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# Stubborn Observers for Infinite-dimensional Systems in Abstract Form with Application to Parabolic PDEs with Boundary Measurement

Salim Zekraoui, Daniele Astolfi, Vincent Andrieu

► To cite this version:

Salim Zekraoui, Daniele Astolfi, Vincent Andrieu. Stubborn Observers for Infinite-dimensional Systems in Abstract Form with Application to Parabolic PDEs with Boundary Measurement. Systems & Control Letters, 2025, 10.1016/j.sysconle.2025.106323. hal-05427697

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Submitted on 20 Dec 2025

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# Stubborn Observers for Infinite-dimensional Systems in Abstract Form with Application to Parabolic PDEs with Boundary Measurement

Salim Zekraouiª, Daniele Astolfiª, and Vincent Andrieuª

ªUniversité Lyon 1, Villeurbanne, France - CNRS, UMR 5007, LAGEPP, France.

## Abstract

Building upon the recently developed “stubborn” observer construction for finite-dimensional systems, we present a novel observer structure for abstract linear infinite-dimensional systems. A key feature of our design is the use of nonlinear output injection terms with an adjustable dynamical saturation function. This approach not only ensures the input-to-state stability (ISS) of the error system in the presence of measurement disturbances but also demonstrates significant resilience to impulsive noise. We illustrate the effectiveness of our method by applying it to a class of parabolic partial differential equations (PDEs).

## 1. Introduction

State estimation for dynamical systems is one of the fundamental topics in control engineering, as system states are often not fully available for measurement due to physical limitations and/or cost-related constraints. This lack of information results in the inapplicability of state-feedback control designs, making estimation techniques essential. To address this challenge, observers—widely studied in both continuous and discrete setups—have been developed to estimate system states based on model information and sensor measurements [1], and thus enabling practical implementation of state-feedback control laws. However, one of the primary issues of observers is sensitivity to measurement noise, which creates a trade-off between convergence rate and noise attenuation. Depending on the nature of the noise, different methods can be employed to enhance observer performance. For example, when output measurements are affected by outliers—impulsive perturbations occurring over short time—most existing methods focus on discrete-time representations and primarily address identification problems, see e.g. [2, 3]. A notable exception is [4], where the authors introduced a novel approach for mitigating the effects of outliers for linear time-invariant systems. Their method consists in adding an adaptive saturation to the output injection error in the observer dynamics so as to reduce the sensitivity to measurement outliers. This design was later extended to various other scenarios, including discrete-time systems [5], high-gain observers [6], multi-agent system synchronization [7], and output-feedback controllers [8]. Further developments have explored its application to more complex nonlinear systems and broader range of estimators [9].

In this article, we investigate similar problems in the context of infinite dimensional systems. In fact, many physical processes are modeled by infinite-dimensional systems, such as partial differential equations (PDEs) and time-delay systems (TDS). State estimation for such systems is a challenging problem that has gained significant attention over the past decade. Two primary approaches have been developed to address this issue: semigroup-based methods (see, e.g., [10–12]) and backstepping-based techniques (see, e.g., [13–15]). The semigroup-based approach reformulates infinite-dimensional systems in an abstract framework using operator theory in Hilbert spaces. This formulation allows the system dynamics to be expressed in a structure similar to the finite-dimensional systems, facilitating the extension of well-established finite-dimensional observer designs through semigroup theory. In contrast, the backstepping approach represents the infinite-dimensional system as a set of coupled PDEs. By applying invertible transformations, it converts the original observer error dynamics into a structured system chosen to ensure the desired stability. The observer gains are derived from the transformation itself, and the stability properties of the target system are mapped back to the original error dynamics using the inverse transformation. As in the finite-dimensional case, most existing observer designs for infinite-dimensional systems struggle with sensitivity to measurement noise, particularly outliers. To address this issue, we extend the stubborn observer framework to the infinite-dimensional case. We begin by formulating the approach in an abstract operator setting and then specialize it to a class of parabolic PDEs.

The paper is organized as follows. Section 2 is preserved for observer construction for abstract systems. In Subsection 2.1, we describe the class of systems we deal with and

<footnote>ªResearch partially funded by the ANR Alligator project. Email addresses: salim.zekraoui@univ-lyon1.fr (Salim Zekraoui), daniele.astolfi@univ-lyon1.fr (Daniele Astolfi), vincent.andrieu@univ-lyon1.fr (and Vincent Andrieu)</footnote>
introduce its associated stubborn observer construction in Subsection 2.2. The main results are presented successively in Subsections 2.3-2.5. Proofs of main results are postponed to Appendix C. Detailed Application and numerical simulation of the proposed approach are given for a class of parabolic PDEs in Section 3. Conclusions and perspectives are given in Section 4. In Appendix A and Appendix B, we give some preliminary results related to input-to-state stability (ISS) and exact observability for infinite-dimensional systems.

## Notation

We denote by $\mathbb{R}$ the set of real numbers and $\mathbb{N}$ the set of positive natural numbers; Let $\mathbb{R}_{\ge 0} = [0, +\infty)$ and $\mathbb{R}_{> 0} = (0, +\infty)$. For any Hilbert space $\mathcal{H}$, we denote by $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ its associated scalar product and $\| \cdot \|_{\mathcal{H}}$ its associated norm; Given two Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, we denote by $\mathcal{L}(\mathcal{H}_1, \mathcal{H}_2)$ the set of bounded linear operators $\mathcal{T}: \mathcal{H}_1 \to \mathcal{H}_2$ and by $\|\mathcal{T}\|_{\mathcal{L}(\mathcal{H}_1, \mathcal{H}_2)} = \sup_{x \in \mathcal{H}_1: \|x\|_{\mathcal{H}_1} = 1} \|\mathcal{T}x\|_{\mathcal{H}_2}$ its associated norm. For any Hilbert space $\mathcal{H}$, we define $L^m((0, d), \mathcal{H})$, $m \in \mathbb{N}$, as the space of functions that are $m$-integrable on $(0, d)$ with $d \in [0, +\infty)$, equipped with the norm $\|f\|_{L^m((0, d), \mathcal{H})} := \left(\int_0^d \|f(s)\|_{\mathcal{H}}^m ds\right)^{1/m}$. We denote by $H^m((0, d), \mathcal{H})$ the space of functions defined on $[0, d]$ such that their derivatives up to order $m$ belong to $L^2((0, d), \mathcal{H})$. We denote by $\mathcal{C}^m([0, d], \mathcal{H})$, $m \in \mathbb{N}$, the set of functions that have continuous derivatives up to the $m$-th order on the closed interval $[0, d]$ with $d \in [0, +\infty)$; Given a function $f: \mathbb{R}_{\ge 0} \to \mathcal{H}$, we denote $\|f\|_{\infty} = \sup_{s \in \mathbb{R}_{\ge 0}} \|f(s)\|_{\mathcal{H}}$. Given $\sigma \in \mathbb{R}_{\ge 0}$, we denote by the saturation function $\text{sat}_{\sigma}: \mathbb{R} \to \mathbb{R}$ as $\text{sat}_{\sigma}(s) = \max\{-\sigma, \min\{s, \sigma\}\}$ and the dead-zone function $\text{dz}_{\sigma}: \mathbb{R} \to \mathbb{R}$ as $\text{dz}_{\sigma}(s) = s - \text{sat}_{\sigma}(s)$. A function $\alpha: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is said to be a class-K function ($\alpha \in \mathcal{K}$) if it is continuous, zero-at-zero (i.e. $\alpha(0) = 0$), and strictly increasing. A continuous function $\beta: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ belongs to the class-KL ($\beta \in \mathcal{KL}$) if $\beta(\cdot, t) \in \mathcal{K}$ for each fixed $t \in \mathbb{R}_{\ge 0}$, and $\beta(r, \cdot)$ is decreasing and $\lim_{t \to +\infty} \beta(r, t) = 0$ for each fixed $r \in \mathbb{R}_{\ge 0}$.

# 2. Stubborn Observer for Abstract Systems

## 2.1. Problem Statement

Let $\mathcal{U}$, $\mathcal{Y}$ and $\mathcal{H}$ be three Hilbert spaces. Consider the following class of linear abstract systems:

$$ \dot{x}(t) = Ax(t) + Bu(t), \quad x(0) = x_0 \in \mathcal{H}, \quad (1a) $$

$$ y(t) = Cx(t) + d(t), \quad (1b) $$

where for any $t \in \mathbb{R}_{\ge 0}$, $x(t) \in \mathcal{H}$ is the state, $u(t) \in \mathcal{U}$ is the control input, $y(t) \in \mathcal{Y}$ is the measured output, $d(t) \in \mathcal{Y}$ is a bounded continuously-defined measurement noise (i.e., $d \in L^{\infty}((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$), and $x_0 \in \mathcal{H}$ is the initial state. Throughout this paper, $\mathcal{A}: \mathcal{D}(\mathcal{A}) \subset \mathcal{H} \to \mathcal{H}$ is the infinitesimal generator of a strongly continuous

linear semigroup $(e^{At})_{t \in \mathbb{R}_{\ge 0}}$ with growth bound $\omega_0 \in \mathbb{R}$ on $\mathcal{L}(\mathcal{H}, \mathcal{H})$, i.e., for all $\lambda_0 > \omega_0$ there exists $M_0 \ge 1$ such that

$$ \|e^{At}\|_{\mathcal{L}(\mathcal{H}, \mathcal{H})} \le M_0 e^{\lambda_0 t}, \quad \text{for all } t \in \mathbb{R}_{\ge 0}. \quad (2) $$

Recall that $\mathcal{D}(\mathcal{A})$ is dense in $\mathcal{H}$. We assume that operators $\mathcal{B}: \mathcal{U} \to \mathcal{H}$ and $\mathcal{C}: \mathcal{H} \to \mathcal{Y}$ are linear such that $\mathcal{C} \in \mathcal{L}(\mathcal{H}, \mathcal{Y})$. In addition, we assume that for any initial state $x_0 \in \mathcal{H}$ (and appropriate input $u$), system (1a) admits a unique continuous (mild) solution $x: \mathbb{R}_{\ge 0} \to \mathcal{H}$.

For systems of the form (1), it is well known that an asymptotic (resp. exponential) linear observer can be designed as

$$ \dot{\hat{x}}_{\ell}(t) = \mathcal{A}\hat{x}_{\ell}(t) + \mathcal{B}u(t) + \mathcal{L}(y(t) - \mathcal{C}\hat{x}_{\ell}(t)), \quad (3a) $$

$$ \hat{x}_{\ell}(0) = \hat{x}_{0} \in \mathcal{H}, \quad (3b) $$

where $\hat{x}_{\ell}(t) \in \mathcal{H}$ is the observer state and $\mathcal{L} \in \mathcal{L}(\mathcal{Y}, \mathcal{H})$ is an observer gain operator ensuring that $\mathcal{A} - \mathcal{L}\mathcal{C}$, the error operator associated with (1) and (3), is an infinitesimal generator of a strongly (resp. exponentially) stable semigroup. Due to the linearity of $\mathcal{A} - \mathcal{L}\mathcal{C}$, input-to-state stability (ISS) with respect to measurement noise $d(t)$ is guaranteed; however, the linear gain structure directly propagates the large errors induced by the (impulsive-like) disturbances $d(t)$, leading to unsatisfactory performance. Inspired by the stubborn observer construction, originally proposed for finite-dimensional systems [4, 6, 9], this paper aims to enhance the performance of linear infinite-dimensional observers (3) in the presence of impulsive measurement noise.

## 2.2. Observer Architecture and Error System

The stubborn observer construction introduced below relies on nonlinear injection terms rather than linear ones (see, e.g., [10, 11]). These nonlinear injection terms involves a dynamical saturation function which not only ensures input-to-state stability of the corresponding error dynamics with respect to measurement noise $d(t)$ but also improve performances in the presence of impulsive measurement noise. The formulation is given as follows:

$$ \dot{\hat{x}}(t) = \mathcal{A}\hat{x}(t) + \mathcal{B}u(t) + \mathcal{L}\text{sat}_{\sqrt{|\sigma(t)|}}(y(t) - \mathcal{C}\hat{x}(t)), \quad (4a) $$

$$ \hat{x}(0) = \hat{x}_0 \in \mathcal{H}, \quad (4b) $$

where $\hat{x}(t) \in \mathcal{H}$ is the stubborn observer state, $\mathcal{L} \in \mathcal{L}(\mathcal{Y}, \mathcal{H})$ given as in (3), $p \in \mathbb{N}$ is an integer and $\sigma(t) \in \mathbb{R}$ is a dynamical saturation level satisfying

$$ \dot{\sigma}(t) = -\lambda\sigma(t) + R \|y(t) - \mathcal{C}\hat{x}(t)\|_{\mathcal{Y}}^{p}, \quad \sigma(0) = \sigma_0 \in \mathbb{R}, \quad (5) $$

where $R \in \mathbb{R}_{> 0}$ and $\lambda \in \mathbb{R}_{> 0}$ are parameters to be chosen to guarantee the desired stability property for (4).

To prove the effectiveness of our observer (4), we define the error state $e(t) := x(t) - \hat{x}(t)$. Differentiating $e(t)$
yields the following system:

$$
\begin{align}
\dot{e}(t) &= \mathcal{A}e(t) - \mathcal{L} \operatorname{sat}_{\sqrt{|\sigma(t)|}} (Ce(t) + d(t)), \tag{6a} \\
&= (\mathcal{A} - \mathcal{L}\mathcal{C})e(t) + \mathcal{L} \left( dz \sqrt{|\sigma(t)|} (Ce(t) + d(t)) - d(t) \right), \tag{6b}
\end{align}
$$

$$
\dot{\sigma}(t) = -\lambda\sigma(t) + R \|Ce(t) + d(t)\|_{y}^{p}. \quad (6c)
$$

To assess the stability and performance of system(6), we first establish its well-posedness and prove its corresponding **CEP** and **BRS** properties<fnref n="1" />.

**Proposition 1.** For all initial conditions ($e_0 := x_0 - \hat{x}_0, \sigma_0$) in $\mathcal{H} \times \mathbb{R}$ and all measurement noise $d \in L^\infty((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$, the error system (6) admits a global mild solution $(e, \sigma) \in \mathcal{C}(\mathbb{R}_{\ge 0}, \mathcal{H} \times \mathbb{R})$, if one of the following conditions holds:

a) $p=1$.

b) $p > 1$ and the semigroup $(e^{At})_{t \in \mathbb{R}_{\ge 0}}$ is compact.

In particular, when $p=1$, the solution is uniquely defined.
In addition, if the initial conditions are in $\mathcal{D}(\mathcal{A}) \times \mathbb{R}$ and $d$ is Lipschitz continuous, then the solution is defined in a strong sense, i.e.:

1) $(e, \sigma) \in \mathcal{C}(\mathbb{R}_{\ge 0}, \mathcal{D}(\mathcal{A}) \times \mathbb{R})$;

2) For all $t \in \mathbb{R}_{\ge 0}$, the following inequality holds:

$$
\limsup_{h \to 0^+} \frac{e(t+h) - e(t)}{h} = \mathcal{A}e(t) - \mathcal{L} \operatorname{sat}_{|\sigma(t)|} (Ce(t) + d(t)).
$$

**Proof.** See Appendix C.1. □

**Remark 1.** Proposition 1 provides sufficient, but not nec-
essary, conditions for the well-posedness of the error sys-
tem (6). For $p > 1$, strong well-posedness could likely be
established under stricter conditions on $\mathcal{A}$ (e.g., genera-
tion of an analytic semigroup). For the subsequent analy-
sis, we will therefore assume **strong well-posedness** for
$p > 1$, provided the initial condition $(e_0 := x_0 - \hat{x}_0, \sigma_0)$
is in $\mathcal{D}(\mathcal{A}) \times \mathbb{R}$ and the measurement noise $d$ is Lipschitz
continuous.

**Proposition 2.** The error system (6) satisfies the **CEP** and **BRS** properties on **H**.

**Proof.** See Appendix C.2. □

**2.3. Global Asymptotic Stability Analysis**

In this subsection, we show that the stubborn observer
(4) is asymptotically stable in the absence of measurement
noise ($d \equiv 0$). Further analysis of the observer’s behavior
in the presence of measurement noise is provided in the
next section. To this end, we suppose throughout this
subsection that the following assumptions are satisfied.

**Assumption 1.** There exist $\tau \in \mathbb{R}_{\ge 0}$ and $\gamma_1(\tau) \in \mathbb{R}_{> 0}$ satisfying

$$
\gamma_1(\tau) \|x_0\|_{\mathcal{H}}^p \le \int_0^\tau \|C e^{A s} x_0\|_y^p ds, \quad \forall x_0 \in \mathcal{D}(\mathcal{A}).
$$

**Assumption 2.** ($\mathcal{A}-\mathcal{L}\mathcal{C}$) is an infinitesimal generator of a strongly stable semigroup.

**Remark 2.** Assumption 1 implies that the pair $(\mathcal{A}, \mathcal{C})$ is exactly observable on $\mathcal{H}$ in time $\tau$ in the $L^p$-sense (see, e.g., [16–18], [19, 20]). In general, exact observability ensures that the entire state can be determined from partial measurements over a sufficiently long time interval. This property, however, is not guaranteed in all cases. Notably, when $\mathcal{H}$ is infinite-dimensional and the output space $\mathcal{Y}$ is finite-dimensional, finite-time exact observability is never achieved (see, e.g., Theorem 6.2.3 in [21]).

We state now our first main stability result.

**Theorem 1.** Suppose that Assumptions 1-2 hold. Then, for every $\lambda \in \mathbb{R}_{>0}$ there exists $R_*(\lambda) \in \mathbb{R}_{>0}$ such that for all $R \ge R_*(\lambda)$, the error system (6), when $d \equiv 0$, is globally stable and attractive on $\mathcal{H} \times \mathbb{R}_{\ge 0}$ in the following sense: for every $\epsilon \in \mathbb{R}_{>0}$, there exists $\delta \in \mathbb{R}_{>0}$ such that for all $(e_0 := x_0 - \hat{x}_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{\ge 0}$, we have

$$
\|e_0\|_{\mathcal{H}} + |\sigma_0| \le \delta \implies \|e(t)\|_{\mathcal{H}} + |\sigma(t)| \le \epsilon, \quad \forall t \in \mathbb{R}_{\ge 0},
$$

and

$$
\lim_{t \to +\infty} \|e(t)\|_{\mathcal{H}} + |\sigma(t)| = 0.
$$

Moreover, if $(\mathcal{A} - \mathcal{L}\mathcal{C})$ generates an exponentially stable semigroup, then $\|e(t)\|_{\mathcal{H}} + |\sigma(t)|$ is exponentially converging to the origin.

**Proof.** See Appendix C.3. □

**2.4. Lyapunov Construction and ISS result**

In the presence of the measurement noise $d$, we give a
construction of a non-coercive Lyapunov functional assess-
ing the **ISS** property of the error system (6) on $\mathcal{H}$. To this
end, we state the following assumption:

**Assumption 3.** There exists $\mathcal{P} \in \mathcal{L}(\mathcal{H}, \mathcal{H})$ a definite-positive operator ($\mathcal{P} > 0$) such that

$$
\underline{p} \|Cx\|_{y}^{2} \leq \langle x, \mathcal{P}x \rangle_{\mathcal{H}} \leq \bar{p} \|x\|_{\mathcal{H}}^{2}, \quad \forall x \in \mathcal{H}, \quad (7)
$$

for some $\bar{p} > 0$ and $\underline{p} > 0$, and

$$
\langle (\mathcal{A} - \mathcal{L}\mathcal{C})x, \mathcal{P}x \rangle_{\mathcal{H}} + \langle x, \mathcal{P}(\mathcal{A} - \mathcal{L}\mathcal{C})x \rangle_{\mathcal{H}} \leq -\epsilon \|x\|_{\mathcal{H}}^{2}, \quad (8)
$$

for all $x \in \mathcal{H}$ and for some $\epsilon > 0$.

<footnote n="1">¹**BRS** and **CEP** are important properties needed to apply the non-coercive Lyapunov characterization of **ISS** (see Appendix A).</footnote>
**Remark 3.** Inequality (8) ensures the exponential stability of $(\mathcal{A} - \mathcal{LC})$ while inequality 3 ensures that $\mathcal{P}$ is non-coercive with an output-based lower bound. Note that Assumption 3 is independent of Assumptions 1 and 2. Notably, when $\mathcal{H}$ is infinite-dimensional and $\mathcal{Y}$ is finite-dimensional, Assumption 1 is never ensured (see Remark 2), whereas Assumption 3 can still be satisfied (see Section 3 for an example).

Thus, we obtain our second main result.

**Theorem 2.** Suppose that Assumption 3 holds. Then, for every $\lambda \in \mathbb{R}_{>0}$ there exists $R^*(\lambda) \in \mathbb{R}_{>0}$ such that for all $R \ge R^*(\lambda)$, the error dynamics (6) is globally ISS on $\mathcal{H} \times \mathbb{R}_{\ge 0}$ with respect to measurement noises $d \in L^{\infty}((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$, i.e., there exist $\beta \in \mathcal{KL}$ and $\alpha \in \mathcal{K}$ such that

$$ \|e(t)\|_{\mathcal{H}}^p + |\sigma(t)| \le \beta (\|e_0\|_{\mathcal{H}}^p + |\sigma_0|, t) + \alpha (\|d\|_{\infty}), $$

for all $(t, e_0, \sigma_0) \in \mathbb{R}_{\ge 0} \times \mathcal{H} \times \mathbb{R}_{\ge 0}$. In addition, if $\mathcal{P}$ is coercive then (6) is globally exponentially ISS.

**Proof.** See Appendix C.4. □

**Remark 4.** Notice that the bounds $R^*(\lambda)$ and $R_*(\lambda)$ of the two theorems are in general different from each other and both are by their construction conservative. In this paper, we are not seeking optimality but only existence of such bounds.

### 2.5. Performance analysis

Inspired by the result of [9], we characterize the effect of impulsive measurement noises $d \in L^{\infty}((0, \infty), \mathcal{Y})$ on the observer system (4) (or equivalently the error system (6)) where we model $d$ as a piece-wise constant perturbation of the form

$$ d(t) := \delta_{\tau}(t) = \begin{cases} \frac{1}{\tau}, & t \in [0, \tau], \\ 0, & t \in (\tau, +\infty), \end{cases} \qquad (9) $$

converging to the Dirac delta function as $\tau \to 0^+$.

**Proposition 3.** Suppose that Assumption 3 holds and consider the error system (6) with initial conditions $(e_0, \sigma_0) = 0$ and consider $d(t)$ given by (9). Then, for every $\tau \in \mathbb{R}_{\ge 0}$ small enough, there exists $M_d \in \mathbb{R}_{>0}$ such that the following estimates holds:

$$ \|e(\tau)\|_{\mathcal{H}} \le M_d \tau^{\frac{1}{p}} \left(1 + o\left(\tau^{\frac{1}{p}}\right)\right), \qquad (10) $$

where $o(s)$ denotes small terms in $s^r$, $r \ge 1$. In particular, $\|e(\tau)\|_{\mathcal{H}}$ converges to zero when $\tau \to 0^+$.

**Proof.** See Appendix C.5 □

## 3. Stubborn observer for a parabolic system

In this section, we provide an illustrative example involving a boundary observation operator satisfying $\mathcal{C} \in \mathcal{L}(\mathcal{H}, \mathcal{Y})$ where $\mathcal{H} = \{v \in H^1((0, 1), \mathbb{R}) : v(1) = 0\}$ and $\mathcal{Y} = \mathbb{R}$. More precisely, we consider the following class of parabolic systems:

$$ \partial_t z(t, s) = a \partial_{ss} z(t, s) + cz(t, s), \qquad (11a) $$

$$ \partial_s z(t, 0) = 0, \qquad (11b) $$

$$ z(t, 1) = u(t), \qquad (11c) $$

$$ y(t) = z(t, 0) + d(t), \qquad (11d) $$

where for $(t, s) \in \mathbb{R}_{\ge 0} \times [0, 1]$, $z(t, s) \in \mathbb{R}$ is the state, $u(t) \in \mathbb{R}$ is the control input, $y(t) \in \mathbb{R}$ is the measured output, $d(t) \in \mathbb{R}$ is a bounded measurement noise, and $z_0 \in \mathcal{H}$ is the initial state at time $t = 0$. We assume that $a \in \mathbb{R}_{>0}$ and $c \in \mathbb{R}_{>0}$. We assume that for a suitable input $u$, system (11) admits a unique (mild) solution.

We recall that by the Backstepping approach in [13], we can construct a linear observer for (11) of the form

$$ \partial_t \tilde{z}_\ell(t, s) = a \partial_{ss} \tilde{z}_\ell(t, s) + c \tilde{z}_\ell(t, s) + L(s) \tilde{y}_\ell(t), \qquad (12a) $$

$$ \partial_s \tilde{z}_\ell(t, 0) = l_0 \tilde{y}_\ell(t), \qquad (12b) $$

$$ \tilde{z}_\ell(t, 1) = u(t), \qquad (12c) $$

$$ \tilde{y}_\ell(t) = z(t, 0) + d(t) - \tilde{z}_\ell(t, 0), \qquad (12d) $$

which if combined with (11) yields the error system:

$$ \partial_t \tilde{z}_\ell(t, s) = a \partial_{ss} \tilde{z}_\ell(t, s) + c \tilde{z}_\ell(t, s) - L(s) \tilde{y}_\ell(t), \qquad (13a) $$

$$ \partial_s \tilde{z}_\ell(t, 0) = -l_0 \tilde{y}_\ell(t), \qquad (13b) $$

$$ \tilde{z}_\ell(t, 1) = 0, \qquad (13c) $$

$$ \tilde{y}_\ell(t) = \tilde{z}_\ell(t, 0) + d(t). \qquad (13d) $$

where $\tilde{z}_\ell(t, \cdot) := z(t, \cdot) - \tilde{z}_\ell(t, \cdot)$. The observer gains $L: [0, 1] \to \mathbb{R}$ and $l_0 \in \mathbb{R}$ are chosen such that $l_0 = P(0, 0)$ and $L(\cdot) = -aP_y(\cdot, 0)$ with kernel $P$ generated from

$$ P(s, y) = -c(1 - s) \frac{I_1(\sqrt{c(2 - s - y)(s - y)})}{\sqrt{c(2 - s - y)(s - y)}}, \qquad (14) $$

for all $(s, y) \in S_P := \{(s, y) \in [0, 1] \times [0, 1] : y \le s\}$ (see [14] for more details). Furthermore, the observer gains in this case are given by

$$ L(s) = \frac{c(1 - s)}{s(2 - s)} I_2 \left( \sqrt{cs(2 - s)} \right), \quad \text{and} \quad l_0 = -\frac{c}{2}, \qquad (15) $$

where we recall that $I_m$, $m \in \mathbb{N}$, denotes the modified Bessel function of the m-th order.

**Proposition 4.** For any given initial condition $\tilde{z}_0 \in H^2((0, 1), \mathbb{R}) \cap H_0^1((0, 1), \mathbb{R})$, the error system (13) is well-posed and globally exponentially ISS on $H^1((0, 1), \mathbb{R})$ with respect to measurement noise $d(t)$.
**Proof.** The proof follows similar lines of [14, 22], where by using the invertible backstepping transformation

$$ \tilde{z}_{\ell}(t, s) = \tilde{\xi}_{\ell}(t, s) - \int_{0}^{s} P(s, y) \tilde{\xi}_{\ell}(t, s) ds, \quad (16) $$

with $P$ defined as in (14), we can transform (13) into the following class of parabolic systems:

$$ \partial_t \tilde{\xi}_\ell(t, s) = a \partial_{ss} \tilde{\xi}_\ell(t, s) - \tilde{L}(s) d(t), \quad (17a) $$

$$ \partial_s \tilde{\xi}_\ell(t, 0) = -\tilde{l}_0 d(t), \quad (17b) $$

$$ \tilde{\xi}_\ell(t, 1) = 0, \quad (17c) $$

for some continuous $\tilde{L}(\cdot) : [0, 1] \to \mathbb{R}$ and $\tilde{l}_0 \in \mathbb{R}$. System (17) is proven to be exponentially **ISS** on $H^1((0, 1), \mathbb{R})$, provided that the initial conditions are chosen sufficiently smooth, via the Lyapunov functional

$$ \overline{V}_1(\xi) = \|\xi\|_{H^1(0,1)}^2 = \|\xi\|_{L^2(0,1)}^2 + \|\xi_s\|_{L^2(0,1)}^2, $$

for all $\xi \in H^1((0, 1), \mathbb{R})$ (see Lemma 3.1 in [22] for a similar result). Consequently, by applying the inverse of transformation (16), we ensure the same stability property for the original system (13). Furthermore, we can express the Lyapunov functional $\overline{V}_1$ in terms of the original dynamics $\tilde{z}_\ell(t, \cdot)$, yielding a coercive Lyapunov functional for the original system (13). For future reference, we denote this functional as $V_1(\tilde{z}_\ell(t, \cdot))$. $\square$

Now, based on the stubborn formulation, we propose the following observer:

$$ \partial_t \hat{z}(t, s) = a \partial_{ss} \hat{z}(t, s) + c \hat{z}(t, s) + L(s) \tilde{y}_{\text{sat}}(t), \quad (18a) $$

$$ \partial_s \hat{z}(t, 0) = l_0 \tilde{y}_{\text{sat}}(t), \quad (18b) $$

$$ \hat{z}(t, 1) = 0, \quad (18c) $$

$$ \dot{\sigma}(t) = -\lambda\sigma(t) + R \left| z(t, 0) - \hat{z}(t, 0) + d(t) \right|, \quad (18d) $$

$$ \tilde{y}_{\text{sat}}(t) = \text{sat}_{\sigma(t)} \left( z(t, 0) - \hat{z}(t, 0) + d(t) \right). \quad (18e) $$

which leads to the following error system:

$$ \partial_t \tilde{z}(t, s) = a \partial_{ss} \tilde{z}(t, s) + c \tilde{z}(t, s) - L(s) \tilde{y}_{\text{sat}}(t), \quad (19a) $$

$$ \partial_s \tilde{z}(t, 0) = -l_0 \tilde{y}_{\text{sat}}(t), \quad (19b) $$

$$ \tilde{z}(t, 1) = 0, \quad (19c) $$

$$ \tilde{y}_{\text{sat}}(t) = \text{sat}_{|\sigma(t)|} \left( \tilde{z}(t, 0) + d(t) \right), \quad (19d) $$

$$ \dot{\sigma}(t) = -\lambda\sigma(t) + R \left| \tilde{z}(t, 0) + d(t) \right|. \quad (19e) $$

**Corollary 1.** For any given initial condition $\tilde{z}_0 \in H^2((0, 1), \mathbb{R}) \cap H_0^1((0, 1), \mathbb{R})$, the error system (19) associated to system (11) and observer (14), (18) is globally exponentially **ISS** on $H^1((0, 1), \mathbb{R})$ with respect to measurement noise $d(t)$.

**Proof.** We rewrite system (19) into an abstract framework with state space $\mathcal{H} = \{v \in H^1((0, 1), \mathbb{R}) : v(1) = 0\}$ and

operators

$$ \begin{aligned} \mathcal{A} &= a \partial_{ss}(\cdot) + c \mathrm{Id}, & \mathcal{B} &= \delta_0(\cdot), \\ \mathcal{C} &= \mathcal{C}_{x=0}(\cdot), & \mathcal{L} &= L - l_0 \delta_0(\cdot), \end{aligned} $$

with $\mathcal{D}(\mathcal{A}) = \{v \in H^2((0, 1), \mathbb{R}) : v(1) = v'(0) = 0\}$ and $\mathcal{C}_{x=0}(f) = f(0)$ for all $f \in \mathcal{H}$.

By Agmon's inequality, the output operator $\mathcal{C}$ is indeed bounded from $\mathcal{H}$ into $\mathbb{R}$ and satisfies

$$ \begin{aligned} |\mathcal{C}e|^2 &:= |e(0)|^2 \le 2\|e\|_{L^2(0,1)} \|e'\|_{L^2(0,1)}, \\ &\le \|e\|_{L^2(0,1)}^2 + \|e'\|_{L^2(0,1)}^2 \le \|e\|_{\mathcal{H}}^2, \end{aligned} $$

for all $e \in \mathcal{D}(\mathcal{A})$, and thus there exists $p \in \mathbb{R}_{>0}$ such that

$$ |\mathcal{C}e|^2 \le p V_1(e), \quad \forall e \in \mathcal{D}(\mathcal{A}), \quad (20) $$

where $V_1$ is the functional from Proposition 4. Because $V_1$ is an exponential **ISS**-Lyapunov functional, an application of Theorem 2 confirms that system (19) is globally exponentially **ISS** on $H^1((0, 1), \mathbb{R})$ with respect to $d(t)$. $\square$

### 3.1. Numerical Simulations

In this subsection, we numerically illustrate the effectiveness of the stubborn observer construction (18) for the system (11) under a suitable output-feedback control design. We take the system parameters as $a = 1$ and $c = 15$, the gains $L(\cdot)$ and $l_0$ as in (15), and the initial conditions as $z_0(s) = (s^2 - s^3)$, $\hat{z}_0 = 0$, and $\sigma_0 = 0$. The stubborn observer parameters are set to $\lambda = R = 10$. The control input $u(t)$ is an exponentially stabilizing output-feedback controller for (11), as given in [14]:

$$ u(t) = \int_{0}^{1} K(1, y) \hat{z}(t, y) dy, \quad (21) $$

where $\hat{z}(t, \cdot)$ is generated from the stubborn observer (18) and the kernel $K$ is given by:

$$ K(s, y) = -(c+c_0) s \frac{I_1(\sqrt{(c+c_0)(s^2-y^2)})}{\sqrt{(c+c_0)(s^2-y^2)}}, \quad \forall (s, y) \in S_P, $$

with tuning parameter $c_0 = 15$. The measurement noise $d(t)$ is taken zero until $t = 7$, when an impulsive disturbance of amplitude 10 and width 3 occurs. It remains zero afterwards until $t = 24$, where a second impulse of amplitude 20 and width 2 is introduced. It remains zero afterwards until $t = 39$, where a last impulse of amplitude 30 and width 1.5 is introduced. For $t > 60$, a sinusoidal noise $f(t) = 0.1 \sin(4t)$ is injected. For the numerical implementation, we employ an explicit finite-difference scheme with space and time discretization steps of $\Delta x = 10^{-2}$ and $\Delta t = 4 \times 10^{-5}$, respectively.

Figures 1 and 2 show the evolution of the state variable $z(t, s)$ under the output-feedback controller (21) and the state error $\tilde{z}(t, s)$ corresponding to the stubborn observer, respectively. Figure 3 provides a comparison between the
<figure><img src="image_3.png" /><figcaption>Figure 1: The evolution of $z(t,s)$ the state of the parabolic PDE (11) using the output-feedback controller $u(t)$ in (21)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2: The evolution of $\tilde{z}(t,s)$ the state of the nonlinear error system (13).</figcaption></figure>

norms of the two systems with linear output injection (red) and the proposed nonlinear stubborn injection (blue). As expected, the stubborn observer demonstrates superior performance the higher the amplitude and the shorter the duration of the impulsive noise (in other terms the closer the noise is to a Dirac impulse). Moreover, the stubborn observer shows comparable, if not better, performance in the presence of other types of measurement noise.

## 4. Conclusions

In this paper, we extend the stubborn observer construction, originally proposed for finite-dimensional systems with measurement outliers, to the infinite-dimensional setting under certain boundedness conditions on the observation operator $C$. We illustrate our results with a class of parabolic PDEs featuring boundary measurements. Future work will focus on relaxing some of the assumptions on $C$ and extending these results to more complex nonlinear infinite-dimensional systems.

<figure><img src="image_5.png" /><figcaption>Figure 3: Comparison between the norms of the nonlinear error system (19) with the norm of its corresponding closed-loop system (11) (in blue) and the linear error system (13) alongside the norm of its corresponding closed-loop system (11) (in red).</figcaption></figure>

## Appendix A. Further Results on ISS for Abstract Systems

In this section, we refer to an abstract system by the triple $\Sigma = (\mathcal{H}, \mathcal{U}, \phi)$ as in Definition 1.3 of [23] and we suppose that the system $\Sigma$ is forward complete in the sense of Definition 1.4 of [23]. We present a sufficient condition for system $\Sigma$ to satisfy the Continuity at Equilibrium Point (CEP) and Bounded Reachability Sets (BRS) given in [23] in Definitions 2.1 and 2.2, respectively. Inspired by Lemmas 3 and 11 in [24], we propose the following technical result:

**Lemma 1.** Consider a forward complete system $\Sigma = (\mathcal{H}, \mathcal{U}, \phi)$. Suppose that there exists a continuous, increasing function $\mu : \mathbb{R}_{\ge 0}^3 \to \mathbb{R}_{\ge 0}$ such that $\mu(0, 0, \cdot) \equiv 0$, and $\|\phi(t, x, u)\|_{\mathcal{H}} \le \mu (\|x\|_{\mathcal{H}}, \|u\|_{\mathcal{U}}, t)$, $\forall (t, x, u) \in \mathbb{R}_{\ge 0} \times \mathcal{H} \times \mathcal{U}$. Then, the system satisfies the CEP and BRS properties.

**Proof.** The proof is a direct application of the definitions of CEP and BRS in [23] and Lemmas 3 and 11 in [24]. □

Inspired by Theorem 2.19 of [23], we propose the following Theorem on ISS for abstract systems.

**Theorem 3.** Consider a forward complete system $\Sigma = (\mathcal{H}, \mathcal{U}, \phi)$. Suppose there exist a $\mu$ satisfying the conditions of Lemma 1 and a continuous function $V : \mathcal{H} \to \mathbb{R}_{\ge 0}$ satisfying $0 \le V(x) \le \psi_1 (\|x\|_{\mathcal{H}})$, for all $x \in \mathcal{H}$, and

$$ \dot{V}_u(x) \le -\psi_2 (\|x\|_{\mathcal{H}}) + \psi_3 (\|u\|_{\mathcal{U}}), \quad \forall (x, u) \in \mathcal{H} \times \mathcal{U}, $$

$$ \dot{V}_u(x(t)) := \limsup_{t \to 0^+} \frac{V(x(t)) - V(x(0))}{h}. $$

<footnote>$^2$ $\dot{V}_u(x)$ denotes the Lie derivative of $V$ at $x$ corresponding to the input $u$ along the solutions of $\Sigma$. More precisely, it is defined as</footnote>
for some $\psi_1, \psi_2 \in \mathcal{K}_\infty$, and $\psi_3 \in \mathcal{K}$. Then, $\Sigma$ is ISS.

## Appendix B. Observability for Abstract Systems

In this section, we present a relevant result on exact observability for nonlinear abstract systems of the form

$$ 
\begin{aligned}
\dot{x}(t) &= \overbrace{Ax(t) + \mathcal{F}(t, y(t))}^{\mathcal{G}(t)x(t)}, \quad x(0) = x_0 \in \mathcal{H}, \quad (\text{B.1}) \\
y(t) &= \mathcal{C}x(t) \in \mathcal{Y},
\end{aligned}
 $$

where for any $t \in \mathbb{R}_{\ge 0}$, $x(t) \in \mathcal{H}$ is the state, $y(t) \in \mathcal{Y}$ is the measured output, $\mathcal{A}: \mathcal{H} \to \mathcal{H}$ is the linear system operator generating a strongly continuous semigroup $(e^{At})_{t \in \mathbb{R}_{\ge 0}}$, $\mathcal{C} \in \mathcal{L}(\mathcal{H}, \mathcal{Y})$ is the output operator, and $\mathcal{F}: \mathbb{R}_{\ge 0} \times \mathcal{Y} \to \mathcal{H}$ is a nonlinear globally uniformly Lipschitz continuous (in the second argument) operator ensuring the existence of unique continuous (mild) solution $x: \mathbb{R}_{\ge 0} \to \mathcal{H}$ for (B.1) depending on the initial condition $x_0$. Inspired by [19, 25], we define the nonlinear semigroup $(e^{G(t)})_{t \in \mathbb{R}_{\ge 0}}$ associated to the solution $x(t)$ of (B.1), i.e. $x(t) = e^{G(t)}x_0$ for every $(t, x_0) \in \mathbb{R}_{\ge 0} \times \mathcal{H}$ and present its following properties:

**Definition 1.** We say that the pair $(\mathcal{G}, \mathcal{C})$ is exact observable on $\mathcal{H}$ in time $\tau \in \mathbb{R}_{\ge 0}$ in the $L^p$-norm sense, if there exists $\gamma_1(\tau) > 0$ such that for all $x_0, x'_0 \in \mathcal{D}(\mathcal{A})$

$$ \int_0^\tau \|Ce^{G(s)}x_0 - Ce^{G(s)}x'_0\|_y^p \mathrm{d}s \geq \gamma_1(\tau)\|x_0 - x'_0\|_{\mathcal{H}}^p. \quad (\text{B.2}) $$

**Lemma 2.** If the pair $(\mathcal{A}, \mathcal{C})$ is exact observable on $\mathcal{H}$ in time $\tau \in \mathbb{R}_{\ge 0}$, then the pair $(\mathcal{G}, \mathcal{C})$ is exact observable on $\mathcal{H}$ in time $\tau$ as well.

**Proof.** We prove the exact observability of $(\mathcal{G}, \mathcal{C})$ by contradiction. Assume that the pair $(\mathcal{G}, \mathcal{C})$ is not exactly observable in time $\tau$. This implies that for every $\gamma_1(\tau) > 0$, there exist $x_0, x'_0 \in \mathcal{D}(\mathcal{A})$ such that

$$ \int_0^\tau \|Ce^{G(s)}x_0 - Ce^{G(s)}x'_0\|_y^p \mathrm{d}s < \gamma_1(\tau)\|x_0 - x'_0\|_{\mathcal{H}}^p. \quad (\text{B.3}) $$

We aim to show that $(\mathcal{A}, \mathcal{C})$ is not exactly observable in time $\tau$; that is, for all $\gamma_2(\tau) > 0$, there exists $v_0 \in \mathcal{D}(\mathcal{A})$ such that

$$ \int_0^\tau \|Ce^{As}v_0\|_y^p \mathrm{d}s < \gamma_2(\tau)\|v_0\|_{\mathcal{H}}^p. \quad (\text{B.4}) $$

To this end, we integrate (B.1) from 0 to $t$ and apply $\mathcal{C}$ to the resulting expression as follows

$$ 
\begin{aligned}
\mathcal{C}x(t) &= \mathcal{C}e^{At}x_0 + \mathcal{C}\int_0^t e^{A(t-s)}\mathcal{F}(s, \mathcal{C}e^{G(s)}x_0) \mathrm{d}s, \\
&= \mathcal{C}e^{G(t)}x_0,
\end{aligned}
 $$

which holds for all $(t, x_0) \in \mathbb{R}_{\ge 0} \times \mathcal{H}$. Therefore, for any $x_0, x'_0 \in \mathcal{D}(\mathcal{A})$, by means of $|a+b|^p \le 2^{p-1}(|a|^p + |b|^p)$, it follows that

$$ 
\begin{aligned}
\int_0^\tau \|Ce^{At}(x_0-x'_0)\|_y^p \mathrm{d}t &\le 2^{p-1} \int_0^\tau \|Ce^{G(t)}x_0-Ce^{G(t)}x'_0\|_y^p \mathrm{d}t \\
&\quad + 2^{p-1} \int_0^\tau \left\| \mathcal{C} \int_0^t e^{A(t-s)} \Delta_{\mathcal{F}}(s, x_0, x'_0) \mathrm{d}s \right\|_y^p \mathrm{d}t,
\end{aligned}
 $$

where $\Delta_{\mathcal{F}}: R_{\ge 0} \times \mathcal{D}(\mathcal{A}) \times \mathcal{D}(\mathcal{A}) \to \mathcal{H}$ is defined as

$$ \Delta_{\mathcal{F}}(s, x_0, x'_0) = \mathcal{F}(s, Ce^{G(s)}x_0) - \mathcal{F}(s, Ce^{G(s)}x'_0). $$

Applying Cauchy-Schwartz and Jensen inequalities on $\mathcal{J}(t)$, we obtain

$$ \mathcal{J}(t) \le t^{p-1} \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, \mathcal{Y})}^p \int_0^t \|e^{A(t-s)}\|_{\mathcal{L}(\mathcal{H}, \mathcal{H})}^p \|\Delta_{\mathcal{F}}(s, x_0, x'_0)\|_{\mathcal{H}}^p \mathrm{d}s, $$

for all $t \in [0, \tau]$. Making use of the growth-rate condition of $(e^{At})_{t \in \mathbb{R}_{\ge 0}}$ given in (2) with a fixed $\lambda_0 > \omega_0$ and by the global uniformly Lipschitz continuousness with Lipschitz constant $\ell > 0$, we have

$$ \mathcal{J}(t) \le \tau^{p-1} \ell^p \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, \mathcal{Y})}^p M_0^p e^{p\lambda_0\tau} \int_0^\tau \|Ce^{G(s)}x_0 - Ce^{G(s)}x'_0\|_y^p \mathrm{d}s, $$

for all $t \in [0, \tau]$. Consequently, it follows that

$$ \int_0^\tau \|Ce^{At}(x_0-x'_0)\|_y^p \mathrm{d}t \le k(\tau) \int_0^\tau \|Ce^{G(t)}x_0-Ce^{G(t)}x'_0\|_y^p \mathrm{d}t, $$

where $k(\tau) := 2^{p-1} + 2^{p-1}\tau^p\ell^p \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, \mathcal{Y})}^p M_0^p e^{p\lambda_0\tau} > 0$.

Now, combining this result with our initial assumption of non-observability (B.3), we conclude that for all $\gamma_2(\tau) = k(\tau)\gamma_1(\tau) > 0$ there exists $v_0 = x_0 - x'_0 \in \mathcal{D}(\mathcal{A})$ such that

$$ \int_0^\tau \|Ce^{As}v_0\|_y^p \mathrm{d}s < \gamma_2(\tau)\|v_0\|_{\mathcal{H}}^p. \quad (\text{B.5}) $$

Therefore, $(\mathcal{A}, \mathcal{C})$ is not finite-time exact observable. By the principle of contraposition, the pair $(\mathcal{G}, \mathcal{C})$ is exact observable in time $\tau$. $\square$

## Appendix C. Proof of the main results

### Appendix C.1. Proof of Proposition 1

When $p > 1$, the existence of a solution $(e, \sigma) \in \mathcal{C}(\mathbb{R}_{\ge 0}, \mathcal{H} \times \mathbb{R})$ is guaranteed for all initial conditions in $\mathcal{H} \times \mathbb{R}$ and any measurement noise $d \in L^\infty(\mathbb{R}_{\ge 0}, \mathcal{Y}) \cap \mathcal{C}(\mathbb{R}_{\ge 0}, \mathcal{Y})$, by Theorem 6.2.2 in [26] in conjunction with the linear boundedness of the saturation function and an application of Gronwall's Lemma.

In the case where $p = 1$, the existence of a unique mild solution is guaranteed by Theorem 6.1.2 in [26] for all initial conditions in $\mathcal{H} \times \mathbb{R}$ (in particular in $\mathcal{D}(\mathcal{A}) \times \mathbb{R}$) and any continuous bounded measurement noise $d \in L^\infty((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$. Taking initial conditions
$(e_0, \sigma_0) \in \mathcal{D}(\mathcal{A}) \times \mathbb{R}$, we aim to show that $(e(t), \sigma(t)) \in \mathcal{D}(\mathcal{A}) \times \mathbb{R}$ for all $t$ and the state $(e, \sigma)$ is right-differentiable in the Dini-sense. Inspired by Theorem 9 in [27], we define $(e(\cdot + h), \sigma(\cdot + h)) \in \mathcal{C}(\mathbb{R}_{\ge 0}, \mathcal{H} \times \mathbb{R})$, $h \in \mathbb{R}_{>0}$, as the mild solution of the same system (6), but with the shifted initial condition $(e(h), \sigma(h)) \in \mathcal{H} \times \mathbb{R}$ and the shifted disturbance $d(\cdot + h) \in L^{\infty}((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$. Note that, by continuity of mild solutions, $\sigma$ is differential in the classical sense $(\sigma \in \mathcal{C}^1(\mathbb{R}_{\ge 0}, \mathbb{R}))$. On the other hand, note that by the integral solution formula, we have

$$ \frac{e(h) - e_0}{h} = \frac{e^{Ah}e_0 - e_0}{h} - \int_0^h \frac{e^{A(h-s)}}{h} \mathcal{L} \mathrm{sat}_{|\sigma(s)|} (Ce(s) + d(s)) \mathrm{d}s, $$

which clearly converges to $\mathcal{A}e_0 - \mathcal{L}\mathrm{sat}_{|\sigma_0|}(Ce_0 + d(0))$ as $h \to 0^+$ since $e_0 \in \mathcal{D}(\mathcal{A})$ and $\sigma, e$ and $d$ are continuous. By the global Lipschitz property of the saturation function with respect to all variables and using the growth-rate condition of the strongly continuous semigroup $(e^{At})_{t \in \mathbb{R}_{\ge 0}}$ given in (2), we have

$$ \begin{aligned} \|e(t+h) - e(t)\|_{\mathcal{H}} &\le M_0 e^{\lambda_0 t} \|e(h) - e_0\|_{\mathcal{H}} \\ &\quad + \int_0^t \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)} M_0 e^{\lambda_0 (t-s)} \|e(s+h) - e(s)\|_{\mathcal{H}} \mathrm{d}s \\ &\quad + \int_0^t \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} M_0 e^{\lambda_0 (t-s)} |\sigma(s+h) - \sigma(s)| \mathrm{d}s \\ &\quad + \int_0^t \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} M_0 e^{\lambda_0 (t-s)} \|d(s+h) - d(s)\|_{y} \mathrm{d}s, \end{aligned} $$

for all $t \in \mathbb{R}_{\ge 0}$. Letting $\bar{\lambda}_0 = M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)}$ and using Gronwall's inequality, we obtain

$$ \begin{aligned} \|e(t+h) - e(t)\|_{\mathcal{H}} &\le M_0 e^{(\lambda_0 + \bar{\lambda}_0)t} \left[ \|e(h) - e_0\|_{\mathcal{H}} \right. \\ &\qquad \left. + \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \int_0^t |\sigma(s+h) - \sigma(s)| \mathrm{d}s \right] \quad (\text{C.1}) \\ &\quad + \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \int_0^t \|d(s+h) - d(s)\|_{y} \mathrm{d}s \right], \end{aligned} $$

for all $t \in \mathbb{R}_{\ge 0}$. Now, by letting $d$ be globally Lipschitz continuous with Lipschitz constant $\ell_d \in \mathbb{R}_{>0}$ and using the global Lipschitz property of $\sigma$ with constant $\ell_\sigma \in \mathbb{R}_{>0}$, we obtain

$$ \|e(t+h) - e(t)\|_{\mathcal{H}} \le M_0 e^{(\lambda_0 + \bar{\lambda}_0)t} \left[ \|e(h) - e_0\|_{\mathcal{H}} + h \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} (\ell_{\sigma} + \ell_d)t \right], $$

for all $t \in \mathbb{R}_{\ge 0}$, and therefore, we deduce that

$$ \limsup_{h \to 0^+} \left\| \frac{e(t+h) - e(t)}{h} \right\|_{\mathcal{H}} < \infty. \quad (\text{C.2}) $$

By definition of the mild solution, we have that

$$ \frac{e^{Ah}e(t) - e(t)}{h} = \frac{e(t+h) - e(t)}{h} + \frac{1}{h} \int_t^{t+h} e^{A(t+h-s)} \mathcal{L} \mathrm{sat}_{|\sigma(s)|} (Ce(s) + d(s)) \mathrm{d}s. $$

Or, by continuity of variables $\sigma, x$ and $d$, we have that

$$ \begin{aligned} & \limsup_{h \to 0^+} \frac{1}{h} \int_{t+h}^t e^{A(t+h-s)} \mathcal{L} \mathrm{sat}_{|\sigma(s)|} (Ce(s) + d(s)) \mathrm{d}s \\ &= \mathcal{L} \mathrm{sat}_{|\sigma(t)|} (Ce(t) + d(t)), \end{aligned} $$

from which we deduce that

$$ e(t) \in \left\{ z \in \mathcal{H} : \limsup_{h \to 0^+} \frac{1}{h} \left\| e^{Ah} z - z \right\|_{\mathcal{H}} < \infty \right\}, $$

and thus $e(t) \in \mathcal{D}(\mathcal{A})$ for all $t \in \mathbb{R}_{\ge 0}$, and $e$ is right differentiable with respect to $t$ such that

$$ \limsup_{h \to 0^+} \frac{e(t+h) - e(t)}{h} = \mathcal{A}e(t) - \mathcal{L} \mathrm{sat}_{|\sigma(t)|} (Ce(t) + d(t)). $$

This concludes the proof. $\square$

## Appendix C.2. Proof of Proposition 2

We start by noticing that if we integrate (6) and use the global lipschitzness of the $\mathrm{sat}_\sigma$ function and the growth rate condition in (2), we have

$$ \begin{aligned} \sigma(t) &= e^{-\lambda t} \sigma_0 + R \int_0^t e^{-\lambda(t-s)} \|Ce(s) + d(s)\|_y^2 \mathrm{d}s, \\ &\le \sigma_0 + 2R \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)}^2 \int_0^t \|e(s)\|_{\mathcal{H}}^2 \mathrm{d}s + \frac{2R}{\lambda} \|d\|_{\infty}^2, \end{aligned} $$

and

$$ \begin{aligned} \|e(t)\|_{\mathcal{H}} &\le \left\| e^{At} e_0 + \int_0^t e^{A(t-s)} \mathcal{L} \mathrm{sat}_{\sqrt{|\sigma(s)|}} (Ce(s) + d(s)) \mathrm{d}s \right\|_{\mathcal{H}} \\ &\le M_0 e^{|\lambda_0|t} \|e_0\|_{\mathcal{H}} + M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \int_0^t e^{|\lambda_0|(t-s)} \|Ce(s)\|_y \mathrm{d}s \\ &\quad + M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \int_0^t e^{|\lambda_0|(t-s)} \|d(s)\|_y \mathrm{d}s, \\ &\le M_0 e^{|\lambda_0|t} \|e_0\|_{\mathcal{H}} + M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \int_0^t e^{|\lambda_0|(t-s)} \|d(s)\|_y \mathrm{d}s \\ &\quad + M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)} \int_0^t e^{|\lambda_0|(t-s)} \|e(s)\|_{\mathcal{H}} \mathrm{d}s. \end{aligned} $$

Letting $\bar{\lambda}_0 = M_0 \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)}$ and using Gronwall's lemma, we get

$$ \|e(t)\|_{\mathcal{H}} \le M_0 e^{(|\lambda_0| + \bar{\lambda}_0)t} \|e_0\|_{\mathcal{H}} + \frac{e^{(|\lambda_0| + \bar{\lambda}_0)t}}{\|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, \mathbb{R})}} \|d\|_{\infty}, \quad (\text{C.3}) $$

and

$$ \sigma(t) \le \sigma_0 + \frac{2RM_0^2 \|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, y)}^2}{|\lambda_0| + \bar{\lambda}_0} e^2 (|\lambda_0| + \bar{\lambda}_0)t \|e_0\|_{\mathcal{H}}^2 \\ + 2R \left[ \frac{e^2 (|\lambda_0| + \bar{\lambda}_0)t}{|\lambda_0| + \bar{\lambda}_0} + \frac{1}{\lambda} \right] \|d\|_{\infty}^2, \quad (\text{C.4}) $$

for all $t \in \mathbb{R}_{\ge 0}$. From these two last inequalities and by Lemma 1, we get the **CEP** and **BRS** properties on $\mathcal{H}$. $\square$
Appendix C.3. Proof of Theorem 1

**Idea of the proof:** Based on Assumptions 1-2, we show that for any $\lambda \in \mathbb{R}_{>0}$ there exists $R_*(\lambda) \in \mathbb{R}_{>0}$ such that for any $R \ge R_*(\lambda)$ the state $(e, \sigma)$ of (6), when $d \equiv 0$, enters at $t = \tau$ the set defined by

$$S_c := \{(e, \sigma) \in \mathcal{H} \times \mathbb{R}_{\ge 0} : \|C\|_{\mathcal{L}(\mathcal{H}, y)}^p \|e\|_{\mathcal{H}}^p \le \sigma\}.$$

Once inside of the set $S_c$, we prove that the state remains inside, and consequently remains inside of the larger set

$$\hat{S}_c := \{(e, \sigma) \in \mathcal{H} \times \mathbb{R}_{\ge 0} : \|Ce\|_y^p \le \sigma\},$$

where the error system (6) reduces to

$$\dot{e} = (A - LC)e, \quad \dot{\sigma} = -\lambda\sigma + R \|Ce\|_y^p. \quad (C.5)$$

The stability and the asymptotic (resp. exponential) convergence properties of (6) then follows from Assumption 2, the cascade structure of (C.5) and inequalities (C.3)-(C.4).

**Proof:** To start, in view of (6c) and the positivity of the initial conditions $\sigma_0 \in \mathbb{R}_{\ge 0}$, we have

$$\sigma(\tau) = e^{-\lambda\tau}\sigma_0 + R \int_0^\tau e^{-\lambda(\tau-s)} \|Ce(s)\|_y^p ds,$$

and thus

$$\int_0^\tau \|Ce(s)\|_y^p ds \le \frac{e^{\lambda\tau}}{R} \sigma(\tau), \quad \forall (e_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{>0}. \quad (C.6)$$

Moreover, using the integral solution formula of (6b), the sector condition $\|dz_{\sigma(t)}(Ce(t))\|_y \le \|Ce(t)\|_y$, and the growth-rate condition of $(e^{\lambda t})_{t \in \mathbb{R}_{\ge 0}}$ with fixed $\lambda_0 > \omega_0$, we get

$$\|e(\tau)\|_{\mathcal{H}}^p \le (2\tau)^{p-1} \|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}^p M_0^p e^{p|\lambda_0|\tau} \int_0^\tau \|Ce(s)\|_y^p ds \\ + 2^{p-1} M_0^p e^{p|\lambda_0|\tau} \|e_0\|_{\mathcal{H}}^p,$$

for every $(e_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{>0}$. Invoking the exact observability of $(A, C)$ over the time interval $\tau$, and utilizing Lemma 2 with $(s, r) \mapsto \mathcal{F}(s, r) = \mathcal{L} \text{sat } \sqrt{\sigma(s)}(r)$, we have

$$\gamma_1(\tau) \|e_0\|_{\mathcal{H}}^p \le \int_0^\tau \|Ce(s)\|_y^p ds, \quad \forall (e_0, \sigma_0) \in \mathcal{D}(A) \times \mathbb{R}_{\ge 0}.$$

This allows us to deduce that

$$e^{\lambda\tau} \|C\|_{\mathcal{L}(\mathcal{H}, y)}^p \|e(\tau)\|_{\mathcal{H}}^p \le R_*(\lambda) \int_0^\tau \|Ce(s)\|_y^p ds,$$

holds for every $(e_0, \sigma_0) \in \mathcal{D}(A) \times \mathbb{R}_{\ge 0}$, where $R^*(\lambda) > 0$ is given by

$$R_*(\lambda) = \|C\|_{\mathcal{L}(\mathcal{H}, y)}^p M_0^p \left[ \frac{2^{p-1}}{\gamma_1(\tau)} + \frac{\|\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}^p}{(2\tau)^{1-p}} \right] e^{(\lambda+p|\lambda_0|)\tau}.$$

Finally, combining this last inequality with (C.6), we get

$$\|C\|_{\mathcal{L}(\mathcal{H}, y)}^p \|e(\tau)\|_{\mathcal{H}}^p \le \frac{R_*(\lambda)}{R} \sigma(\tau), \quad (C.7)$$

for every $(e_0, \sigma_0) \in \mathcal{D}(A) \times \mathbb{R}_{\ge 0}$, and consequently for all $(e_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{\ge 0}$ due to the density of $\mathcal{D}(A)$ in $\mathcal{H}$. Hence, if $R \ge R^*(\lambda)$, then $(e(\tau), \sigma(\tau)) \in S_c$.

To conclude, we note that for any $\epsilon \in \mathbb{R}_{>0}$, the shifted solution $(e(\cdot + \epsilon), \sigma(\cdot + \epsilon))$ satisfies the same differential equation as (6), but with the initial condition $(e(\epsilon), \sigma(\epsilon))$. Since the preceding arguments depend only on the structure of the system's dynamics, they apply equally to this shifted trajectory. This directly implies that $(e(\tau + \epsilon), \sigma(\tau + \epsilon)) \in S_c$, whenever $R \ge R^*(\lambda)$. This concludes the proof. $\square$

Appendix C.4. Proof of Theorem 2

Let us start by rewriting (6) as follows:

$$\dot{e}(t) = (A - LC)e(t) + L \, dz_{\sqrt{\sigma(t)}}(Ce(t)) + L \Delta_L(t), \quad (C.8a)$$

$$\dot{\sigma}(t) = -\lambda\sigma(t) + R\|Ce(t)\|_y^p + R\Delta_R(t), \quad (C.8b)$$

where

$$\Delta_L(t) = \text{sat}_{\sqrt{\sigma(t)}}(Ce(t)) - \text{sat}_{\sqrt{\sigma(t)}}(Ce(t) + d(t)), \quad (C.9a)$$

$$\Delta_R(t) = \|Ce(t) + d(t)\|_y^p - \|Ce(t)\|_y^p, \quad (C.9b)$$

Notice that

$$\begin{aligned} |\Delta_R| &\le 2^{p-1} \|d\|_y^p + (2^{p-1} - 1) \|Ce\|_y^p, \\ \|\Delta_L\|_y &\le \|d\|_y, \text{ and } \|dz_{\sqrt{\sigma}(Ce)}\|_y \le \|Ce\|_y. \end{aligned} \quad (C.10)$$

Let us consider the following Lyapunov functional:

$$W(e, \sigma) = V_1(e) + (\zeta - \rho\eta)\sigma + \eta V_2(e, \sigma), \quad (C.11)$$

where $V_1$ and $V_2$ are given by

$$V_1(e) = \langle e, Pe \rangle_{\mathcal{H}}^{\frac{p}{2}}, \quad V_2(e, \sigma) = \max\{V_1(e), \rho\sigma\}, \quad (C.12)$$

with parameters $\rho \in \mathbb{R}_{>0}$, $\zeta \in \mathbb{R}_{>0}$ and $\eta \in \mathbb{R}_{>0}$ to be selected below, operator $\mathcal{P}$ given as in Assumption 3. The functional $W$ is positive definite on $\mathcal{H} \times \mathbb{R}_{\ge 0}$ and non-coercive, i.e.: there exist $\underline{q} \in \mathbb{R}_{>0}$ and $\bar{q} \in \mathbb{R}_{>0}$ such that

$$\underline{q}(\|Ce\|_y^p + \sigma) \le W(e, \sigma) \le \bar{q}(\|e\|_{\mathcal{H}}^p + \sigma), \quad \forall (e, \sigma) \in \mathcal{H} \times \mathbb{R}_{\ge 0}.$$

However, $W$ is not differentiable on the set of measure zero $\{(e, \sigma) : \langle e, Pe \rangle_{\mathcal{H}}^p - \rho^2\sigma^2 = 0\}$. Nevertheless, $W$ is continuous and uniformly continuous, allowing us to define its upper right Dini derivative along the solutions of (C.8) and to provide an **ISS** Lyapunov-like analysis for (C.8). To achieve this, we adapt Lemma 1.28 in [28] and analyze three main cases. In each case, we restrict our attention to initial conditions within $\mathcal{D}(A) \times \mathbb{R}_{\ge 0}$ and disturbances
to Lipschitz continuous functions $d \in L^\infty((0, \infty), \mathcal{Y})$, and upon obtaining the **ISS** inequality on $\mathcal{D}(\mathcal{A}) \times \mathbb{R}_{\ge 0}$, we employ density arguments to generalize the results to $\mathcal{H} \times \mathbb{R}_{\ge 0}$ and to functions $d \in L^\infty((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$. To simplify the analysis, we focus on the case $p = 1$. The other scenarios follows similar analysis of the case $p = 1$ and [4, 6].

**Case 1:** If $V_1(e) < \rho\sigma$, then there exists a constant $\rho > 0$ such that $\|Ce\|_y \le \sigma$. In this scenario, the error system (C.8) is simplified to

$$ 
\begin{aligned}
\dot{e}(t) &= (\mathcal{A} - \mathcal{L}\mathcal{C})e(t) + \mathcal{L}\Delta_{\mathcal{L}}(t), \\
\dot{\sigma}(t) &= -\lambda\sigma(t) + R\|Ce(t)\|_y + R\Delta_R(t),
\end{aligned}
\quad (C.13) $$

and the Lyapunov functional $W$ reduces to

$$ W(e, \sigma) = V_1(e) + \zeta\sigma. $$

Consider initial conditions $(e_0, \sigma_0) \in \mathcal{D}(\mathcal{A}) \times \mathbb{R}_{\ge 0}$ and global Lipschitz continuous disturbance $d \in L^\infty((0, \infty), \mathcal{Y})$. The upper right Dini derivative of $W$ along the solutions of (C.8), denoted for simplicity as $D^+W(e, \sigma)^3$ where we omit the time dependency, is given by

$$ D^+W(e, \sigma) = \frac{\langle \mathcal{A} - \mathcal{L}\mathcal{C} \rangle_{\mathcal{H}} e, \mathcal{P}e \rangle_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} + \frac{\langle e, \mathcal{P}\mathcal{L}\Delta_{\mathcal{L}} \rangle_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} - \lambda\zeta\sigma \\ + R\zeta\|Ce\|_y + R\zeta\Delta_R. $$

Making use of (C.9b) and Assumption 3, we obtain

$$ D^+W \le -\varepsilon \frac{\|e\|_{\mathcal{H}}^2}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} + \|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \frac{\|e\|_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} \|d\|_y \\ - \lambda\zeta\sigma + R\zeta\|C\|_{\mathcal{L}(y, \mathcal{H})} \|e\|_{\mathcal{H}} + R\zeta\|d\|_y. $$

Using inequality (7) and taking

$$ b_1 = \frac{\varepsilon}{\sqrt{\bar{p}}} - R\zeta\|C\|_{\mathcal{L}(y, \mathcal{H})}, \quad b_2 = \frac{\lambda\zeta}{2} > 0, \quad (C.14a) $$

$$ b_3 = \frac{\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}}{\sqrt{\bar{p}}} + R\zeta > 0, \quad (C.14b) $$

we obtain

$$ D^+W \le -b_1\|e\|_{\mathcal{H}} - b_2\sigma + b_3\|d\|_y, \quad (C.15) $$

where $b_1$ is positive if we choose the parameters $\lambda \in \mathbb{R}_{>0}$, $R \in \mathbb{R}_{>0}$, $\zeta \in \mathbb{R}_{>0}$, and $\eta \in \mathbb{R}_{>0}$ such that

$$ 0 < \zeta < \frac{\varepsilon}{\sqrt{\bar{p}}R\|C\|_{\mathcal{L}(y, \mathcal{H})}}. \quad (C.16) $$

$$ D^+W(e(t), \sigma(t)) = \limsup_{h \to 0^+} \frac{W(e(t+h), \sigma(t+h)) - W(e(t), \sigma(t))}{h} $$

**Case 2:** If $V_1(e) > \rho\sigma$, then the error system (C.8) reduces to

$$ 
\begin{aligned}
\dot{e}(t) &= (\mathcal{A} - \mathcal{L}\mathcal{C})e(t) + \mathcal{L} \mathrm{d}z_{\sigma(t)}(Ce(t)) + \mathcal{L}\Delta_{\mathcal{L}}(t), \\
\dot{\sigma}(t) &= -\lambda\sigma(t) + R\|Ce(t)\|_y + R\Delta_R(t),
\end{aligned}
\quad (C.17) $$

and the Lyapunov functional $W$ is simplified to

$$ W(e, \sigma) = (1 + \eta)V_1(e) - (\rho\eta - \zeta)\sigma, \text{ with } \rho\eta - \zeta > 0. $$

Taking initial conditions $(e_0, \sigma_0) \in \mathcal{D}(\mathcal{A}) \times \mathbb{R}_{\ge 0}$ and global Lipschitz continuous disturbances $d \in L^\infty((0, \infty), \mathcal{Y})$, and recomputing $D^+W(e, \sigma)$ in this case yields

$$ 
\begin{aligned}
D^+W = (1 + \eta) \frac{\langle (\mathcal{A} - \mathcal{L}\mathcal{C})e, \mathcal{P}e \rangle_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} + (1 + \eta) \frac{\langle e, \mathcal{P}\mathcal{L}\Delta_{\mathcal{L}} \rangle_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} \\ 
+ (1 + \eta) \frac{\langle e, \mathcal{P}\mathcal{L} \mathrm{d}z_{\sigma}(Ce) \rangle_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} + \lambda(\rho\eta - \zeta)\sigma \\ 
- R(\rho\eta - \zeta)\|Ce\|_y - R(\rho\eta - \zeta)\Delta_R.
\end{aligned}
 $$

Using (C.9b) and Assumption 3 and applying Cauchy-Schwartz inequality, we obtain

$$ D^+W \le -\varepsilon(1 + \eta) \frac{\|e\|_{\mathcal{H}}^2}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} + (1 + \eta)\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})} \\ \times \left[ \frac{\|e\|_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} \|\mathrm{d}z_{\sigma}(Ce)\|_y + \frac{\|e\|_{\mathcal{H}}}{\sqrt{\langle e, \mathcal{P}e \rangle_{\mathcal{H}}}} \|\Delta_{\mathcal{L}}\|_y \right] \\ + \lambda(\rho\eta - \zeta)\sigma - R(\rho\eta - \zeta)\|Ce\|_y + R(\rho\eta - \zeta)\|d\|_y. $$

Employing the condition of Case 2 alongside inequality (C.10) and by taking

$$ c_1 = \frac{\varepsilon}{\sqrt{\bar{p}}} (1 + \eta) - \lambda(\rho\eta - \zeta) \frac{\sqrt{\bar{p}}}{\rho}, \quad (C.18a) $$

$$ c_2 = R(\rho\eta - \zeta) - (1 + \eta) \frac{\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}}{\sqrt{\bar{p}}}, \quad (C.18b) $$

$$ c_3 = (1 + \eta) \frac{\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}}{\sqrt{\bar{p}}} + R(\rho\eta - \zeta) > 0, \quad (C.18c) $$

we get

$$ D^+W \le -\frac{c_1}{2} \|e\|_{\mathcal{H}} - \frac{c_1 \rho}{2\sqrt{\bar{p}}} \sigma - c_2 \|Ce\|_y + c_3 \|d\|_y, $$

where $c_1$ and $c_2$ are positive if we choose the parameters $\lambda \in \mathbb{R}_{>0}$, $R \in \mathbb{R}_{>0}$, $\zeta \in \mathbb{R}_{>0}$, and $\eta \in \mathbb{R}_{>0}$ such that

$$ 0 < \eta < \frac{\varepsilon}{\lambda\bar{p}}, \quad R \ge (1 + \eta) \frac{\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}}{\sqrt{\bar{p}}\rho\eta}, \quad (C.19a) $$

$$ 0 < \zeta < \rho\eta - (1 + \eta) \frac{\|\mathcal{P}\mathcal{L}\|_{\mathcal{L}(y, \mathcal{H})}}{\sqrt{\bar{p}}R}. \quad (C.19b) $$

**Case 3:** If $V_1(e) = \rho\sigma$, then the error system (C.8) becomes equivalent to both (C.13) and (C.17). Moreover,

<footnote>$^3D^+W(e(t), \sigma(t))$ denotes the upper right Dini derivative along $(e, \sigma)$ the solutions of (6), for a given $d(t)$ (the dependency on $d$ is removed to simplify the presentation). More precisely, it is defined as</footnote>
by adapting Lemma 1.28 in [28] to Hilbert spaces, the upper right Dini derivative of W is given by

$$D^+W(e, \sigma) = \dot{V}_1(e) + (\zeta - \rho\eta)\dot{\sigma} + \eta \max\{\dot{V}_1(e), \rho\dot{\sigma}\}, \\ = \max\{\overbrace{\dot{V}_1(e) + \zeta\sigma}^{\cdot}, \overbrace{(1 + \eta)V_1(e) - (\rho\eta - \zeta)\sigma}^{\cdot}\},$$

where by observing the two expressions within the max function, we recognize the upper right Dini derivatives of W obtained in Case 1 and Case 2, respectively. Making use of these two scenarios, we consider the larger of the two upper bounds, leading to

$$D^+W \leq -\min\{b_1, c_1\} \|e\|_{\mathcal{H}} - \min\{b_2, c_2\} \sigma + \max\{b_3, c_3\} \|d\|_{\infty}$$

where the positivity of $\min\{b_1, c_1\}$ and $\min\{b_2, c_2\}$ is guaranteed by taking the parameters $\lambda \in \mathbb{R}_{>0}$, $R \in \mathbb{R}_{>0}$, $\zeta \in \mathbb{R}_{>0}$, and $\eta \in \mathbb{R}_{>0}$ such that

$$0 < \eta < \frac{\varepsilon}{\lambda\bar{p}}, R > R^*(\lambda) := (1 + \eta) \frac{\|\mathcal{PL}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})}}{\sqrt{\bar{p}\rho\eta}}, \\ 0 < \zeta < \min \left\{ \rho\eta - (1 + \eta) \frac{\|\mathcal{PL}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})}}{\sqrt{\bar{p}R}}, \frac{\varepsilon}{\sqrt{\bar{p}R}\|\mathcal{C}\|_{\mathcal{L}(\mathcal{H}, \mathcal{Y})}} \right\}.$$

Combining all the analysis in Cases 1-3, we obtain

$$D^+W \leq -d_1W + d_2\|d\|_y,$$

which, by integration, see Corollary A.5.45 in [21], implies that

$$W(e(t), \sigma(t)) - W(e_0, \sigma_0) \leq - \int_0^t d_1 W(e(s), \sigma(s))ds \\ + \int_0^t d_2 \|d(s)\|_y ds, \qquad (C.20)$$

for all $(t, e_0, \sigma_0) \in \mathbb{R}_{\geq 0} \times \mathcal{D}(\mathcal{A}) \times \mathbb{R}_{\geq 0}$ and all global Lipschitz continuous functions $d \in L^{\infty}((0, \infty), \mathcal{Y})$, with some constants $d_1, d_2 > 0$.

By the density of $\mathcal{D}(\mathcal{A})$ in $\mathcal{H}$ and of bounded Lipschitz functions in continuous bounded functions, the integral inequality (C.20) extends to all $(e_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{\geq 0}$ and all $d \in L^{\infty}((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$. As in [27], this leads to the following **ISS** Lyapunov characterization:

$$\dot{W}_d(e_0, \sigma_0) \leq -d_1 W(e_0, \sigma_0) + d_2 \|d(0)\|_y,$$

for all $(e_0, \sigma_0) \in \mathcal{H} \times \mathbb{R}_{\geq 0}$ and all $d \in L^{\infty}((0, \infty), \mathcal{Y}) \cap \mathcal{C}((0, \infty), \mathcal{Y})$. Thus, using this last property, Proposition 2 and Theorem 2.19. in [23], we get the **ISS** of (6). Moreover, when $\mathcal{P}$ is coercive operator, W becomes coercive as well and thus this last inequality implies exponential **ISS** of (6) by means of Theorem 2.18. in [23] $\square$

## Appendix C.5. Proof of Proposition 3

We start by computing $e(\tau)$ as follows

$$e(\tau) = \int_{0}^{\tau} e^{A(\tau-s)} \mathcal{L} \sqrt{\sigma(s)} ds, \qquad (C.21)$$

with $p \geq 1$ and $\sigma(s)$ given by

$$\sigma(s) = \int_{0}^{s} e^{-\lambda(s-r)} R \|\mathcal{C}e(r) + \delta_{\tau}(r)\|_{y}^{p} dr, \quad \forall s \in [0, \tau].$$

By integrating this inequality with $\sigma(0) = 0$ and using the expression of $\delta_{\tau}(r)$ in (9), we obtain

$$\sigma(s) \leq \frac{2^{p-1}R}{\lambda} (1 - e^{-\lambda s}) \left( \frac{1}{\tau^p} + c \right), \\ \leq 2^{p-1}R (s + o(s^2)) \left( \frac{1}{\tau^p} + c \right), \\ \leq 2^{p-1}R (\tau^{1-p} + o(\tau^{2-p})),$$

which holds for all $s \in [0, \tau]$, where $c := \sup_{s \in [0, \tau]} \|\mathcal{C}e(s)\|_{y}^{p}$ and $e^{-\lambda\tau}$ is approximated by $1 - \lambda\tau + o(\tau^2)$ for small $\tau \in \mathbb{R}_{>0}$. Next, using this inequality, we get

$$\|e(\tau)\|_{\mathcal{H}} \leq \|\mathcal{L}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})} \int_{0}^{\tau} \left\| e^{A(\tau-s)} \right\|_{\mathcal{L}(\mathcal{H}, \mathcal{H})} \sqrt{\sigma(s)} ds, \\ \leq \sqrt{2^{p-1}RM_0} \|\mathcal{L}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})} \frac{(e^{|\lambda_0|\tau} - 1)}{|\lambda_0|} \sqrt{\tau^{1-p} + o(\tau^{2-p})}, \\ \leq \sqrt{2^{p-1}RM_0} \|\mathcal{L}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})} (\tau + o(\tau^2)) (\tau^{\frac{1-p}{p}} + o(\tau^{\frac{2-p}{p}})), \\ \leq \sqrt{2^{p-1}RM_0} \|\mathcal{L}\|_{\mathcal{L}(\mathcal{Y}, \mathcal{H})} \tau^{\frac{1}{p}} (1 + o(\tau^{\frac{1}{p}})).$$

Consequently, when $\tau \to 0^+$, namely when $\delta_{\tau}$ tends to a Dirac impulse, it follows that $\|e(\tau)\|_{\mathcal{H}} \to 0$. $\square$

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