# Feedback stabilization of some fourth-order nonlinear parabolic equations with saturared controls

Patricio Guzmán, Felipe Labra, Hugo Parada

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Patricio Guzmán, Felipe Labra, Hugo Parada. Feedback stabilization of some fourth-order nonlinear parabolic equations with saturared controls. 2025. hal-05398937v1

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Preprint submitted on 4 Dec 2025 (v1), last revised 18 Dec 2025 (v2)

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# FEEDBACK STABILIZATION OF SOME FOURTH-ORDER NONLINEAR PARABOLIC EQUATIONS WITH SATURATED CONTROLS

PATRICIO GUZMÁN, FELIPE LABRA, AND HUGO PARADA

ABSTRACT. In this work, we analyze the internal and boundary stabilization of the Cahn-Hilliard and Kuramoto-Sivashinsky equations under saturated feedback control. We conduct our study through the spectral analysis of the associated linear operator. We identify a finite number of eigenvalues related to the unstable part of the system and then design a stabilization strategy based on modal decomposition, linear matrix inequalities (LMIs), and geometric conditions on the saturation function. Local exponential stabilization in $H^2$ is established.

## 1. INTRODUCTION

We address the problem of feedback stabilization for two prototype fourth-order nonlinear parabolic equations: the Cahn-Hilliard (CH) and Kuramoto-Sivashinsky (KS) equations, both considered under saturated control actions. These models arise in different physical contexts but share similar mathematical structures and stabilization challenges. Regarding each model, we briefly mention: The CH equation was introduced in [5, 6, 7] as a continuum model describing phase separation in an isotropic binary mixture when it is sufficiently cooled. Typical examples include binary alloys and binary solutions. Further details on the model can be found in [21, 45, 42] and the references therein. The KS equation was proposed in [29, 30] as a model for phase turbulence in reaction-diffusion systems, and later in [41, 51] to describe the propagation of plane flames. It also captures the onset of instabilities in several physical and chemical systems; see, for instance, [13, 26].

Let us now introduce the mathematical model. Consider $L > 0$ as the length of the spatial domain and $\lambda > 0$ as the anti-diffusion parameter. Define the nonlinear term

$$ \mathcal{N}(y) = \delta y \partial_x y - \nu \partial_x^2 (y^3), $$

which allows us to recover CH term $\partial_x^2(y^3)$ and the KS term $y \partial_x y$. The main system studied in this work is therefore given by

$$ \begin{cases} \partial_t y + \lambda \partial_x^2 y + \partial_x^4 y + \mathcal{N}(y) = \mathcal{F}(y), & t > 0, x \in (0, L), \\ y(0, x) = y_0(x), & x \in (0, L), \end{cases} \quad (1.1) $$

where $\mathcal{F}(y)$ denotes a suitable feedback control law with saturation. Note that the system involves two nonlinear components $\mathcal{N}$ and $\mathcal{F}$, the latter arising from the saturation effect. The above system is complemented with one of the following three sets of boundary conditions

$$ y(t, 0) = y(t, L) = \partial_x y(t, 0) = \partial_x y(t, L) = 0, \quad (1.2a) $$

$$ y(t, 0) = y(t, L) = \partial_x^2 y(t, 0) = \partial_x^2 y(t, L) = 0, \quad (1.2b) $$

$$ \partial_x y(t, 0) = \partial_x y(t, L) = \partial_x^3 y(t, 0) = \partial_x^3 y(t, L) = 0. \quad (1.2c) $$

In the literature, the first set of boundary conditions (1.2a) is commonly referred to as *clamped*, while (1.2b) is known as *hinged*. Both are standard choices when analyzing KS equation. For CH equation, (1.2c) is the boundary condition usually adopted, as it ensures mass conservation in the model describing phase separation of an isotropic binary mixture under negligible cooling effects and zero mass flux at the boundaries. An important part of the analysis will be devoted to the *linear* version of (1.1), which is

<footnote>P. Guzmán received partial financial support from FONDECYT 11240290 and H. Parada is currently supported by the Agence Nationale de la Recherche through the QuBiCCS project (ANR-24-CE40-3008).</footnote>
$$ \left\{ \begin{array}{ll} \partial_t y + \lambda \partial_x^2 y + \partial_x^4 y = \mathcal{F}(y), & t > 0, x \in (0, L), \\ y(0, x) = y_0(x), & x \in (0, L). \end{array} \right. \qquad (1.3) $$

The main idea relies on the spectral analysis of the associated spatial linear operator. This approach is commonly referred to in the literature as spectral reduction or modal decomposition. Typically, this framework is applied to Sturm-Liouville operators to stabilize the finite-dimensional unstable subspace. Such a strategy has been successfully employed in several contexts, for instance, for the semilinear one-dimensional heat equation [19], the multidimensional heat equation with disturbances [25], CH equation [3, 44, 16], KS equation [24, 28, 31], and other related cases [20, 33, 27]. For the linear KS equation (1.3) with clamped boundary conditions, [10] established null controllability in the case $L = 1$ if and only if $\lambda \notin \mathcal{N}$, where

$$ \mathcal{N} = \{ \pi^2(k^2 + \ell^2), k, \ell \in \mathbb{N}, 1 \le k < \ell, k \text{ and } l \text{ have the same parity} \}. \qquad (1.4) $$

In the same work, when $\lambda \notin \mathcal{N}$, an explicit feedback law was designed to ensure exponential stabilization in the $L^2(0, 1)$ norm. Exponential stability for the hinged case was later analyzed in [28]. Additional controllability results were obtained in [12], and null controllability on star-shaped networks was addressed in [9]. From the stabilization viewpoint, several related studies have treated KS equation, including [15, 14, 37, 24, 31]. Control problems for CH type systems have been extensively investigated in the literature, mainly in the context of optimal and distributed control formulations [22, 18, 17]. However, feedback stabilization of such models has received comparatively little attention. Among the few contributions in this direction, the work [3] studied the local stabilization of a nonisothermal CH system around equilibrium states using a Riccati-based feedback law constructed from the unstable spectral modes of the associated linearized operator. These results were further extended in [38]. We also mention the recent works [23, 1], which address local controllability and trajectory stabilization, respectively.

In practical applications, actuator saturation is an unavoidable feature arising from physical or technological limitations. The analysis of such constraints in infinite-dimensional systems has attracted growing attention in recent years. For instance, [49] investigated wave equations with both distributed and boundary saturated feedback laws, while more recent works have developed systematic Lyapunov-based approaches for saturated boundary stabilization of PDEs [34, 35]. Similar ideas have also been explored for dispersive models, such as the Korteweg-de Vries equation [39, 47, 46, 48]. To the best of our knowledge, however, no contribution has yet addressed the stabilization of fourth-order parabolic equations under input saturation. The closest related studies concern linear reaction-diffusion systems [43] and the nonlinear Schlögl model [1]. For a recent and comprehensive review of saturation-based stabilization methods for PDEs, we refer to [36]. In this work, we consider the following saturation map. Let $\ell > 0$ be the saturation level. We define the pointwise saturation map,

$$ \mathbf{sat}(s) = \begin{cases} s, & \text{if } |s| \le \ell, \\ \ell \operatorname{sgn}(s), & \text{if } |s| \ge \ell. \end{cases} \qquad (1.5) $$

This saturation was considered in [43, 39]. The main contribution of this work is the analysis of CH and KS equations under saturated feedback, acting either internally or at the boundary, including stability results in $L^2$ and $H^2$ under mild controllability assumptions on the finite-dimensional unstable subspace. Furthermore, we establish the existence and uniqueness of solutions for the corresponding closed-loop systems. Most of the analysis is carried out in detail for the case of internal saturation, while the boundary case follows by adapting the same arguments. To the best of our knowledge, this is the first work addressing these equations under saturated control actions. The rest of the paper is organized as follows. In Section 2, we introduce the notation and some preliminary results. Section 3 is devoted to $L^2$ stabilization for the internal feedback case, while Section 4 establishes $H^2$ stabilization, which allows us to address the fully nonlinear problem. In Section 5, we briefly explain how the same ideas can be extended to boundary feedback. Finally, some remarks and concluding perspectives are presented in Section 6.

## 2. PRELIMINARIES

We begin this section by introducing the linear operator associated with the spatial part of the system and recall some of its fundamental spectral properties:

$$ A : y \in D(A) \subset L^2(0, L) \longrightarrow -y''' - \lambda y'' \in L^2(0, L), $$
$$D(\mathcal{A}) = \{y \in L^2(0, L), \mathcal{A}y \in L^2(0, L) \text{ and } y \text{ satisfies (1.2)}\}.$$

In what follows, whenever no ambiguity arises, we omit the explicit mention of the boundary conditions. Hence, unless otherwise stated, all results apply to any of the admissible boundary conditions in (1.2). By standard arguments in spectral theory, the operator $\mathcal{A}$ is self-adjoint and has a compact resolvent. Consequently, its spectrum $\sigma(\mathcal{A})$ consists only of real eigenvalues $\{\sigma_k\}_{k \in \mathbb{N}}$ satisfying $\sigma_k \to -\infty$ as $k \to \infty$. The associated eigenfunctions $\{e_k\}_{k \in \mathbb{N}} \subset D(\mathcal{A})$ form an orthonormal basis of $L^2(0, L)$.

**Remark 2.1.** For the hinged boundary conditions (1.2b), the spectral analysis is straightforward. Indeed, in this case $\mathcal{A} = -\mathcal{L}^2 - \lambda\mathcal{L}$, where $\mathcal{L}$ denotes the Laplacian with Dirichlet boundary conditions. Thus, the eigenvalues and eigenfunctions are given by:

$$\sigma_k = \left(\frac{k\pi}{L}\right)^2 \left(\lambda - \left(\frac{k\pi}{L}\right)^2\right), \quad e_k(x) = \sin\left(\frac{k\pi x}{L}\right).$$

A similar characterization holds when considering the Neumann Laplacian and boundary conditions (1.2c).

Throughout this work, we identify matrices $M_{m \times n}$ with $\mathbb{R}^{m \times n}$ and are denoted by bold capital letters, $\mathbf{A} \in \mathbb{R}^{n \times n}$, $\mathbf{B} \in \mathbb{R}^{n \times m}$. The identity matrix is denoted by $\mathbf{I}_n \in \mathbb{R}^{n \times n}$. We denote by $\mathbb{S}^n$ the space of $n \times n$ symmetric matrices. If $\mathbf{A} \in \mathbb{S}^n$, then $\mathbf{A}$ is orthonormal diagonalizable, and we can order its eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. In particular, we have

$$\det \mathbf{A} = \prod_{k=1}^{n} \lambda_k, \quad \mathrm{tr} \, \mathbf{A} = \sum_{k=1}^{n} \lambda_k.$$

We consider $\mathbb{S}^n$ endowed with the Frobenius norm:

$$\|\mathbf{A}\| = \left(\sum_{k=1}^{n} \lambda_k^2\right)^{1/2}.$$

For $z \in \mathbb{R}^n$, we consider the Euclidean norm $\|\cdot\|$. In that case, we have the following expressions for $\lambda_1$ and $\lambda_n$:

$$\lambda_1 = \sup_{z \neq 0} \frac{z^\top \mathbf{A} z}{|z|^2}, \quad \lambda_n = \inf_{z \neq 0} \frac{z^\top \mathbf{A} z}{|z|^2}, \quad \text{and} \quad \lambda_n |z|^2 \le z^\top \mathbf{A} z \le \lambda_1 |z|^2, \quad \forall z \in \mathbb{R}^n.$$

A matrix $\mathbf{A} \in \mathbb{S}^n$ is positive semi-definite if $z^\top \mathbf{A} z \ge 0, \forall z \in \mathbb{R}^n$. Given $\mathbf{A}, \mathbf{B} \in \mathbb{S}^n$, we use the Loewner partial order [50, Section 1.10] to write $\mathbf{A} \succeq \mathbf{B} \iff \mathbf{A} - \mathbf{B} \succeq 0$, which is equivalent to ask the eigenvalues of $\mathbf{A} - \mathbf{B}$ be non-negative. If $\mathbf{A} \in \mathbb{S}^n$ is positive semi-definite, we denote by $\mathbf{A} \succeq 0$. We say that $\mathbf{A} \in \mathbb{S}^n$ is positive definite, if $z^\top \mathbf{A} z > 0, \forall z \in \mathbb{R}^n \setminus \{0\}$. Take $\mathbf{B} \in \mathbb{S}^n$ positive definite, then we have $\mathbf{A} \succ \mathbf{B} \iff \mathbf{A} - \mathbf{B} \succ 0$. We say that $\mathbf{A} \in \mathbb{R}^{n \times n}$ is Hurwitz if all its eigenvalues have negative real part.

We consider the space $L^2(0, L)$ of square integrable functions on $(0, L)$, endowed with the inner product $\langle f, g \rangle_{L^2(0,L)} = \int_0^L f(x)g(x)dx$ with associated norm, denoted by $\|\cdot\|_{L^2(0,L)}$. For an integer $s \ge 1$, the $s$-order Sobolev space is denoted by $H^s(0, L)$ and is endowed with its usual norm, denoted by $\|\cdot\|_{H^s(0,L)}$. Let $X_n \subset L^2(0, L)$ be the subspace spanned by $(e_j)_{j=1}^n$ and $\Pi_n$ be the orthogonal projection onto $X_n$, that is

$$\Pi_n y(t, \cdot) = \sum_{j=1}^{n} y_j(t, \cdot) e_j, \quad y_j(t, \cdot) = \langle y(t, \cdot), e_j \rangle_{L^2(0,L)}.$$

Let $X_n^\perp$ denotes the orthogonal complement of $X_n$ in $L^2(0, L)$ and $\iota: \mathbb{R}^n \to X_n$ be the isomorphism defined by $\iota(e^j) = e_j$, where $(e^j)_{j=1}^n$ is the canonical basis of $\mathbb{R}^n$. We recall that

$$\ell^2(\mathbb{N}, \mathbb{R}) = \left\{ (x_k)_{k \in \mathbb{N}} \in \mathbb{R}^\mathbb{N} : \sum_{k=1}^\infty |x_k|^2 < \infty \right\}.$$

We are going to use the isometric representation of $L^2(0, L)$ as $\ell_2(\mathbb{N}, \mathbb{R})$ obtained by the isomorphism induced by $e_j(\cdot) \to e^j$, where $(e^j)_{j \in \mathbb{N}}$ is the standard basis vectors in $\ell_2(\mathbb{N}, \mathbb{R})$ and endowed with the standard $\ell_2$ norm. In this sense, we can write $L^2(0, L) = X_n \oplus X_n^\perp$, where $\oplus$ is the orthogonal sum of subspaces. We denote $\ell_2(\mathbb{N}, \mathbb{R}) = \mathbb{R}^n \oplus \ell_{2,j>n}$, where we identify $\mathbb{R}^n$ with the sequences with support in $\{1, \dots, n\}$ and $\ell_{2,j>n}$ is the set of sequences in $\ell_2(\mathbb{N}, \mathbb{R})$ which are 0 in the first $n$ entries.
To conclude this section, we state a Gronwall-type inequality that will be useful in the proof of stability for the fully nonlinear system. Let $b, k: [0, \infty) \to \mathbb{R}$ be two continuous functions. Let $p \ge 0$ be a constant with $p \ne 1$. Let $v(t)$ be a positive differentiable function such that

$$v'(t) \le b(t)v(t) + k(t)v(t)^p, \quad t \in [0, \infty). \qquad (2.1)$$

The following result can be found in [2, Lemma 4.1].

**Lemma 2.1.** Let $q = 1 - p$ and consider,

$$w(t) = v(0)^q + q \int_0^t k(s) \exp \left( -q \int_0^s b(\tau) d\tau \right) ds. \qquad (2.2)$$

If $w(t) > 0$ for all $t \in [0, \infty)$, then

$$v(t) \le \exp \left( \int_0^t b(s) \, ds \right) w(t)^{1/q}, \quad t \in [0, \infty). \qquad (2.3)$$

### 3. INTERNAL STABILIZATION WITH SATURATED CONTROL

In this section, we design an internal feedback law, ensuring exponential stabilization of the linearized system under input saturation. The nonlinear effects will be treated later as perturbations of this linear dynamics in Section 4. Inspired by the framework introduced in [43], we consider the controlled linear system

$$\left\{ \begin{array}{ll} \partial_t y + \lambda \partial_x^2 y + \partial_x^4 y = \displaystyle\sum_{k=1}^m b_k(x) \mathbf{sat}(u_k(t)), & t > 0, x \in (0, L), \\ y(0, x) = y_0(x), & x \in (0, L). \end{array} \right. \qquad (3.1)$$

Here $u: (0, \infty) \to \mathbb{R}^m$ denotes the control vector $u(t) = (u_1(t), \dots, u_m(t))^\top$, and the vector function $b(x) = (b_1(x), \dots, b_m(x))$, where $b_k \in L^2(0, L)$ describes the localization of the $k$-component of the control $u_k \in \mathbb{R}$. The saturation map acts component-wise, i.e.,

$$\mathbf{sat}(u(t)) = (\mathbf{sat}(u_1(t)), \dots, \mathbf{sat}(u_m(t)))^\top.$$

Formally, $y(t, \cdot) \in D(\mathcal{A})$ can be written as

$$y(t, \cdot) = \sum_{j=1}^{\infty} y_j(t) e_j(\cdot),$$

where $y_j(t)$ are the associated Fourier components given by $\langle y(t, \cdot), e_j(\cdot) \rangle_{L^2(0,L)}$. Similarly, since $b_k \in L^2(0, L)$ for $k = 1, \dots, m$

$$b_k(\cdot) = \sum_{j=1}^{\infty} b_{jk} e_j(\cdot), \quad b_{jk} = \langle b_k(\cdot), e_j(\cdot) \rangle_{L^2(0,L)}, \quad \forall j \in \mathbb{N}, \forall k = 1, \dots, m.$$

Now, observe that (3.1) can be written as

$$\partial_t y = \mathcal{A} y + \sum_{k=1}^{m} b_k(x) \mathbf{sat}(u_k(t)). \qquad (3.2)$$

Using that $e_j$ are eigenfunctions of $\mathcal{A}$ we obtain the following infinite-dimensional system of ODEs

$$\dot{y}_j(t) = \sigma_j y_j(t) + \mathbf{b}_j \mathbf{sat}(u(t)), \quad \forall j \in \mathbb{N}, \qquad (3.3)$$

where $\mathbf{b}_j = (b_{j1} \cdots b_{jm}) \in \mathbb{R}^{1 \times m}$. By the asymptotic behaviour of the eigenvalues, we can choose $n \in \mathbb{N}$ as the number of non-negative eigenvalues and select $\eta > 0$ such that

$$\sigma_j < -\eta < 0, \quad \forall j \ge n+1. \qquad (3.4)$$

We then consider the projection of $y$ onto the unstable space. Let $z$, $\mathbf{A}$, and $\mathbf{B}$ be defined as follows

$$z := \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}, \quad \mathbf{A} := \begin{pmatrix} \sigma_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sigma_n \end{pmatrix}, \quad \mathbf{B} := \begin{pmatrix} b_{11} & \cdots & b_{1m} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nm} \end{pmatrix}. \qquad (3.5)$$
Then, the dynamics of the first $n$ Fourier modes are governed by

$$ \dot{z}(t) = \mathbf{A}z(t) + \mathbf{B}\mathbf{sat}(u(t)). \qquad (3.6) $$

We make the following assumption

**Assumption 1.** The pair $(\mathbf{A}, \mathbf{B})$ is stabilizable, i.e. there exists $\mathbf{K} \in \mathbb{R}^{m \times n}$, such that $\mathbf{A} + \mathbf{B}\mathbf{K}$ is Hurwitz.

This assumption can be verified in many practical cases (see the discussion in Section 3.4). In [52, Example 1.1], can be observed that even if $\mathbf{A} + \mathbf{B}\mathbf{K}$ is Hurwitz, the presence of saturation in the feedback system $\dot{z}(t) = \mathbf{A}z(t) + \mathbf{B}\mathbf{sat}(\mathbf{K}z)$ may lead to the appearance of new equilibrium points and even diverging trajectories.

**Remark 3.1.** Note that even though the unstable part of the system (3.1) is finite-dimensional, the control acts on all the Fourier modes of the solution.

We now aim to show that the exponential stability of (3.3) follows from the exponential stabilization of (3.6). We look for the control $u$ in the form $u = \mathbf{K}z(t)$, and define the matrices $\mathbf{K}_j := \mathbf{K}\iota^{-1}e_j$, $j = 1, \dots, n$ and $\mathbf{K} := (\mathbf{K}_1, \dots, \mathbf{K}_n) \in \mathbb{R}^{m \times n}$, with this choice, we obtain

$$ \begin{aligned} u(t) &= \mathbf{K}\iota^{-1}(\Pi_n y(\cdot, t)) \\ &= \sum_{j=1}^{n} y_j \mathbf{K}\iota^{-1} e_j(\cdot) \\ &= \sum_{j=1}^{n} y_j \mathbf{K}_j(\cdot) \\ &= \mathbf{K}z(t). \end{aligned} $$

With this feedback law, system (3.3) becomes the following infinite-dimensional ODE in $\ell_2(\mathbb{N}, \mathbb{R})$

$$ \dot{y}_j(t) = \sigma_j y_j(t) + \mathbf{b}_j \mathbf{sat}(\mathbf{K}z(t)), \quad \forall j \in \mathbb{N}, \qquad (3.7) $$

and can be split as the finite-dimensional subsystem (the projection onto the unstable space)

$$ \dot{z}(t) = \mathbf{A}z(t) + \mathbf{B}\mathbf{sat}(\mathbf{K}z(t)), \qquad (3.8) $$

and the infinite-dimensional subsystem (the projection onto the stable space)

$$ \dot{y}_j(t) = \sigma_j y_j(t) + \mathbf{b}_j \mathbf{sat}(\mathbf{K}z(t)), \quad j \ge n+1. \qquad (3.9) $$

In this direction, (3.7) is decomposed into a cascade system. The strategy is first stabilize the finite-dimensional part (3.6) and then use this information in the infinite-dimensional part (3.9).

**Remark 3.2.** System (3.7) can also be written as

$$ \dot{y}_j(t) = \sigma_j y_j(t) + \mathbf{b}_j \mathbf{sat}(\mathbf{K}\iota^{-1}(\Pi_n y)), \quad \forall j \in \mathbb{N}. $$

Before proceeding with the proof of exponential stability, we recall some definitions concerning the region of attraction and local exponential stability.

**Definition 3.1 (Region of attraction).** Assume that $\mathbf{K}$ is chosen such that $0$ is a locally asymptotically stable equilibrium of (3.8). We say that $S \subset \mathbb{R}^n$ is a region of attraction of $0$ if

(1) $0 \in \text{int}(S)$;

(2) for any $z_0 \in S$ the corresponding solution of (3.6) satisfies $z(t, z_0) \to 0$ as $t \to \infty$;

(3) $S$ is forward invariant, i.e., for any $z_0 \in S$ it holds that $z(t, z_0) \in S$ for all $t \ge 0$.

By considering the partial order induced by set inclusion, the largest set satisfying the above properties is called the maximal region of attraction

**Definition 3.2.** System (3.8) is called a locally exponentially in 0 with region of attraction $S$, if the following conditions hold:

(1) there exist $\varepsilon, M, \alpha > 0$ such that for all initial conditions with $|z_0| \le \varepsilon$, it holds

$$ |z(t, z_0)| \le Me^{-\alpha t}|z_0|, \quad \forall t \ge 0. \qquad (3.10) $$

(2) $\overline{B_\varepsilon(0)} \subset S$ and $S$ is a region of attraction of (3.8).
The above definitions can be analogously extended to system (3.9) and our PDE model (3.1). With these tools at hand, we are now in position to state our main stability result of this section: consider the closed-loop system

$$ egin{cases} 
\partial_t y + \lambda \partial_x^2 y + \partial_x^4 y = \sum_{k=1}^{m} b_k(x) \mathbf{sat}((\mathbf{K}_t^{-1}(\Pi_n y(\cdot, t))))_k, & t > 0, x \in (0, L), \\
y(0, x) = y_0(x), & x \in (0, L).
\end{cases} 
\quad (3.11) $$

**Theorem 3.1.** Let $\mathbf{K}$ be as in Assumption 1. For all $T > 0$ and $y_0 \in H^2(0, L)$, there exists a unique solution $y \in C([0, T]; H^2(0, L)) \cap L^2(0, T; H^4(0, L))$ of the closed-loop system (3.11). Moreover, this system is locally exponentially stable at 0 with a region of attraction $\iota(\mathcal{R}) \times X_n^\perp$ in the $L^2$-norm, where

$$ \mathcal{R} := \{z \in \mathbb{R}^n : z^\top \mathbf{P} z \le 1\}, \quad \text{for some } \mathbf{P} \in \mathbb{R}^{n \times n}. $$

The proof of Theorem 3.1 is divided into two parts: in Section 3.1, we address the local-in-time well-posedness using fixed-point arguments, Section 3.2 is devoted to the global in time well-posedness and in Section 3.3, we establish the exponential stability of the closed-loop system.

### 3.1. Local well-posedness.
In this section, we study the well-posedness of the closed-loop system (3.11). We start by considering the following linear system with a source term:

$$ egin{cases} 
\partial_t y + \partial_x^4 y = f, & t \in (0, T), x \in (0, L), \\
y(0, x) = y_0(x), & x \in (0, L).
\end{cases} 
\quad (3.12) $$

Let $B_T := C([0, T]; H^2(0, L)) \cap L^2(0, T; H^4(0, L))$, endowed with the norm

$$ \|f\|_{B_T} = \left( \|f\|_{C([0,T];H^2(0,L))}^2 + \|f\|_{L^2(0,T;H^4(0,L))}^2 \right)^{1/2}. $$

With some slight modifications of [40, Proposition 1], we have

**Proposition 3.1.** Let $T > 0$, $y_0 \in H^2(0, L)$ and $f \in L^2(0, T; L^2(0, L))$, then equation (3.12) has a unique (mild) solution $y \in B_T$ and there exists $C$ independent of $T$, such that

$$ \|y\|_{B_T} \le C \sqrt{1 + T^2} \left( \|y_0\|_{H^2(0,L)} + \|f\|_{L^2((0,T);L^2(0,L))} \right). \quad (3.13) $$

Moreover, if $y_0 \in D(\mathcal{A})$ and $f \in C^1([0, T]; L^2(0, L))$, then $y$ is a strong solution and $y \in C([0, T]; D(\mathcal{A})) \cap C^1([0, T]; L^2(0, L))$.

Now, for $\delta, \nu = 0, 1$, we consider the following system

$$ egin{cases} 
\partial_t y + \lambda \partial_x^2 y + \partial_x^4 y + \delta y \partial_x y - \nu \partial_x^2 (y^3) = \sum_{k=1}^{m} b_k(x) \mathbf{sat}((\mathbf{K}_t^{-1}(\Pi_n y(\cdot, t))))_k, & t > 0, x \in (0, L), \\
y(0, x) = y_0(x), & x \in (0, L).
\end{cases} 
\quad (3.14) $$

System (3.14) reduces to (3.11) when $\delta = \nu = 0$ and coincides with the fully nonlinear closed-loop system (4.16) studied in Section 4.1. We first show local in time well-posedness of (3.14).

**Proposition 3.2.** Let $y_0 \in H^2(0, L)$, then there exists $T^* > 0$ such that equation (3.14) has a unique (mild) solution $y \in B_{T^*}$.

**Proof:** Throughout this part, $C$ will denote a positive constant, which may vary from line to line, but does not depend on $T$. Let $\theta \in (0, T]$ (to be chosen later). For the moment, assume that $y_0 \in D(\mathcal{A})$ and $f \in C^1([0, T]; L^2(0, L))$. Let $w \in B_\theta$ and consider the nonlinear operators

$$ 
\begin{aligned}
\mathcal{F}(w) &= \sum_{k=1}^{m} b_k(x) \mathbf{sat}((\mathbf{K}_t^{-1}(\Pi_n w(\cdot, t))))_k, \quad \mathcal{G}(w) = -\lambda \partial_x^2 w - \delta w \partial_x w + \nu \partial_x^2 (y^3), \\
\mathcal{H}(w) &= \mathcal{F}(w) + \mathcal{G}(w).
\end{aligned}
 $$

Define the map $\Phi: B_\theta \to B_\theta$ by $\Phi(w) = y$, where $y$ is the unique solution of

$$ 
\begin{cases} 
\partial_t y = \mathcal{A}_0 y + \mathcal{H}(w), & t \in (0, \theta), \\
y(0, \cdot) = y_0,
\end{cases} 
$$
where $A_0y = -y'''$ and $D(A_0) = D(A)$. Note that $y$ is well-defined. Indeed, if $w \in B_\theta$, using (A.1) and (A.2) we obtain $H(w) \in L^2(0, \theta; L^2(0, L))$ and

$$ \|H(w)\|_{L^2(0,\theta;L^2(0,L))} \le C\sqrt{\theta} (\|w\|_{B_\theta} + \delta\|w\|_{B_\theta}^2 + \nu\|w\|_{B_\theta}^3) \quad (3.15) $$

Using, (3.13) and (3.15), we get

$$ \begin{aligned} \|\Phi(w)\|_{B_T} &\le C\sqrt{1+\theta^2} (\|y_0\|_{H^2(0,L)} + \|H(w)\|_{L^2((0,\theta);L^2(0,L))}) \\ &\le C\sqrt{1+\theta^2}\|y_0\|_{H^2(0,L)} + C\sqrt{1+\theta^2}\sqrt{\theta} (\|w\|_{B_\theta} + \delta\|w\|_{B_\theta}^2 + \nu\|w\|_{B_\theta}^3), \end{aligned} $$

Now, we take $R = 4C\|y_0\|_{H^2(0,L)}$ and consider $\Phi$ restricted to the ball $B_{\theta,R} := \{w \in B_\theta, \|w\|_{B_\theta} \le R\}$. If $\theta$ is small enough such that

$$ \begin{cases} \sqrt{1+\theta^2} < 2, \\ C\sqrt{\theta}(1+\delta R + \nu R^2) < \frac{1}{2}, \end{cases} $$

we get $\|\Phi(w)\|_{B_T} \le R$, i.e., $\Phi$ maps $B_{\theta,R}$ into itself. Now, for $w_1, w_2 \in B_{\theta,R}$, define $\Delta = \Phi(w_1) - \Phi(w_2)$, then $\Delta$ satisfies

$$ \left\{ \begin{array}{ll} \partial_t \Delta = A_0 \Delta + H(w_1) - H(w_2), & t \in (0, \theta), \\ \Delta(0, \cdot) = 0. & \end{array} \right. \quad (3.16) $$

Observe that

$$ \begin{aligned} H(w_1) - H(w_2) ={}& F(w_1) - F(w_2) - \lambda \partial_x^2 \Delta - \delta \Delta \partial_x w_2 - \delta w_1 \partial_x \Delta + 3\nu(w_1)^2 \partial_x^2 \Delta \\ & + 3\nu \Delta \partial_x^2 w_2 (w_1 + w_2) + 6\nu \Delta (\partial_x w_1)^2 + 6\nu \partial_x \Delta \partial_x (w_1 + w_2), \end{aligned} $$

therefore, using that the saturation map is Lipschitz continuous and similar to the previous estimates

$$ \|H(w_1) - H(w_2)\|_{L^2(0,\theta;L^2(0,L))} \le C\sqrt{\theta} \left[ \|\Delta\|_{B_\theta} + \delta\|\Delta\|_{B_\theta} (\|w_1\|_{B_\theta} + \|w_2\|_{B_\theta}) + \nu\|\Delta\|_{B_\theta} (\|w_1\|_{B_\theta} + \|w_2\|_{B_\theta})^2 \right]. $$

Using this estimate and (3.13) for system (3.16) we see

$$ \begin{aligned} \|\Phi(w_1) - \Phi(w_2)\|_{B_\theta} &\le C\sqrt{1+\theta^2}\sqrt{\theta} \left\| \left[ 1 + \delta (\|w_1\|_{B_\theta} + \|w_2\|_{B_\theta}) + \nu (\|w_1\|_{B_\theta} + \|w_2\|_{B_\theta})^2 \right] \Delta \right\|_{B_\theta} \\ &\le 2C\sqrt{\theta}(1 + 2R + 4R^2) \|w_1 - w_2\|_{B_\theta}. \end{aligned} $$

By choosing $\theta$ small enough such that $2C\sqrt{\theta}(1 + 2R + 4R^2) < 1$, we get that $\Phi$ is a contraction. Banach fixed-point theorem ensures the existence of unique solution $y \in B_\theta$ of (3.14). $\square$

**Remark 3.3.** A similar analysis can be done in a less regular framework, namely $y_0 \in L^2(0, L)$ and $y \in C([0, T]; L^2(0, L)) \cap L^2(0, T; H^2(0, L))$. However, since our focus is on the stability of the fully nonlinear system, we need solutions in $y \in C([0, T]; H^2(0, L))$.

3.2. **Global well-posedness.** In this part, we use some a priori and stability estimates to extend the solutions for all positive time. Let $T > 0$, we multiply (3.14) by $y$ and integrating over the spatial domain

$$ \frac{1}{2} \frac{d}{dt} \left( \int_0^L |y|^2 dx \right) + \int_0^L |\partial_x^2 y|^2 dx = -\lambda \int_0^L y \partial_x^2 y dx + \int_0^L F(y) y dx - \delta \int_0^L y^2 \partial_x y dx + \nu \int_0^L \partial_x^2 (y^3) y dx. \quad (3.17) $$

Observe that, due to the boundary conditions, (1.2) the last term is non-positive,

$$ \begin{aligned} \int_0^L \partial_x^2 (y^3) y dx &= 6 \int_0^L (\partial_x y)^2 y^2 dx + 3 \int_0^L \partial_x^2 y y^3 dx \\ &= 6 \int_0^L (\partial_x y)^2 y^2 dx - 9 \int_0^L (\partial_x y)^2 y^2 dx \\ &\le 0. \end{aligned} $$

For the reminding nonlinear term, we perform the computation under boundary conditions (1.2c). In the other cases, the analysis is simpler because

$$ \int_0^L y^2 \partial_x y dx = \int_0^L \partial_x (y)^3 dx = 0. $$
First and second terms of (3.17) are estimated as in Section 3.1. For the third one, since $\partial_x y(t, \cdot) \in H_0^1(0, L)$, we obtain

$$ 
\begin{aligned}
\delta \int_0^L y^2 \partial_x y dx &\le \|\partial_x y\|_{L^\infty(0,L)} \delta \int_0^L y^2 dx \\
&\le C \|\partial_x^2 y\|_{L^2(0,L)} \delta \int_0^L y^2 dx \\
&\le \frac{1}{3} \|\partial_x^2 y\|_{L^2(0,L)}^2 + C \delta^2 \left( \int_0^L y^2 dx \right)^2.
\end{aligned}
$$

Gathering all the estimates and multiplying by 2, we observe

$$ \frac{d}{dt} \left( \int_0^L |y|^2 dx \right) + \frac{2}{3} \int_0^L |\partial_x^2 y|^2 dx \le C \int_0^L |y|^2 dx + C \delta^2 \left( \int_0^L y^2 dx \right)^2. \quad (3.18) $$

Now multiply (3.14) by $\partial_x^4 y$ to obtain

$$ \frac{1}{2} \frac{d}{dt} \left( \int_0^L |\partial_x^2 y|^2 dx \right) + \int_0^L |\partial_x^4 y|^2 dx = -\lambda \int_0^L \partial_x^2 y \partial_x^4 y dx + \int_0^L \mathcal{F}(y) \partial_x^4 y dx - \delta \int_0^L y \partial_x y \partial_x^4 y dx + \nu \int_0^L \partial_x^2(y^3) \partial_x^4 y dx. \quad (3.19) $$

We now estimate right-hand side terms of the previous equality. For the first one

$$ -\lambda \int_0^L y \partial_x^2 \partial_x^4 y dx \le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + \frac{5\lambda^2}{2} \int_0^L |\partial_x^2 y|^2 dx. $$

For the second one

$$ 
\begin{aligned}
\int_0^L \mathcal{F}(y) \partial_x^4 y dx &\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + \frac{5}{2} \int_0^L |\mathcal{F}(y)|^2 dx \\
&\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + C \int_0^L |y|^2 dx \\
&\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + C \|y\|_{H^2(0,L)}^2.
\end{aligned}
$$

The third one, by the boundary conditions, either $y(t, \cdot) \in H_0^1(0, L)$ or $\partial_x y(t, \cdot) \in H_0^1(0, L)$

$$ 
\begin{aligned}
-\delta \int_0^L y \partial_x y \partial_x^4 y dx &\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + \frac{5\delta^2}{2} \int_0^L |y \partial_x y|^2 dx \\
&\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + C \|y\|_{H^2(0,L)}^4.
\end{aligned}
$$

For the last one,

$$ 
\begin{aligned}
\nu \int_0^L \partial_x^2(y^3) \partial_x^4 y dx &\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + \frac{5\nu^2}{2} \int_0^L |\partial_x^2(y^3)|^2 dx \\
&\le \frac{1}{5} \int_0^L |\partial_x^4 y|^2 dx + C \|y\|_{H^2(0,L)}^6.
\end{aligned}
$$

Combining these estimates, and multiplying by 2 we observe

$$ \frac{d}{dt} \left( \int_0^L |\partial_x^2 y|^2 dx \right) + \frac{2}{5} \int_0^L |\partial_x^2 y|^2 dx \le C \|y\|_{H^2(0,L)}^2 \left( 1 + \|y\|_{H^2(0,L)}^2 + \|y\|_{H^2(0,L)}^4 \right). \quad (3.20) $$

By adding (3.18) and (3.20)

$$ \frac{d}{dt} \|y\|_{H^2(0,L)}^2 + \int_0^L |\partial_x^2 y|^2 dx \le C \|y\|_{H^2(0,L)}^2 \left( 1 + \|y\|_{H^2(0,L)}^2 + \|y\|_{H^2(0,L)}^4 \right). \quad (3.21) $$

Since the differential inequality (3.21) prevents any finite-time blow-up, the local solution can be extended globally, yielding well-posedness for all $T > 0$.
### 3.3. Exponential stability.
We now address the stability part of Theorem 3.1. We start with the following proposition that ensures that thanks to the cascade structure, it suffices to stabilize the finite-dimensional subsystem (3.6) to conclude the stability in the state space $\ell_2(\mathbb{N}, \mathbb{R})$ for (3.7) and $L^2(0, L)$ for (3.11).

**Proposition 3.3.** Let $K$ be as in Assumption 1 and $S \subset \mathbb{R}^n$ the region of attraction of (3.8) such that the system is locally exponentially stable in 0. Then:

(1) System (3.7) (resp. (3.11)) with the feedback law $u(t) = Kz(t)$ is locally exponentially stable in 0 with region of attraction $S \times \ell_{2, j > N}$ (resp. $\iota(S) \times X_N^\perp$).

(2) In addition, for any closed and bounded set $G \subset \text{int}(\iota(S) \times X_N^\perp)$, there exist two positive constants $C$, $\alpha$ such that for any initial condition $y_0 \in G$, the solution $y$ of (3.11) satisfies

$$ \|y(t, \cdot)\|_{L^2(0,L)} \leq Me^{-\alpha t}\|y_0\|_{L^2(0,L)}. \quad (3.22) $$

**Proof:** Let $K$ as in Assumption 1 and $S \subset \mathbb{R}^n$ the region of attraction of (3.6) such that the system is locally exponentially stable in 0. Take $G' \subset \text{int}(S)$ bounded and closed. Then, there exist $M, a > 0$ such that for all $z_0 \in G'$ the solution of (3.6) satisfies

$$ |z(t)| \leq Me^{-at}|z_0|, \quad t \geq 0. \quad (3.23) $$

Applying the feedback law $u(t) = Kz(t)$ and Duhamel's formulae for (3.9)

$$ y_j(t) = e^{\sigma_j t} y_j(0) + \mathbf{b}_j \int_0^t e^{\sigma_j(t-s)} \mathbf{sat}(\mathbf{K}z(s))ds, \quad \forall j \geq n+1. $$

Observe that, using the definition of the saturation map (1.5) we get $|\mathbf{sat}(\mathbf{K}z)| \leq |\mathbf{K}z| \leq \|\mathbf{K}\|\|z\|$. Therefore, for all $j \geq n+1$, using (3.4) and (3.23) we get

$$ \begin{aligned} |y_j(t)| &\leq e^{-\eta t} |y_j(0)| + M|\mathbf{b}_j| \int_0^t e^{-\eta(t-s)} e^{-as} \|\mathbf{K}\| |z_0| ds, \\ &= e^{-\eta t} |y_j(0)| + M|\mathbf{b}_j| \|\mathbf{K}\| |z_0| \left( \frac{e^{-at} - e^{-\eta t}}{\eta - a} \right), \end{aligned} $$

where the last term is positive due to the monotonicity of the exponential function. Therefore,

$$ |y_j(t)| \leq 2e^{-2\eta t} |y_j(0)|^2 + 2(M\|\mathbf{K}\|)^2 |\mathbf{b}_j|^2 |z_0|^2 \left( \frac{e^{-at} - e^{-\eta t}}{\eta - a} \right)^2. $$

Without loss of generality, assume $\eta > a$,

$$ \left( \frac{e^{-at} - e^{-\eta t}}{\eta - a} \right)^2 \leq \frac{e^{-2at}}{(\eta - a)^2}. $$

Summing over, $j \geq n+1$, we obtain the following estimate

$$ \sum_{j \geq n+1} |y_j(t)| \leq 2e^{-2at} \sum_{j \geq n+1} |y_j(0)|^2 + 2 \left( \frac{M \|\mathbf{K}\|}{(\eta - a)} \right)^2 \left( \sum_{j \geq n+1} |\mathbf{b}_j|^2 \right) e^{-2at} |z_0|^2. $$

By definition and Bessel inequality

$$ \sum_{j \geq n+1} |\mathbf{b}_j|^2 = \sum_{j \geq n+1} \sum_{k=1}^{m} |b_{jk}|^2 \leq \|\mathbf{b}\|_{(L^2(0,L))^m}^2. $$

Finally, recalling (3.23)

$$ |z(t)|^2 = \sum_{j=1}^{n} |y_j(t)|^2, \quad \text{and} \quad |z_0|^2 = \sum_{j=1}^{n} |y_j(0)|^2, $$

we deduce

$$ \sum_{j \in \mathbb{N}} |y_j(t)| \leq C e^{-2at} \sum_{j \in \mathbb{N}} |y_j(0)|, \quad C = \max \left\{ 2, M^2, 2 \left( \frac{M \|\mathbf{K}\| \|\mathbf{b}\|_{(L^2(0,L))^m}^2}{(\eta - a)} \right)^2 \right\}. $$

This corresponds to exponential decay in the state space $\ell_2(\mathbb{N}, \mathbb{R})$ for initial data in $G' \times \ell_{2, j > n}$. To conclude the proof, it is enough to take $G' \subset \text{int}(S)$ such that $0 \in \text{int}(G')$ and thus $0 \in \text{int}(G') \times \ell_{2, j > n}$. For the second
part, using the isomorphism $\iota : \mathbb{R}^n \to X_n$, the isometric isomorphism between $\ell_{2,j>n}$ and $X_n^\perp$ and Parseval inequality we obtain the local exponential decay in $0 \in L^2(0, L)$ with region of attraction $\iota(G') \times X_n^\perp \subset L^2(0, L)$. $\square$

**Remark 3.4.** The previous proof is analogues if we have now $\eta < a$ considering $e^{-at} < e^{-\eta t}$. In that case, we get

$$ \sum_{j \in \mathbb{N}} |y_j(t)| \le Ce^{-2\eta t} \sum_{j \in \mathbb{N}} |y_j(0)|. $$

If $\eta = a$, since the spectrum of $\mathcal{A}$ is discrete, we can always consider $\tilde{\eta}$ close to $\eta$ such that $\tilde{\eta} \neq a$, and (3.4) still holds with $\tilde{\eta}$.

The main hypothesis in the proof of the previous proposition is that (3.8) is locally exponentially stable. As we already mentioned, even if $\mathbf{A} + \mathbf{B}\mathbf{K}$ is Hurwitz, (3.8) could not be globally stable. To show that (3.8) is locally exponentially stable, we use the theory of LMIs and appropriate geometric properties of the saturation. This strategy was used in [43], since our system is different from the one analyzed in [43], the matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{K}$ are not the same too, but they have the same structure. The spirit of the proof is the following. It is well known that to ensure the exponential stability of a linear system $\dot{z} = \mathbf{J}z$ it is enough to find a positively defined matrix $\mathbf{P}$ satisfying the Lyapunov inequality,

$$ \mathbf{J}^\top \mathbf{P} + \mathbf{P}\mathbf{J} < -\mathbf{I}_n, $$

which corresponds to a system of $\frac{n(n-1)}{2}$ equations that can be solved analytically and numerically. The action of the saturation map makes the task more intricate yet still allows us to guarantee local stability.

For the saturation map **sat**, we introduce the deadzone nonlinearity $\phi : \mathbb{R}^m \to \mathbb{R}^m$ by

$$ \phi(u) = \mathbf{sat}(u) - u, \quad \forall u \in \mathbb{R}^m, $$

we also use the following generalized sector condition [52, Lemma 1.6].

**Lemma 3.1.** If for some $\mathbf{C} \in \mathbb{R}^{m \times n}$, $z \in \mathbb{R}^n$ and $j \in \{1, \dots, m\}$ it holds that $|((\mathbf{K} - \mathbf{C})z)_j| \le \ell$, then

$$ \phi_j(\mathbf{K}z)(\phi_j(\mathbf{K}z) + (\mathbf{C}z)_j) \le 0. $$

As was noticed in [52, Appendix C.5], the above inequality remains true when we multiply by positive scalars. In particular, for $\mathbf{D} \in \mathbb{R}^{m \times m}$ a diagonal, positive defined matrix, we have

$$ \phi_j(\mathbf{K}z)\mathbf{D}(\phi_j(\mathbf{K}z) + (\mathbf{C}z)_j) \le 0. $$

The motivation behind using the generalized sector condition is that allows to construct a Lyapunov functional that will help us to study exponential decay thanks to the appearance of an augmented matrix involving LMIs. Another useful tool is the Schur complement [52, Appendix C.5].

**Lemma 3.2.** (Schur's lemma) Let $\mathbf{A} \in \mathbb{R}^{n \times n}$, $\mathbf{B} \in \mathbb{R}^{m \times n}$, $\mathbf{C} \in \mathbb{R}^{m \times m}$ and consider

$$ \mathbf{M} := \begin{pmatrix} \mathbf{A} & \mathbf{B}^\top \\ \mathbf{B} & \mathbf{C} \end{pmatrix}. $$

If $\mathbf{C}$ is positive definite, then $\mathbf{M}$ is positive definite if and only if its Schur complement $\mathbf{M} \setminus \mathbf{C} := \mathbf{A} - \mathbf{B}^\top \mathbf{C}^{-1} \mathbf{B}$ is positive semi-definite.

The next proposition shows the local exponential decay of (3.8) and gives us an explicit region of attraction [43, Proposition 2].

**Proposition 3.4.** Let $\mathbf{K}$ be as in Assumption 1. Let $\mathbf{P} \in \mathbb{R}^{n \times n}$ be symmetric and positive definite, and $\mathbf{D} \in \mathbb{R}^{n \times n}$ be diagonal and positive definite and $\mathbf{C} \in \mathbb{R}^{m \times n}$ such that

$$ M_1 := \begin{pmatrix} (\mathbf{A} + \mathbf{B}\mathbf{K})^\top \mathbf{P} + \mathbf{P}(\mathbf{A} + \mathbf{B}\mathbf{K}) & \mathbf{P}\mathbf{B} - (\mathbf{D}\mathbf{C})^\top \\ (\mathbf{P}\mathbf{B})^\top - \mathbf{D}\mathbf{C} & -2\mathbf{D} \end{pmatrix} < 0, \quad (3.24) $$

and

$$ M_2 := \begin{bmatrix} \mathbf{P} & (\mathbf{K} - \mathbf{C})^\top \\ \mathbf{K} - \mathbf{C} & \ell^2 \mathbf{I}_m \end{bmatrix} \succeq 0. \quad (3.25) $$
Then (3.8) is locally exponentially stable in 0 with region of attraction $\mathcal{R}$ given by

$$ \mathcal{R} := \{z \in \mathbb{R}^n : z^\top \mathbf{P} z \le 1\}. \quad (3.26) $$

Moreover, in $\mathcal{R}$, the function $V_1(z) := z^\top \mathbf{P} z$, $z \in \mathbb{R}^n$, decreases exponentially fast to 0 along the solutions to (3.8), i.e., there is a constant $\alpha > 0$ so that

$$ \dot{V}_1(z) \le -\alpha |z|^2, \quad z \in \mathcal{R}. \quad (3.27) $$

**Proof:** Let us start by proving that the LMIs (3.24) and (3.25) are feasible, i.e., that if we take $\mathbf{K}$ as in Assumption 1, then it is possible to find $\mathbf{P} \in \mathbb{R}^{m \times n}$ symmetric positive definite, $\mathbf{D} \in \mathbb{R}^{m \times m}$ diagonal, positive definite and $\mathbf{C} \in \mathbb{R}^{m \times n}$ such that (3.24) and (3.25) hold. As $\mathbf{A} + \mathbf{B}\mathbf{K}$ is Hurwitz, by the Lyapunov inequality [4, equation (1.2)], there exists a symmetric positive definite matrix $\mathbf{P} \in \mathbb{R}^{m \times n}$ such that

$$ (\mathbf{A} + \mathbf{B}\mathbf{K})^\top \mathbf{P} + \mathbf{P} (\mathbf{A} + \mathbf{B}\mathbf{K}) < 0. $$

To ensure the existence of such a $\mathbf{C}$, take $\mathbf{C} = \mathbf{0} \in \mathbb{R}^{m \times n}$ and observe that $\ell^2 \mathbf{I}_m > 0$. Consider $\mathbf{P} \in \mathbb{R}^{m \times n}$ with $\|\mathbf{P}\|$ sufficiently large such that

$$ \mathbf{P} - \mathbf{K}^\top \frac{1}{\ell^2} \mathbf{I}_m \mathbf{K} \ge 0. $$

and $\mathbf{D} \in \mathbb{R}^{m \times m}$ with $\|\mathbf{D}\|$ large enough. It follows from (3.2) that (3.24) and (3.25) are feasible. As (3.24) and (3.25) are feasible, we can consider $\mathbf{P}, \mathbf{D}, \mathbf{C}$, not necessarily the same as before satisfying (3.24) and (3.25). Then,

$$ \begin{aligned} \dot{V}_1(z) &= \dot{z}^\top \mathbf{P} z + z^\top \mathbf{P} \dot{z} \\ &= (\mathbf{A}z + \mathbf{B} \mathbf{sat} \mathbf{K} z)^\top \mathbf{P} z + z^\top \mathbf{P} (\mathbf{A}z + \mathbf{B} \mathbf{sat} \mathbf{K} z), \end{aligned} $$

introducing the deadzone nonlinearity, we get

$$ \mathbf{sat} \mathbf{K} z = \phi(\mathbf{K} z) + \mathbf{K} z, $$

thus

$$ \dot{V}_1(z) = z^\top [(\mathbf{A} + \mathbf{B}\mathbf{K})^\top \mathbf{P} + \mathbf{P}(\mathbf{A} + \mathbf{B}\mathbf{K})] z + 2z^\top \mathbf{P} \mathbf{B} \phi(\mathbf{K} z), $$

where we have used $z^\top \mathbf{P} \mathbf{B} \phi(\mathbf{K} z) = \phi(\mathbf{K} z)^\top (\mathbf{P} \mathbf{B})^\top z$. Suppose that $|((\mathbf{K} - \mathbf{C})z)_j| \le \ell$ for all $j \in \{1, \dots, m\}$ by Lemma 3.1 we obtain

$$ 0 \le -2\phi(\mathbf{K} z)^\top \mathbf{D}(\phi(\mathbf{K} z) + \mathbf{C} z)), $$

therefore

$$ \begin{aligned} \dot{V}_1(z) &= z^\top [(\mathbf{A} + \mathbf{B}\mathbf{K})^\top \mathbf{P} + \mathbf{P}(\mathbf{A} + \mathbf{B}\mathbf{K})] z + 2z^\top \mathbf{P} \mathbf{B} \phi(\mathbf{K} z) \\ &\le z^\top [(\mathbf{A} + \mathbf{B}\mathbf{K})^\top \mathbf{P} + \mathbf{P}(\mathbf{A} + \mathbf{B}\mathbf{K})] z + 2z^\top \mathbf{P} \mathbf{B} \phi(\mathbf{K} z) - 2\phi(\mathbf{K} z)^\top \mathbf{D}(\phi(\mathbf{K} z) + \mathbf{C} z)) \\ &= \begin{pmatrix} z \\ \phi(\mathbf{K} z) \end{pmatrix}^\top \mathbf{M}_1 \begin{pmatrix} z \\ \phi(\mathbf{K} z) \end{pmatrix}. \end{aligned} $$

As $\mathbf{M}_1$ is negative definite, we can take $-\alpha < 0$ as the largest eigenvalue of $\mathbf{M}_1$, it follows that $\mathbf{M}_1 \preceq -\alpha \mathbf{I}_{n+m}$, from where we get

$$ \dot{V}_1(z) \le -\alpha |z|^2. \quad (3.28) $$

Let $\beta_{\max} > 0$ be the largest eigenvalue of $\mathbf{P} \in \mathbb{R}^{n \times n}$,

$$ -\alpha |z|^2 \le -\frac{\alpha}{\beta_{\max}} z^\top \mathbf{P} z = -\frac{\alpha}{\beta_{\max}} V_1(z). $$

We conclude

$$ \dot{V}_1(z) \le -\frac{\alpha}{\beta_{\max}} V_1(z). $$

Finally, we must prove that $|((\mathbf{K} - \mathbf{C})z)_j| \le \ell$ for all $j \in \{1, \dots, m\}$. In that direction, we force to the attraction region to be included in the generalized sector

$$ \mathcal{R} = \{z \in \mathbb{R}^n : z^\top \mathbf{P} z \le 1\} \subseteq \{z \in \mathbb{R}^n : |(\mathbf{K} - \mathbf{C})z| \le \ell\}. $$

This can be rewritten as
$$ \{z \in \mathbb{R}^n : z^\top \mathbf{P} z \le 1\} \subseteq \{z \in \mathbb{R}^n : z^\top (\mathbf{K} - \mathbf{C})^\top \frac{1}{\ell^2} \mathbf{I}_m (\mathbf{K} - \mathbf{C}) z \le 1\}. \quad (3.29) $$

As both sets are ellipsoids centered at $0 \in \mathbb{R}^n$, [8, Proposition 1] (3.29) is equivalent to the following matrix inequality

$$ \mathbf{M}_2 \setminus \ell^2 \mathbf{I}_m = \mathbf{P} - (\mathbf{K} - \mathbf{C})^\top \frac{1}{\ell^2} \mathbf{I}_m (\mathbf{K} - \mathbf{C}) \geq 0, \quad (3.30) $$

where $\mathbf{M}_2 \setminus \ell^2 \mathbf{I}_m$ denotes the Schur's complement. As $\ell^2 \mathbf{I}_m$ is positive definite, by Lemma 3.2, (3.30) is equivalent to ask $\mathbf{M}_2$ satisfies (3.25), i.e.,

$$ \mathbf{M}_2 = \begin{pmatrix} \mathbf{P} & (\mathbf{K} - \mathbf{C})^\top \\ \mathbf{K} - \mathbf{C} & \ell^2 \mathbf{I}_m \end{pmatrix} \geq 0. $$

□

We can now easily deduce our stabilization result for the system (3.1). Let $\mathbf{K}$ as in Assumption 1, then taking $\mathbf{P}$, $\mathbf{D}$ and $\mathbf{C}$ as in Proposition 3.4, then (3.8) is locally exponentially stable in $\mathcal{R} := \{z \in \mathbb{R}^n : z^\top \mathbf{P} z \le 1\}$. By Proposition 3.3 (3.11) is locally exponentially stable in $0 \in L^2(0, L)$ with region of attraction $\iota(\mathcal{R}) \times X_n^\perp$. This completes the proof of Theorem 3.1. □

**3.4. On the controllability Assumption 1.** In the context of control involving Sturm-Liouville operators, it is standard to work under Assumption 1, i.e., the pair $(\mathbf{A}, \mathbf{B})$ is stabilizable (controllable). In this part, we wish to go a bit further in this assumption in the specific case of the spatial operator associated to the linear KS equation with clamped boundary conditions (1.2a).

(1) Case $m = 1$. With a single spatial localization $b$, we have

$$ \mathbf{B} = \begin{pmatrix} b_1 \\ \vdots \\ b_n \end{pmatrix}. $$

Therefore, the Kalman condition for the pair $(\mathbf{A}, \mathbf{B})$ is seen as

$$ \begin{aligned} \det (\mathbf{B}, \mathbf{A}\mathbf{B}, \dots, \mathbf{A}^{n-1}\mathbf{B}) &= \det \begin{pmatrix} b_1 & \sigma_1 b_1 & \cdots & \sigma_1^{n-1} b_1 \\ \vdots & \vdots & \ddots & \vdots \\ b_n & \sigma_n b_n & \cdots & \sigma_n^{n-1} b_n \end{pmatrix} \\ &= \prod_{j=1}^n b_j \det \begin{pmatrix} 1 & \sigma_1 & \cdots & \sigma_1^{n-1} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & \sigma_n & \cdots & \sigma_n^{n-1} \end{pmatrix} \\ &= \prod_{j=1}^n b_j \mathrm{Vdm}_{\sigma_1, \dots, \sigma_n}. \end{aligned} $$

where $\mathrm{Vdm}_{\sigma_1, \dots, \sigma_n} = \prod_{1 \le i < k \le n} (\sigma_k - \sigma_i)$ corresponds to the Vandermonde determinant. Hence, controllability of the pair $(\mathbf{A}, \mathbf{B})$ holds if $b_j \ne 0$ for all $j \in \{1, \dots, n\}$ and when the eigenvalues are simple. This last condition holds provided that $\lambda \notin \mathcal{N}$.

(2) Case $m \ge 2$ and simple eigenvalues. Using the Popov-Belevitch-Hautus test to study the block associated to each eigenvalue. In this case the matrix $\mathbf{B}$ has the form

$$ \mathbf{B} = \begin{pmatrix} b_{11} & \cdots & b_{1m} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nm} \end{pmatrix}, $$

For each eigenvalue, we aim to obtain

$$ \mathrm{Rank} [\mathbf{A} - \sigma_j \mathbf{I}_m \mid \mathbf{B}] = n, \quad \forall j \in \{1, \dots, n\}. $$

Thus, the pair $(\mathbf{A}, \mathbf{B})$ is controllable if

$$ \forall j \in \{1, \dots, n\}, \exists k \in \{1, \dots, m\} : b_{jk} \ne 0. $$
(3) Case $m \ge 1$ with repeated eigenvalues. When $m = 1$, the controllability may fail when the Kalman matrix

$$ \mathbf{K}_{\mathbf{A},\mathbf{B}} = [\mathbf{B} | \mathbf{A}\mathbf{B} | \cdots | \mathbf{A}^{n-1}\mathbf{B}] \in \mathbb{R}^{n \times nm}, $$

satisfies $\mathrm{rank}(\mathbf{K}) < n$. If we consider the localizations of the controller as design functions, a possible approach is to introduce additional control localizations, i.e., to consider (in the unsaturated case)

$$ \partial_t y = \mathcal{A}y + \sum_{k=1}^{m+N_0} b_k u_k. $$

In that framework, the Kalman matrix takes the form

$$ \tilde{\mathbf{K}}_{\mathbf{A},\mathbf{B}} = [\mathbf{B} | \mathbf{A}\mathbf{B} | \cdots | \mathbf{A}^{n-1}\mathbf{B}] \in \mathbb{R}^{n \times n(m+N_0)}, $$

where $N_0 \in \mathbb{N}$ is the number of additional localizations. Since the column operations do not change the rank of the matrix, we can assume that

$$ \tilde{\mathbf{K}}_{\mathbf{A},\mathbf{B}} = [\mathbf{K}_{\mathbf{A},\mathbf{B}} | \mathbf{N}]. $$

We need an appropriated number of additional localization to complement the rank of $\mathbf{K}_{\mathbf{A},\mathbf{B}}$. As shown in [32], one possible choice is

$$ N_0 = \max\{d_1, \dots, d_n\}, $$

where $d_j$ is the algebraic multiplicity of $\sigma_j$.

**Remark 3.5.** The previous condition is clearly not sharp. For instance take,

$$ \mathbf{A} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}, $$

where we have $\sigma_1 = \sigma_2 = 2$ and $\mathrm{rank}(\mathbf{K}_{\mathbf{A},\mathbf{B}}) = 2$. By the proposed strategy, we take $N_0 = 2$ and the matrix

$$ \mathbf{B}_1 = \begin{bmatrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ 4 & 1 & 1 \end{bmatrix} \quad \Rightarrow \quad \mathrm{rank}(\mathbf{K}_{\mathbf{A},\mathbf{B}_1}) = 3, $$

i.e., the pair $(\mathbf{A}, \mathbf{B})$ is controllable. But for instance, we can also take

$$ \mathbf{B}_2 = \begin{bmatrix} 2 & 1 \\ 3 & 1 \\ 4 & 1 \end{bmatrix} \quad \Rightarrow \quad \mathrm{rank}(\mathbf{K}_{\mathbf{A},\mathbf{B}_2}) = 3. $$

Repeating the same idea with

$$ \mathbf{B}_3 = \begin{bmatrix} 2 & 1 \\ 3 & 0 \\ 4 & 0 \end{bmatrix}, \quad \mathbf{B}_4 = \begin{bmatrix} 2 & 0 \\ 3 & 0 \\ 4 & 1 \end{bmatrix}, $$

we get $\mathrm{rank}(\mathbf{K}_{\mathbf{A},\mathbf{B}_3}) = 3$ and $\mathrm{rank}(\mathbf{K}_{\mathbf{A},\mathbf{B}_4}) = 2$.

To conclude this part, we study a very common situation in control theory for PDEs. Can we take a localization of the form, $b(x) = 1_\omega$ where $\omega \subset (0, L)$ is a non-empty open set? We just mention that in the case $m = 1$ and simple eigenvalues, this is not always possible with this approach. Following [10, Lemma 2.1], we know that for $\sigma > 0$ the eigenfunctions of $\mathcal{A}$ are of the form

$$ e(x) = A \cos \left( \alpha \left( x - \frac{L}{2} \right) \right) + B \sin \left( \alpha \left( x - \frac{L}{2} \right) \right) + C \cos \left( \beta \left( x - \frac{L}{2} \right) \right) + D \sin \left( \beta \left( x - \frac{L}{2} \right) \right), $$

where $\alpha, \beta \in \mathbb{R}$ are given by

$$ \alpha = \sqrt{\frac{-\lambda - \sqrt{\lambda^2 - 4\sigma}}{2}}, \quad \beta = \sqrt{\frac{-\lambda + \sqrt{\lambda^2 - 4\sigma}}{2}}. $$

By imposing the boundary conditions, we focus on the case

$$ \beta \cos(\beta/2) \sin(\alpha/2) = \alpha \cos(\alpha/2) \sin(\beta/2), \quad \sin(\alpha/2) \neq 0. $$
Thus, we get

$$e_j(x) = C \left[ -\frac{\sin(\beta/2)}{\sin(\alpha/2)} \sin\left(\alpha\left(x - \frac{L}{2}\right)\right) + \sin\left(\beta\left(x - \frac{L}{2}\right)\right) \right].$$

The localization coefficients are then given by $b_j(x) = \langle 1_\omega, e_j \rangle_{L^2(0,L)} = \int_\omega e_j(x)dx$. If we take $\omega = (\frac{L}{4}, \frac{3L}{4}) \subset (0, L)$, and observing the symmetry of $e_j$ with respect to $x = \frac{L}{2}$ we get

$$b_j(x) = \int_{L/4}^{3L/4} e_j(x)dx = 0, \quad \det(\mathbf{K}_{A,B}) = 0.$$

This shows that, even when the eigenvalues are simple, the choice of $\omega \subset (0, L)$ depends on the structure of the eigenfunctions.

## 4. $H^2$ AND FULLY NONLINEAR STABILITY

In the previous sections, we dealt with the exponential stability in the $L^2$-norm. To obtain a stability result for the fully nonlinear system (which includes the nonlinearity $N(y)$), we now prove exponential stability in the $H^2$-norm. Motivated by [24], we consider the following Lyapunov functional:

$$V_2(t) = \frac{M}{2}V_1(t) - \langle y, Ay \rangle_{L^2(0,L)} = \frac{M}{2}z^\top \mathbf{P}z - \langle y, Ay \rangle_{L^2(0,L)}, \quad (4.1)$$

where $z = \iota(\Pi_n y)$ corresponds to the finite dimensional component of $y$ solution of (3.1) and $M > 0$ big enough to fix later.

**Remark 4.1.** The term $-\langle y, Ay \rangle_{L^2(0,L)}$ may seem unusual in the structure of the Lyapunov functional $V_2$. This Lyapunov functional has a frequency-based; that is, it separates low and high frequency behaviour. For instance if, $\mathbf{P} = \mathbf{I}$ and $y = \sum_{j=1}^\infty y_j e_j$, we get

$$V_2(t) = \frac{M}{2} \sum_{j=1}^{n} \left( \frac{M}{2} - \sigma_j \right) y_j^2 + \sum_{j=1}^{\infty} (-\sigma_j) y_j^2.$$

The goal of this section is to establish the following result:

**Theorem 4.1.** Let $\mathbf{K}$, $\mathbf{P}$ be as in Proposition 3.4. For any $y_0 \in H^2(0, L)$ such that $\iota(\Pi_n y_0) \subset S = \mathcal{R}$, where $\mathcal{R}$ is the region of attraction of the finite dimensional system. The unique solution of the closed-loop system (3.11) satisfies

$$\|y(t, \cdot)\|_{H^2(0,L)} \le Ce^{-at}\|y_0\|_{H^2(0,L)}, \quad t > 0.$$

**Proof:** We start by noting that, due to the boundary conditions (1.2), it is enough to show the exponential decay of $\|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}$. Indeed, we have

$$\begin{aligned} \|\partial_x y(t, \cdot)\|_{L^2(0,L)}^2 &= \int_0^L \partial_x y(t, \cdot) \partial_x y(t, \cdot) dx \\ &= - \int_0^L y(t, \cdot) \partial_x^2 y(t, \cdot) dx \\ &\le \frac{1}{2} \left( \|y(t, \cdot)\|_{L^2(0,L)}^2 + \|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2 \right). \end{aligned} \quad (4.2)$$

**Lemma 4.1.** There exist $C_1, C_2 > 0$ such that

$$\frac{C_1}{2}|z|^2 + \frac{C_1}{2C_2} \|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)} \le V_2(t). \quad (4.3)$$

**Proof:** Let $\beta_{\min}, \beta_{\max} > 0$ be the minimum and maximum eigenvalues of the positive defined and symmetric matrix $\mathbf{P}$. Therefore, we have $V_1(t) \ge \beta_{\min}|z|^2$. On the other hand, since $\sigma_1 \ge \sigma_j$ for all $j = 1, \dots, n$ and that
$z$ is solution of finite dimensional (3.8)

$$ 
\begin{aligned}
-\frac{1}{2} \sum_{j=1}^{\infty} \sigma_j y_j(t)^2 &= -\frac{1}{2} \sum_{j=1}^{n} \sigma_j y_j(t)^2 - \frac{1}{2} \sum_{j=n+1}^{\infty} \sigma_j y_j(t)^2 \\
&\geq -\frac{\sigma_1}{2} |z|^2 - \frac{1}{2} \sum_{j=n+1}^{\infty} \sigma_j y_j(t)^2.
\end{aligned}
$$

Therefore,

$$ 
\begin{aligned}
V_2(t) &= -\frac{1}{2} \sum_{j=1}^{\infty} \sigma_j y_j(t)^2 + \frac{M}{2} V_1(t) \\
&\geq \frac{(M\beta_{\min} - \sigma_1)}{2} |z|^2 - \frac{1}{2} \sum_{j=n+1}^{\infty} \sigma_j y_j(t)^2 \qquad (4.4) \\
&\geq C_1 \left( |z|^2 - \sum_{j=n+1}^{\infty} \sigma_j y_j(t)^2 \right), \quad C_1 = \min \left\{ \frac{(M\beta_{\min} - \sigma_1)}{2}, \frac{1}{2} \right\}.
\end{aligned}
$$

Similarly to before, and recalling that $-e_j''' - \lambda e_j'' = \sigma_j e_j$, for all $j \in \mathbb{N}$

$$ 
\begin{aligned}
\|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2 &= \sum_{(j,k) \in \mathbb{N}^2} y_j(t)y_k(t) \langle e_j'', e_k'' \rangle_{L^2(0,L)} \\
&= \sum_{(j,k) \in \mathbb{N}^2} y_j(t)y_k(t) \langle e_j, e_k''' \rangle_{L^2(0,L)} \\
&= \sum_{(j,k) \in \mathbb{N}^2} y_j(t)y_k(t) \langle e_j, \lambda e_k'' - \sigma_k e_k \rangle_{L^2(0,L)} \\
&= -\lambda \langle y(t, \cdot), \partial_x^2 y(t, \cdot) \rangle_{L^2(0,L)} - \sum_{k=1}^{\infty} \sigma_j y_j^2(t) \\
&\leq \frac{1}{2} \left( \lambda^2 \|y(t, \cdot)\|_{L^2(0,L)}^2 + \|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2 \right) - \sum_{k=1}^{\infty} \sigma_j y_j^2(t).
\end{aligned}
$$

Next, we find

$$ 
\begin{aligned}
\|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2 &\leq \lambda^2 \|y(t, \cdot)\|_{L^2(0,L)}^2 - 2 \sum_{k=1}^{\infty} \sigma_j y_j^2(t) \\
&\leq \lambda^2 \sum_{k=1}^{\infty} y_j^2(t) - 2 \sum_{k=n+1}^{\infty} \sigma_j y_j^2(t) \\
&= \lambda^2 \sum_{k=1}^{n} y_j^2(t) + \sum_{k=n+1}^{\infty} (\lambda^2 - 2\sigma_j) y_j^2(t) \\
&= \lambda^2 |z|^2 + \sum_{k=n+1}^{\infty} (\lambda^2 - 2\sigma_j) y_j^2(t).
\end{aligned}
$$

Now, for $j \geq n + 1$ we have $\sigma_j < 0$ and $0 < -\sigma_{n+1} \leq -\sigma_j$, thus

$$ 0 < \lambda^2 - 2\sigma_j = -\sigma_j \left( 2 - \frac{\lambda^2}{\sigma_j} \right) \leq -\sigma_j \left( 2 - \frac{\lambda^2}{\sigma_{n+1}} \right). $$

Therefore, we get the following estimate

$$ \|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2 \leq C_2 \left( |z|^2 - \sum_{j=n+1}^{\infty} \sigma_j y_j^2(t) \right), $$

where $C_2 = \max \left\{ \lambda^2, 2 - \frac{\lambda^2}{\sigma_{n+1}} \right\} > 0$. Similarly
$$ -\sum_{j=n+1}^{\infty} \sigma_j y_j^2(t) \geq \frac{\|\partial_x^2 y(t, \cdot)\|_{L^2(0,L)}^2}{C_2} - |z|^2. \quad (4.5) $$

Finally, from the previous computations we obtain

$$ \begin{aligned} V_2(t) &\geq C_1 |z|^2 - C_1 \sum_{j=n+1}^{\infty} \sigma_j y_j^2(t) \\ &\geq C_1 |z|^2 - \frac{C_1}{2} \sum_{j=n+1}^{\infty} \sigma_j y_j^2(t) \\ &\geq C_1 |z|^2 + \frac{C_1 \|\partial_x^2 y\|_{L^2(0,L)}^2}{2C_2} - \frac{C_1 |z|^2}{2} \\ &= \frac{C_1}{2} |z|^2 + \frac{C_1}{2C_2} \|\partial_x^2 y\|_{L^2(0,L)}, \end{aligned} $$

from where we deduce (4.3). □

**Lemma 4.2.** *There exists a > 0 such that*

$$ V_2(t) \leq e^{-at} V_2(0), \quad \forall t \geq 0. $$

**Proof:** From Proposition 3.4, we know that the finite dimensional system (3.8) satisfies (3.27). On the other hand, as $\mathcal{A}$ is self-adjoint we have

$$ \frac{d}{dt} \langle \mathcal{A}y, y \rangle_{L^2(0,L)} = 2 \langle \mathcal{A}y, \partial_t y \rangle_{L^2(0,L)}. $$

Using (3.2) with control $u = \mathbf{K}z$ we see

$$ \begin{aligned} -\frac{d}{dt} \langle \mathcal{A}y, y \rangle_{L^2(0,L)} &= -2 \left\langle \mathcal{A}y, \mathcal{A}y + \sum_{k=1}^{m} b_k \cdot \mathbf{sat}(\mathbf{K}z)_k \right\rangle_{L^2(0,L)} \\ &= -2 \|\mathcal{A}y\|_{L^2(0,L)}^2 - 2 \left\langle \mathcal{A}y, \sum_{k=1}^{m} b_k \cdot \mathbf{sat}(\mathbf{K}z)_k \right\rangle_{L^2(0,L)}. \end{aligned} \quad (4.6) $$

The second term in the above inequality can be estimated as follows

$$ \begin{aligned} -2 \left\langle \mathcal{A}y, \sum_{k=1}^{m} b_k \cdot \mathbf{sat}(\mathbf{K}z)_k \right\rangle_{L^2(0,L)} &\le \|\mathcal{A}y\|_{L^2(0,L)}^2 + |b_j|^2 \cdot |\mathbf{sat} \mathbf{K}z|^2 \\ &\le \|\mathcal{A}y\|_{L^2(0,L)}^2 + \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 |z|^2. \end{aligned} $$

Using this information in (4.6)

$$ \begin{aligned} -\frac{d}{dt} \langle \mathcal{A}y, y \rangle_{L^2(0,L)} &\le -\|\mathcal{A}y\|_{L^2(0,L)}^2 + \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 |z|^2. \quad (4.7) \\ &= -\sum_{j=1}^{\infty} \sigma_j^2 y_j^2(t) + \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 |z|^2. \end{aligned} $$

Observe that the eigenvalues $\{\sigma_j\}_{j \in \mathbb{N}}$ satisfy the following

$$ \begin{cases} \sigma_1 > \sigma_j \ge 0, & j \le n \\ 0 > \sigma_{n+1} \ge \sigma_j, & j \ge n+1. \end{cases} $$

Moreover, if $j \ge n+1$ we have
$$ -\sigma_j \leq \frac{\sigma_j^2}{-\sigma_{n+1}}, \quad \forall j \geq n+1, \qquad (4.8) $$

and therefore, taking

$$ M \geq -\frac{1}{\sigma_{n+1}}, \qquad (4.9) $$

we see from (4.9) and (4.8) that

$$ \begin{aligned} M \|A y\|_{L^2(0,L)}^2 &= M \sum_{j=1}^{\infty} \sigma_j^2 y_j^2(t) \\ &\geq M \sum_{j=n+1}^{\infty} \sigma_j^2 y_j^2(t) \\ &\geq -\frac{1}{\sigma_{n+1}} \sum_{j=n+1}^{\infty} \sigma_j^2 y_j^2(t) \\ &\geq -\sum_{j=n+1}^{\infty} \sigma_j y_j^2(t). \end{aligned} $$

From the last line we deduce

$$ -\|A y\|_{L^2(0,L)}^2 \leq \frac{1}{M} \sum_{j=n+1}^{\infty} \sigma_j y_j^2(t) \leq \frac{1}{M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t). \qquad (4.10) $$

Gathering (4.10), (4.7) and (3.27)

$$ \begin{aligned} \dot{V}_2(t) &= -\frac{1}{2} \frac{d}{dt} \langle A y, y \rangle_{L^2(0,L)} + \frac{M}{2} \dot{V}_1(t) \\ &\leq -\frac{1}{2} \sum_{j=1}^{\infty} \sigma_j^2 y_j^2(t) + \frac{\left( \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 \right)}{2} |z|^2 - \frac{\alpha M}{2} |z|^2 \\ &= -\frac{1}{2} \|A y\|_{L^2(0,L)}^2 + \frac{\left( \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 - \alpha M \right)}{2} |z|^2 \\ &\leq \frac{1}{2M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) + \frac{\left( \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 - \alpha M \right)}{2} |z|^2. \end{aligned} \qquad (4.11) $$

As $\alpha > 0$, we can take

$$ C_3 > \frac{1}{2\alpha}, \qquad (4.12) $$

and $M$ such that

$$ \|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 - \alpha M < -\frac{M}{2C_3}, \qquad (4.13) $$

to obtain

$$ \dot{V}_2(t) \leq \frac{1}{2M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) - \frac{M}{4C_3} |z|^2. \qquad (4.14) $$

**Remark 4.2.** From (4.12) and estimate (4.13), we deduce the inequality

$$ 0 < \frac{\|\mathbf{K}\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2}{\left(\alpha - \frac{1}{2C_3}\right)} < M, $$
that ensure that the value of $C_3 > 0$ is feasible and do not affect the sign of $M$.

On the other hand, by definition of $\beta_{\max}$

$$ -|z|^2 \leq -\frac{1}{\beta_{\max}} z^\top \mathbf{P} z, $$

using this in (4.14), we get

$$ \begin{aligned} \dot{V}_2(t) &\leq \frac{1}{2M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) - \frac{M}{4C_3} |z|^2 \\ &\leq \frac{1}{2M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) - \frac{1}{2C_3 \beta_{\max}} \frac{M}{2} |z|^2. \end{aligned} $$

Finally, if $M \geq 2C_3\beta_{\max}$, we obtain

$$ \begin{aligned} \dot{V}_2(t) &\leq \frac{1}{2M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) - \frac{1}{2C_3 \beta_{\max}} \frac{M}{2} |z|^2 \\ &\leq \frac{1}{2C_3 \beta_{\max}} \left( \frac{1}{2} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) - \frac{M}{2} z^\top \mathbf{P} z \right) \tag{4.15} \\ &= -\frac{1}{2C_3 \beta_{\max}} \left( -\frac{1}{2} \langle \mathcal{A} y, y \rangle_{L^2(0,L)} + \frac{M}{2} V_1(t) \right) \\ &= -\frac{1}{2C_3 \beta_{\max}} V_2(t). \end{aligned} $$

which finishes the proof of Lemma 4.2 with $a = \frac{1}{2C_3\beta_{\max}}$.

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

We conclude the proof of the exponential stability of $V_2$ by invoking the following lemma,

**Lemma 4.3.** *There exists $C_4 > 0$ such that*

$$ V_2(0) \leq C_4 \|y_0\|_{H^2(0,L)}^2. $$

**Proof:** Using the expression of $\mathcal{A}$ and its boundary conditions (1.2) we observe that

$$ \begin{aligned} V_2(t) &= -\frac{1}{2} \langle \mathcal{A} y, y \rangle_{L^2(0,L)} + \frac{M}{2} V_1(t) \\ &\leq \frac{1}{2} \| \partial_x^2 y(t, \cdot) \|_{L^2(0,L)}^2 - \frac{\lambda}{2} \| \partial_x y(t, \cdot) \|_{L^2(0,L)}^2 + \frac{M \beta_{\max}}{2} |z|^2 \\ &\leq \frac{1}{2} \| \partial_x^2 y(t, \cdot) \|_{L^2(0,L)}^2 + \frac{M \beta_{\max}}{2} \| y(t, \cdot) \|_{L^2(0,L)}^2 \\ &\leq C_4 \| y(t, \cdot) \|_{H^2(0,L)}^2. \end{aligned} $$

We conclude by taking $t = 0$.

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

Finally, Theorem 4.1 comes immediately from Lemmas 4.1 to 4.3.

<figure><img src="image_5.png" /></figure>

**4.1. Nonlinear case.** In this section, we study the stabilization of the nonlinear system (1.1) under saturated feedback law. More specifically, we show the local exponential stability of the following system

$$ \left\{ \begin{array}{ll} \partial_t y + \lambda \partial_x^2 y + \partial_x^4 y + \mathcal{N}(y) = \sum_{k=1}^{m} b_k(x) \mathbf{sat}((\mathbf{K}_l^{-1}(\Pi_n y(\cdot, t)))_k, & t > 0, x \in (0, L), \\ y(0, x) = y_0(x), & x \in (0, L). \end{array} \right. \quad (4.16) $$

Similarly to the linear case expanded every solution $y(t, \cdot) \in D(\mathcal{A})$ of (4.16) as a series on the eigenfunctions, $e_j(\cdot)$ we get that (4.16) is equivalent to the next infinity dimensional control system
$$ \dot{y}_j(t) = \sigma_j y_j(t) y + \mathbf{b}_j \mathbf{sat}((\mathbf{K}_t^{-1}(\Pi_n y(\cdot, t))) + f_j(t), \quad j \in \mathbb{N}, \qquad (4.17) $$

where $f_j(t) = \langle -\mathcal{N}(y), e_j \rangle_{L^2(0,L)}$. Recalling that $n \in \mathbb{N}$ is the number of unstable eigenvalues of $\mathcal{A}$, we define

$$ \mathbf{F}(t) = \begin{pmatrix} f_1(t) \\ \vdots \\ f_n(t) \end{pmatrix}, \qquad (4.18) $$

we can construct the next finite dimensional system:

$$ \dot{z}(t) = \mathbf{A}z(t) + \mathbf{B}\mathbf{sat}(\mathbf{K}z(t)) + \mathbf{F}(t). \qquad (4.19) $$

Now we state our main result.

**Theorem 4.2.** Consider the system (4.16) and let $\mathbf{K}$, $\mathbf{P}$ as in Proposition 3.4, then there exists $\varepsilon > 0$ such that for all $y_0 \in D(\mathcal{A})$ satisfying $\|y_0\|_{H^2(0,L)} \le \varepsilon$, the unique solution $y \in C([0,T]; H^2(0,L)) \cap L^2(0,T, H^4(0,L))$ for all $T > 0$ satisfy

$$ \|y(t, \cdot)\|_{H^2(0,L)} \le C e^{-at} \|y_0\|_{H^2(0,L)}, \qquad (4.20) $$

for $C > 0$.

**Proof:** By Section 3.1, we know that (4.16) correspond to (3.14) with $\delta = 1$. Then, well-posedness part is a consequence of Proposition 3.2 and (3.21). The proof of the stability part of Theorem 4.2 is based on the Lyapunov function (4.1). Let $T > 0$, we calculate the derivative of $V_2$ along the trajectories of (4.17)

$$ \dot{V}_2(t) = \frac{M}{2} \frac{d}{dt} (z^{\top} \mathbf{P} z) - \frac{d}{dt} \langle y, \mathcal{A} y \rangle_{L^2(0,L)} $$

We will analyze these two terms. First using (4.19)

$$ \frac{d}{dt} (z^{\top} \mathbf{P} z) = (\mathbf{A} z + \mathbf{B} \mathbf{sat}(\mathbf{K} z) + \mathbf{F})^{\top} \mathbf{P} z + z^{\top} \mathbf{P} (\mathbf{A} z + \mathbf{B} \mathbf{sat}(\mathbf{K} z) + \mathbf{F}). $$

By Proposition (3.4) we know that in the set $\mathcal{R}$

$$ (\mathbf{A} z + \mathbf{B} \mathbf{sat}(\mathbf{K} z))^{\top} \mathbf{P} z + z^{\top} \mathbf{P} (\mathbf{A} z + \mathbf{B} \mathbf{sat}(\mathbf{K} z)) \le -2\alpha |z|^2. $$

Thus, we get

$$ \frac{d}{dt} (z^{\top} \mathbf{P} z) \le -\alpha |z|^2 + 2 \mathbf{F}^{\top} \mathbf{P} z. \qquad (4.21) $$

Then, using Young's inequality for $\varepsilon_1 > 0$ to define later

$$ |\mathbf{F}^{\top} \mathbf{P} z| \le \|\mathbf{P}\| \|\mathbf{F}\| |z| \le \frac{\|\mathbf{P}\|}{2} \left( \frac{1}{\varepsilon_1} |\mathbf{F}|^2 + \varepsilon_1 |z|^2 \right). \qquad (4.22) $$

Using the orthonormality of $e_j$, we observe

$$ \begin{aligned} |f_j(t)| &= \left| \int_0^L f(t, x) e_j(x) dx \right| \\ &\le \|f(t, \cdot)\|_{L^2(0,L)} \\ &= \|y \partial_x y\|_{L^2(0,L)}. \end{aligned} $$
By the continuous injection from $H^1(0, L)$ to $L^\infty(0, L)$ and (4.2)

$$ \begin{aligned} \|y \partial_x y\|_{L^2(0,L)} &\le \|y\|_{L^\infty(0,L)} \|\partial_x y\|_{L^2(0,L)} \\ &\le \|y\|_{H^1(0,L)}^2 \\ &= C \left( \|\partial_x y\|_{L^2(0,L)}^2 + \|y\|_{L^2(0,L)}^2 \right) \\ &\le \frac{C}{2} \left( \|\partial_x^2 y\|_{L^2(0,L)}^2 + 3 \|y\|_{L^2(0,L)}^2 \right) \\ &= \frac{C}{2} \left( \|\partial_x^2 y\|_{L^2(0,L)}^2 + 3|z|^2 + 3 \sum_{j=n+1}^{\infty} y_j^2(t) \right), \end{aligned} $$

the term $\sum_{j=n+1}^{\infty} y_j^2(t)$ can be estimated as (4.10) and using (4.4) obtaining

$$ \begin{aligned} \sum_{j=n+1}^{\infty} y_j^2(t) &= \sum_{j=n+1}^{\infty} \frac{-\sigma_j}{-\sigma_j} y_j^2(t) \\ &\le \sum_{j=n+1}^{\infty} \frac{-\sigma_j}{-\sigma_{n+1}} y_j^2(t) \\ &\le \frac{1}{-\sigma_{n+1} C_1} V_2 - \frac{1}{-\sigma_{n+1}} |z|^2 \end{aligned} $$

Finally, joining these previous estimates with (4.3) we get the existence of $C_5 > 0$ such that

$$ |f_j(t)| \le C_5 V_2(t). \quad (4.23) $$

**Remark 4.3.** If we consider boundary conditions such that $e_j \in H_0^1(0,1)$, the term $f_j$ can be alternatively estimated as follows

$$ \begin{aligned} f_j(t) &= \int_0^L y \partial_x y e_j dx \\ &= \frac{1}{2} \int_0^L (\partial_x |y|^2) e_j dx \\ &= -\frac{1}{2} \int_0^L |y|^2 e_j' dx. \end{aligned} $$

Thus, $|f_j(t)| \le \|e_j\|_{W^{1,\infty}(0,L)} \|y(t, \cdot)\|_{L^2(0,L)}^2$.

Using (4.23), (4.21) and (4.22) we obtain

$$ \begin{aligned} \frac{d}{dt} (z^\top \mathbf{P} z) &\le -\alpha |z|^2 + \frac{\|\mathbf{P}\| C_5^2}{2\varepsilon_1} V_2^2 + \frac{\|\mathbf{P}\| \varepsilon_1}{2} |z|^2 \\ &\le -\frac{\alpha}{2} |z|^2 + \frac{\|\mathbf{P}\|^2 C_5^2}{2\alpha} V_2^2, \end{aligned} \quad (4.24) $$

where, we have taken $\varepsilon_1 = \frac{\alpha}{\|\mathbf{P}\|}$. On the other hand, similar to (4.6), we have that

$$ -\frac{1}{2} \frac{d}{dt} \langle y, \mathcal{A} y \rangle_{L^2(0,L)} = - \|\mathcal{A} y\|_{L^2(0,L)}^2 - \left\langle \mathcal{A} y, \sum_{k=1}^{m} b_k \cdot \mathbf{sat}(\mathbf{K} z)_k \right\rangle_{L^2(0,L)} - \langle f, \mathcal{A} y \rangle_{L^2(0,L)}, $$

Using the same ideas as for the first term and with (4.14), we get from Cauchy inequality
$$ 
\begin{aligned}
-\frac{1}{2} \frac{d}{dt} \langle y, Ay \rangle_{L^2(0,L)} &\leq -\frac{1}{2} \|Ay\|_{L^2(0,L)}^2 + \frac{\left( \|K\|^2 \sum_{k=1}^m \|b_k\|_{L^2(0,L)}^2 \right)}{2} |z|^2 + \frac{C_5^2}{2} V_2^2 \\
&\leq \frac{1}{4M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) + \frac{\left( \|K\|^2 \sum_{k=1}^m \|b_k\|_{L^2(0,L)}^2 \right)}{2} |z|^2 + \frac{C_5^2}{2} V_2^2.
\end{aligned}
\quad (4.25) $$

Combining (4.24) and (4.25) we derive

$$ \dot{V}_2(t) \leq \frac{1}{4M} \sum_{j=1}^{\infty} \sigma_j y_j^2(t) + \frac{\left( 2 \|K\|^2 \sum_{k=1}^{m} \|b_k\|_{L^2(0,L)}^2 - M\alpha \right)}{4} |z|^2 + \frac{C_5^2}{2\alpha} (\|P\|^2 + \alpha) V_2^2. $$

Then, if we take $M$ big enough, and following the (4.13), (4.14) and (4.15), we can conclude the existence of $A$, $B > 0$ such that

$$ \dot{V}_2(t) \leq -AV_2(t) + BV_2^2(t), \text{ for all } t \in [0, T). $$

We can argue that if $V_2(0) \leq A/(2B)$, with $V_2(0) \neq 0$, then $V_2(t) \leq (A/B)e^{-At}$ for all $t \in [0, \infty)$. To that end, we apply Lemma 2.1 with $V_2(t) = v(t)$, $b(t) = -A$, $k(t) = B$ and $p = 2$. Then, in virtue of (2.2) we get

$$ w(t) \geq V_2(0)^{-1} - (B/A)(1 - e^{-At}) \geq V(0)^{-1} - B/A. $$

Accordingly, if we choose $V_2(0) > 0$ so that $V_2(0)^{-1} - B/A \geq B/A$, which happens when $V_2(0) \leq A/(2B)$, it follows that $w(t) \geq B/A > 0$ for all $t \in [0, \infty)$. Therefore, from (2.3) we deduce the required inequality:

$$ V_2(t) \leq (A/B)e^{-At}, \quad t \in [0, T). \quad (4.26) $$

To finish the proof of the stabilization, note that by Lemma 4.3

$$ V_2(0) \leq C_4 \|y_0\|_{H^2(0,L)}^2. $$

Finally taking $\varepsilon > 0$ small enough such that if $\|y_0\|_{H^2(0,L)}^2 \leq \varepsilon$ then $z(0) \in \mathcal{R}$ and $V_2(0) \leq A/(2B)$ we conclude (4.20). This stability estimate gives us if $\|y_0\|_{H^2(0,L)}^2 \leq \varepsilon$, then $\|y(t, \cdot)\|_{L^2(0,L)}^2 \leq C \|y_0\|_{H^2(0,L)}^2$. Thus, from (3.21) we get the global in time well-posedness in $C([0, T]; H^2(0, L)) \cap L^2(0, T; H^4(0, L))$.

□

## 5. BOUNDARY FEEDBACK STABILIZATION

In this section, we briefly outline how the same ideas extend to boundary feedback. The analysis is similar to the internal case, with only minor modifications concerning the boundary operator. Again, the main result of this section holds for the three class of boundary conditions. For simplicity, we write the proof in the case *clamped* case and KS nonlinearity. We consider the following system

$$ 
\left\{
\begin{array}{ll}
\partial_t y + \lambda \partial_x^2 y + \partial_x^4 y + y \partial_x y = 0, & t > 0, x \in (0, L), \\
y(t, 0) = y(t, L) = \partial_x y(t, L) = 0, & t > 0, \\
\partial_x y(t, 0) = u(t), & t > 0, \\
y(0, x) = y_0(x), & x \in (0, L), \\
\dot{u}(t) = \mathbf{sat}(h(t)), & t > 0.
\end{array}
\right.
\quad (5.1) $$
We also consider the linearized system subject to boundary saturation

$$ egin{cases} 
\partial_t y + \lambda \partial_x^2 y + \partial_x^4 y = 0, & t > 0, x \in (0, L), \\
y(t, 0) = y(t, L) = \partial_x y(t, L) = 0, & t > 0, \\
\partial_x y(t, 0) = u(t), & t > 0, \\
y(0, x) = y_0(x), & x \in (0, L), \\
\dot{u}(t) = \mathbf{sat}(h(t)), & t > 0.
\end{cases} 
\quad (5.2) $$

Note that in the systems (5.1)-(5.2) the input control is $h(t)$ but acts on the system by $\dot{u}(t) = \mathbf{sat}(h(t))$ that means that our control acts as an integrator. The main result of this section, namely Theorem 5.1, states the local exponential stability of (5.1) with saturated boundary control. Many of the computation of this section are quite similar to those presented in Section 3 and for that reason are omitted.

Inspired by [10, 24] we define $d(x) = (\frac{x^3}{L^2} - \frac{2x^2}{L} + x)$ and we make the change of variable $w(t, x) = y(t, x) - d(x)u(t)$. From now, we suppose that $u \in H^1(0, T)$ for all $T > 0$. Easy calculations show that $w$ satisfies the following system

$$ egin{cases} 
w_t = \mathcal{A}w + a(x)u(t) + b(x) \mathbf{sat}(h(t)), & t > 0, x \in (0, L), \\ w(t, 0) = w(t, L) = 0, & t > 0, \\ \partial_x w(t, 0) = \partial_x w(t, L) = 0, & t > 0, \\ w(0, x) = y_0 - d(x)u(0), & x \in (0, L), \\ \dot{u}(t) = \mathbf{sat}(h(t)), & t > 0, \end{cases} 
\quad (5.3) $$

where $a(x) = -\lambda d''(x)$ and $b(x) = -d(x)$.

**Remark 5.1.** Due to the parabolic behaviour, there is no a big difference if the control acts at left or right boundary, this differs for instance the case of dispersive equations [11]. In the case of different boundary conditions, we can consider as lifting function. In general, the lifting function is built as a third-order polynomial that vanish at the not acted boundary conditions, and it is equal to 1 where the control acts.

**Remark 5.2.** We take $\dot{u}(t) = \mathbf{sat}(h(t))$ and not directly $u(t) = \mathbf{sat}(h(t))$. This is motivated because $\mathbf{sat}$ operator is not enough smooth for making the change of variable $w = y - du$.

Similar to Section 3, expanding $w(t, \cdot) \in D(\mathcal{A})$ of (5.3) as a series in the eigenfunctions basis $\{e_j\}_{j \in \mathbb{N}}$, we observe that (5.3) is equivalent to the next infinity dimensional control system

$$ egin{cases} 
\dot{u}(t) = \mathbf{sat}(h(t)), \\ 
\dot{w}_j(t) = \sigma_j w_j(t) + a_j u(t) + b_j \mathbf{sat}(h(t)), & j \in \mathbb{N}, \end{cases} 
\quad (5.4) $$

where $a_j = \langle a, e_j \rangle_{L^2(0,L)}$ and $b_j = \langle b, e_j \rangle_{L^2(0,L)}$. Let $n \in \mathbb{N}$ the number of unstable eigenvalues of $\mathcal{A}$ and $\eta > 0$ as in (3.4). Defining the matrices:

$$ z(t) = \begin{pmatrix} u(t) \\ w_1(t) \\ \vdots \\ w_n(t) \end{pmatrix}, \quad \mathbf{A} = \begin{pmatrix} 0 & \cdots & \cdots & 0 \\ a_1 & \sigma_1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_n & 0 & \cdots & \sigma_n \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 \\ b_1 \\ \vdots \\ b_n \end{pmatrix}. 
\quad (5.5) $$

We can construct the next unstable finite dimensional system:

$$ \dot{z}(t) = \mathbf{A}z(t) + \mathbf{B}\mathbf{sat}(h(t)). 
\quad (5.6) $$

In this case, we have the following proposition about the controllability of the pair $(\mathbf{A}, \mathbf{B})$

**Proposition 5.1.** (Proposition 1, [10]) If $\lambda \notin \mathcal{N}$, then the pair $(\mathbf{A}, \mathbf{B})$ is stabilizable.

Along this section, we work under the assumption

**Assumption 2.** The anti-difussion parameter $\lambda \notin \mathcal{N}$, where, $\mathcal{N}$ is defined in (1.4).
As the pair $(A, B)$ is stabilizable, there exists $K = (k_0 \ k_1 \ \dots \ k_n) \in \mathbb{R}^{1 \times (n+1)}$, such that $A + BK$ is Hurwitz. Proposition 5.1 motivates the choice of the control $h(t) = Kz(t)$.

$$ \dot{z}(t) = Az(t) + B\mathbf{sat}(Kz(t)), \qquad (5.7) $$

As in the internal case, we can prove the local exponential stability

**Proposition 5.2.** There exists $P \in \mathbb{R}^{n+1 \times n+1}$ a symmetric positive definite, such that (5.7) is locally exponentially stable in 0 with region of attraction $\mathcal{R}$ given by

$$ \mathcal{R} := \{z \in \mathbb{R}^{n+1} : z^\top P z \le 1\}. \qquad (5.8) $$

Moreover, in $\mathcal{R}$, the function $V_1(z) := z^\top P z$, $z \in \mathbb{R}^{n+1}$, decreases exponentially fast to 0 along the solutions to (5.7), i.e., there is a constant $\alpha > 0$ so that

$$ \dot{V}_1(z) \le -\alpha|z|^2, \quad z \in \mathcal{R}. \qquad (5.9) $$

**Remark 5.3.** Note that, the control expression is not too explicit at the first glance. In fact $\dot{u}(t) = \mathbf{sat}(Kz(t)) = \mathbf{sat}\left(k_0u(t) + \sum_{j=1}^{n}k_jw_j(t)\right)$. But the system (5.7) is well-defined because $\mathbf{sat}$ is Lipschitz continuous.

In this direction, we can prove the following result with respect to the exponential stability of (5.4)

**Proposition 5.3.** Let $K$ as in Proposition 5.1 and $S = S_0 \times S_n \subset \mathbb{R}^{n+1}$ the region of attraction of (5.7) such that the system is locally exponentially stable in 0, where $S_0 \subset \mathbb{R}$ and $S_n \subset \mathbb{R}^n$. Then:

(1) system (5.4) with the feedback law $h(t) = Kz(t)$ is locally exponentially stable in 0 with region of attraction $S \times \ell_{2,j>n}$.

(2) system (5.3) with the feedback law $h(t) = Kz(t)$ is locally exponentially stable in 0 with region of attraction $S_0 \times \iota(S_n) \times X_n^\perp$.

In addition, for any closed and bounded set $G \subset \text{int}(S_0 \times \iota(S_n) \times X_n^\perp)$, there exists two positive values $M$, $\alpha$ such that for any initial condition $w(0, \cdot) \in G$, the solution $(u, w)$ of (5.3) with controller $h(t) = Kz(t)$ satisfies

$$ |u(t)| + \|w(t, \cdot)\|_{L^2(0,L)} \le Ce^{-at}\|w(0, \cdot)\|_{L^2(0,L)}. \qquad (5.10) $$

**Proof:** The proof is straightforward, following Section 3. $\square$

**Remark 5.4.** We search for region of attraction of the form $S = S_0 \times S_n$ for simply the notation. In [43] and, the past section works with internal controller, which implies that the state is only given by the projections $w_1, \dots, w_n$.

**Remark 5.5.** Our control is given by $\dot{u}(t) = Kz(t)$, thus we need to prescribe the initial condition $u(0)$. Without loss of generality, we can take $u(0) = 0$, which implies $w(0, \cdot) = y_0$.

By the previous remark, from (5.10) we obtain the $L^2$ decay of $y = w + ud$

$$ \begin{aligned} \|y(t, \cdot)\|_{L^2(0,L)} &\le \|w(t, \cdot)\|_{L^2(0,L)} + |u(t)|\|d\|_{L^2(0,L)} \\ &\le \max\{1, \|d\|_{L^2(0,L)}\} (w(t, \cdot)\|_{L^2(0,L)} + |u(t)|) \\ &\le C \max\{1, \|d\|_{L^2(0,L)}\} e^{-at} \|w(0, \cdot)\|_{L^2(0,L)} \\ &= C \max\{1, \|d\|_{L^2(0,L)}\} e^{-at} \|y_0\|_{L^2(0,L)}. \end{aligned} $$

Following Theorem 4.1, the $H^2$ stability can also be shown in the boundary case. This allows us to prove the main result of this section.

**Theorem 5.1.** Let $K$ as in Proposition 5.1, $P$ as in (5.8) and $h(t) = Kz$, then there exists $\varepsilon > 0$ such that for all $y_0 \in H_0^2(0, L)$ satisfying $\|y_0\|_{H_0^2(0,L)} \le \varepsilon$, the unique solution $y \in C([0, T]; H^2(0, L)) \cap L^2(0, T, H^4(0, L))$ for all $T > 0$ satisfy
$$ \|y(t, \cdot)\|_{H_0^2(0,L)} \le Ce^{-\alpha t}\|y_0\|_{H_0^2(0,L)}, \quad (5.11) $$

for $C > 0$.

## 6. CONCLUSIONS AND PERSPECTIVES

The present work establishes a framework for the stabilization of fourth-order parabolic equations with input saturation for internal and boundary feedback laws. Our approach relied on spectral decomposition and modal stabilization using LMIs. We established local exponential stability in the $L^2$ and $H^2$ norms. Using this high regularity decay, we prove that the fully nonlinear systems can be stabilized with saturated actuator. A natural perspective for future research is to extend these results to boundary actuation without relying on spectral lifting arguments. This could be achieved by exploiting the theory of admissible control operators for analytic semigroups, which provides a direct way to handle unbounded boundary inputs. Such an approach would allow the development of a unified methodology applicable to saturated stabilization for general parabolic operators, including the multidimensional case.

# APPENDIX A. ESTIMATES LOCAL IN TIME WELL POSEDNESS

Let $w \in B_\theta$, then

$$ \begin{aligned} \|\mathcal{F}(w)\|_{L^2(0,\theta;L^2(0,L))}^2 &= \int_0^\theta \int_0^L \left( \sum_{k=1}^m b_k(x) \mathbf{sat} \left( (\mathbf{K}_\iota^{-1}(\Pi_n w(\cdot, t)))_k \right) \right)^2 dxdt \\ &\le \sum_{k=1}^m \|b_k\|_{L^2(0,L)}^2 \int_0^\theta \left( \mathbf{sat} \left( (\mathbf{K}_\iota^{-1}(\Pi_n w(\cdot, t)))_k \right) \right)^2 dt \tag{A.1} \\ &\le \left( \sum_{k=1}^m \|b_k\|_{L^2(0,L)}^2 \right) \|\mathbf{K}\|^2 \int_0^\theta |\iota^{-1}(\Pi_n w(\cdot, t))|^2 dt \\ &\le C\theta \|w\|_{L^2(0,\theta;L^2(0,L))}^2, \\ &\le C\theta \|w\|_{B_\theta}^2. \end{aligned} $$

For the nonlinear term $G$, we have

$$ \|\mathcal{G}(w)\|_{L^2(0,\theta;L^2(0,L))} \le \lambda \|\partial_x^2 w\|_{L^2(0,\theta;L^2(0,L))} + \delta \|w \partial_x w\|_{L^2(0,\theta;L^2(0,L))} + \delta \|\partial_x^2(w^3)\|_{L^2(0,\theta;L^2(0,L))}, $$

we study these three terms separately

* For the first one,

$$ \begin{aligned} \|\partial_x^2 w\|_{L^2(0,\theta;L^2(0,L))}^2 &= \int_0^\theta \int_0^L |\partial_x^2 w|^2 dxdt \\ &\le \theta \|w\|_{C^2([0,\theta];H^2(0,L))}^2 \\ &\le \theta \|w\|_{B_\theta}^2. \end{aligned} $$

* For the second one, using the injection of $H^1(0, L)$ into $L^\infty(0, L)$

$$ \begin{aligned} \|w \partial_x w\|_{L^2(0,\theta;L^2(0,L))}^2 &= \int_0^\theta \int_0^L |w \partial_x w|^2 dxdt \\ &\le \int_0^\theta \|\partial_x w(t, \cdot)\|_{L^\infty(0,L)}^2 \|w(t, \cdot)\|_{L^\infty(0,L)}^2 dt \\ &\le \int_0^\theta \|w(t, \cdot)\|_{H^2(0,L)}^4 dt \\ &\le C\theta \|w\|_{C^2([0,\theta];H^2(0,L))}^4 \\ &\le C\theta \|w\|_{B_\theta}^4. \end{aligned} $$
• For the last one,

$$
\begin{align*}
\|\partial_x^2(w^3)\|_{L^2(0,\theta;L^2(0,L))}^2 &= \int_0^\theta \int_0^L |6w(\partial_x w)^2 + 3w^2 \partial_x^2 w|^2 dxdt \\
&\le C \left( \int_0^\theta \int_0^L w^2(\partial_x w)^4 dxdt + \int_0^\theta \int_0^L w^4(\partial_x^2 w)^2 dxdt \right) \\
&\le C \left( \int_0^\theta \|w(t,\cdot)\|_{L^\infty(0,L)}^2 \|\partial_x w(t,\cdot)\|_{L^\infty(0,L)}^4 dt \right. \\
&\qquad \left. + \int_0^\theta \|w(t,\cdot)\|_{L^\infty(0,L)}^4 \|\partial_x^2 w(t,\cdot)\|_{L^2(0,L)}^2 dt \right) \\
&\le C \int_0^\theta \|w(t,\cdot)\|_{H^2(0,L)}^6 dt \\
&\le C\theta \|w\|_{C^2([0,\theta];H^2(0,L))}^6 \\
&\le C\theta \|w\|_{B_\theta}^6.
\end{align*}
$$

Joining all the estimates, we get

$$
\|\mathcal{G}(w)\|_{L^2(0,\theta;L^2(0,L))} \le C\sqrt{\theta} (\|w\|_{B_\theta} + \delta\|w\|_{B_\theta}^2 + \nu\|w\|_{B_\theta}^3). \quad (A.2)
$$

REFERENCES

[1] Behzad Azmi, Karl Kunisch, and Sérgio S Rodrigues. Saturated feedback stabilizability to trajectories for the Schlögl parabolic equation. *IEEE Transactions on Automatic Control*, 68(12):7089–7103, 2023.

[2] D. Bainov and P. Simeonov. *Integral Inequalities and Applications*, volume 57 of Mathematics and Its Applications. Springer, 1992.

[3] Viorel Barbu, Pierluigi Colli, Gianni Gilardi, and Gabriela Marinoschi. Feedback stabilization of the Cahn-Hilliard type system for phase separation. *Journal of Differential Equations*, 262(3):2286–2334, 2017.

[4] Stephen Boyd. *Linear matrix inequalities in system and control theory*. Society for Industrial & Applied, 1 1994.

[5] John Cahn. On spinodal decomposition. *Acta Metallurgica*, 9(9):795–801, 1961.

[6] John Cahn and John Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. *Journal of Chemical Physics*, 28(2):258–267, 1958.

[7] John Cahn and John Hilliard. Spinodal decomposition. a reprise. *Acta Metallurgica*, 19(2):151–161, 1971.

[8] Julien Calbert, Lucas Egidio, and Raphaël Jungers. An efficient method to verify the inclusion of ellipsoids. *IFAC-PapersOnLine*, 56(2):1958–1963, 1 2023.

[9] Cristian M Cazacu, Liviu I Ignat, and Ademir F Pazoto. Null-controllability of the linear Kuramoto–Sivashinsky equation on star-shaped trees. *SIAM Journal on Control and Optimization*, 56(4):2921–2958, 2018.

[10] Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. *Communications on Pure & Applied Analysis*, 9(1):91–102, 2010.

[11] Eduardo Cerpa and Jean-Michel Coron. Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition. *IEEE Transactions on Automatic Control*, 58(7):1688–1695, 2013.

[12] Eduardo Cerpa, Patricio Guzmán, and Alberto Mercado. On the control of the linear Kuramoto-Sivashinsky equation. *ESAIM: Control, Optimisation and Calculus of Variations*, 23(1):165–194, nov 2016.

[13] Liang-Heng Chen and Hsueh-Chia Chang. Nonlinear waves on liquid film surfaces—II. Bifurcation analyses of the long-wave equation. *Chemical Engineering Science*, 41(10):2477–2486, 1986.

[14] Panagiotis D. Christofides and Antonios Armaou. Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. *Systems & Control Letters*, 39(4):283–294, apr 2000.

[15] P.D. Christofides. Feedback control of the Kuramoto-Sivashinsky equation. In *Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171)*. IEEE, 2000.

[16] Pierluigi Colli, Gianni Gilardi, and Ionut Munteanu. Stabilisation of a linearised Cahn-Hilliard system for phase separation by proportional boundary feedbacks. *International Journal of Control*, 91(2):452–460, 2021.

[17] Pierluigi Colli, Gianni Gilardi, Andrea Signori, and Jürgen Sprekels. Optimal temperature distribution for a nonisothermal cahn-hilliard system with source term. *Applied Mathematics and Optimization*, 88:Paper No. 68, 31, 2023.

[18] Pierluigi Colli, Gianni Gilardi, and Jürgen Sprekels. Optimal velocity control of a viscous cahn-hilliard system with convection and dynamic boundary conditions. *SIAM Journal on Control and Optimization*, 56(3):1665–1691, 2018.

[19] Jean-Michel Coron and Emmanuel Trélat. Global steady-state controllability of one-dimensional semilinear heat equations. *SIAM journal on control and optimization*, 43(2):549–569, 2004.

[20] Jean-Michel Coron and Emmanuel Trélat. Global steady-state stabilization and controllability of 1d semilinear wave equations. *Communications in Contemporary Mathematics*, 8(04):535–567, 2006.

[21] Charles Elliott. *The Cahn-Hilliard model for the kinetics of phase separation*, volume 88 of *Mathematical Models for Phase Change Problems, International Series on Numerical Mathematics*. Birkhauser, 1989.
[22] Sara Frigeri, Elisabetta Rocca, and Jürgen Sprekels. Optimal distributed control of a nonlocal Cahn–Hilliard/Navier–Stokes system in two dimensions. *SIAM Journal on Control and Optimization*, 54(1):221–250, 2016.

[23] Patricio Guzmán. Local exact controllability to the trajectories of the cahn–hilliard equation. *Applied Mathematics & Optimization*, 82(1):279–306, 2020.

[24] Patricio Guzmán, Swann Marx, and Eduardo Cerpa. Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control. *IFAC-PapersOnLine*, 52(2):70–75, 2019.

[25] Patricio Guzmán and Christophe Prieur. Rapid stabilization of a reaction-diffusion equation with distributed disturbance. In *2020 59th IEEE Conference on Decision and Control (CDC)*, pages 666–671. IEEE, 2020.

[26] A. P. Hooper and R. Grimshaw. Two-dimensional disturbance growth of linearly stable viscous shear flows. *Physics of Fluids*, 8(6):1424–1432, jun 1996.

[27] Rami Katz and Emilia Fridman. Constructive method for finite-dimensional observer-based control of 1-d parabolic PDEs. *Automatica*, 122:109285, 2020.

[28] Rami Katz and Emilia Fridman. Regional stabilization of the 1-d Kuramoto-Sivashinsky equation via modal decomposition. *IEEE Control Systems Letters*, 6:1814–1819, 2022.

[29] Y. Kuramoto and T. Tsuzuki. On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach. *Progress of Theoretical Physics*, 54(3):687–699, sep 1975.

[30] Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. *Progress of Theoretical Physics*, 55(2):356–369, feb 1976.

[31] Hugo Lhachemi. Local output feedback stabilization of a nonlinear Kuramoto-Sivashinsky equation. *IEEE Transactions on Automatic Control*, 68(12):8165–8170, 2023.

[32] Hugo Lhachemi, Ionut Munteanu, and Christophe Prieur. Boundary output feedback stabilisation for 2-D and 3-D parabolic equations. *arXiv (Cornell University)*, 2 2023.

[33] Hugo Lhachemi and Christophe Prieur. Feedback stabilization of a class of diagonal infinite-dimensional systems with delay boundary control. *IEEE Transactions on Automatic Control*, 66(1):105–120, 2020.

[34] Hugo Lhachemi and Christophe Prieur. Local output feedback stabilization of a reaction-diffusion equation with saturated actuation. *IEEE Transactions on Automatic Control*, 68(1):564–571, 2022.

[35] Hugo Lhachemi and Christophe Prieur. Nonlinear boundary output feedback stabilization of reaction-diffusion equations. *Systems & Control Letters*, 166:105301, 2022.

[36] Hugo Lhachemi and Christophe Prieur. Saturated boundary stabilization of partial differential equations using control-lyapunov functions. *Control of Partial Differential Equations*, pages 81–164, 2023.

[37] Wei-Jiu Liu and Miroslav Krstić. Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation. *Non-linear Analysis: Theory, Methods & Applications*, 43(4):485–507, 2001.

[38] G Marinoschi. Internal feedback stabilization of cahn–hilliard system with viscosity effect. *Pure Appl. Funct. Anal*, 3(1):107–135, 2018.

[39] Swann Marx, Eduardo Cerpa, Christophe Prieur, and Vincent Andrieu. Global stabilization of a Korteweg–de Vries equation with saturating distributed control. *SIAM Journal on Control and Optimization*, 55(3):1452–1480, 2017.

[40] Patricio Guzmán Meléndez. Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation. *Journal of Mathematical Analysis and Applications*, 408(1):275–290, 2013.

[41] Daniel Michelson and Gregory Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames ii: Numerical experiments. *Acta Astronautica*, 4(11-12):1207–1221, 1977.

[42] Alain Miranville. *The Cahn-Hilliard Equation: Recent Advances and Applications*, volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, 2019.

[43] Andrii Mironchenko, Christophe Prieur, and Fabian Wirth. Local stabilization of an unstable parabolic equation via saturated controls. *IEEE Transactions on Automatic Control*, pages 1–1, 2020.

[44] Ionut Munteanu. *Boundary Stabilization of Parabolic Equations*, volume 93 of Progress in Nonlinear Differential Equations and Their Applications, PNLDE Subseries in Control. Birkhauser, 2019.

[45] Amy Novick-Cohen. *The Cahn-Hilliard equation*, volume 4 of Handbook of Differential Equations: Evolutionary Equations. Elsevier, 2008.

[46] Hugo Parada. Stability analysis of a Korteweg-de vries equation with saturated boundary feedback. *IFAC-PapersOnLine*, 55(26):1–6, 2022.

[47] Hugo Parada, Emmanuelle Crépeau, and Christophe Prieur. Global well-posedness of the KdV equation on a star-shaped network and stabilization by saturated controllers. *SIAM Journal on Control and Optimization*, 60(4):2268–2296, 2022.

[48] Hugo Parada and Subrata Majumdar. Stabilization of Kawahara equation with input saturation and saturated boundary feedback. Accepted in Mathematical Control and Related Fields, 2024.

[49] Christophe Prieur, Sophie Tarbouriech, and Joao M. Gomes da Silva. Wave equation with cone-bounded control laws. *IEEE Transactions on Automatic Control*, 61(11):3452–3463, nov 2016.

[50] Friedrich Pukelsheim. Optimal design of experiments. *Choice Reviews Online*, 31(02):31–0986, 10 1993.

[51] Gregory Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames i: Derivation of basic equations. *Acta Astronautica*, 4(11-12):1177–1206, 1977.

[52] Sophie Tarbouriech, Germain Garcia, João Manoel Gomes da Silva Jr, and Isabelle Queinnec. *Stability and stabilization of linear systems with saturating actuators*. Springer Science & Business Media, 2011.
(P. Guzmán) DEPARTAMENTO DE MATEMÁTICA, UNIVERSIDAD TÉCNICA FEDERICO SANTA MARÍA, VALPARAÍSO, CHILE.

Email address: patricio.guzmanm@usm.cl

(F. Labra) FACULTAD DE MATEMÁTICAS, PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE, SANTIAGO, CHILE.

Email address: flabrai@estudiante.uc.cl

(H. Parada) UNIVERSITÉ DE LORRAINE, CNRS, INRIA, IECL, NANCY, FRANCE.

Email address: hugo.parada@inria.fr