max\{t_j, t - \frac{1}{\lambda_{\min}}\} \le s \le t} \left( |\beta_j(s, 0)| e^{-\nu(t-s)} \right). \end{aligned} \quad (A.14) $$

Finally, using a Lyapunov analysis, we estimate the $L^p(0, 1)$ norm of $\bar{\alpha}_{j,3}(t, x)$. The function $\bar{\alpha}_{j,3}(t, x)$ verifies $\partial_t \bar{\alpha}_{j,3}(t, x) + \lambda(x) \partial_x \bar{\alpha}_{j,3}(t, x) = e_{j,1}(t, x)$ with the boundary condition $\bar{\alpha}_{j,3}(t, 0) = 0$ for all $t \in [t_j, t_{j+1})$ and the initial data $\bar{\alpha}_{j,3}(t_j, x) = 0$ for all $x \in [0, 1]$. Define

$$ \mathcal{V}_p(\bar{\alpha}_{j,3}) = \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^p dx, \quad (A.15) $$

with $\bar{\omega}_1 > 0$ to be selected later on. The following holds for all $t \in (t_j, t_{j+1})$:

$$ \begin{aligned} \dot{\mathcal{V}}_p(t) = &-p \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^{p-1} \operatorname{sgn}(\bar{\alpha}_{j,3}(t, x)) \\ & \times \lambda(x) \frac{\partial \bar{\alpha}_{j,3}(t, x)}{\partial x} dx \\ &+ p \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^{p-1} \operatorname{sgn}(\bar{\alpha}_{j,3}(t, x)) e_{j,1}(t, x) dx. \end{aligned} $$

Integrating by parts, we obtain the following estimate for all $t \in (t_j, t_{j+1})$:

$$ \begin{aligned} \dot{\mathcal{V}}_p(t) \le &-e^{-\bar{\omega}_1} |\bar{\alpha}_{j,3}(t, 1)|^p \lambda(1) \\ &-\bar{\omega}_1 \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^p \lambda(x) dx \\ &+ \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^p \lambda'(x) dx \\ &+ p \int_0^1 e^{-\bar{\omega}_1 x} |\bar{\alpha}_{j,3}(t, x)|^{p-1} |e_{j,1}(t, x)| dx. \end{aligned} \quad (A.16) $$
Using Young's inequality on the last term, i.e.,
$$|\bar{\alpha}_{j,3}(t,x)|^{p-1}|e_{j,1}(t,x)| \le \frac{p-1}{p}\epsilon_1^{p-1}|\bar{\alpha}_{j,3}(t,x)|^p + \frac{1}{p}\epsilon_1^{-p}|e_{j,1}(t,x)|^p \quad \text{for all } \epsilon_1 > 0; \text{ and defining}$$
$$\lambda'_{\max} := \max_{x \in [0,1]} \lambda'(x), \text{ together with (42), we obtain from (A.16), that for all } t \in (t_j, t_{j+1}), \text{ and all } \epsilon_1 > 0, \text{ the following estimate:}$$

$$\dot{V}_p(t) \le - \left( \varpi_1 \lambda_{\min} - \lambda'_{\max} - (p-1) \epsilon_1^{\frac{p}{p-1}} \right) V_p(t) + \epsilon_1^{-p} \|e_{j,1}(t, \cdot)\|_{L^p}^p. \quad (A.17)$$

Using the comparison principle and the fact that $V_p(t_j) = 0$ (since $\bar{\alpha}_{j,3}(t_j, x) = 0$), we get for all $t \in [t_j, t_{j+1})$

$$V_p(t) \le \epsilon_1^{-p} \int_{t_j}^t e^{-(\varpi_1 \lambda_{\min} - \lambda'_{\max} - (p-1)\epsilon_1^{p/p-1}) (t-s)} \times \|e_{j,1}(s, \cdot)\|_{L^p}^p ds. \quad (A.18)$$

Moreover, for every $\nu > 0$, we obtain from (A.18), the following estimate, for all $t \in [t_j, t_{j+1})$:

$$V_p(t) \le \epsilon_1^{-p} \int_{t_j}^t e^{-(\varpi_1 \lambda_{\min} - \lambda'_{\max} - (p-1)\epsilon_1^{p/p-1} - p\nu)(t-s)} ds \times \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{L^p}^p e^{-p\nu(t-s)}). \quad (A.19)$$

From the definition (A.15) and using (A.19), we obtain

$$\|\bar{\alpha}_{j,3}(t, \cdot)\|_{L^p} \le e^{\varpi_1/p} \epsilon_1^{-1} \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \times \left( \int_{t_j}^t e^{-(\varpi_1 \lambda_{\min} - \lambda'_{\max} - (p-1)\epsilon_1^{p/p-1} - p\nu)(t-s)} ds \right)^{1/p}. \quad (A.20)$$

Selecting $\varpi_1 = \frac{p\nu + \lambda'_{\max} + p\lambda_{\min}}{\lambda_{\min}}$ and $\epsilon_1 = (\lambda_{\min})^{(p-1)/p}$, we get from (A.20) the following estimate:

$$\|\bar{\alpha}_{j,3}(t, \cdot)\|_{L^p} \le e^{\left(1 + \frac{\nu}{\lambda_{\min}} + \frac{\lambda'_{\max}}{p\lambda_{\min}}\right) (\lambda_{\min})^{-(p-1)/p}} \times \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \left(\int_{t_j}^t e^{-\lambda_{\min}(t-s)} ds\right)^{1/p} \le \frac{1}{\lambda_{\min}} e^{\left(1 + \frac{\nu}{\lambda_{\min}} + \frac{\lambda'_{\max}}{p\lambda_{\min}}\right)} \max_{t_j \le s \le t} (\|e_{j,1}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) .$$

Since $\bar{\alpha}_{j,3}(t, \cdot)$ depends on $e_{j,1}(s, \cdot)$ (see (A.6) and (A.9)) with $s \in [\max\{t_j, t - \phi_1(x)\}, t]$ and (A.12) holds true, we

further obtain

$$\|\bar{\alpha}_{j,3}(t, \cdot)\|_{L^p} \le \frac{1}{\lambda_{\min}} e^{\left(1 + \frac{\nu}{\lambda_{\min}} + \frac{\lambda'_{\max}}{p\lambda_{\min}}\right)} \times \max_{\max\{t_j, t - \frac{1}{\lambda_{\min}}\} \le s \le t} \left(\|e_{j,1}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}\right). \quad (A.21)$$

Finally, using (A.3), together with (A.10), (A.14) and (A.21), and letting $p \to \infty$ so that $\lim_{p \to \infty} (\|\alpha_j(t, \cdot)\|_{L^p}) = \|\alpha_j(t, \cdot)\|_{\infty}$, with initial data $\|\alpha_j(t_j, \cdot)\|_{L^p} \le \|\alpha_j(t_j, \cdot)\|_{\infty}$ we obtain the ISS estimate (59).

On the other hand, the explicit solution of (30) with boundary condition (31) and initial data $\bar{\beta}_j(t, x)$, $x \in [0, 1]$ for all $t \in [t_j, t_{j+1})$, is given as follows:

$$\bar{\beta}_j(t, x) = \bar{\beta}_j(t_j, \phi_2^{-1}(\phi_2(x) + t_j)) + \int_{t_j}^t e_{j,2}(\tau, \phi_2^{-1}(\phi_2(x) + t - \tau)) d\tau \quad (A.22)$$

for $t_j \le t \le t_j + \phi_2(1) - \phi_2(x)$; and

$$\beta(t, x) = \rho \alpha_j (t - (\phi_2(1) - \phi_2(x)), 1) + \int_{t-(\phi_2(1)-\phi_2(x))}^t e_{j,2}(\tau, \phi_2^{-1}(\phi_2(x) + t - \tau)) d\tau \quad (A.23)$$

for $t_j + \phi_2(1) - \phi_2(x) \le t \le t_{j+1}$; where $\phi_2(x)$ is the monotonically increasing function of $x$ defined in (36). Similarly, from (A.22) and (A.23), it holds that

$$\bar{\beta}_j(t, x) = \bar{\beta}_{j,1}(t, x) + \bar{\beta}_{j,2}(t, x) + \bar{\beta}_{j,3}(t, x) \quad (A.24)$$

where

$$\bar{\beta}_{j,1}(t, x) := \bar{\beta}_j(t_j, \phi_2^{-1}(\phi_2(x) + t - t_j)), \quad (A.25)$$

$$\bar{\beta}_{j,2}(t, x) := 0, \quad (A.26)$$

$$\bar{\beta}_{j,3}(t, x) := \int_{t_j}^t e_{j,2}(\tau, \phi_2^{-1}(\phi_2(x) + t - \tau)) d\tau, \quad (A.27)$$

for $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j \le t \le t_j + \phi_2(1) - \phi_2(x)$;

$$\bar{\beta}_{j,1}(t, x) := 0, \quad (A.28)$$

$$\bar{\beta}_{j,2}(t, x) := \rho \alpha_j (t - (\phi_2(1) - \phi_2(x)), 1), \quad (A.29)$$

$$\bar{\beta}_{j,3}(t, x) := \int_{t-(\phi_2(1)-\phi_2(x))}^t e_{j,2}(\tau, \phi_2^{-1}(\phi_2(x) + t - \tau)) d\tau, \quad (A.30)$$

for $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j + \phi_2(1) - \phi_2(x) \le t \le t_{j+1}$.

We now estimate the $L^p(0, 1)$ norm of $\bar{\beta}_{j,1}$, with $p \in [1, \infty)$. From (A.25), the following estimate holds for every $\nu > 0$:

$$|\bar{\beta}_{j,1}(t, x)|^p = |\bar{\beta}_j(t_j, \phi_2^{-1}(\phi_2(x) + t - t_j))|^p \le e^{-p\nu(t-t_j)} e^{p\nu\phi_2(1)} |\bar{\beta}_j(t_j, \phi_2^{-1}(\phi_2(x) + t - t_j))|^p,$$
for all $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j \le t \le t_j + \phi_2(1) - \phi_2(x)$. Therefore, we get

$$ \|\bar{\beta}_{j,1}(t, \cdot)\|_{L^p} \le e^{-\nu(t-t_j)} e^{\nu\phi_2(1)} \\ \times \left( \int_0^{\phi_2^{-1}(\phi_2(1)-t-t_j)} |\beta_j(t_j, \phi_2^{-1}(\phi_2(x)+t-t_j))|^p dx \right)^{1/p} \quad (A.31) $$

Using the change of variable $\xi = \phi_2^{-1}(\phi_2(x)+t-t_j)$ together with the facts that $\frac{\partial}{\partial x}(\phi_2^{-1}(\phi_2(x)+t-t_j)) = \frac{\mu(\phi_2^{-1}(\phi_2(x)+t-t_j))}{\mu(x)}$ (which follows from the inverse function theorem), and $\phi_2^{-1}(t-t_j) \ge 0$, we get

$$ \|\bar{\beta}_{j,1}(t, \cdot)\|_{L^p} \le e^{-\nu(t-t_j)} e^{\nu\phi_2(1)} \\ \times \left( \int_0^1 |\beta_j(t_j, \xi)|^p \frac{\mu(x)}{\mu(\xi)} d\xi \right)^{1/p} . \quad (A.32) $$

Therefore, we further obtain

$$ \|\bar{\beta}_{j,1}(t, \cdot)\|_{L^p} \le e^{-\nu(t-t_j)} e^{\nu\phi_2(1)} \left(\frac{\mu_{\max}}{\mu_{\min}}\right)^{1/p} \|\beta_j(t_j, \cdot)\|_{L^p}, \quad (A.33) $$

with $\mu_{\min}$ and $\mu_{\max}$ defined in (42) and (43), respectively.

We estimate now the $L^p(0, 1)$ norm of $\bar{\beta}_{j,2}(t, x)$. From (A.29), for every $\nu > 0$, we get

$$ \begin{aligned} |\bar{\beta}_{j,2}(t,x)|^p &= |\rho|^p |\alpha_j(t-(\phi_2(1)-\phi_2(x)),1)|^p \\ &= e^{p\nu\phi_2(x)} e^{-p\nu\phi_2(1)} e^{-p\nu(t-(t+\phi_2(1)-\phi_2(x)))} \\ & \quad \times |\rho|^p |\alpha_j(t-(\phi_2(1)-\phi_2(x)),1)|^p \\ & \le e^{p\nu\phi_2(x)} |\rho|^p \max_{t-(\phi_2(1)-\phi_2(x)) \le s \le t} (|\alpha_j(s,1)|^p e^{-p\nu(t-s)}), \end{aligned} \quad (A.34) $$

for all $t \in [t_j, t_{j+1})$, $x \in [0, 1]$ with $t_j + \phi_2(1) - \phi_2(x) \le t \le t_{j+1}$. Notice that

$$ \max\{t_j, t - \phi_2(1)\} \le t - \phi_2(1) + \phi_2(x) \le t. \quad (A.35) $$

Therefore from (A.34), we obtain the following estimate: $|\bar{\beta}_{j,2}(t,x)|^p \le e^{p\nu\phi_2(x)} |\rho|^p \max_{\max\{t_j, t - \frac{1}{\mu_{\min}}\} \le s \le t} (|\alpha_j(s,1)|^p e^{-p\nu(t-s)})$.

Hence, we obtain for every $\nu > 0$, $p \in [1, \infty)$ and $t \ge t_j$

$$ \|\bar{\beta}_{j,2}(t, \cdot)\|_{L^p} \le e^{\nu\phi_2(1)} |\rho| \\ \times \max_{\max\{t_j, t - \frac{1}{\mu_{\min}}\} \le s \le t} \left( |\alpha_j(s,1)| e^{-\nu(t-s)} \right) \quad (A.36) $$

We now estimate the $L^p(0, 1)$ norm of $\bar{\beta}_{j,3}(t, x)$. We have $\partial_t \bar{\beta}_{j,3}(t, x) - \mu(x) \partial_x \bar{\beta}_{j,3}(t, x) = e_{j,2}(t, x)$ with boundary

condition $\bar{\beta}_{j,3}(t, 1) = 0$ for all $t \in [t_j, t_{j+1})$ and initial data $\bar{\beta}_{j,3}(t_j, x) = 0$ for all $x \in [0, 1]$. Define

$$ W_p(\bar{\beta}_{j,3}) = \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t, x)|^p dx, \quad (A.37) $$

with $\bar{\omega}_2 > 0$ to be selected later on. For all $t \in (t_j, t_{j+1})$, we have:

$$ \begin{aligned} \dot{W}_p(t) = & p \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t, x)|^{p-1} \operatorname{sgn}(\bar{\beta}_{j,3}(t, x)) \\ & \times \mu(x) \frac{\partial \bar{\beta}_{j,3}(t, x)}{\partial x} dx \\ & + p \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t, x)|^{p-1} \operatorname{sgn}(\bar{\beta}_{j,3}(t, x)) e_{j,2}(t, x) dx. \end{aligned} $$

Integrating by parts, we obtain for all $t \in (t_j, t_{j+1})$:

$$ \begin{aligned} \dot{W}_p(t) \le & -|\bar{\beta}_{j,3}(t,0)|^p \mu(0) - \bar{\omega}_2 \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t,x)|^p \mu(x) dx \\ & - \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t,x)|^p \mu'(x) dx \\ & + p \int_0^1 e^{\bar{\omega}_2 x} |\bar{\beta}_{j,3}(t,x)|^{p-1} |e_{j,2}(t,x)| dx. \end{aligned} \quad (A.38) $$

Using the Young's inequality on the last term and defining $\mu'_{\min} := \min_{x \in [0,1]} \mu'(x)$, together with (42), we obtain from (A.38) for all $t \in (t_j, t_{j+1})$, and all $\epsilon_2 > 0$, the following estimate:

$$ \dot{W}_p(t) \le - \left( \bar{\omega}_2 \mu_{\min} + \mu'_{\min} - (p-1) \epsilon_2^{\frac{p}{p-1}} \right) W_p(t) \\ + \epsilon_2^{-p} \|e_{j,2}(t, \cdot)\|_{L^p}^p. $$

Using the comparison principle and the fact that $W_p(t_j) = 0$ (since $\bar{\beta}_{j,3}(t_j, x) = 0$), we get for all $t \in [t_j, t_{j+1})$, $W_p(t) \le \epsilon_2^{-p} \int_{t_j}^t e^{-(\bar{\omega}_2 \mu_{\min} + \mu'_{\min} - (p-1)\epsilon_2^{p/p-1})(t-s)} \|e_{j,2}(s, \cdot)\|_{L^p}^p ds$. This implies that for every $\nu > 0$, all $t \in [t_j, t_{j+1})$:

$$ W_p(t) \le \epsilon_2^{-p} \int_{t_j}^t e^{-(\bar{\omega}_2 \mu_{\min} + \mu'_{\min} - (p-1)\epsilon_2^{p/p-1} - p\nu)} (t-s) ds \\ \times \max_{t_j \le s \le t} \left( \|e_{j,2}(s, \cdot)\|_{L^p}^p e^{-p\nu(t-s)} \right). \quad (A.39) $$

From the definition (A.37) and (A.39), we get

$$ \begin{aligned} \|\bar{\beta}_{j,3}(t, \cdot)\|_{L^p} \le & e^{\bar{\omega}_2/p} \epsilon_2^{-1} \max_{t_j \le s \le t} (\|e_{j,2}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \\ & \times \left( \int_{t_j}^t e^{-(\bar{\omega}_2 \mu_{\min} + \mu'_{\min} - (p-1)\epsilon_2^{p/p-1} - p\nu)} (t-s) ds \right)^{1/p}. \end{aligned} \quad (A.40) $$
Selecting $\bar{\omega}_2 = \frac{p\nu - \mu_{\min}'}{\mu_{\min}} + p\mu_{\min}$, and $\epsilon_2 = (\bar{\omega}_2 p^{-1}\mu_{\min} - \nu - p^{-1}\mu_{\min}')^{(p-1)/p} = (\mu_{\min})^{(p-1)/p}$, we

get from (A.40) the following estimate:

$$ \begin{aligned} \|\bar{\beta}_{j,3}(t, \cdot)\|_{L^p} &\le e \left(1 + \frac{\nu}{\mu_{\min}} - \frac{\mu_{\min}'}{p\mu_{\min}}\right) (\mu_{\min})^{-(p-1)/p} \\ &\quad \times \max_{t_j \le s \le t} (\|e_{j,2}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \left(\int_{t_j}^t e^{-\mu_{\min}(t-s)} ds\right)^{1/p} \\ &\le \frac{1}{\mu_{\min}} e \left(1 + \frac{\nu}{\mu_{\min}} - \frac{\mu_{\max}'}{p\mu_{\min}}\right) \max_{t_j \le s \le t} (\|e_{j,2}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \end{aligned} $$

Since $\bar{\beta}_{j,3}(t, \cdot)$ depends on $e_{j,2}(s, \cdot)$ (see (A.27) and (A.30)) with $s \in [\max\{t_j, t - (\phi_2(1) - \phi_2(x))\}, t]$ and (A.35) holds true, we further obtain

$$ \begin{aligned} \|\bar{\beta}_{j,3}(t, \cdot)\|_{L^p} &\le \frac{1}{\mu_{\min}} e \left(1 + \frac{\nu}{\mu_{\min}} - \frac{\mu_{\min}'}{p\mu_{\min}}\right) \quad (A.41) \\ &\quad \times \max_{\max\{t_j, t - \frac{1}{\mu_{\min}}\} \le s \le t} (\|e_{j,2}(s, \cdot)\|_{L^p} e^{-\nu(t-s)}) \end{aligned} $$

Finally, using (A.24), together with (A.10), (A.36) and (A.41), the triangle inequality and letting $p \to \infty$ so that $\lim_{p \to \infty} (\|\beta_j(t, \cdot)\|_{L^p}) = \|\beta_j(t, \cdot)\|_{\infty}$, with initial data $\|\beta_j(t_j, \cdot)\|_{L^p} \le \|\beta_j(t_j, \cdot)\|_{\infty}$ we obtain the ISS estimate (60). This concludes the proof.

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