# Rapid stabilization of the heat equation with localized disturbance

Patricio Guzmán, Hugo Parada, Christian Calle-Cárdenas

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Patricio Guzmán, Hugo Parada, Christian Calle-Cárdenas. Rapid stabilization of the heat equation with localized disturbance. 2025. hal-05421864

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# Rapid stabilization of the heat equation with localized disturbance

Christian Calle * Patricio Guzmán * and Hugo Parada †

## Abstract

This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation.

**2020 Mathematics Subject Classification:** 35B40, 34G20, 35K05, 93D15, 93D23.

**Keywords:** Heat equation, locally distributed control, disturbance, feedback stabilization, Lyapunov techniques, exponential stability.

## 1 Introduction

Partial differential equations (PDEs) play a central role in the mathematical modeling of a wide range of physical phenomena, such as heat distribution in solids and fluids, wave propagation, and the lateral deflection of strings and beams. Once a model is formulated, a fundamental objective in control theory is to design feedback laws that stabilize the system's state, either toward an equilibrium or a desired trajectory. The traditional stabilization analysis often proceeds under idealized assumptions, with the absence of external disturbances. However, in practical applications, systems are subject to disturbances arising from unmodeled fast dynamics, parameter uncertainties, or fluctuating environmental loads, for instance. Those disturbances can affect the stability of the system. Similarly, control designs that require actuation across the entire spatial domain are often physically unrealizable. Instead, controls might be applied through a specific subregion of the domain or on a section of its boundary. Consequently, a central and challenging objective in PDE control is to develop stabilization strategies that are both robust to disturbances and spatially localized in their actuation.

Stabilization problems for one-dimensional PDEs, such as controlling a string or a rod, are well-studied. There are numerous results for various boundary conditions and actuator configurations, where it is common to employ powerful methods like backstepping, which has proven effective for many one-dimensional models [10, 11, 13, 14, 17, 35, 40, 43, 44, 49]. However, it remains a challenging open problem

<footnote>*Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile.
E-mail: patricio.guzmann@usm.cl, ccalle@usm.cl (corresponding author)
†Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France.
E-mail: hugo.parada@inria.fr</footnote>
to introduce the backstepping method to general multi-dimensional models. In contrast, controlling multi-
dimensional PDEs, such as the heat equation or the wave equation, presents profoundly greater analytical
and geometrical challenges. Control design becomes more difficult due to the complex interplay between
the system’s geometry, the spectrum of its spatial differential operator, and the actuator’s location.

This work focuses on the multi-dimensional heat equation, which is a well-studied subject in the control theory. It is important to consider the seminal works of Lions [33, 34], who established a functional analysis framework for controllability and stabilization of distributed parameter systems. A step forward was achieved by Triggiani [45, 46], who proved abstract stabilizability results and boundary feedback stabilization for parabolic systems, via compact resolvent and spectral arguments. Subsequently, the works of Barbu and Triggiani [6] and by Barbu and Wang [7] extended this theory to nonlinear and semilinear parabolic systems, establishing internal stabilization by finite-dimensional controllers. Breiten and Kunisch [8] proposed a Riccati feedback framework for reaction-diffusion systems arising in cardiac electrophysiology, illustrating the robustness of such designs in multi-dimensional domains. Recently, Badra and Takahashi [2] presented the Fattorini criterion for approximate controllability and stabilization of parabolic systems, providing refined spectral characterizations.

In the multi-dimensional context, numerous studies have addressed systems subject to disturbances and developed diverse methods to counteract such disturbances, each with distinct advantages and applicability conditions. The choice of method depends on the nature of the disturbance considered and its structural relationship to the control input. When disturbance is a constant, it can be followed by traditional and well-known methods, which include but are not limited to the Spectral (Pole Placement) Method, the Riccati-Based Method, and the Backstepping Method (one-dimensional case) [12, 28]. When the disturbance is not constant and time-dependent is more general and of significant interest. When the disturbance enters the system through the same channel as the control input (a *matched condition*), robust methods like Sliding Mode Control (SMC) are highly effective. SMC drives the system trajectory onto a predetermined sliding manifold in finite time, inducing invariance to a class of matched disturbances [3, 19, 22]. To handle the general and challenging case of bounded, unmatched disturbances, Active Disturbance Rejection Control (ADRC) offers a powerful solution. Its core principle is to treat all unknown dynamics and disturbances as a *total disturbance*. This quantity is estimated in real-time by an Extended State Observer (ESO) and actively canceled by the control law [51, 52]. This provides robustness without requiring a precise model of the disturbance itself.

It can also be mentioned the recent work of Balogoun, Marx, and Plestan [4], where well-posedness and global stabilization results for infinite-dimensional systems subject to disturbances and admissible control operators, using SMC, were established. Their control design relies on the sliding variable $\sigma(t) = \langle\phi, z(t)\rangle_H$, where $\phi$ is an eigenfunction of the adjoint operator $A_L^*$ associated with the closed-loop generator $A_L = A + BL$. In our setting, $A$ denotes the Laplacian and $B$ represents a localized internal control operator, we find the assumptions of [4] are satisfied. However, the explicit computation of such eigenfunctions is considerably more involved. In contrast, our approach relies solely on the spectral properties of the Laplacian, thus avoiding the construction of $A_L$ and leading to a direct variational formulation of the feedback law. In the recent paper due to Labbadi and Roman [29], they achieved finite- and fixed-time stabilization by means of set-valued feedback of maximal monotone type. Although the localized control operator proposed here satisfies the same structural assumptions, their feedback involves additional nonlinear power terms. In contrast, the present method employs a simpler monotone law that preserves the natural dissipativity of the Laplacian. Finally, a relevant approach is found in Xiang [41], where a localized finite-dimensional stabilizer for the multi-dimensional heat equation was built, via the Frequency Lyapunov method, a localized finite-dimensional stabilizer for the multi-dimensional heat equation using spectral arguments and well-chosen Lyapunov functions. That work is the starting point of
our study.

The contribution of the present work is a novel control framework for the robust stabilization of a multi-dimensional heat equation subject to general bounded disturbances. It's designed a feedback control law that acts only on an arbitrarily small subdomain and incorporates a disturbance rejection mechanism to achieve robustness against a broad class of time- and space-varying unknown disturbances, ensuring a decay rate as large as desired. Providing a rigorous stability analysis for the resulting closed-loop system, establishing the well-posedness and exponential stability of the state to the desired equilibrium. The stabilization of the multi-dimensional heat equation with localized disturbance has been previously studied in [52], where only asymptotic stability via the ADRC approach was considered. While they get the eventual decay of the system's energy, the rate of convergence remains undetermined and potentially slow. In contrast, the present work yields a significantly stronger result, the exponential stability. Specifically, we prove the existence of constants $C \ge 1$, such that for every $\lambda > 0$ the system's energy $E(t)$ satisfies

$$E(t) \le Ce^{-\lambda t}E(0), \quad \forall t > 0.$$

This represents a qualitative improvement over asymptotic decay, as it provides a robust and fast rate of stabilization.

The remainder of this paper is organized as follows: Section 2 formalizes the problem statement and presents the necessary mathematical preliminaries. Section 3 is devoted to the main tool of the present work, the Spectral inequality. The main stability theorem and its detailed proof are given in Section 4. Section 5 presents the well-posedness of the resulting closed-loop system via maximal monotone operator theory. Finally, Section 6 offers concluding remarks and directions for future research.

## 2 Problem Statement

In the following, we present a precise formulation of the stabilization problem considered in this article. Let $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) be an open domain with smooth boundary $\partial\Omega$. Let $\omega \subset \Omega$ be a nonempty open subset of positive Lebesgue measure (i.e., $|\omega| > 0$). In this article, we focus our interest on a multi-dimensional heat equation controlled and perturbed on a subdomain:

$$ \left\{ \begin{array}{ll} y_t - \Delta y = \chi_{\omega}(u+d), & (t,x) \in (0, \infty) \times \Omega \\ y(t,x) = 0, & (t,x) \in (0, \infty) \times \partial\Omega \\ y(0,x) = y_0(x), & x \in \Omega \end{array} \right. \qquad (P) $$

where $\chi_{\omega}$ denotes the characteristic function on $\omega$, that is to say, $\chi_{\omega}(x) = 1$ if $x \in \omega$ and $\chi_{\omega}(x) = 0$ if $x \notin \omega$. The aim is to achieve exponential stabilization of system (P) by employing a distributed feedback control law $u = u(t,x)$ that acts only on the interior subdomain $\omega \subset \Omega$. The control must simultaneously suppress the effects of an unknown distributed disturbance $d = d(t,x)$.

Regarding the undisturbed case ($d = 0$) the problem under consideration has been solved in Xiang [41]. The stabilization problem for partial differential equations subjected to unknown disturbances, acting either in the domain or at the boundary, has been object of recent interest. In Table 1 we present, without being exhaustive, some of the concerned literature.
<table><caption>Table 1: Stabilization of partial differential equations subjected to unknown disturbances.</caption><thead><tr><th>Equation</th><th>Distributed disturbance</th><th>Boundary disturbance</th><th>Multidimensional</th></tr></thead><tbody><tr><td>Heat</td><td>[51]</td><td>[15, 23, 24, 51]</td><td>[25, 41, 52, 51]</td></tr><tr><td>Wave</td><td>[16, 38]</td><td>[19, 20, 37, 36, 53]</td><td>[18, 27, 30, 47, 48]</td></tr><tr><td>Beam</td><td>[1, 26]</td><td>[21]</td><td>-</td></tr></tbody></table>

Throughout this work, we adopt standard notation for Sobolev and Hilbert spaces. We denote by $L^2(\Omega)$ the usual Hilbert space of square-integrable functions on $\Omega$, equipped with the inner product

$$ (u, v)_{L^2(\Omega)} = \int_{\Omega} uv \, dx, \quad \text{and norm} \quad \|u\|_{L^2(\Omega)}^2 = (u, u)_{L^2(\Omega)}. $$

For $k \in \mathbb{N}$, the Sobolev space $H^k(\Omega)$ denotes the space of functions in $L^2(\Omega)$ with weak derivatives up to order $k$ in $L^2(\Omega)$, endowed with the norm

$$ \|u\|_{H^k(\Omega)}^2 = \sum_{|\alpha| \le k} \|D^\alpha u\|_{L^2(\Omega)}^2. $$

We write $H_0^1(\Omega)$ for the subspace of $H^1(\Omega)$ consisting of functions with zero trace on $\partial\Omega$. We introduce the second-order elliptic operator $\mathcal{A}$ given by

$$ \left\{ \begin{array}{l} \mathcal{A} : D(\mathcal{A}) \subset L^2(\Omega) \rightarrow L^2(\Omega) \\ D(\mathcal{A}) = \{ \phi \in H_0^1(\Omega) / \Delta\phi \in L^2(\Omega) \} \\ \mathcal{A}\phi = -\Delta\phi. \end{array} \right. \qquad (\text{Op}) $$

We note that $\mathcal{A}$ is selfadjoint and has compact resolvent. Hence, the spectrum of $\mathcal{A}$ consists of only isolated eigenvalues with finite multiplicity. Furthermore, there exists a Hilbert orthogonal basis $\{e_i\}_{i \in \mathbb{N}}$ of $D(\mathcal{A})$ consisting of eigenfunctions of $\mathcal{A}$, associated with the sequence of eigenvalues $\{\tau_i\}_{i \in \mathbb{N}}$. Note that

$$ \begin{gathered} 0 < \tau_1 \le \tau_2 \le \tau_3 \le \dots \le \tau_i \le \dots < +\infty \quad \text{and} \quad \tau_i \xrightarrow{i \to +\infty} \infty. \tag{2.1} \\ -\Delta e_i = \tau_i e_i \quad \text{with} \quad e_i|_{\partial\Omega} = 0. \end{gathered} $$

Different eigenvalues $\tau_i$ may coincide, but each eigenvalue only has finite algebraic multiplicity.

Given $\lambda > 0$, let $N(\lambda) := \#\{i \in \mathbb{N} : \tau_i \le \lambda\}$, i.e., $\tau_{N(\lambda)} \le \lambda < \tau_{N(\lambda)+1}$. Then Weyl's law [50] gives $N(\lambda) \sim (2\pi)^{-n}\omega_n \text{vol}(\Omega) \lambda^{n/2}$, where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$. Henceforth, for the sake of notational convenience, we shall denote $N(\lambda)$ simply by $N$.

For any $y \in L^2(\Omega)$, its eigenfunction expansion reads

$$ y(x) = \sum_{i=1}^{\infty} y_i e_i(x), \quad y_i := (y, e_i)_{L^2(\Omega)}. $$

We define the orthogonal projection onto the span of the first $N$ eigenfunctions by $P_N y := \sum_{i=1}^{N} y_i e_i$, and $P_N^\perp$ the co-projection.

In this setting, both the control input and the disturbance are expanded in the eigenfunction basis. Specifically, the control and disturbance terms localized to the control region $\omega$ take the form
$$ \chi_{\omega} d(t, x) = \chi_{\omega} \sum_{i=1}^{\infty} d_{i}(t) e_{i}(x), \quad \chi_{\omega} u(t, x) = \chi_{\omega} \sum_{i=1}^{\infty} u_{i}(t) e_{i}(x). $$

Although the disturbance is assumed to be unknown, we ask it to satisfy the following two assumptions, which are the standard ones that can be found in the literature.

(A1) $d \in L^1(0, \infty; L^2(\Omega))$.

(A2) There exists $D \in (0, \infty)$ such that $\|d(t, \cdot)\|_{L^2(\Omega)} \le D$, for every $t \in [0, \infty)$.

In order to reject the effects of the disturbance, we use the sign multivalued operator in a Hilbert space $H$, $\text{sign}_H(\cdot) : H \to 2^H$ ($2^H$ denotes the power set of $H$), given by

$$ \text{sign}_H(f) = \begin{cases} \frac{f}{\|f\|_H}, & \text{if } f \neq 0 \\ \{g \in H / \|g\|_H \le 1\}, & \text{if } f = 0. \end{cases} \qquad (\text{sign}) $$

To that end, we employ the property of the multivalued sign operator in the Hilbert space $L^2(\Omega)$.

$$ \int_{\Omega} \theta f \, dx = \|f\|_{L^2(\Omega)}, \quad \forall f \in L^2(\Omega), \quad \forall \theta \in \text{sign}_{L^2(\Omega)}(f). $$

Our main result is the following one:

**Theorem 2.1.** Let us assume (A1) and (A2). Let $y_0$ in $L^2(\Omega)$ be the initial condition. Let $\lambda$ in $(0, \infty)$ be the desired decay rate. Then, there exists a feedback law $\mathcal{G}_{\lambda}: L^2(\Omega) \to L^2(\Omega)$ such that (P) is exponentially stable in $L^2(\Omega)$, with decay rate $\lambda$. Being more explicit, (P) with the feedback law $u = \mathcal{G}_{\lambda}$ has a unique weak solution $y$ in $C([0, \infty); L^2(\Omega))$, and it satisfies

$$ \|y(t, \cdot)\|_{L^2(\Omega)} \le e^{-\lambda t} \|y_0\|_{L^2(\Omega)}, \quad t \in [0, \infty) \qquad (2.2) $$

**Remark 2.1.** Assumption (A1) is required for the well-posedness part of Theorem 1, and assumption (A2) is needed for the construction of the feedback law.

## 3 Spectral Inequality

A crucial tool in the feedback design is the spectral inequality, which ensures that the modal energy in the controlled region is sufficiently observable. For instance, concerning the eigenfunctions $\{e_i\}_{i \in \mathbb{N}}$ one has the following result:

**Proposition 3.1.** The eigenfunctions $\{e_i\}_{i \in \mathbb{N}}$ satisfy

1. Orthonormal basis: $(e_i, e_j)_{L^2(\Omega)} = \delta_{ij}$.

2. (Unique continuation) The symmetric matrix $J_N$ given below is invertible [6],

$$ J_N := ((e_i, e_j)_{L^2(\omega)})_{i,j=1}^{N}. $$

3. (Weak Spectral inequality) There exist $C_{\lambda} > 0$ such that for $Y_N = (a_1, \dots, a_N) \in \mathbb{R}^N$, we have
$$Y_N^T J_N Y_{N(x)} = \left\| \sum_{n=1}^{N} a_n e_n \right\|^2_{L^2(\Omega)} \geq C_\lambda \sum_{n=1}^{N} a_n^2. \quad (3.1)$$

**Remark 3.1.** The weak spectral inequality follows as a consequence of the unique continuation property.

**Remark 3.2.** For the Laplacian operator, a more precise spectral inequality - known as the quantitative spectral inequality - was introduced by Lebeau and Robbiano [31], and previously discussed in [32]. This inequality was later used by Xiang [41] to establish quantitative rapid stabilization results. In this work, we employ the weak spectral inequality instead, as our focus is on rapid stabilization rather than its quantitative version. As a result, the constant in our spectral inequality does not depend on the decay rate parameter $\lambda$.

# 4 Feedback Design

A main objective in controlling distributed parameter systems is the development of feedback laws which ensure the system state converges to a prescribed target while maintaining robustness in the presence of external disturbances. In this section, we prove our main result related with the exponential stabilization of the heat equation with localized control and bounded perturbation. We base our ideas on [41]. We present the construction of such a feedback law for problem (P), assuming suitable regularity of the solution. To clarify the main ideas underlying the feedback design, we decompose the control input $u$ as

$$u = \tilde{u} + \hat{u},$$

where the term $\tilde{u}$ is designed to achieve the desired decay rate of the state, while $\hat{u}$ will be designed separately to mitigate the impact of the disturbance, $d$, while preserving the stabilizing effect of $\tilde{u}$.

For completeness, we recall several useful identities that will be used in the subsequent analysis. First, the projection of the eigenfunction $e_j$ onto the control region is given by

$$\chi_{\omega} e_j = \sum_{i=1}^{\infty} (\chi_{\omega} e_j, e_i)_{L^2(\Omega)} e_i = \sum_{i=1}^{\infty} (e_i, e_j)_{L^2(\omega)} e_i.$$

The time derivative and Laplacian terms satisfy the standard orthogonality properties:

$$\int_{\Omega} y_t e_j \, dx = \sum_{i=1}^{\infty} \frac{d}{dt} y_i \int_{\Omega} e_i e_j \, dx = \frac{d}{dt} y_j,$$

$$\int_{\Omega} \Delta y e_j \, dx = \int_{\Omega} \sum_{i=1}^{\infty} y_i (\Delta e_i, e_j) \, dx = - \sum_{i=1}^{\infty} y_i \tau_i \int_{\Omega} e_i e_j \, dx = - \tau_j y_j,$$

where $\tau_j$ is the eigenvalue associated with $e_j$.

Finally, the projections of the feedback and disturbance components are given by

$$\int_{\Omega} \chi_{\omega} \left( \sum_{i=1}^{N} e_i \tilde{u}_i(t) \right) e_j \, dx = \sum_{i=1}^{N} \tilde{u}_i(t) (e_i, e_j)_{L^2(\Omega)},$$
$$ \int_{\Omega} \chi_{\omega} \left( \sum_{i=1}^{N} e_i [\hat{u}_i(t) + d_i(t)] \right) e_j \, dx = \sum_{i=1}^{N} [\hat{u}_i(t) + d_i(t)] (e_i, e_j)_{L^2(\omega)}. $$

Then, we deduce

$$ \left\{ \begin{aligned} y_1'(t) &= -\tau_1 y_1(t) + \sum_{i=1}^{N} \tilde{u}_i(t) (e_i, e_1)_{L^2(\omega)} + \sum_{i=1}^{N} (\hat{u}_i(t) + d_i(t)) (e_i, e_1)_{L^2(\omega)} \\ \vdots &\vdots \\ y_N'(t) &= -\tau_N y_N(t) + \sum_{i=1}^{N} \tilde{u}_i(t) (e_i, e_N)_{L^2(\omega)} + \sum_{i=1}^{N} (\hat{u}_i(t) + d_i(t)) (e_i, e_N)_{L^2(\omega)}. \end{aligned} \right. \quad (4.1) $$

Thus, with the aid of the matrices

$$ X_N(t) := \begin{pmatrix} y_1(t) \\ y_2(t) \\ \vdots \\ y_N(t) \end{pmatrix}, \quad U_N(t) := \begin{pmatrix} \tilde{u}_1(t) \\ \tilde{u}_2(t) \\ \vdots \\ \tilde{u}_N(t) \end{pmatrix}, \quad A_N := \begin{pmatrix} -\tau_1 & & & \\ & -\tau_2 & & \\ & & \ddots & \\ & & & -\tau_N \end{pmatrix} \\ F_N := \begin{pmatrix} \hat{u}_1(t) + d_1(t) \\ \hat{u}_2(t) + d_2(t) \\ \vdots \\ \hat{u}_N(t) + d_N(t) \end{pmatrix}, $$

and using the definition of $J_N$, we can construct a finite system

$$ \dot{X}_N(t) = A_N X_N(t) + J_N U_N(t) + J_N F_N. $$

## 4.1 Design of $\tilde{u}$

In this section, we present the construction of the feedback control law and introduce a Lyapunov function designed for stability analysis. For a given parameter $\lambda$ (and consequently, a fixed $N$), we select constants $\gamma_{\lambda}, \mu_{\lambda} > 0$, which will be specified later. Following the approach suggested in [41], we define the feedback control law as:

$$ U_N(y(t)) := -\gamma_{\lambda} X_N(t). $$

To analyze the stability of the closed-loop system, we introduce the following Lyapunov function, referred to as the Frequency Lyapunov function

$$ V(y) := \mu_{\lambda} \|X_N\|^2 + \|P_N^{\perp} y\|_{L^2(\Omega)}^2, \quad \forall y \in L^2(\Omega). $$

Here, $\|X_N\|^2$ denotes the Euclidean norm $X_N^T X_N = \sum_{i=1}^N y_i^2$, which is equivalent to $\|P_N y\|_{L^2(\Omega)}^2$. For any initial state $y_0 \in H_0^1(\Omega)$, the time derivative of $V(y(t))$ is computed as follows
$$
\begin{align*}
\frac{d}{dt} V(y(t)) &= \frac{d}{dt} (\mu_{\lambda} \|X_N\|^2) + \frac{d}{dt} (P_N^{\perp} y, P_N^{\perp} y)_{L^2(\Omega)} \\
&= \mu_{\lambda} \frac{d}{dt} \sum_{i=1}^{N} y_i^2 + \frac{d}{dt} \left( \sum_{i=N+1}^{\infty} y_i e_i, \sum_{i=N+1}^{\infty} y_i e_i \right)_{L^2(\Omega)} \\
&= \mu_{\lambda} \frac{d}{dt} \|X_N\|^2 + 2 \left\langle P_N^{\perp} y, \frac{d}{dt} y \right\rangle_{H_0^1(\Omega), H^{-1}(\Omega)}.
\end{align*}
$$

Expanding the derivative of ‖X_N‖^2 further, we obtain

$$
\begin{align*}
\mu_{\lambda} \frac{d}{dt} \|X_N\|^2 &= \mu_{\lambda} (\dot{X}_N^{\top} X_N + X_N^{\top} \dot{X}_N) \\
&= \mu_N \left[ (A_N X_N - \gamma_{\lambda} J_N X_N + J_N F_N)^{\top} X_N + X_N^{\top} (A_N X_N - \gamma_{\lambda} J_N X_N + J_N F_N) \right] \\
&= \mu_{\lambda} \left[ X_N^{\top} (2A_N - 2\gamma_{\lambda} J_N) X_N + 2F_N^{\top} J_N X_N \right] \\
&= 2\mu_{\lambda} X_N^{\top} A_N X_N - 2\mu_{\lambda} \gamma_{\lambda} X_N^{\top} J_N X_N + 2\mu_{\lambda} F_N^{\top} J_N X_N.
\end{align*}
$$

By using (3.1), we have

$$
\begin{align*}
\mu_{\lambda} \frac{d}{dt} \|X_N\|_2^2 &\le -2\mu_{\lambda}\gamma_{\lambda}X_N^{\top}J_NX_N + 2\mu_{\lambda}F_N^{\top}J_NX_N \\
&\le -2\mu_{\lambda}\gamma_{\lambda}C_{\lambda}\|X_N\|^2 + 2\mu_{\lambda}\sum_{j=1}^N \sum_{i=1}^N (\hat{u}_i(t) + d_i(t)) (e_i, e_j)_{L^2(\omega)} y_j(t) \\
&= -2\mu_{\lambda}\gamma_{\lambda}C_{\lambda}\|X_N\|^2 + 2\mu_{\lambda}((\hat{u} + d), \chi_{\omega}P_Ny)_{L^2(\Omega)}.
\end{align*}
$$

On the other hand, by using the Cauchy–Schwarz inequality, weak spectral inequality (2.1), and Cauchy-$\epsilon$
inequality with $\epsilon = \frac{\gamma_{\lambda}}{\lambda} > 0$, it follows
$$
\begin{align*}
& 2 \left\langle P_N^{\perp} y, \frac{d}{dt} y \right\rangle_{H_0^1(\Omega) \times H^{-1}(\Omega)} \\
&= 2 \left\langle P_N^{\perp} y, \Delta y - \gamma_{\lambda} \chi_{\omega} \left( \sum_{i=1}^N e_i y_i(t) \right) + \chi_{\omega}(\hat{u} + d) \right\rangle_{H_0^1(\Omega) \times H^{-1}(\Omega)} \\
&= 2 \left( P_N^{\perp} y, \Delta y \right)_{L^2(\Omega)} - 2\gamma_{\lambda} \left( P_N^{\perp} y, \chi_{\omega} \left( \sum_{i=1}^N e_i y_i \right) \right)_{L^2(\Omega)} + 2 \left( P_N^{\perp} y, \chi_{\omega}(\hat{u} + d) \right)_{L^2(\Omega)} \\
&= 2 \sum_{j=N+1}^{\infty} y_j^2 (e_j, \Delta e_i)_{L^2(\Omega)} - 2\gamma_{\lambda} \left( P_N^{\perp} y, \chi_{\omega} P_N y \right)_{L^2(\Omega)} + 2 \left( P_N^{\perp} y, \chi_{\omega}(\hat{u} + d) \right)_{L^2(\Omega)} \\
&\leq -2 \sum_{j=N+1}^{\infty} y_j^2 \tau_j + 2\gamma_{\lambda} \left( P_N^{\perp} y, P_N y \right)_{L^2(\omega)} + 2 \left( P_N^{\perp} y, \chi_{\omega}(\hat{u} + d) \right)_{L^2(\Omega)} \\
&\leq -2 \sum_{j=N+1}^{\infty} \lambda y_j^2 + 2\gamma_{\lambda} \left\| P_N^{\perp} y \right\|_{L^2(\omega)} \| P_N y \|_{L^2(\omega)} + 2 \left( P_N^{\perp} y, \chi_{\omega}(\hat{u} + d) \right)_{L^2(\Omega)} \\
&\leq -2\lambda \left\| P_N^{\perp} y \right\|_{L^2(\Omega)}^2 + \lambda \left\| P_N^{\perp} y \right\|_{L^2(\Omega)}^2 + \frac{\gamma_{\lambda}^2}{\lambda} \| P_N y \|_{L^2(\Omega)}^2 + 2 \left( \chi_{\omega} P_N^{\perp} y, (\hat{u} + d) \right)_{L^2(\Omega)} \\
&= -\lambda \left\| P_N^{\perp} y \right\|_{L^2(\Omega)}^2 + \frac{\gamma_{\lambda}^2}{\lambda} \| X_N \|_{L^2(\Omega)}^2 + 2 \left( (\hat{u} + d), \chi_{\omega} P_N^{\perp} y \right)_{L^2(\Omega)}.
\end{align*}
$$

Therefore,

$$
\frac{d}{dt} V(y(t)) \leq \left(-2\mu_{\lambda}\gamma_{\lambda}C_{\lambda} + \frac{\gamma_{\lambda}^2}{\lambda}\right) \|X_N\|^2 - \lambda \left\|P_N^{\perp}y\right\|_{L^2(\Omega)}^2 + 2\left((\hat{u} + d), \mu_{\lambda}\chi_{\omega}P_Ny + \chi_{\omega}P_N^{\perp}y\right)_{L^2(\Omega)}.
$$

4.2 Design of $\hat{u}$

To attenuate the effect of the unknown disturbance $d(t, x)$, we now construct the control term $\hat{u}$. Let $H = L^2(\Omega)$, and recall the key property of the multivalued sign operator in $L^2(\Omega)$

$$
\int_{\Omega} \theta f \, dx = \|f\|_{L^2(\Omega)}, \quad \forall f \in L^2(\Omega), \quad \forall \theta \in \operatorname{sign}_{L^2(\Omega)}(f).
$$

Since the disturbance is supposed to be unknown, we can not choose $\hat{u}(t, x) = -d(t, x)$. Therefore, thanks
to the Cauchy-Schwarz inequality, we have

$$
\begin{align*}
& 2 \left( (\hat{u} + d), \mu_{\lambda} \chi_{\omega} P_N y + \chi_{\omega} P_N^{\perp} y \right)_{L^2(\Omega)} \\
&= 2 \left( \hat{u}, \chi_{\omega} (\mu_{\lambda} P_N y + P_N^{\perp} y) \right)_{L^2(\Omega)} + 2 \left( d, \chi_{\omega} (\mu_{\lambda} P_N y + P_N^{\perp} y) \right)_{L^2(\Omega)} \\
&\leq \left( \hat{u}, \chi_{\omega} (\mu_{\lambda} P_N y + P_N^{\perp} y) \right)_{L^2(\Omega)} + 2 \|d(t, \cdot)\|_{L^2(\Omega)} \left\| \chi_{\omega} (\mu_{\lambda} P_N y + P_N^{\perp} y) \right\|_{L^2(\Omega)}
\end{align*}
$$

Under assumption (A2), the disturbance satisfies $\|d(t, \cdot)\|_{L^2(\Omega)} \le D$. Then, it suffices to consider the
feedback law

$$ \hat{u}(t, x) = -D \operatorname{sign}_{L^2(\Omega)} \left[ \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right], $$

thereby guaranteeing that

$$ \begin{aligned} & 2 \left( (\hat{u} + d), \mu_{\lambda} \chi_{\omega} P_N y + \chi_{\omega} P_N^{\perp} y \right)_{L^2(\Omega)} \\ & \leq -2D \left( \operatorname{sign}_{L^2(\Omega)} \left[ \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right], \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right)_{L^2(\Omega)} + 2D \left\| \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right\|_{L^2(\Omega)} \\ & \leq -2D \left\| \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right\|_{L^2(\Omega)} + 2D \left\| \chi_{\omega}(\mu_{\lambda} P_N y + P_N^{\perp} y) \right\|_{L^2(\Omega)} = 0. \end{aligned} $$

From the previous derivations, we obtain

$$ \frac{d}{dt} V(y(t)) \leq \left( -2\mu_{\lambda}\gamma_{\lambda}C_{\lambda} + \frac{\gamma_{\lambda}^2}{\lambda} \right) \|X_N\|^2 - \lambda \|P_N^{\perp}y\|_{L^2(\Omega)}^2. \quad (4.2) $$

We now fix the parameters

$$ \gamma_{\lambda} := \frac{\lambda}{C_{\lambda}}, \quad \mu_{\lambda} := \frac{\gamma_{\lambda}^2}{\lambda^2} = C_{\lambda}^{-2}, $$

which, when substituted into (4.2), yield

$$ \begin{aligned} \frac{d}{dt} V(y(t)) & \leq (-2\mu_{\lambda}\lambda + \mu_{\lambda}\lambda) \|X_N\|^2 - \lambda \|P_N^{\perp}y\|_{L^2(\Omega)}^2 \\ & = -\mu_{\lambda}\lambda \|X_N\|^2 - \lambda \|P_N^{\perp}y\|_{L^2(\Omega)}^2 \\ & = -\lambda (\mu_{\lambda} \|X_N\|^2 + \|P_N^{\perp}y\|_{L^2(\Omega)}^2) = -\lambda V(y(t)). \end{aligned} $$

This differential inequality implies the exponential decay of $V(y(t))$. Indeed, we observe that

$$ \frac{d}{dt} \left( e^{\lambda t} V(y(t)) \right) = e^{\lambda t} \left( \frac{d}{dt} V(y(t)) + \lambda V(y(t)) \right) \leq 0. $$

Upon integration, this becomes

$$ V(y(t)) \leq e^{-\lambda t} V(y(0)). $$

Let us introduce the constants

$$ \alpha := \min \{1, \sqrt{\mu_{\lambda}}\} = \min \left\{1, \frac{1}{C_{\lambda}}\right\}, \quad \beta := \max \{1, \sqrt{\mu_{\lambda}}\} = \max \left\{1, \frac{1}{C_{\lambda}}\right\}. $$

Using these constants, we can bound the Lyapunov functional as follows

$$ \begin{aligned} V(y(t)) &= \mu_{\lambda} \|P_N y(t)\|_{L^2(\Omega)}^2 + \|P_N^{\perp} y(t)\|_{L^2(\Omega)}^2 \\ &\geq \alpha^2 (\|P_N y(t)\|_{L^2(\Omega)}^2 + \|P_N^{\perp} y(t)\|_{L^2(\Omega)}^2) = \alpha^2 \|y(t)\|_{L^2(\Omega)}^2, \end{aligned} $$
and similarly,

$$V(y(0)) \leq \left( \beta^2 \|P_N y(0)\|_{L^2(\Omega)}^2 + \beta^2 \|P_N^{\perp} y(0)\|_{L^2(\Omega)}^2 \right) = \beta^2 \|y(0)\|_{L^2(\Omega)}^2.$$

Combining these bounds with the exponential decay of $V(y(t))$, we conclude that

$$\alpha^2 \|y(t)\|_{L^2(\Omega)}^2 \leq V(y(t)) \leq e^{-\lambda t} V(y(0)) \leq \beta^2 e^{-\lambda t} \|y(0)\|_{L^2(\Omega)}^2.$$

Taking square roots, we obtain the exponential stability estimate

$$\|y(t)\|_{L^2(\Omega)} \leq \frac{\beta}{\alpha} e^{-\frac{\lambda}{2}t} \|y(0)\|_{L^2(\Omega)}.$$

## 4.3 Feedback law

The goal of this section is to derive the feedback law. We begin by constructing the associated feedback operator. The feedback control system is defined through the following operators:

1. The linear feedback operator $\mathcal{F}: L^2(\Omega) \to L^2(\Omega)$:

$$\mathcal{F}\phi = -\gamma_{\lambda} \left( \sum_{m=1}^{N} (\phi, e_i)_{L^2(\Omega)} e_i \right) = -\gamma_{\lambda} P_N \phi, \text{ with } \gamma_{\lambda} = C_{\lambda}^{-1} \lambda.$$

2. The nonlinear set-valued operator $\mathcal{B}: L^2(\Omega) \to 2^{L^2(\Omega)}$:

$$\mathcal{B}\phi := -D \operatorname{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}\phi), \quad \text{where } \mathcal{C}\phi := \mu_{\lambda}P_N y + P_N^{\perp}y.$$

Finally, recalling that the total control is given by the decomposition $u = \tilde{u} + \hat{u}$, given by

$$\tilde{u}(t, x) = \mathcal{F}(y(t, x)), \quad \hat{u}(t, x) = \mathcal{B}(y(t, x)),$$

we obtain the explicit form of the feedback law

$$u(t, x) = -\gamma_{\lambda} P_N y - D \operatorname{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}y). \qquad (4.3)$$

Then, the closed-loop System is given by

$$\left\{
\begin{array}{ll}
\partial_t y + \mathcal{A} y - \chi_{\omega} \mathcal{F} y - \chi_{\omega} \mathcal{B} y \ni \chi_{\omega} d(t) & \text{in } (0, \infty) \times \Omega \\
y = 0 & \text{on } (0, \infty) \times \partial \Omega \\
y(0, x) = y_0(x) & \text{in } \Omega.
\end{array}
\right.
\qquad (4.4)$$

## 5 Well-Posedness

In this section, we present the well-posedness of the closed-loop system (4.4), which is a differential inclusion, through maximal monotone operator theory. To this end, we reformulate the system in an abstract setting by introducing the multivalued operator
$$ egin{cases} A : D(A) \subset L^2(\Omega) \longrightarrow 2^{L^2(\Omega)} \\ D(A) = D(\mathcal{A}) \\ A(y) = -\Delta y + \gamma_{\lambda}\chi_{\omega}P_Ny + \chi_{\omega}D \operatorname{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}y). \end{cases} \qquad (5.1) $$

It follows that the resulting closed-loop system is the differential inclusion

$$ egin{cases} y'(t) + Ay(t) \ni \chi_{\omega}d(t), & t > 0 \\ y(0) = y_0. \end{cases} \qquad (5.2) $$

We define the inner product $\langle \cdot, \cdot \rangle_{\mu}$ on $L^2(\Omega)$ by

$$ (u, v)_{\mu} = \mu(P_N u, P_N v) + (P_N^{\perp} u, P_N^{\perp} v). $$

This bilinear form induces a norm $\|u\|_{\mu}^2 = (u, u)_{\mu} = \mu\|P_N u\|^2 + \|P_N^{\perp} u\|^2$, which is equivalent to the original norm on $L^2(\Omega)$. Since $\mu > 1$, we have $\|u\|_{L^2(\Omega)}^2 \le (u, u)_{\mu} \le \mu\|u\|_{L^2(\Omega)}^2$, it follows that $(L^2(\Omega), \langle \cdot, \cdot \rangle_{L^2(\Omega)})$ and $(L^2(\Omega), \langle \cdot, \cdot \rangle_{\mu})$ define the same topology. Moreover, notice that $V(y(t)) = \|u\|_{\mu}^2 = \mu(u, P_N v) + (u, P_N^{\perp} v) = (u, \mathcal{C}v)$, which makes natural the choice of the inner product and will play a central role in this section.

To establish the well-posedness of the system, we will employ the maximal monotone operator theory. In this framework, we present two key results: the first one (Proposition 5.1) states that the operator $A$ is monotone, while the second one (Proposition 5.2) states that the operator $I + A$ is surjective.

**Proposition 5.1.** The operator $A$ defined by (5.1) is monotone.

*Proof.* Let $y_1, y_2 \in D(A)$, there exists $\eta_i \in \operatorname{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}y_i)$, $i = 1, 2$. Note that we can decompose $A$ into linear and nonlinear parts. Its linear part is given by

$$ A_1 y := -\Delta y + \gamma_{\lambda}\chi_{\omega}P_N y, $$

and the nonlinear part is

$$ By = \chi_{\omega}D \operatorname{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}y). \qquad (5.3) $$

We want to prove

$$ \langle A y_1 - A y_2, y_1 - y_2 \rangle_{\mu} = \langle A_1 y_1 - A_1 y_2, y_1 - y_2 \rangle_{\mu} + \langle B y_1 + B y_2, y_1 - y_2 \rangle_{\mu} \ge 0. $$

To analyze the linear part, we consider $z = y_1 - y_2$. This gives us

$$ \begin{aligned} \langle A_1 z, z \rangle_{\mu} = & -\mu_{\lambda} \langle \Delta z, P_N z \rangle_{L^2(\Omega)} - \langle \Delta z, P_N^{\perp} z \rangle_{L^2(\Omega)} + \mu_{\lambda}\gamma_{\lambda} \langle \chi_{\omega}P_N z, P_N z \rangle_{L^2(\Omega)} \\ & + \gamma_{\lambda} \langle \chi_{\omega}P_N z, P_N^{\perp} z \rangle_{L^2(\Omega)}. \end{aligned} \qquad (5.4) $$

Notice that, by using integration by parts in the first and second term of (5.4), we get
$$
\begin{align*}
& - \mu_{\lambda} \langle \Delta z, P_N z \rangle_{L^2(\Omega)} - \langle \Delta z, P_N^{\perp} z \rangle_{L^2(\Omega)} \\
&= \mu_{\lambda} \sum_{i=1}^{N} z_i^2 (\nabla e_i, \nabla e_i)_{L^2(\Omega)^n} - \frac{1}{2} \sum_{i=N+1}^{\infty} y_i^2 (\Delta e_i, e_i)_{L^2(\Omega)} + \frac{1}{2} \sum_{i=N+1}^{\infty} z_i^2 (\nabla e_i, \nabla e_i)_{L^2(\Omega)^n} \\
&= \mu_{\lambda} \|\nabla(P_N z)\|_{L^2(\Omega)}^2 + \frac{1}{2} \sum_{i=N+1}^{\infty} \tau_i z_i^2 + \|\nabla(P_N^{\perp} z)\|_{L^2(\Omega)}^2 \\
&\geq \mu_{\lambda} \|\nabla(P_N z)\|_{L^2(\Omega)}^2 + \frac{1}{2} \|\nabla(P_N^{\perp} z)\|_{L^2(\Omega)}^2 + \frac{\lambda}{2} \|P_N^{\perp} z\|_{L^2(\Omega)}^2 \\
&\geq \frac{1}{2} \|\nabla z\|_{L^2(\Omega)}^2 + \frac{\lambda}{2} \|P_N^{\perp} z\|_{L^2(\Omega)}^2. \tag{5.5}
\end{align*}
$$

Furthermore, using (3.1) and the Cauchy-$\epsilon$ inequality in the third and fourth term of (5.4), taking $\epsilon = \frac{\sqrt{2}}{C_\lambda}$, it yields

$$
\begin{align*}
& \mu_{\lambda}\gamma_{\lambda} \langle \chi_{\omega}P_N z, P_N z \rangle_{L^2(\Omega)} \\
& \geq \mu_{\lambda}\gamma_{\lambda}\|P_N y\|_{L^2(\omega)}^2 - \gamma_{\lambda}\|P_N y\|_{L^2(\Omega)}\|P_N^{\perp} y\|_{L^2(\omega)} \\
& \geq \mu_{\lambda}\gamma_{\lambda}C_{\lambda}\|P_N y\|_{L^2(\Omega)}^2 - \frac{\gamma_{\lambda}\epsilon}{2}\|P_N y\|_{L^2(\Omega)}^2 - \frac{\gamma_{\lambda}}{2\epsilon}\|P_N^{\perp} y\|_{L^2(\Omega)}^2 \\
& = \left( \mu_{\lambda}\gamma_{\lambda}C_{\lambda} - \frac{\gamma_{\lambda}\sqrt{2}}{2C_{\lambda}} \right) \|P_N y\|_{L^2(\Omega)}^2 - \frac{\gamma_{\lambda}C_{\lambda}}{2\sqrt{2}} \|P_N^{\perp} y\|_{L^2(\Omega)}^2. \tag{5.6}
\end{align*}
$$

Now, using (5.5) and (5.6) in (5.4), gathering terms and recalling that $\gamma_{\lambda} = \frac{\lambda}{C_{\lambda}}$ and $\mu_{\lambda} = \frac{1}{C_{\lambda}^2}$. It follows

$$
\begin{align}
\langle A_1 z, z \rangle_{\mu} &\ge \frac{1}{2} \|\nabla z\|_{L^2(\Omega)}^2 + \left( \frac{\lambda}{C_{\lambda}^2} - \frac{\lambda \sqrt{2}}{2C_{\lambda}^2} \right) \|P_N z\|_{L^2(\Omega)}^2 + \left( \frac{\lambda}{2} - \frac{\lambda \sqrt{2}}{4} \right) \|P_N^{\perp} z\|_{L^2(\Omega)}^2 \nonumber \\
&\ge \frac{1}{2} \|\nabla z\|_{L^2(\Omega)}^2 + \frac{\lambda}{C_{\lambda}^2} \left( 1 - \frac{\sqrt{2}}{2} \right) \|P_N z\|_{L^2(\Omega)}^2 + \frac{\lambda}{2} \left( 1 - \frac{\sqrt{2}}{2} \right) \|P_N^{\perp} z\|_{L^2(\Omega)}^2 \tag{5.7} \\
&= C \left( \|\nabla z\|_{L^2(\Omega)}^2 + \|z\|_{L^2(\Omega)}^2 \right) = C \|z\|_{H_0^1(\Omega)}^2 \ge 0, \nonumber
\end{align}
$$

where $C = \min \left\{ \frac{1}{2}, \frac{\lambda}{C_{\lambda}^2} \left( 1 - \frac{\sqrt{2}}{2} \right), \frac{\lambda}{2} \left( 1 - \frac{\sqrt{2}}{2} \right) \right\}$.

For the nonlinear part, we have
$$
\begin{align*}
\langle By_1 - By_2, y_1 - y_2 \rangle_{\mu} &= \mu_{\lambda} \langle \chi_{\omega} \eta_1 - \chi_{\omega} \eta_2, P_N(y_1 - y_2) \rangle_{L^2(\Omega)} + \langle \chi_{\omega} \eta_1 - \chi_{\omega} \eta_2, P_N^{\perp}(y_1 - y_2) \rangle_{L^2(\Omega)} \\
&= \langle \eta_1 - \eta_2, \chi_{\omega} \mu P_N(y_1 - y_2) \rangle_{L^2(\Omega)} + \langle \eta_1 - \eta_2, \chi_{\omega} P_N^{\perp}(y_1 - y_2) \rangle_{L^2(\Omega)} \\
&= \langle \eta_1 - \eta_2, \chi_{\omega} \mu_{\lambda} P_N(y_1 - y_2) + \chi_{\omega} (P_N^{\perp}(y_1 - y_2)) \rangle_{L^2(\Omega)} \\
&= \langle \eta_1 - \eta_2, \chi_{\omega} \mathscr{C} y_1 - \chi_{\omega} \mathscr{C} y_2 \rangle_{L^2(\Omega)} \geq 0.
\end{align*}
$$

The last inequality is a consequence of the multivalued sign operator.

Therefore, we get

$$
\langle Ay_1 - Ay_2, y_1 - y_2 \rangle_{\mu} \geq \langle A_1 y_1 - A_1 y_2, y_1 - y_2 \rangle_{\mu} + \langle By_1 + By_2, y_1 - y_2 \rangle_{\mu} \geq 0. \quad (5.8)
$$

<figure><img src="image_2.png" /></figure>

**Remark 5.1.** The choice of the $\mu$-inner product is particularly useful in the nonlinear part of the proof.
This definition yields a natural monotonicity property, which is key to handling the sign operator.

**Proposition 5.2.** The operator $A$ defined by 5.1 satisfies $R(I + A) = L^2(\Omega)$.

*Proof.* Given a $f \in L^2(\Omega)$, we need to show the existence of $y \in D(A)$ such that
$$y + Ay \ni f,$$

or equivalently

$$
y - \Delta y + \gamma_{\lambda} \chi_{\omega} P_{N}(y) + \chi_{\omega} D \mathrm{sign}_{L^{2}(\Omega)}(\chi_{\omega} \mathcal{C} y) \ni f. \quad (5.9)
$$

To handle the set-valued signum nonlinearity, we employ a regularization argument based on the Yosida approximation. Let $\varphi(y) = D\|y\|_{L^2(\Omega)}$, whose subdifferential is $\partial\varphi(y) = D \cdot \mathrm{sign}_{L^2(\Omega)}(y)$. For $\sigma > 0$, the Moreau Regularization [42, Chapter IV, Proposition 1.8] of $\varphi(y)$ is given by

$$
\begin{align*}
\varphi_{\sigma}(y) &= \min \left\{ \frac{1}{2\sigma} \|y-z\|^2 + \varphi(z) : z \in L^2(\Omega) \right\} \\
&= \frac{1}{\sigma} \|y - J_{\sigma}y\|^2 + \varphi(J_{\sigma}(y)) \\
&= \frac{\sigma}{2} \|\alpha_{\sigma}(y)\|^2 + \varphi(J_{\sigma}(y)),
\end{align*}
$$

where $\alpha_{\sigma}$ is the Yosida approximation and $J_{\sigma}$ is the resolvent. Moreover, we have $\alpha_{\sigma}(y) = (\partial\varphi)_{\sigma}(y) = \nabla\varphi_{\sigma}(y)$. Now, we define the regularized operator

$$
B_{\sigma}(y) := D\chi_{\omega}\alpha_{\sigma}(\chi_{\omega}\mathcal{C}y).
$$

Note that $B_{\sigma} : L^2(\Omega) \to L^2(\Omega)$ is single-valued and satisfies $\|B_{\sigma}(y)\|_{L^2(\Omega)} \le D$ for all $y \in L^2(\Omega)$.
Indeed, by the properties of the Yosida approximation, we have $\|\alpha_{\sigma}(y)\|_{L^2(\Omega)} = \|\partial\varphi_{\sigma}(y)\|_{L^2(\Omega)}$ and
$\|\partial\varphi_{\sigma}(y)\|_{L^2(\Omega)} \le \|\partial\varphi(y)\|_{L^2(\Omega)}$. Then, using the definition of the multivalued sign operator, for any
$\eta \in \mathrm{sign}_{L^2(\Omega)}(\chi_{\omega}\mathcal{C}u)$, we have to consider two cases

1. If $\chi_{\omega} \mathscr{C} u \neq 0$ in $L^2(\Omega)$, then
$$ \eta = \frac{\chi_{\omega} \mathcal{C} u}{\| \chi_{\omega} \mathcal{C} u \|_{L^2(\Omega)}}. $$

By construction, $\|\eta\|_{L^2(\Omega)} = 1$.

2. If $\chi_{\omega} \mathcal{C} u = 0$ in $L^2(\Omega)$, then:

$$ \eta \in \{h \in L^2(\Omega) \mid \|h\|_{L^2(\Omega)} \le 1\}. $$

Thus, $\|\eta\|_{L^2(\Omega)} \le 1$.

Therefore,

$$ \|D\chi_{\omega}\eta\|_{L^2(\Omega)} = D\left(\int_{\omega} |\eta|^2 dx\right)^{1/2} \le D\left(\int_{\Omega} |\eta|^2 dx\right)^{1/2} \le D. $$

We now consider the regularized equation: find $y_{\sigma} \in H_0^1(\Omega)$ such that

$$ y_{\sigma} - \Delta y_{\sigma} + \gamma_{\lambda} \chi_{\omega} P_N(y_{\sigma}) = f - B_{\sigma}(y_{\sigma}). \qquad (5.10) $$

We proceed via the Schauder Fixed Point Theorem. Consider the map

$$ T_{\sigma}: L^2(\Omega) \to L^2(\Omega) \\ u \mapsto T(u) = y_{\sigma, u}, $$

where $y_{\sigma, u}$ is solution of

$$ y_{\sigma, u} - \Delta y_{\sigma, u} + \gamma_{\lambda} \chi_{\omega} P_N(y_{\sigma, u}) = f - B_{\sigma}(u). \qquad (5.11) $$

For a regular solution $y$ of (5.11) we take the $\mu$-inner product with a test function $z$, to derive the variational formulation $a(y, z) = (f - B_{\sigma}, z)_{\mu}$, where the bilinear form $a: H_0^1(\Omega) \times H_0^1(\Omega) \to \mathbb{R}$ is given by

$$ a(y, z) = (y, z)_{\mu} + (\nabla y, \nabla z)_{\mu} + (\gamma_{\lambda} \chi_{\omega} P_N(y), z)_{\mu}, $$

We have that $a(\cdot, \cdot)$ is continuous and coercive on $H_0^1(\Omega)$. Indeed, for $y \in H_0^1(\Omega)$, we have

$$ a(y, y) = \|y\|_{\mu}^{2} + \|\nabla y\|_{\mu}^{2} + \gamma_{\lambda}\mu\|P_N y\|_{L^2(\omega)}^{2} + \gamma_{\lambda} \int_{\omega} P_N(y) P_{\perp}^{\perp}(y) dx. $$

By using (5.7), we have

$$ a(y, y) \ge \|y\|_{\mu}^{2} + C \|y\|_{H_{0}^{1}(\Omega)}^{2} \ge \beta \|y\|_{H_{0}^{1}(\Omega)}^{2}, $$

where $\beta = \min\{1, C\}$.

For any $u \in L^2(\Omega)$, the right-hand side of (5.11), $f - B_{\sigma}(u)$, belongs to $L^2(\Omega)$. Thus, by the Lax-Milgram Theorem, there exists a unique solution $y_{\sigma, u} \in H_0^1(\Omega)$ to the variational formulation

$$ (y, z)_{\mu} + (\nabla y, \nabla z)_{\mu} + (\gamma_{\lambda} \chi_{\omega} P_N(y), z)_{\mu} = (f - B_{\sigma}(u), z)_{\mu}, \quad \forall z \in H_{0}^{1}(\Omega). $$
By definition on $\mu$-inner product, we can note that

$$ (w, v)_{\mu} = \mu_{\lambda}(w, P_{N}v)_{L^{2}(\Omega)} + (w, P_{N}^{\perp}v)_{L^{2}(\Omega)} = (w, \mu_{\lambda}P_{N}v + P_{N}^{\perp}v)_{L^{2}(\Omega)} = (w, \mathscr{C}v)_{L^{2}(\Omega)}. $$

Now, since $\nabla$ and $\mathscr{C}$ commute, we have

$$ (y, \mathscr{C}z)_{L^2(\Omega)} + (\nabla y, \nabla \mathscr{C}z)_{L^2(\Omega)} + (\gamma_{\lambda}\chi_{\omega}P_{N}(y), \mathscr{C}z)_{L^2(\Omega)} = (f - B_{\sigma}(u), \mathscr{C}z)_{L^2(\Omega)}, \quad \forall z \in H_{0}^{1}(\Omega). $$

Notice that $\mathscr{C}: H_0^1(\Omega) \to H_0^1(\Omega)$ is an isomorphism (Appendix C). Then the variational formulation

$$ a(y, \mathscr{C}z) = F(\mathscr{C}z), \quad \forall z \in H_0^1 $$

is equivalent to

$$ a(y, w) = F(w), \quad \forall w \in H_0^1, $$

by taking $w = \mathscr{C}z$ and noting $z = \mathscr{C}^{-1}w \in H_0^1$. It follows that

$$ (y, w)_{L^2(\Omega)} + (\nabla y, \nabla w)_{L^2(\Omega)} + \gamma_{\lambda}(\chi_{\omega}P_{N}(y), w)_{L^2(\Omega)} = (f - B_{\sigma}(u), w)_{L^2(\Omega)}, \quad \forall w \in H_{0}^{1}(\Omega). $$

From which we have $y_{\sigma,u} - \Delta y_{\sigma,u} + \gamma_{\lambda}\chi_{\omega}P_{N}(y_{\sigma,u}) = f - B_{\sigma}(u)$, in the sense of distributions.

Since $y_{\sigma,u} \in H_0^1(\Omega)$, the term $\gamma_{\lambda}\chi_{\omega}P_{N}(y_{\sigma,u})$ belongs to $L^2(\Omega)$, as $P_{N}(y_{\sigma,u}) \in L^2(\Omega)$ and $\chi_{\omega}$ is bounded.

Therefore, the entire right-hand side $f - B_{\sigma}(u) - \gamma_{\lambda}\chi_{\omega}P_{N}(y_{\sigma,u})$ lies in $L^2(\Omega)$. By standard elliptic regularity theory [5, Theorem 9.25], we conclude that $y_{\sigma,u} \in H^2(\Omega)$. Moreover, since $y_{\sigma,u} \in H^2(\Omega)$, we get

$$ -\Delta y_{\sigma,u} + y_{\sigma,u} + \gamma_{\lambda}\chi_{\omega}P_{N}y_{\sigma,u} = f - B_{\sigma}(u) \quad \text{a.e. on } \Omega. \qquad (5.12) $$

Now, Let $M > 0$, to be selected later, and define $K_M := \{v \in H_0^1(\Omega) \cap H^2(\Omega) : \|v\|_{H^2(\Omega)} \le M\}$.

* Upper bound for $\|y_{\sigma,u}\|_{L^2(\Omega)}$ independent of $\sigma \in (0, \infty)$.

Let $y_{\sigma,u} \in K_M$, testing (5.11) with $y_{\sigma,u}$ in the $\mu$-inner product gives

$$ (y_{\sigma,u}, y_{\sigma,u})_{\mu} + (-\Delta y_{\sigma,u}, y_{\sigma,u})_{\mu} + \gamma_{\lambda}(\chi_{\omega}P_{N}y_{\sigma,u}, y_{\sigma,u})_{\mu} = (f - B_{\sigma}(u), y_{\sigma,u})_{\mu}. \qquad (5.13) $$

By using Cauchy-Schwarz and Holder inequalities in the right-hand side of (5.13), we obtain the a priori estimate

$$ \begin{aligned} (f - B_{\sigma}(u), y_{\sigma,u})_{\mu} &\le \mu (f - B_{\sigma}(u), P_{N}y_{\sigma,u})_{L^2(\Omega)} + (f - B_{\sigma}(u), P_{N}^{\perp}y_{\sigma,u})_{L^2(\Omega)} \\ &\le \mu \|f - B_{\sigma}(u)\|_{L^2(\Omega)} \|P_{N}y_{\sigma,u}\|_{L^2(\Omega)} + \|f - B_{\sigma}(u)\|_{L^2(\Omega)} \|P_{N}^{\perp}y_{\sigma,u}\|_{L^2(\Omega)} \\ &\le \left(\frac{\mu}{2} + \frac{1}{2}\right) \|f - B_{\sigma}(u)\|_{L^2(\Omega)}^2 + \frac{\mu_{\lambda}}{2} \|P_{N}y_{\sigma,u}\|_{L^2(\Omega)}^2 + \frac{1}{2} \|P_{N}^{\perp}y_{\sigma,u}\|_{L^2(\Omega)}^2 \\ &\le \frac{1}{2}(\mu_{\lambda} + 1) \left(2\|f\|_{L^2(\Omega)}^2 + 2\|B_{\sigma}(u)\|_{L^2(\Omega)}^2\right) + \frac{\mu_{\lambda}}{2} \|P_{N}y_{\sigma,u}\|_{L^2(\Omega)}^2 + \frac{1}{2} \|P_{N}^{\perp}y_{\sigma,u}\|_{L^2(\Omega)}^2. \end{aligned} \qquad (5.14) $$

On the other hand, note that $(y_{\sigma,u}, y_{\sigma,u})_{\mu} = \mu_{\lambda}\|P_{N}y_{\sigma,u}\|_{L^2(\Omega)}^2 + \|P_{N}^{\perp}y_{\sigma,u}\|_{L^2(\Omega)}^2$, and by using (5.7) in (5.13)
we get $(-\Delta y_{\sigma,u}, y_{\sigma,u})_\mu + \gamma_\lambda(\chi_\omega P_N y_{\sigma,u}, y_{\sigma,u})_\mu \ge 0$. Then, it follows

$$ 
\begin{aligned}
& \mu_\lambda \|P_N y_{\sigma,u}\|_{L^2(\Omega)}^2 + \|P_N^\perp y_{\sigma,u}\|_{L^2(\Omega)}^2 \\
& \le (\mu_\lambda + 1) \left( \|f\|_{L^2(\Omega)}^2 + \|B_\sigma(u)\|_{L^2(\Omega)}^2 \right) + \frac{\mu_\lambda}{2} \|P_N y_{\sigma,u}\|_{L^2(\Omega)}^2 + \frac{1}{2} \|P_N^\perp y_{\sigma,u}\|_{L^2(\Omega)}^2.
\end{aligned}
$$

Then

$$ \frac{\mu_\lambda}{2} \|P_N y_{\sigma,u}\|_{L^2(\Omega)}^2 + \frac{1}{2} \|P_N^\perp y_{\sigma,u}\|_{L^2(\Omega)}^2 \le (\mu_\lambda + 1) \left( \|f\|_{L^2(\Omega)}^2 + \|B_\sigma(u)\|_{L^2(\Omega)}^2 \right). $$

Since $0 < C < 1$ and $\mu_\lambda = \frac{1}{C^2} > 1$, then $\mu_\lambda > 1$, and by using the boundedness of $B_\sigma$, we get

$$ \|y_{\sigma,u}\|_{L^2}^2 \le 4\mu_\lambda (\|f\|_{L^2(\Omega)}^2 + D^2). \qquad (5.15) $$

• Upper bound for $\|y_{\sigma,u}\|_{H^1(\Omega)}$ independent of $\sigma \in (0, \infty)$.

Testing (5.12) with $y_{\sigma,u}$ in the $L^2$- inner product gives

$$ (y_{\sigma,u}, y_{\sigma,u}) + (-\Delta y_{\sigma,u}, y_{\sigma,u}) = (f - B_{\sigma}(u), y_{\sigma,u}) - \gamma_{\lambda}(\chi_{\omega} P_N y_{\sigma,u}, y_{\sigma,u}). \qquad (5.16) $$

On the left side, by integrating by parts, we have

$$ (y_{\sigma,u}, y_{\sigma,u}) + (-\Delta y_{\sigma,u}, y_{\sigma,u}) = \|y_{\sigma,u}\|_{L^2(\Omega)}^2 + \|\nabla y_{\sigma,u}\|_{L^2(\Omega)}^2 = \|y\|_{H^1(\Omega)}^2. $$

On the right side, we have

$$ 
\begin{aligned}
(f - B_\sigma(u), y_{\sigma,u}) - \gamma_\lambda(\chi_\omega P_N y_{\sigma,u}, y_{\sigma,u}) &\le \|f - B_\sigma(u)\|_{L^2(\Omega)} \|y_{\sigma,u}\|_{L^2(\Omega)} + \gamma_\lambda \|P_N y_{\sigma,u}\|_{L^2(\omega)} \|y_{\sigma,u}\|_{L^2(\omega)} \\
&\le \frac{1}{2} \|f - B_\sigma(u)\|_{L^2(\Omega)}^2 + \frac{1}{2} \|y_{\sigma,u}\|_{L^2(\Omega)}^2 + \gamma_\lambda \|y_{\sigma,u}\|_{L^2(\Omega)}^2 \\
&\le \|f\|_{L^2(\Omega)}^2 + \|B_\sigma(u)\|_{L^2(\Omega)}^2 + \left(\frac{1}{2} + \gamma_\lambda\right) \|y_{\sigma,u}\|_{L^2(\Omega)}^2.
\end{aligned}
$$

Therefore, using the previous inequalities, the bound of $B_\sigma(u)$ and (5.15), we have

$$ 
\begin{aligned}
\|y_{\sigma,u}\|_{H^1(\Omega)}^2 &\le \|f\|_{L^2(\Omega)}^2 + D^2 + 4\mu_\lambda \left(\frac{1}{2} + \gamma_\lambda\right) \left(\|f\|_{L^2(\Omega)}^2 + D^2\right) \\
&= \left(1 + 4\mu_\lambda \left(\frac{1}{2} + \gamma_\lambda\right)\right) \left(\|f\|_{L^2(\Omega)}^2 + D^2\right).
\end{aligned}
$$

• Upper bound for $\|y_{\sigma,u}\|_{H^2(\Omega)}$ independent of $\sigma \in (0, \infty)$.

By using (5.12) and integrating on $\Omega$, we get

$$ 
\begin{aligned}
\|\Delta y_{\sigma,u}\|_{L^2(\Omega)}^2 &= \|y_{\sigma,u} + \gamma_{\lambda}\chi_{\omega}P_N y_{\sigma,u} + B_{\sigma}(y_{\sigma,u}) - f\|_{L^2(\Omega)}^2 \\
&\le 4 \left( \|y_{\sigma,u}\|_{L^2(\Omega)}^2 + \gamma_{\lambda}^2 \|\chi_{\omega}P_N y_{\sigma,u}\|_{L^2(\Omega)}^2 + \|f\|_{L^2(\Omega)}^2 + \|B_{\sigma}(y_{\sigma,u})\|_{L^2(\Omega)}^2 \right) \\
&\le 4 \left( (1 + \gamma_{\lambda}^2) \|y_{\sigma,u}\|_{L^2(\Omega)}^2 + \|f\|_{L^2(\Omega)}^2 + \|B_{\sigma}(y_{\sigma})\|_{L^2(\Omega)}^2 \right).
\end{aligned}
$$
Therefore, by using the bound of $B_\sigma$ and $y_\sigma$, we get

$$ \begin{aligned} \|\Delta y_\sigma\|_{L^2(\Omega)}^2 &\le 16(1 + \gamma_\lambda^2)\mu_\lambda (\|f\|_{L^2(\Omega)}^2 + D^2) + 4 (\|f\|_{L^2(\Omega)}^2 + D^2) \\ &= 4(4(1 + \gamma_\lambda^2)\mu_\lambda + 1) (\|f\|_{L^2(\Omega)}^2 + D^2). \end{aligned} $$

Choosing $M \ge 4(4(1 + \gamma_\lambda^2)\mu_\lambda + 1) (\|f\|_{L^2(\Omega)}^2 + D^2)$, we have $T_\sigma(K_M) \subset K_M$. Since $\{y_{\sigma,u}\} \subset K_M$ is bounded in $H^2(\Omega)$, by the Banach-Alaoglu theorem, there exists a subsequence, still denoted $y_{\sigma,u}$, converging weakly in $H^2(\Omega)$ to some $w \in H^2(\Omega)$, i.e., $y_{\sigma,u} \to w$ in $H^2(\Omega)$. By Rellich-Kondrachov theorem [9, Theorem 9.16, page 285], we know that the embedding $H^2(\Omega) \hookrightarrow L^2(\Omega)$ is compact, so $y_{\sigma,u} \to w$ strongly in $L^2(\Omega)$. Moreover, since $y_{\sigma,u} \in H_0^1(\Omega)$ and $H_0^1(\Omega)$ is a closed subspace of $H^1(\Omega)$, and strong $L^2$ convergence with boundedness in $H^1$ implies weak $H^1$ convergence to the same limit, we get $w \in H_0^1(\Omega)$. Therefore, $w \in K_M$, so $K_M$ is closed in $L^2(\Omega)$. Thus, we conclude that $K_M$ is compact $L^2(\Omega)$. By the Schauder Fixed Point Theorem, $T_\sigma$ has a fixed point $y_\sigma \in K_M \subset H^2(\Omega) \cap H_0^1(\Omega)$, which is a solution to (5.11).

Accordingly, we have shown

**Lemma 5.1.** For any $\sigma \in (0, \infty)$ there exists $y_\sigma \in H^2(\Omega) \cap H_0^1(\Omega)$ such that $y_\sigma - \Delta y_\sigma = f - B\sigma(y_\sigma) - \gamma_\lambda \chi_\omega P_N y_\sigma$ for almost every $x \in \Omega$.

Let us consider the $y_\sigma$ given by Lemma 5.1. We proceed to prove that $R(I + A) = L^2(\Omega)$ by analyzing what happens to $y_\sigma$ as $\sigma \to 0^+$. By using a priori estimates as before, we obtain that the sequence $\{y_\sigma\}$ is uniformly bounded in $H_0^1(\Omega)$, i.e.,

$$ \|y_\sigma\|_{H^1(\Omega)}^2 \le M. $$

Therefore, there exists a subsequence (still denoted by $\{y_\sigma\}$) and a function $y \in H_0^1(\Omega)$ such that

$$ y_\sigma \to y \quad \text{in} \quad H_0^1(\Omega), $$

$$ y_\sigma \to y \quad \text{in} \quad L^2(\Omega). $$

By the continuity of the operators $\mathcal{C}$ and $\chi_\omega$, it follows that $\chi_\omega \mathcal{C} y_\sigma \to \chi_\omega \mathcal{C} y$ in $L^2(\Omega)$.

From the bound $\|B_\sigma(y_\sigma)\|_{L^2(\Omega)} \le D$, there exists a $g \in L^2(\Omega)$ and a further subsequence such that

$$ B_\sigma(y_\sigma) \to g \quad \text{in} \quad L^2(\Omega). $$

We can now pass to the limit in the weak formulation of (5.10). For any test function $z \in H_0^1(\Omega)$, we have

$$ \int_{\Omega} y_{\sigma} z + \int_{\Omega} \nabla y_{\sigma} \nabla z \, dx + \gamma_{\lambda} \int_{\omega} P_{N}(y_{\sigma}) z \, dx = \int_{\Omega} (f - B_{\sigma}(y_{\sigma})) z \, dx. $$

Taking the limit $\sigma \to 0^+$, we obtain

$$ \int_{\Omega} yz + \nabla y \cdot \nabla z \, dx + \gamma_{\lambda} \int_{\omega} P_{N}(y) z \, dx = \int_{\Omega} (f - g) z \, dx. $$

This implies the equation holds in the sense of distributions, we have
$$y - \Delta y + \gamma_{\lambda} \chi_{\omega} P_N y = f - g. \qquad (5.17)$$

Accordingly, Lemma 5.1 and the previous arguments yield

**Lemma 5.2.** There exists $y \in H^2(\Omega) \cap H_0^1(\Omega)$ such that $y - \Delta y + \gamma_{\lambda} \chi_{\omega} P_N y = f - g$, for almost every $x \in \Omega$.

In view of Lemma 5.2, we see that in order to complete the proof of $R(I + A) = L^2(0, L)$ it remains to show that $g \in D\chi_{\omega} \text{sign}_{L^2(\Omega)}(\chi_{\omega} Cy)$. Let $\xi_{\sigma} = \alpha_{\sigma}(\chi_{\omega} Cy_{\sigma})$, so that $B_{\sigma}(y_{\sigma}) = D\chi_{\omega}\xi_{\sigma}$. By the properties of the Yosida approximation, $\xi_{\sigma} \in \partial\varphi(J_{\sigma}(\chi_{\omega} Cy_{\sigma})) = \text{sign}_{L^2(\Omega)}(J_{\sigma}(\chi_{\omega} Cy_{\sigma}))$. Consequently, $\|\xi_{\sigma}\|_{L^2(\Omega)} \le 1$. Thus, on a subsequence,

$$\xi_{\sigma} \to \xi \text{ in } L^2(\Omega), \text{ with } \|\xi\|_{L^2(\Omega)} \le 1.$$

Furthermore, we have the estimate

$$\|\chi_{\omega} Cy_{\sigma} - J_{\sigma}(\chi_{\omega} Cy_{\sigma})\|_{L^2(\Omega)} = \sigma \|\alpha_{\sigma}(\chi_{\omega} Cy_{\sigma})\|_{L^2(\Omega)} = \sigma \|\xi_{\sigma}\|_{L^2(\Omega)} \le \sigma \to 0,$$

and since $\chi_{\omega} Cy_{\sigma} \to \chi_{\omega} Cy$, it follows that $J_{\sigma}(\chi_{\omega} Cy_{\sigma}) \to \chi_{\omega} Cy$ in $L^2(\Omega)$. Indeed,

$$\begin{align}
\| J_{\sigma}(\chi_{\omega} Cy_{\sigma}) - \chi_{\omega} Cy \|_{L^2(\Omega)} &= \| J_{\sigma}(\chi_{\omega} Cy_{\sigma}) - \chi_{\omega} Cy_{\sigma} + \chi_{\omega} Cy_{\sigma} - \chi_{\omega} Cy \|_{L^2(\Omega)} \nonumber \\
&\le \| J_{\sigma}(\chi_{\omega} Cy_{\sigma}) - \chi_{\omega} Cy_{\sigma} \|_{L^2(\Omega)} + \| \chi_{\omega} Cy_{\sigma} - \chi_{\omega} Cy \|_{L^2(\Omega)} \tag{5.18} \\
&\to 0. \nonumber
\end{align}$$

Now, by using [42, Proposition 1.6.] we have that if $\xi_{\sigma} \in \text{sign}(J_{\sigma}(\chi_{\omega} Cy_{\sigma}))$, $\xi_{\sigma} \to \xi$, and $J_{\sigma}(\chi_{\omega} Cy_{\sigma}) \to \chi_{\omega} Cy$, it follows that $\xi \in \text{sign}_{L^2(\Omega)}(\chi_{\omega} Cy)$. Finally, since $B_{\sigma}(y_{\sigma}) = D\chi_{\omega}\xi_{\sigma} \to D\chi_{\omega}\xi$ and we have $B_{\sigma}(y_{\sigma}) \to g$, the uniqueness of the weak limit implies $g = D\chi_{\omega}\xi$. Substituting into (5.17), we conclude that $y$ satisfies (5.9), completing the proof. ■

**Remark 5.2.** An interesting approach to proving that the operator (5.1) is maximal monotone is presented in [29, Lemma 3.], which consist in proving that $B \text{sign}(B)$ is maximal monotone, where the linear closed operator $B$ considered in [29], has to satisfies $B^2 = B$ and $B = B^*$. However, this method fails in our setting, since $\chi_{\omega} C$ under consideration in our work fails to satisfy the requisite conditions.

In order to complete the proof of Theorem 2.1 we proceed to show that (P) with the feedback law (4.3) has a unique weak solution $y \in C([0, \infty); L^2(\Omega))$ and it satisfies (2.2).

Let us return to (5.2). In view of Proposition 5.1 and Proposition 5.2, we can apply [42, Chapter IV, Lemma 1.3] to conclude that the operator $A$ defined in (5.1) is maximal monotone. For the moment, let us assume that $d \in W^{1,1}(0, \infty; L^2(\Omega))$ and $y_0 \in D(A)$. Then, [42, Chapter IV, Theorem 4.1] gives the existence of a unique $y \in W^{1,1}(0, \infty; L^2(\Omega))$ such that

$$\left\{
\begin{array}{l}
y'(t) + Ay(t) \ni \chi_{\omega}d(t) \text{ for almost every } t > 0 \\
y(t) \in D(A) \text{ for every } t \ge 0 \\
y(0) = y_0.
\end{array}
\right.
\qquad (5.19)$$
Given $(d, \hat{d}) \in W^{1,1}(0, \infty; L^2(\Omega))^2$ and $(y_0, \hat{y}_0) \in D(A)^2$, let us take the corresponding unique solution $(y, \hat{y}) \in W^{1,1}(0, \infty; L^2(\Omega))^2$. Then, by [42, Chapter IV, (4.12)], we have

$$ \|y(t, \cdot) - \hat{y}(t, \cdot)\|_{L^2(\Omega)} \le \|y_0 - \hat{y}_0\|_{L^2(\Omega)} + \int_0^t \|d(s, \cdot) - \hat{d}(s, \cdot)\|_{L^2(\omega)} ds, \quad t \ge 0. \quad (5.20) $$

We may use (5.20) to define, as in [42, Page 183] for instance, the notion of weak solution of (5.2).

**Definition 5.1.** A generalized solution of (5.2) is a function $y \in C([0, T], L^2(\Omega))$ for which there exists a sequence of (absolutely continuous) solutions $y_n$ of

$$ \frac{dy_n}{dt} + A(y_n) \ni \chi_\omega d_n, \quad n \ge 1 $$

with $d_n \to d$ in $L^1(0, T; L^2(\Omega))$ and $y_n \to y$ in $C([0, T], L^2(\Omega))$.

Taking into account the density of $W^{1,1}(0, \infty; L^2(\Omega))$ in $L^1(0, \infty; L^2(\Omega))$ and of $D(A)$ in $L^2(\Omega)$, we have that generalized solution is well defined.

Therefore, in virtue of [42, Chapter IV, Theorem 4.1A] we have that (4.4) has a unique weak solution $y \in C([0, \infty); L^2(\Omega))$ provided that $d \in L^1(0, \infty; L^2(\Omega))$ and $y_0 \in L^2(\Omega)$. Moreover, all the formal computations done in Section 4.3 make sense, implying that (2.2) is satisfied.

Accordingly, we have shown

**Proposition 5.3.** Let $d \in L^1(0, \infty; L^2(\Omega))$ and $y_0 \in L^2(\Omega)$. Then, there exists a unique $y = y(t, x)$ such that:

1. $y \in C([0, \infty); L^2(\Omega))$.

2. $y(0) = y_0$.

3. Any two generalized solutions of the problem (5.2) satisfies the estimate (5.20).

4. $y(t, x)$ satisfies the exponential decay (2.2) of Theorem 2.1.

Concluding the proof of the well-posedness part of Theorem 2.1.

# 6 Concluding Remarks and Perspectives

In this work, we presented a rapid stabilization strategy for the heat equation under unknown disturbances with a localized internal feedback law. We have employed the method introduced by [41] combined with the sign multivalued operator (sign) to design suitable feedback laws that guarantee exponential decay with an arbitrary decay rate. The assumptions made on the unknown disturbance (Assumptions (A1) and (A2)) are the standard ones that can be found in the literature. The corresponding closed-loop system (4.4) is formulated as a differential inclusion, and its well-posedness is proved via maximal monotone operator theory. The main difficulty in the application of the maximal monotone operator theory comes from the nonlinearity of the sign operator. To deal with the monotonicity, a new inner product related to the Frequency Lyapunov Functional was used; the maximality was treated by regularizing the operator via the Yosida Approximation and then applying a fixed-point argument.

Following the steps of the previous section, we could prove the exponential stability in the case where a potential is included that makes the system unstable, i.e., the equation
$$ egin{cases} y_t - \Delta y + a(x)y = \chi_{\omega}(u+d), & (t,x) \in (0, \infty) \times \Omega \\ y(t,x) = 0, & (t,x) \in (0, \infty) \times \partial \Omega \\ y(0,x) = y_0(x), & x \in \Omega \end{cases} \qquad (6.1) $$

where $c \in L^\infty(\Omega)$ is an extra source of instability. In that case, we get a section of the eigenvalues to be non-positive. The calculations follow in a similar way to the case without a potential, the election of the same feedback law and Lyapunov functional works, considering the same total control as before, given by the decomposition $u(t,x) = \tilde{u} + \hat{u}$. The crucial step in this case is the election of the constants $\mu_\lambda$ and $\gamma_\lambda$, where the proposed election is

$$ \gamma_{\lambda} := \frac{1}{C}(\lambda - \tau_{1}), \qquad \mu_{\lambda} := \frac{\gamma_{\lambda}^{2}}{\lambda^{2}} = \frac{(\lambda - \tau_{1})^{2}}{\lambda^{2}C^{2}}, \qquad (6.2) $$

where $\tau_1$ is the first of the non-positive eigenvalues, i.e., $-\tau_1 \ge \cdots \ge -\tau_j \ge 0$. Another interesting approach is to study the problem in divergence form, i.e.,

$$ egin{cases} y_t - \operatorname{div}(\gamma(x) \cdot \nabla y) = \chi_{\omega}(u+d), & (t,x) \in (0, \infty) \times \Omega, \\ y(t,x) = 0, & (t,x) \in (0, \infty) \times \partial \Omega, \\ y(0,x) = y_0(x), & x \in \Omega, \end{cases} $$

where $\gamma \in C^2(\bar{\Omega})$ with $\gamma(x) \ge \gamma_0 > 0$ in $\Omega$, for which spectral inequalities remain fundamentally important. In particular, a case of considerable interest is based on the spectral inequality introduced by Osses and Triki [39], which refines the classical spectral inequality of [31], also employed in [41]. Their result not only recovers the classical base case, but the bound in terms of the frequency number is more precise since it depends on the coefficients of the linear combination of the eigenfunctions.

The analysis developed in this paper left some interesting open problems, and one immediate extension would be to develop general results for parabolic systems with analytic semigroup generators. There could be also interesting to consider systems of coupled PDEs. Another challenging direction involves the rapid stabilization of the multidimensional wave equation under geometric control conditions, where only observability inequalities are available.

## Conflicts of Interest

Te authors declare no conflicts of interest

## Acknowledgments

Christian Calle received financial support by ANID BECAS/DOCTORADO NACIONAL 21252433 and PIIC 048/2025 provided by the Universidad Técnica Federico Santa María. Patricio Guzmán was supported by FONDECYT 11240290. Hugo Parada is currently supported by the Agence Nationale de la Recherche through the QuBiCCS project ANR-24-CE40-3008.

## A Weak Spectral Inequality Proof

*Proof of the weak spectral inequality (3.1).* We argue by contradiction. Let's assume that (3.1) is false, i.e,
$$ \forall m \in \mathbb{N}, \exists a^m := (a_1^m, a_2^m, \dots, a_n^m) : \left\| \sum_{n=1}^{N} a_n^m e_n \right\|_{L^2(\omega)}^2 < \frac{1}{m} \sum_{n=1}^{N} (a_n^m)^2. \quad (A.1) $$

Then, let's consider a sequence $v^m$ (renormalize)

$$ v^m := \frac{a^m}{\|a^m\|_{\mathbb{R}^N}}, \quad m = 1, 2, \dots $$

Note that

$$ \|v^m\|_{\mathbb{R}^N} = \frac{\|a^m\|_{\mathbb{R}^N}}{\|a^m\|_{\mathbb{R}^N}} = 1 \quad (A.2) $$

and

$$ \left\| \sum_{n=1}^{N} v_n^m e_n \right\|_{L^2(\omega)} = \left\| \sum_{n=1}^{N} \frac{a_n^m e_n}{\|a^m\|_{\mathbb{R}^N}} \right\|_{L^2(\omega)}^2 = \frac{1}{\|a^m\|_{\mathbb{R}^N}^2} \left\| \sum_{n=1}^{N} a_n^m e_n \right\| < \frac{1}{m \|a^m\|_{\mathbb{R}^N}^2} \|a^m\|^2 = \frac{1}{m}. \quad (A.3) $$

In particular, the functions {$v^m$}$_{m \in \mathbb{N}}$ are bounded in $L^2(\omega)$. Then there exists a convergent subsequence {$v^{m_j}$}$_{j \in \mathbb{N}} \subset \{v^m\}$$_{m \in \mathbb{N}}$ such that

$$ v^{m_j} \to v \quad \text{as} \quad m \to \infty \quad \text{in} \quad \mathbb{R}^N, $$

by using (A.2), we have that

$$ \lim_{m \to \infty} \|v^m\|_{\mathbb{R}^N} = \|v\|_{\mathbb{R}^N} = 1. \quad (A.4) $$

In a similar way, taking the limit as $m \to \infty$ in (A.3), we get

$$ \sum_{n=1}^{N} v_n e_n = 0, \quad \text{in } \omega. $$

Since $e_n$ is a continuous function, we have that there exists an uncountable set $M \subset \omega$ such that

$$ \forall x \in M, \exists n = n_x : e_n(x) \neq 0. $$

Note that if we have that $e_n = 0$ in $\omega$, by the unique continuation property, we have that $e_n = 0$ in $\Omega$, which is not possible.

Therefore,

$$ \sum_{n=1}^{N} v_n e_n(x_0) = 0, \quad \forall x_0 \in \mathcal{M}. \quad (A.5) $$

As $e_n(x_0) \neq 0$ for all $n = n_x$, and $v_n$ solves (A.5), we have that

$$ v_1, \dots, v_N = 0, $$

which is a contradiction because by (A.4), we have that $\|v_n\|_{\mathbb{R}^N} = 1$. Thus, we get
$$Y_N^\top J_N Y_N \geq C_\lambda \|Y_N\|_2^2.$$

<figure><img src="image_3.png" /></figure>

## B Inner product

We define the inner product $(\cdot, \cdot)_\mu$ on $L^2(\Omega)$ by

$$(u, v)_\mu = \mu(P_N u, P_N v) + (P_N^\perp u, P_N^\perp v),$$

This bilinear form induces a norm $\|u\|_\mu = \sqrt{(u, u)_\mu} = \sqrt{\mu\|P_N u\|^2 + \|P_N^\perp u\|^2}$, which is equivalent to the original norm on $H$. Since $\mu > 1$, we have $\|u\|_{L^2(\Omega)}^2 \le (u, u)_\mu \le \mu\|u\|_{L^2(\Omega)}^2$, it follows that $(L^2(\Omega), (\cdot, \cdot))_{L^2(\Omega)}$ and $(L^2(\Omega), (\cdot, \cdot))_\mu$ define the same topology. This inner product will be central to the preconditioning strategy developed.

We must check the axioms. Let $u, v, w \in H$ and $\alpha, \beta \in \mathbb{R}$.

**1. Symmetry:** We need to check if $(u, v)_\mu = (v, u)_\mu$.

Since the original $L^2$ inner product $(\cdot, \cdot)$ is symmetric, we have

$$ \begin{aligned} (u, v)_\mu &= \mu(P_N u, P_N v) + (P_N^\perp u, P_N^\perp v) \\ &= \mu(P_N v, P_N u) + (P_N^\perp v, P_N^\perp u) \\ &= (v, u)_\mu. \end{aligned} $$

So, it is symmetric. Note that from this property we can deduce that

$$ \begin{aligned} (u, u)_\mu &= \mu(P_N u, P_N u)_{L^2(\Omega)} + (P_N^\perp u, P_N^\perp u)_{L^2(\Omega)} \\ &= \mu(u, P_N u)_{L^2(\Omega)} + (u, P_N^\perp u)_{L^2(\Omega)} \\ &= (u, \mathscr{C}v)_{L^2(\Omega)}. \end{aligned} $$

**2. Linearity:** This follows directly from the linearity of the inner product and the linearity of the projections $P_N$ and $P_N^\perp$.

$$ \begin{aligned} (\alpha u + \beta v, w)_\mu &= \mu(\alpha u + \beta v, P_N w) + (\alpha u + \beta v, P_N^\perp w) \\ &= \alpha[\mu(u, P_N w) + (u, P_N^\perp w)] + \beta[\mu(v, P_N w) + (v, P_N^\perp w)] \\ &= \alpha(u, w)_\mu + \beta(v, w)_\mu \end{aligned} $$

So, it is linear.

**3. Positive-Definiteness:**

We need $(u, u)_\mu \ge 0$ for all $u \ne 0$.

$$(u, u)_\mu = \mu(P_N u, P_N u) + (P_N^\perp u, P_N^\perp u) = \mu\|P_N u\|^2 + \|P_N^\perp u\|^2$$

Since $\mu > 0$ and norms are non-negative, $(u, u)_\mu \ge 0$. Furthermore, $(u, u)_\mu = 0$ implies $\|P_N^\perp u\| = 0$ and $\|P_N u\| = 0$. This means $P_N u = 0$ and $P_N^\perp u = 0$, which together imply $u = 0$.
C Isomorphism of C

**Proposition C.1.** Let $P_N$ be the $L^2$-orthogonal projection onto the first $N$ eigenfunctions of $-\Delta$ with Dirichlet conditions. For $\mu_\lambda > 0$, define $\mathcal{C} = \mu_\lambda P_N + P_N^\perp$. Then $\mathcal{C}: H_0^1(\Omega) \to H_0^1(\Omega)$ is a bounded linear isomorphism.

*Proof.* In order to prove that $\mathcal{C}$ is a bounded linear isomorphism, we have to verify it is a bounded linear bijection with bounded inverse between Hilbert spaces.

It's clear linear, let's show that it is bounded. Let $y \in H_0^1(\Omega)$, arbitrary. Then

$$ \begin{aligned} \|\mathcal{C}y\|_{H_0^1(\Omega)}^2 &= \mu \sum_{i=1}^N \lambda_i y_i^2 + \sum_{i=N+1}^{\infty} \lambda_i y_i^2 + \mu \|P_N y\|_{L^2(\Omega)}^2 + \|P_N^\perp y\|_{L^2(\Omega)}^2 \\ &\le \max\{\mu_\lambda, 1\} \|y\|_{H_0^1(\Omega)}. \end{aligned} $$

Let's define $C^{-1}: H_0^1(\Omega) \to H_0^1(\Omega)$ given by $C^{-1}y := \frac{1}{\mu_\lambda} P_N y + P_N^\perp y$. Then

$$ \begin{aligned} \|\mathcal{C}^{-1}y\|_{H_0^1(\Omega)}^2 &= \frac{1}{\mu_\lambda} \sum_{i=1}^N \lambda_i y_i^2 + \sum_{i=N+1}^{\infty} \lambda_i y_i^2 + \frac{1}{\mu_\lambda} \|P_N y\|_{L^2(\Omega)}^2 + \|P_N^\perp y\|_{L^2(\Omega)}^2 \\ &\le \max\left\{\frac{1}{\mu_\lambda}, 1\right\} \|y\|_{H_0^1(\Omega)}^2. \end{aligned} $$

We also have that $\mathcal{C}\mathcal{C}^{-1} = I$. Indeed,

$$ \begin{aligned} \mathcal{C}\mathcal{C}^{-1}y &= \mathcal{C}\left(\frac{1}{\mu_\lambda} P_N y + P_N^\perp y\right) \\ &= P_N y + P_N^\perp y = y. \end{aligned} $$

Since both $\mathcal{C}$ and $\mathcal{C}^{-1}$ are bounded, $\mathcal{C}$ is an isomorphism of $H_0^1$. So indeed

$$ w = \mathcal{C}z \in H_0^1 \quad \text{if and only if} \quad z \in H_0^1. $$

<figure><img src="image_4.png" /></figure>

References

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