# Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under
Intermittent Sensing

M. C. Belhadjoudja<EMAIL_ADDRESS>M. Maghenem
<EMAIL_ADDRESS>E. Witrant<EMAIL_ADDRESS>C.
Prieur<EMAIL_ADDRESS>The authors are with the Université
Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France

###### Abstract

We study in this paper boundary stabilization, in the $L^{2}$ sense, of the
one-dimensional Kuramoto-Sivashinsky equation subject to intermittent sensing.
We assume that we measure the state of this spatio-temporal equation on a
given spatial subdomain during certain intervals of time, while we measure the
state on the remaining spatial subdomain during the remaining intervals of
time. As a result, we assign a feedback law at the boundary of the spatial
domain and force to zero the value of the state at the junction of the two
subdomains. Throughout the study, the destabilizing coefficient is assumed to
be space-dependent and bounded but unknown. Adaptive boundary controllers are
designed under different assumptions on the forcing term. In particular, when
the forcing term is null, we guarantee global exponential stability of the
origin. Furthermore, when the forcing term is bounded and admits a known upper
bound, we guarantee input-to-state stability, and only global uniform ultimate
boundedness is guaranteed when the upper bound is unknown. Numerical
simulations are performed to illustrate our results.

###### keywords:

KS equation, intermittent sensing, boundary control, $L^{2}$-stability,
adaptive control.

, , ,

##  1 Introduction

A fundamental constituent of a control loop is the sensor. Sensors have
usually some limitations that need to be taken into account when designing a
controller. In particular, for systems evolving in both space and time, an
important limitation is related to the spatial measurement range. That is, one
should not expect to measure the state at every spatial location for all time.

Systems that evolve in both space and time are usually described by partial
differential equations (PDEs). In the physics community, a certain attention
has been devoted to the stabilization of some PDEs under a spatially-limited
range of sensing. For example, in [29], the authors assume that there is a
finite number of sensors located at periodically separated spatial points and
that each sensor measures the average of the state over a neighborhood of its
location. Under this sensing scenario, a controller is designed and applied at
the location of each sensor. A similar study is in [25], where the state is
measured at a specific spatial location, and a control action is imposed at
that same location. In [11], a time-periodic reset of the state is performed
at uniformly separated spatial points, using data collected at these points.
The aforementioned results are validated via numerical simulations only.
Hence, it is important to study the stability properties of the obtained
closed-loop systems. Our goal here is to study the effect of a specific
spatially-limited sensing scenario on the stabilization of the Kuramoto-
Sivashinsky (KS) equation.

The KS equation was first introduced in late 1970 by Y. Kuramoto [13] and G.
Sivashinsky [26, 27], and is given by

$\displaystyle u_{t}+uu_{x}+\lambda(x)u_{xx}+u_{xxxx}=0,$ (1)

where $(x,t)\in(0,1)\times(0,+\infty)$ and, known as the destabilizing
coefficient, $\lambda:[0,1]\rightarrow\mathbb{R}_{>0}$ creates the anti-
diffusion phenomena. This equation is used to model phase turbulence in
reaction-diffusion systems [13], thermo-diffusive instabilities in laminar
flame fronts [26, 27], fluctuations of fluid films on inclined supports [3,
18], and plasma instabilities [14]. The nonhomogeneous version of (1), also
called the Noisy KS equation (NKS), is given by

$\displaystyle u_{t}+uu_{x}+\lambda(x)u_{xx}+u_{xxxx}=f(x,t),$ (2)

where the right-hand side $f$, we name as the forcing term, accounts for
disturbances, noises, and unmodeled dynamics. Equation (2) is used, for
example, to model surface erosion via Ion sputtering [15, 6], where $f$ in
this case is a Gaussian white noise that represents the fluctuations in the
flux of bombarding particles.

Since the seminal work in [19], various boundary controllers have been
developed for (1), under different assumptions on either the destabilizing
coefficient $\lambda$ or the size of the initial condition. When $\lambda$ is
small and unknown, an adaptive boundary controller is designed in [10]. In
[5], an integral transformation is proposed to achieve exponential
stabilization with an arbitrarily specified decay rate, provided that the
initial condition is sufficiently small and that $\lambda$ does not belong to
a set of critical values. In [16], a boundary controller is designed to
achieve local output feedback stabilization, independently of the size of
$\lambda$. Boundary stabilization of (2) is considered in [4], under the
assumption that $\lambda\in(0,1)$. The aforementioned boundary controllers
either assume $\lambda$ to be sufficiently small or the initial condition
sufficiently close to the origin, in which cases, only boundary measurements
are required. In all the aforementioned control references, $\lambda$ is
assumed to be constant. However, in physics and mathematics literature, it is
usually space-dependent; see, e.g., [7, 1].

Boundary stabilization of (1) (and (2)) independently of the size of $\lambda$
(and $f$) and the initial condition using only boundary measurements remains
open. To the best of our knowledge, the first work that studied boundary
stabilization of (1) under limited (intermittent) sensing range is [20], where
it is assumed that, for some $Y\in(0,1)$, the state $u$ is measured at the
spatial subdomain $(0,Y)$ during certain time intervals, and that it is
measured over the spatial subdomain $(Y,1)$ during the remaining time
intervals. The actuators act at three different locations: at $x=0$, $x=Y$,
and at $x=1$. Feedback controllers are designed at $x=0$ and at $x=1$, and a
zero state is imposed at $x=Y$. The proposed boundary controllers are shown to
guarantee global exponential stability of the origin in the $L^{2}$ sense.
Moreover, the controllers are shown to remain bounded and converge
asymptotically to zero along the closed-loop solutions. However, the
destabilizing coefficient $\lambda$ is assumed to be constant and perfectly
known.

In this paper, we generalize the approach in [20] by designing adaptive
boundary controllers for the NKS equation under different assumptions on $f$
as well as the available measurements. To start, we revisit the problem
studied [20], where the sensing was intermittent and $f=0$. Different from
[20], we assume that $\lambda$, $|\lambda|_{\infty}$, and
$|\lambda^{\prime}|_{\infty}$ are unknown. As a consequence, we design an
adaptive boundary controller guaranteeing global exponential stability of the
origin $u:=0$ in the $L^{2}$ sense. To obtain this result, we exploit the
possibility of achieving exponential convergence of the $L^{2}$ norm of the
measured state to zero at any desired rate. Roughly speaking, during the time
intervals when only $u$ on $(Y,1)$ is measured, we set $u=0$ at $x=0$. Hence,
if the resulting PDE defined on $(0,Y)$ is unstable, the state $u$ at $(0,Y)$
may grow during these time intervals. To still guarantee convergence of $u$ to
zero on the entire domain $(0,1)$, we need to guarantee a sufficiently fast
decay of the $L^{2}$ norm of $u$ at $(0,Y)$ during the time intervals when $u$
at $(0,Y)$ is measured. To achieve a desirable exponential decay, when
$\lambda$ is unknown, we propose a novel adaptive control strategy. Next, we
tackle the case where $f\neq 0$. To illustrate the challenges that
intermittent sensing brings, we start considering the case where the state $u$
is measured everywhere and for all time, but both $\lambda$ and $f$ are
completely unknown. Note that from now on, when we say that $\lambda$ and $f$
are unknown, we mean that $|\lambda|_{\infty}$, $|\lambda^{\prime}|_{\infty}$
and $|f|_{\infty}$ are unknown. In this case, the actuators act at $x=0$ and
$x=1$. We set to zero the input at $x=1$ and design an adaptive feedback
controller at $x=0$ that guarantees global practical attractivity in the
$L^{2}$ sense. Namely, the solutions converge in the $L^{2}$ norm, as
$t\to\infty$, to any chosen neighborhood of the origin. Practical attractivity
cannot be guaranteed under intermittent sensing. Indeed, we show in the latter
case that even if $|f|_{\infty}$ is known, one can guarantee only input-to-
state stability with respect to $|f|_{\infty}$, in the $L^{2}$ sense. Indeed,
on the time intervals when we do not measure the state on $(0,Y)$, as we set
the control input at $x=0$ to zero, the state can reach, independently of its
history, a constant value that is proportional to the supremum of $f$.
Finally, we consider the most general scenario, where the system is subject to
intermittent sensing and $\lambda$ and $f$ are both unknown. In this case, the
only result we can guarantee is global uniform ultimate boundedness in the
$L^{2}$ sense. Namely, we show the existence of a constant, not necessarily
proportional in $f$ and independent of the initial conditions, towards which
the closed-loop solutions converge in finite time. The reaching time depends
on the $L^{2}$ norm of the initial conditions.

A preliminary version of this work is in [2], where the KS equation is
considered when $\lambda$ is unknown and constant. In addition to considering
the perturbed case, namely, the NKS equation, deeper discussions, and detailed
proofs are included here.

The rest of the paper is organized as follows. In Section 2, we describe the
considered intermittent sensing scenario, the boundary control locations, as
well as the hypotheses that hold throughout the paper. In Section 3, we
present our main results. In Section 4, we present the proofs of our main
results. Finally, in Section 5, we illustrate our theoretical results via
numerical simulations.

Notation. Depending on the context, a.e. means either almost every or almost
everywhere. Let $X$ be a Banach space with a norm $|.|_{X}$,
$p\in\\{1,2,...,+\infty\\}$, and let $a,b\in\mathbb{R}$. We denote by
$L^{p}(a,b;X)$ the space of measurable functions $u:[a,b]\to X$, with finite
$p$-norm $|.|_{p}$, where $|u|_{p}^{p}:=\int_{a}^{b}|u(t)|_{X}^{p}dt$ if
$p<\infty$ and $|u|_{\infty}:=\text{ess}\sup_{t\in[a,b]}|u(t)|_{X}$. When
$X=\mathbb{R}$, we just write $L^{p}(a,b)$ instead of $L^{p}(a,b;\mathbb{R})$.
For $k\in\mathbb{N}$, we denote by $H^{k}(a,b)$ the Sobolev space of functions
whose distributional derivatives, up to order $k$, are in $L^{2}(a,b)$. For
$k\in\mathbb{N}\cup\\{\infty\\}$, we denote by $\mathcal{C}^{k}(a,b)$ the
space of $k$-times continuously differentiable functions on $(a,b)$. For
$(x,t)\mapsto f(x,t)$, the partial derivative of $f$ with respect to $t$ is
denoted by $\partial_{t}f$ and the $k^{th}$ partial derivative with respect to
$x$ is denoted by $\partial_{x}^{k}f$. We denote the time derivative of a
function $V$ by either $\frac{d}{dt}V$ or $\dot{V}$. We also denote the
derivative of a function $x\mapsto u(x)$ of a scalar variable by $u^{\prime}$
and its second derivative by $u^{\prime\prime}$. For $(x,t)\mapsto u(x,t)$, we
may write $u(x)$ instead of $u(x,t)$. Finally, for $x\in\mathbb{R}$, $\mbox{
sgn}(x)=1$ if $x>0$, $=0$ if $x=0$ and $=-1$ if $x<0$.

##  2 Context: Sensing and Control Locations

###  2.1 Intermittent Sensing

Consider equations (1) and (2). Following [20], we let $Y\in[0,1]$ and we
assume that $u((0,Y),t)$ is measured during certain intervals of time and that
$u((Y,1),t)$ is measured during the remaining intervals of time. More
precisely, we assume the existence of a sequence
$\\{t_{i}\\}^{\infty}_{i=1}\subset\mathbb{R}_{+}$, with $t_{1}=0$ and
$t_{i+1}>t_{i}$, such that

  1. S1)

$u((0,Y),t)$ is available for all $t\in
I_{1}:=\bigcup^{\infty}_{k=1}[t_{2k-1},t_{2k})$.

  2. S2)

$u((Y,1),t)$ is available for all $t\in
I_{2}:=\bigcup^{\infty}_{k=1}[t_{2k},t_{2k+1})$.

Associated with the proposed intermittent sensing scenario, we consider the
following dwell-time assumption.

###### Assumption 1

There exist four constants $\overline{T}_{1}$, $\overline{T}_{2}$,
$\underline{T}_{1}$, $\underline{T}_{2}>0$ such that, for each
$k\in\mathbb{N}^{*}$, we have

$\displaystyle\underline{T}_{1}\leq
t_{2k}-t_{2k-1}\leq\overline{T}_{1},\quad\underline{T}_{2}\leq
t_{2k+1}-t_{2k}\leq\overline{T}_{2}.$

###  2.2 Control Locations

In the case of intermittent sensing, we propose to control (1) and (2) at
three different locations: at $x=0$, $x=Y$, and $x=1$. We, therefore,
assimilate (1) and (2) to a system of two NKS equations interconnected by
boundary constraints at $x=Y$. That is, we introduce the system of PDEs

$\displaystyle w_{t}+ww_{x}+\lambda w_{xx}+w_{xxxx}$ $\displaystyle=f\qquad
x\in(0,Y),$ (3a) $\displaystyle v_{t}+vv_{x}+\lambda v_{xx}+v_{xxxx}$
$\displaystyle=f\qquad x\in(Y,1),$ $\displaystyle w(Y)$
$\displaystyle=w_{x}(Y)=w_{x}(0)=0,$ (3b) $\displaystyle v(Y)$
$\displaystyle=v_{x}(Y)=v_{x}(1)=0,$ $\displaystyle w(0)$
$\displaystyle=u_{1},~{}v(1)=u_{2},$

where $(u_{1},u_{2})$ are control inputs to be designed.

We define a solution $u:(0,1)\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ to (2)
subject to the boundary conditions

$\displaystyle u(Y)$ $\displaystyle=0$ (4) $\displaystyle u_{x}(0)$
$\displaystyle=u_{x}(Y)=u_{x}(1)=0$ $\displaystyle u(0)$
$\displaystyle=u_{1},~{}u(1)=u_{2},$

for some $(u_{1},u_{2})$, as $u(x,t)=w(x,t)$ a.e. on
$(0,Y)\times\mathbb{R}_{>0}$ and $u(x,t)=v(x,t)$ a.e. on
$(Y,1)\times\mathbb{R}_{>0}$, with $(w,v)$ a strong solution to (3) subject to
the same $(u_{1},u_{2})$.

######  Remark 2.1.

We recall that $(w,v)$ is a strong solution to (3) subject to $(u_{1},u_{2})$
if $w\in L^{2}\left(I,H^{4}(0,Y)\right)$ and $v\in
L^{2}\left(I,H^{4}(Y,1)\right)$ for every compact set
$I\subset\mathbb{R}_{>0}$, and $(w,v)$ verifies (3a) a.e. in space and time
and verifies (3b) a.e. in time; see [8, Chapter 9].

######  Remark 2.2.

A solution to (2) under (4), as previously defined, is not necessarily a
strong solution to (2). Indeed, a strong solution to (2) must satisfy
$u(\cdot,t)\in H^{4}(0,1)$. However, the concatenation of $w$ and $v$ (forming
a strong solution to (3)) satisfies, a priori, only $u(\cdot,t)\in
H^{2}(0,1)$, since $w_{xxx}(Y)$ and $v_{xxx}(Y)$ are not necessarily equal.
Imposing more regularities at $x=Y$ in (3) would lead to an over-determined
system of PDEs.

According to the aforementioned concept of solutions, guaranteeing an
$L^{2}$-stability property for (2) under (4) is equivalent to guaranteeing the
same $L^{2}$-stability property for (3).

Our results in the sequel use the following assumption.

###### Assumption 2

Each one of the maps $f$, $\lambda$, and $\lambda^{\prime}$ has a finite sup
norm.

We stress that the coefficient $\lambda$ in (3a) and the constants
$(\overline{T}_{1},\overline{T}_{2},\underline{T}_{1},\underline{T}_{2})$ in
Assumption 1 are unknown.

##  3 Results

In this section, we design $(u_{1},u_{2})$ to guarantee different
($L^{2}$-stability) properties for (2) under (4) in the following different
situations.

  * •

When $f=0$ and the sensing is intermittent.

  * •

When $f\neq 0$, $|f|_{\infty}$ is unknown, and the entire state is measured
all the time.

  * •

When $f\neq 0$, $|f|_{\infty}$ is known, and the sensing is intermittent.

  * •

When $f\neq 0$, $|f|_{\infty}$ is unknown, and the sensing is intermittent.

Note that a similar control strategy is exploited for the four aforementioned
situations. This approach, however, is shown to guarantee different
($L^{2}$-stability) properties depending on the considered situation.

###  3.1 Intermittent Sensing with $f=0$

We start considering the Lyapunov function candidates

$\displaystyle
V_{1}(w):=\frac{1}{2}\int_{0}^{Y}w(x)^{2}dx,~{}~{}V_{2}(v):=\frac{1}{2}\int_{Y}^{1}v(x)^{2}dx.$
(5)

Furthermore, we let

$\displaystyle(u_{1},u_{2}):=\left\\{\begin{matrix}(\kappa(V_{1},w_{xxx}(0),\hat{\theta}_{1}),0)&\quad\text{on}~{}I_{1},\\\
(0,-\kappa(V_{2},v_{xxx}(1),\hat{\theta}_{2}))&\quad\text{on}~{}I_{2}.\end{matrix}\right.$
(6)

The explicit expression of $\kappa$ in (6) is given by

$\kappa(V,\omega,\hat{\theta}):=\left\\{\begin{aligned} -\mbox{
sgn}(\omega)\sqrt[3]{V}\qquad&\text{if $|\omega|\geq l(V,\hat{\theta})$},\\\
-3(3\hat{\theta}+1)\sqrt[3]{V}\qquad&\text{otherwise,}\end{aligned}\right.$
(7)

where $l(V,\hat{\theta}):=(1/3)(1+3\hat{\theta})V^{2/3}$ and the parameters
$(\hat{\theta}_{1},\hat{\theta}_{2})$ are dynamically updated according to the
following algorithm.

###### Algorithm 1

Given $\Delta_{1}$, $\Delta_{2}$, and $\sigma>0$, we dynamically update
$(\hat{\theta}_{1},\hat{\theta}_{2})$ according to the following rules:

  1. R1)

On every interval $[t_{2k-1},t_{2k})\subset I_{1}$, we set
$\dot{\hat{\theta}}_{2}=0$. Moreover, if
$V_{1}(t_{2k-1})>V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})}$, we set
$\dot{\hat{\theta}}_{1}:=\Delta_{1}$; otherwise, we set
$\dot{\hat{\theta}}_{1}:=0$.

  2. R2)

On every interval $[t_{2k},t_{2k+1})\subset I_{2}$, we set
$\dot{\hat{\theta}}_{1}:=0$. Moreover, if
$V_{2}(t_{2k})>V_{2}(t_{2k-2})\exp^{-\sigma(t_{2k}-t_{2k-2})}$, we set
$\dot{\hat{\theta}}_{2}:=\Delta_{2}$; otherwise, we set
$\dot{\hat{\theta}}_{2}:=0$.

  3. R3)

On $[t_{1},t_{3}]$, $\hat{\theta}_{1}=\hat{\theta}_{1}(0)\geq 0$ and
$\hat{\theta}_{2}=\hat{\theta}_{2}(0)\geq 0$.

In this situation, we show that the closed-loop system enjoys the following
property of $L^{2}$ global exponential stability ($L^{2}$-GES).

######  Property 1 ($L^{2}$-GES)

For each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists $\gamma\geq
1$ such that, along the solutions to (3) subject to (6), we have

$\displaystyle V_{1}(t)+V_{2}(t)\leq\gamma(V_{1}(0)+V_{2}(0))\exp^{-\sigma
t}\quad\forall t\geq 0,$ (8)

where $\sigma>0$ comes from Algorithm 1.

######  Theorem 3.1.

Consider system (3) such that Assumption 2 holds with $f=0$. Consider the
intermittent sensing scenario  S1)- S2) such that Assumption 1 holds. Let
$(V_{1},V_{2})$ be defined in (5), $\kappa$ defined in (7), and the parameters
$(\hat{\theta}_{1},\hat{\theta}_{2})$ governed by Algorithm 1. By letting
$(u_{1},u_{2})$ as in (6), we conclude that, for (3) in closed loop,

  * •

The $L^{2}$-GES property is verified.

  * •

For each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists $M>0$ such
that, for each $(V_{1}(0),V_{2}(0))$,
$|\hat{\theta}_{1}|_{\infty},|\hat{\theta}_{2}|_{\infty}\leq M$.

  * •

The inputs $(u_{1},u_{2})$ remain bounded and converge asymptotically to zero.

Sketch of proof: We start by referring to Lemma 6.2 in the appendix, which
shows that along (3) we have

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+P(u_{1},w_{xxx}(0)),$ (9)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}-P(u_{2},v_{xxx}(1)),$ (10)

where $(\theta_{1},\theta_{2})$ are constant parameters that depend on
$\lambda$ and $\lambda^{\prime}$, and

$\displaystyle P(u,\omega):=\frac{u^{3}}{3}+u\omega.$ (11)

By setting $(u_{1},u_{2})$ as (6) and $\kappa$ as in (7), we obtain

$\displaystyle P(u_{1},w_{xxx}(0))$
$\displaystyle\leq-\hat{\theta}_{1}V_{1}\quad\text{on}~{}I_{1},$ (12)
$\displaystyle-P(u_{2},v_{xxx}(1))$
$\displaystyle\leq-\hat{\theta}_{2}V_{2}\quad\text{on}~{}I_{2}.$

Moreover, since $u_{1}=0$ on $I_{2}$ and $u_{2}=0$ on $I_{1}$, we have

$\displaystyle P(u_{1},w_{xxx}(0))$ $\displaystyle=0\quad\text{on}~{}I_{2},$
$\displaystyle P(u_{2},v_{xxx}(1))$ $\displaystyle=0\quad\text{on}~{}I_{1}.$

As a consequence, $(V_{1},V_{2})$ satisfy

$\displaystyle\left\\{\begin{matrix}\begin{matrix}\dot{V}_{1}\leq&(\theta_{1}-\hat{\theta}_{1})V_{1}\\\
\dot{V}_{2}\leq&\theta_{2}V_{2}\end{matrix}\qquad\text{a.e. on}~{}I_{1},\\\
~{}~{}\\\ \begin{matrix}\dot{V}_{1}\leq&\theta_{1}V_{1}\\\
\dot{V}_{2}\leq&(\theta_{2}-\hat{\theta}_{2})V_{2}\end{matrix}\qquad\text{a.e.
on}~{}I_{2}.\end{matrix}\right.$ (13)

Roughly speaking, if $(\hat{\theta}_{1},\hat{\theta}_{2})\to+\infty$ as
$t\to+\infty$, then the decrease of $V_{1}$ on $I_{1}$ (resp. of $V_{2}$ on
$I_{2}$) would be significant enough to compensate for the eventual increase
of $V_{1}$ on $I_{2}$ (resp. of $V_{2}$ on $I_{1}$). Since the adaptation
parameters need to remain bounded, the idea behind Algorithm 1 is to increase
$\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ and to freeze them whenever a decay
of $(V_{1},V_{2})$ at a desired rate is observed. To detect the decrease of
$V_{1}$, we compare $V_{1}(t_{2k-1})$ with $V_{1}(t_{2k-3})$. The latter two
values are available for design since $t_{2k-1}\in[t_{2k-1},t_{2k})\subset
I_{1}$ and $t_{2k-3}\in[t_{2k-3},t_{2k-2})\subset I_{1}$. The same reasoning
applies for $V_{2}$ mutatis-mutandis.

######  Remark 3.2.

The discontinuity of $\kappa$ is key to verify (12) while guaranteeing that
$u_{1}$ and $u_{2}$ are bounded in $w_{xxx}(0)$ and $v_{xxx}(1)$,
respectively. This allows to guarantee input boundedness along the closed-loop
solutions provided that the same property holds for $\hat{\theta}_{1}$,
$\hat{\theta}_{2}$, $V_{1}$, and $V_{2}$. This discontinuity, however, can
lead to the unsuitable chattering phenomena. Another discontinuity in
$(u_{1},u_{2})$ is due to transitioning from a time interval in $I_{1}$ to a
time interval in $I_{2}$, and vice versa, see (6). Discontinuous boundary
control of PDEs brings challenges both in terms of numerical and practical
implementation and the study of well-posedness, only a few works have been
done in this direction; see, e.g., [31, 17].

######  Remark 3.3.

Since we do not know a subset of $\mathbb{R}$ containing both $\lambda$ and
$f$, the adaptive technique based on batch least-square optimization [9, Eq.
18] cannot be used here. Furthermore, the adaptive approach in [28], although
it does not require any information about $\lambda$ and $f$, does not
necessarily guarantee asymptotic convergence of $V_{1}$ and $V_{2}$ to zero
under the considered intermittent-sensing scenario. Indeed, using such a
technique, we can show that, for a desired $\alpha>0$,
$V_{1}(t)\leq(V_{1}(t_{2k})+|\theta_{1}(t_{2k})-\hat{\theta}_{1}(t_{2k})|)\exp^{-\alpha(t-t_{2k})}$
for all $t\in[t_{2k},t_{2k+1})\subset I_{1}$. If such inequality was verified
for all $t\geq t_{2k}$, then asymptotic convergence of $V_{1}$ to $0$ would
follow. The latter conclusion does not necessarily hold in the context of
intermittent sensing mainly because
$\tilde{\theta}_{1}:=\theta_{1}-\hat{\theta}_{1}$ is not guaranteed to
converge to zero. Finally, the adaptation techniques in [23, 21, 22] guarantee
$V_{1}(t)\leq V_{1}(t_{2k})\exp^{-\alpha(t-t_{2k})}$ for all
$t\in[t_{2k},t_{2k+1})\subset I_{1}$, but not at any desired $\alpha$. Hence,
they cannot be used to compensate for the possible increase of $V_{1}$ on
$I_{2}$.

###  3.2  Full Sensing, $f\neq 0$, and $|f|_{\infty}$ Unknown

In this section, we suppose that the state of the NKS equation is measured
everywhere and for all time. As a result, we assume that we act only at $x=0$
and $x=1$ using the boundary conditions

$\displaystyle u(0)=u_{1},\ \ u(1)=u_{x}(0)=u_{x}(1)=0,$ (14)

where $u_{1}$ is the control input to be designed.

Consider the Lyapunov function candidate

$\displaystyle V(u):=\frac{1}{2}\int_{0}^{1}u(x)^{2}dx,$ (15)

Using Lemma 6.2 in the Appendix, with $(w,Y,V)=(u,1,V_{1})$, we conclude that,
along the solutions to (2) under (14), we have

$\displaystyle\dot{V}\leq\theta
V+C\sqrt{V}+P(u_{1},u_{xxx}(0))\quad\text{a.e.},$ (16)

where
$\theta:=|\lambda^{\prime}|_{\infty}+2\left(|\lambda|_{\infty}+1/2\right)\left(\left(|\lambda|_{\infty}+1/2\right)+12\right)$
and $C:=\sqrt{2}|f|_{\infty}$.

Using the feedback law

$\displaystyle u_{1}:=\kappa(V,u_{xxx}(0),\hat{\theta}),$ (17)

where $\kappa$ is introduced in (7) and $\hat{\theta}$ is an adaptation
parameter to be designed later, we obtain

$\displaystyle\dot{V}\leq(\theta-\hat{\theta})V+C\sqrt{V}\qquad\quad\text{a.e.}$
(18)

Using Young inequality, we obtain that, for each $\epsilon>0$,

$\displaystyle\dot{V}\leq\left(\theta+\frac{C^{2}}{\epsilon}-\hat{\theta}\right)V+\epsilon\quad\text{a.e.}$
(19)

We update the parameter $\hat{\theta}$ as follows.

###### Algorithm 2

Given $\Delta$, $\tau$, $\epsilon$, and $\sigma>0$, the coefficient
$\hat{\theta}$ is dynamically updated, on each interval $[k\tau,(k+1)\tau]$
with $k\in\mathbb{N}^{*}$, according to the following rules:

  1. R1)

For each $t\in[k\tau,(k+1)\tau]$ if

$V(s)\leq V(k\tau)\exp^{-\sigma(s-k\tau)}+\frac{\epsilon}{\sigma}\quad\forall
s\in[k\tau,t],$

then $\dot{\hat{\theta}}(t)=0$; otherwise,

$\displaystyle\hat{\theta}(r)=\hat{\theta}(k\tau)+\Delta\quad\forall
r\in[t,(k+1)\tau].$ (20)

  2. R2)

On the interval $[0,\tau)$, we set $\hat{\theta}=\hat{\theta}(0)\geq 0$.

We will show that the closed-loop system enjoys the property of $L^{2}$ global
practical attractivity ($L^{2}$-GpA).

######  Property 2 ($L^{2}$-GpA)

For any $\eta>0$, we can find $(\tau,\sigma,\epsilon)$ such that, for any
$\hat{\theta}(0)\geq 0$ and for any $V(0)\geq 0$, we have
$\limsup_{t\to+\infty}V(t)\leq\eta$.

######  Theorem 3.4.

Consider equation (2) such that Assumption 2 holds. Consider the boundary
conditions in (14) and let the feedback law $u_{1}:=\kappa$ in (7) with
$\hat{\theta}$ updated according to Algorithm 2. Then, for the resulting
closed-loop system,

  * •

The $L^{2}$-GpA property is verified.

  * •

For each $\hat{\theta}(0)$, there exists $M>0$ such that, for each $V(0)$, we
have $|\hat{\theta}|_{\infty}\leq M$.

  * •

The input $u_{1}$ remains bounded.

######  Remark 3.5.

In view of (16), our design of $u_{1}$ guarantees that
$P(u_{1},u_{xxx}(0))\leq-\hat{\theta}_{1}V$, where $P$ is in (11), and thus
(18) holds. The latter Lyapunov inequality does not allow us to guarantee
$L^{2}$-GES due to the term $C\sqrt{V}$, which becomes dominant when $V$ is
small. If we attempt to compensate the latter term by designing $u_{1}$ such
that

$\displaystyle P(u_{1},u_{xxx}(0))$
$\displaystyle\leq-\hat{\theta}V-\hat{C}\sqrt{V},$ (21)

where $\hat{C}$ is an adaptation parameter, we would obtain

$\displaystyle\dot{V}\leq(\theta-\hat{\theta})V+(C-\hat{C})\sqrt{V}\quad\text{a.e.}$
(22)

In this case, if $\hat{C}>C$ and $\hat{\theta}\geq\theta$, we would conclude
that

$\displaystyle\dot{V}\leq-\alpha\sqrt{V}\quad\text{a.e.},\quad\alpha:=\hat{C}-C>0.$
(23)

Hence, $V$ would reach zero in finite time implying the same for $w$. However,
$w=0$ is not a solution to the NKS equation when $f\neq 0$. Hence, trying to
compensate $C\sqrt{V}$ can lead to an ill-posed closed-loop system.

######  Remark 3.6.

To guarantee the continuity of $u_{1}=\kappa(V,u_{xxx}(0),\hat{\theta})$ with
respect to its arguments and its boundedness along the closed-loop solutions,
one can try to redesign the function $\kappa$ to be a continuous function
solution to $\kappa^{3}+3\kappa u_{xxx}(0)\leq 3\hat{\theta}V$ and guarantee
boundedness of the $H^{1}$ norm of the state $u$ in closed loop. In this case,
input boundedness would follow thanks to Agmon’s inequality [12, Lemma A.2].
While this reasoning can apply to the full-sensing case, it cannot be applied
to the intermittent-sensing case. Indeed, during the time intervals when $w$
(resp. $v$) is not measured, we are not able to upper-bound the $H^{1}$ norm
of $w$ (resp. $v$), as we have no information on the terms $w_{xx}(\\{0,Y\\})$
and $w_{xxx}(\\{0,Y\\})$ (resp. $v_{xx}(\\{Y,1\\})$ and $v_{xxx}(\\{Y,1\\})$)
that affect the $H^{1}$ norm of $w$ (resp. $v$).

###  3.3 Intermittent Sensing with $f\neq 0$ and $|f|_{\infty}$ Known

We suppose here that $|f|_{\infty}\neq 0$ (or an upper bound of
$|f|_{\infty}$) is known. In the context of intermittent sensing, we design
$(u_{1},u_{2})$ according to (6)-(7), with
$(\hat{\theta}_{1},\hat{\theta}_{2})$ therein dynamically updated as follows.

###### Algorithm 3

Given $\Delta_{1},\Delta_{2},\sigma>0$, we update
$(\hat{\theta}_{1},\hat{\theta}_{2})$ according to the following rules:

  1. R1)

On each interval $[t_{2k-1},t_{2k})\subset I_{1}$, we set
$\dot{\hat{\theta}}_{2}:=0$. Moreover, if

$\displaystyle V_{1}($ $\displaystyle
t_{2k-1})>V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})}$
$\displaystyle+\left(C_{1}+\frac{C_{1}^{2}}{4}\right)\exp^{\left(\hat{\theta}_{1}(t_{2k-3})+1\right)\left(t_{2k-1}-t_{2k-3}\right)},$

we set $\dot{\hat{\theta}}_{1}:=\Delta_{1}$; otherwise, we set
$\dot{\hat{\theta}}_{1}:=0$.

  2. R2)

On each interval $[t_{2k},t_{2k+1})\subset I_{2}$, we set
$\dot{\hat{\theta}}_{1}:=0$. Moreover, if

$\displaystyle V_{2}($ $\displaystyle
t_{2k})>V_{2}(t_{2k-2})\exp^{-\sigma(t_{2k}-t_{2k-2})}$
$\displaystyle~{}+\left(C_{2}+\frac{C_{2}^{2}}{4}\right)\exp^{\left(\hat{\theta}_{2}(t_{2k-2})+1\right)\left(t_{2k}-t_{2k-2}\right)},$

we set $\dot{\hat{\theta}}_{2}:=\Delta_{2}$; otherwise, we set
$\dot{\hat{\theta}}_{2}:=0$.

  3. R3)

On $[t_{1},t_{3}]$, $\hat{\theta}_{1}=\hat{\theta}_{1}(0)\geq 0$ and
$\hat{\theta}_{2}=\hat{\theta}_{2}(0)\geq 0$.

We show here that the closed-loop system enjoys the property of $L^{2}$-input-
to-state stability ($L^{2}$-ISS) with respect to the perturbation $f$.

######  Property 3 ($L^{2}$-ISS)

For each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists $\gamma\geq
1$ and a class $\mathcal{K}$ function $\Phi$ such that, for all $t\geq 0$, we
have

$\displaystyle V_{1}(t)+V_{2}(t)\leq\gamma(V_{1}(0)+V_{2}(0))e^{-\sigma
t}+\Phi(|f|_{\infty}),$ (24)

where $\sigma>0$ comes from Algorithm 3.

######  Theorem 3.7.

Consider system (3) such that Assumption 2 holds. Consider the intermittent
sensing scenario in  S1)- S2) such that Assumption 1 holds. Let
$(V_{1},V_{2})$ be defined in (5), $\kappa$ be defined in (7), and the
parameters $(\hat{\theta}_{1},\hat{\theta}_{2})$ governed by Algorithm 3. By
letting $(u_{1},u_{2})$ as in (6), we conclude that, for (3) in closed-loop
system,

  * •

The $L^{2}$-ISS property is verified.

  * •

For each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists $M>0$ such
that for all $(V_{1}(0),V_{2}(0))$, we have
$|\hat{\theta}_{1}|_{\infty},|\hat{\theta}_{2}|_{\infty}\leq M$.

  * •

The inputs $(u_{1},u_{2})$ remain bounded.

######  Remark 3.8.

In Section 3.1 and when $f=0$, to prove that $(V_{1},V_{2})\to 0$ as
$t\rightarrow+\infty$, we used the fact that the maximal value of $V_{1}$
(resp. $V_{2}$) on each interval in $I_{1}$ (resp. $I_{2}$) is smaller than
the value of $V_{1}$ (resp. $V_{2}$) at the beginning of that interval
multiplied by a constant. Moreover, when $f\neq 0$, the maximal value $V_{1}$
(resp. $V_{2}$) on each interval in $I_{1}$ (resp. $I_{2}$) additionally
depends on $|f|_{\infty}$. In other words, there are values, depending on
$|f|_{\infty}$, that $V_{1}$ (resp. $V_{2}$) can reach over each interval in
$I_{1}$ (resp. $I_{2}$) independently on its initial value at the beginning of
that interval. As a consequence, we can only try to decrease $V_{1}$ on
$I_{1}$ and $V_{2}$ on $I_{2}$ significantly enough so that (24) holds.

###  3.4 Intermittent Sensing With $f\neq 0$ and $|f|_{\infty}$ Unknown

We design $(u_{1},u_{2})$ as in (6) with $\kappa$ as in (7) and
$(\hat{\theta}_{1},\hat{\theta}_{2})$ are updated according to the following
algorithm.

###### Algorithm 4

Given $\Delta_{1},\Delta_{2}$, $\sigma>0$, the coefficients
$(\hat{\theta}_{1},\hat{\theta}_{2})$ are dynamically updated as follows:

  1. R1)

On every interval $[t_{2k-1},t_{2k})\subset I_{1}$, we set
$\dot{\hat{\theta}}_{2}:=0$. Moreover, if

$\displaystyle
V_{1}(t_{2k-1})>V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})}$
$\displaystyle+\left(\hat{\theta}_{1}(t_{2k-3})+\frac{\hat{\theta}_{1}(t_{2k-3})^{2}}{4}\right)\exp^{\left(\hat{\theta}_{1}(t_{2k-3})+1\right)\left(t_{2k-1}-t_{2k-3}\right)},$

we set $\dot{\hat{\theta}}_{1}:=\Delta_{1}$; otherwise, we set
$\dot{\hat{\theta}}_{1}:=0$.

  2. R2)

On every interval $[t_{2k},t_{2k+1})\subset I_{2}$, we set
$\dot{\hat{\theta}}_{1}:=0$. Moreover, if

$\displaystyle V_{2}(t_{2k})>V_{2}(t_{2k-2})\exp^{-\sigma(t_{2k}-t_{2k-2})}$
$\displaystyle+\left(\hat{\theta}_{2}(t_{2k-2})+\frac{\hat{\theta}_{2}(t_{2k-2})^{2}}{4}\right)\exp^{\left(\hat{\theta}_{2}(t_{2k-2})+1\right)\left(t_{2k}-t_{2k-2}\right)},$

we set $\dot{\hat{\theta}}_{2}:=\Delta_{2}$; otherwise, we set
$\dot{\hat{\theta}}_{2}:=0$.

  3. R3)

On $[t_{1},t_{3}]$, $\hat{\theta}_{1}=\hat{\theta}_{1}(0)\geq 0$ and
$\hat{\theta}_{2}=\hat{\theta}_{2}(0)\geq 0$.

We will show that the resulting closed-loop system satisfies the property of
$L^{2}$ global uniform ultimate boundedness ($L^{2}$-GUUB).

######  Property 4 ($L^{2}$-GUUB)

For any $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists a constant
$r>0$ such that for any $R>0$, there exists a finite time $T(R)\geq 0$ such
that, along the closed-loop solutions, we have

$\displaystyle V_{1}(0)+V_{2}(0)\leq R\Rightarrow V_{1}(t)+V_{2}(t)\leq
r~{}\forall t\geq T(R).$ (25)

######  Theorem 3.9.

Consider system (3) such that Assumption 2 holds. Consider the intermittent
sensing scenario in  S1)- S2) such that Assumption 1 holds. Let
$(V_{1},V_{2})$ be defined in (5), $\kappa$ be defined in (7), and
$(\hat{\theta}_{1},\hat{\theta}_{2})$ be governed by Algorithm 4. Then, if we
set $(u_{1},u_{2})$ as in (6), for the resulting closed-loop system,

  * •

The $L^{2}$-GUUB property is verified.

  * •

For each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exists $M>0$ such
that for all $(V_{1}(0),V_{2}(0))$, we have
$|\hat{\theta}_{1}|_{\infty},|\hat{\theta}_{2}|_{\infty}\leq M$.

  * •

The control inputs remain bounded.

Sketch of proof: According to Lemma 6.2 in the Appendix, along the closed-loop
solutions, $(V_{1},V_{2})$ verify

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+C_{1}\sqrt{V}_{1}+P(u_{1},w_{xxx}(0))\quad\text{a.e.},$
(26)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}+C_{2}\sqrt{V}_{2}-P(u_{2},v_{xxx}(1))\quad\text{a.e.},$

where $P$ is in (11) and $(C_{1},C_{2})$ are constant parameters that depend
on $|f|_{\infty}$; see (90). Now, by setting $(u_{1},u_{2})$ as in (6), one
can show that

$\displaystyle\left\\{\begin{matrix}\begin{matrix}\dot{V}_{1}\leq&(\theta_{1}+C_{1}-\hat{\theta}_{1})V_{1}+C_{1}\\\
\dot{V}_{2}\leq&\theta_{2}V_{2}+C_{2}\sqrt{V_{2}}\end{matrix}\qquad\text{a.e.
on}~{}I_{1},\\\ ~{}~{}\\\
\begin{matrix}\dot{V}_{1}\leq&\theta_{1}V_{1}+C_{1}\sqrt{V_{1}}\\\
\dot{V}_{2}\leq&(\theta_{2}+C_{2}-\hat{\theta}_{2})V_{2}+C_{2}\end{matrix}\qquad\text{a.e.
on}~{}I_{2}.\end{matrix}\right.$ (27)

Our approach consists of decreasing $V_{1}$ on $I_{1}$ and $V_{2}$ on $I_{2}$
significantly enough so that they remain bounded all the time. To do so, we
can show that by letting $(\hat{\theta}_{1},\hat{\theta}_{2})\to+\infty$ as
$t\to+\infty$, we can guarantee that $V_{1}$ and $V_{2}$ remain bounded at the
price of using unbounded control. To guarantee input boundedness, the idea
behind Algorithm 4 is to increase $\hat{\theta}_{1}$ (resp.
$\hat{\theta}_{2}$) until the inequality in R1) (resp. R2)) is not verified,
and $\hat{\theta}_{1}$ (resp. $\hat{\theta}_{2}$) remains constant as long as
R1) (resp. R2)) is not verified. One can show that there exists a finite time,
from which, the conditions in the inequalities in R1) (resp. R2)) are never
verified. Hence, we can conclude boundedness of $\hat{\theta}_{1}$ (resp.
$\hat{\theta}_{2})$) and thus boundedness of $(u_{1},u_{2})$.

######  Remark 3.10.

In view of (26), we could design $(u_{1},u_{2})$ such that

$\displaystyle P(u_{1},w_{xxx}(0))$
$\displaystyle\leq-\hat{\theta}_{1}V_{1}-\hat{C}_{1}\sqrt{V_{1}},$
$\displaystyle-P(u_{2},v_{xxx}(1))$
$\displaystyle\leq-\hat{\theta}_{2}V_{2}-\hat{C}_{2}\sqrt{V_{2}},$

where $P$ is in (11) and $(\hat{C}_{1},\hat{C}_{2})$ are adaptation
parameters. In this case, we would obtain

$\displaystyle\dot{V}_{1}\leq(\theta_{1}-\hat{\theta}_{1})V_{1}+(C_{1}-\hat{C}_{1})\sqrt{V}_{1}\quad\text{a.e.
on}~{}I_{1}.$
$\displaystyle\dot{V}_{2}\leq(\theta_{2}-\hat{\theta}_{2})V_{2}+(C_{2}-\hat{C}_{2})\sqrt{V}_{2}\quad\text{a.e.
on}~{}I_{2}.$

This choice is not suitable for the same ill-posedness reasons described in
Remark 3.5.

######  Remark 3.11.

In Section 3.4, where $|f|_{\infty}$ (or its upper bound) was known, we were
able to use this knowledge to guarantee the relatively stronger $L^{2}$-ISS
property. Indeed, the parameters $C_{1}$ and $C_{2}$, involving
$|f|_{\infty}$, are used in Algorithm 3 to tune the control parameters until
convergence of $V_{1}$ (resp. $V_{2}$) to a neighborhood of the origin, whose
size is proportional to $C_{1}$ (resp. $C_{2}$), is detected. Here, since
$|f|_{\infty}$ is unknown, we are not able to guarantee that $V_{1}$ (resp.
$V_{2}$), although maintained bounded, is proportional to $C_{1}$ (resp.
$C_{2}$) in steady state.

##  4 Proofs

###  4.1 Proof of Theorem 3.1

The proof of Theorem 3.1 follows in four steps. In Step 1, by setting
$(u_{1},u_{2})$ as in (6), we show that the Lyapunov functions $(V_{1},V_{2})$
in (5), along the closed-loop solutions, verify the inequalities in (13). In
Step 2, we show that $(\hat{\theta}_{1},\hat{\theta}_{2})$ become constant
after some finite time $T>0$. Furthermore, the maximal values that
$(\hat{\theta}_{1},\hat{\theta}_{2})$ can reach do not exceed the maximum
value between $\hat{\theta}_{1}(0)$ and a constant $M_{1}$ and
$\hat{\theta}_{2}(0)$ and a constant $M_{2}$, respectively. The constants
$M_{1}$ and $M_{2}$ depend only on
$(\sigma,\theta_{1},\theta_{2},\overline{T}_{1},\overline{T}_{2},\underline{T}_{1},\underline{T}_{2},\Delta_{1},\Delta_{2})$.
In Step 3, we show that the function $V_{1}+V_{2}$ verifies the inequality
(8), which concludes $L^{2}$-GES of the origin. Finally, we use the structure
of $\kappa$ in (7) guaranteeing that $|u_{1}|\leq
3(3|\hat{\theta}_{1}|_{\infty}+1)\sqrt[3]{V_{1}}$ and $|u_{2}|\leq
3(3|\hat{\theta}_{2}|_{\infty}+1)\sqrt[3]{V_{2}}$, after proving that
$(\hat{\theta}_{1},\hat{\theta}_{2})$ are bounded and $(V_{1},V_{2})$ are
bounded and converge asymptotically to zero, to conclude that $(u_{1},u_{2})$
are bounded and converge asymptotically to zero.

Step 1: Under Lemma 6.2 in the Appendix, $(V_{1},V_{2})$ verify for almost
everywhere the inequalities

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0),$
(28)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}-\frac{u_{2}^{3}}{3}-u_{2}v_{xxx}(1).$
(29)

By setting $u_{1}=0$ on $I_{2}$ and $u_{2}=0$ on $I_{1}$, we obtain

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}\quad\text{a.e. on}~{}I_{2},$ (30)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}\quad\text{a.e. on}~{}I_{1}.$ (31)

Thus, it remains to show that

$\displaystyle\dot{V}_{1}\leq(\theta_{1}-\hat{\theta}_{1})V_{1}\quad\text{a.e.
on}~{}I_{1},$ (32)
$\displaystyle\dot{V}_{2}\leq(\theta_{2}-\hat{\theta}_{2})V_{2}\quad\text{a.e.
on}~{}I_{2}.$ (33)

Let us prove inequality (32), the proof of (33) being similar. To do so, we
distinguish between two cases. When $|w_{xxx}(0)|\geq
l(V_{1},\hat{\theta}_{1})$, then

$\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0)\leq\frac{V_{1}}{3}-\sqrt[3]{V_{1}}l(V_{1},\hat{\theta}_{1})\leq-\hat{\theta}_{1}V_{1}.$

Otherwise, we have

$\displaystyle\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0)$
$\displaystyle\leq-3(3\hat{\theta}_{1}+1)^{3}V_{1}$
$\displaystyle+3(3\hat{\theta}_{1}+1)\sqrt[3]{V_{1}}l(V_{1},\hat{\theta}_{1})\leq-\hat{\theta}_{1}V_{1}.$

Step 2: We first show that $(\hat{\theta}_{1},\hat{\theta}_{2})$ become
constants after a finite time $T>0$. Indeed, proceeding by contradiction, we
assume that there is no such a $T>0$ such that
$\dot{\hat{\theta}}_{1}(t)=\dot{\hat{\theta}}_{2}(t)=0$ for all $t\geq T$.
This means, according to R1)-R2) in Algorithm 1, that there exists an infinite
number of time intervals, each one having a length greater or equal than
$\min\\{\underline{T}_{1},\underline{T}_{2}\\}$, on which, $\hat{\theta}_{1}$
and $\hat{\theta}_{2}$ are linearly increasing, thus,
$\lim_{t\to\infty}\hat{\theta}_{1}(t)=\lim_{t\to\infty}\hat{\theta}_{2}(t)=\infty$.
It follows that there must exists $k^{\prime}>1$ such that, for all $k\geq
k^{\prime}$, we have $\tilde{\theta}_{1}(t_{2k-3})<0$,
$\tilde{\theta}_{2}(t_{2k-2})<0$, and

$\displaystyle\tilde{\theta}_{1}(t_{2k-3})\underline{T}_{1}+\theta_{1}\overline{T}_{1}\leq-\sigma(\overline{T}_{1}+\overline{T}_{2}),$
(34)
$\displaystyle\tilde{\theta}_{2}(t_{2k-2})\underline{T}_{2}+\theta_{2}\overline{T}_{2}\leq-\sigma(\overline{T}_{1}+\overline{T}_{2}).$
(35)

As a consequence, for all $k\geq k^{\prime}$, we have

$\displaystyle\tilde{\theta}_{1}(t_{2k-3})(t_{2k-2}-t_{2k-3})$
$\displaystyle+\theta_{1}(t_{2k-1}-t_{2k-2})$
$\displaystyle\leq-\sigma(t_{2k-1}-t_{2k-3}),$ (36)
$\displaystyle\tilde{\theta}_{2}(t_{2k-2})(t_{2k-1}-t_{2k-2})$
$\displaystyle+\theta_{2}(t_{2k}-t_{2k-1})$
$\displaystyle\leq-\sigma(t_{2k}-t_{2k-2}).$ (37)

On the other hand, using Grönwall-Bellman inequality, we find that, for all
$k\geq k^{\prime}$,

$\displaystyle V_{1}(t_{2k-1})\leq
V_{1}(t_{2k-3})\exp^{\int_{t_{2k-3}}^{t_{2k-2}}\tilde{\theta}_{1}(t)dt+\theta_{1}(t_{2k-1}-t_{2k-2})}$
$\displaystyle\leq
V_{1}(t_{2k-3})\exp^{\tilde{\theta}_{1}(t_{2k-3})(t_{2k-2}-t_{2k-3})+\theta_{1}(t_{2k-1}-t_{2k-2})},$
(38) $\displaystyle V_{2}(t_{2k})\leq
V_{2}(t_{2k-2})\exp^{\int_{t_{2k-2}}^{t_{2k-1}}\tilde{\theta}_{2}(t)dt+\theta_{2}(t_{2k}-t_{2k-1})}$
$\displaystyle\leq
V_{2}(t_{2k-2})\exp^{\tilde{\theta}_{2}(t_{2k-2})(t_{2k-1}-t_{2k-2})+\theta_{2}(t_{2k}-t_{2k-1})}.$
(39)

By combining (36), (37), (38), and (39), we conclude that, for all $k\geq
k^{\prime}$, we have

$\displaystyle V_{1}(t_{2k-1})$ $\displaystyle\leq
V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})},$ (40) $\displaystyle
V_{2}(t_{2k})$ $\displaystyle\leq
V_{2}(t_{2k-2})\exp^{-\sigma(t_{2k}-t_{2k-2})}.$ (41)

From the design of $\hat{\theta}_{1}$ and $\hat{\theta}_{2}$, the inequalities
in (40)-(41) imply that $\hat{\theta}_{1}$ is constant for all $t\geq
t_{2k^{\prime}-1}$ and $\hat{\theta}_{2}$ is constant for all $t\geq
t_{2k^{\prime}}$, which yields to a contradiction.

Now, we show the existence of $M_{1}$, $M_{2}>0$ (functions of
$\sigma,\theta_{1},\theta_{2},\overline{T}_{1},\overline{T}_{2},\underline{T}_{1},\underline{T}_{2},\Delta_{1},\Delta_{2}$)
such that

$\displaystyle|\hat{\theta}_{1}|_{\infty}\leq\max\\{M_{1},\hat{\theta}_{1}(0)\\},~{}|\hat{\theta}_{2}|_{\infty}\leq\max\\{M_{2},\hat{\theta}_{2}(0)\\}.$
(42)

Let us suppose that there exists $T\geq 0$ such that

$\hat{\theta}_{1}(T)=\theta_{1}+\frac{\sigma\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{1}\overline{T}_{1}}{\underline{T}_{1}}.$
(43)

We can always pick such a $T$ to be in $I_{1}$, as $\hat{\theta}_{1}$ is
constant over each interval in $I_{2}$. Thus, we let
$T\in[t_{2k^{\prime}-3},t_{2k^{\prime}-2})\subset I_{1}$ for some
$k^{\prime}>1$. Using R1), we conclude that
$0\leq\dot{\hat{\theta}}(t)\leq\Delta_{1}$ for all $t\geq 0$. As a result,

$\hat{\theta}_{1}(T)\leq\hat{\theta}_{1}(t_{2k^{\prime}+1})\leq\hat{\theta}_{1}(T)+2\Delta_{1}\overline{T}_{1}.$

At the same time, under (43) and the monotonicity of $\hat{\theta}_{1}$, we
conclude that

$\displaystyle V_{1}(t)\leq
V_{1}(t_{2k^{\prime}-1})\exp^{-\sigma(t-t_{2k^{\prime}-1})}\quad\forall t\geq
t_{2k^{\prime}+1}.$

According to R1), the latter inequality implies that
$\dot{\hat{\theta}}_{1}(t)=0$ for all $t\geq t_{2k^{\prime}+1}$. As a result,

$\displaystyle\hat{\theta}_{1}(t)\leq\hat{\theta}_{1}(T)+2\Delta_{1}\overline{T}_{1}\quad\forall
t\geq t_{2k^{\prime}+1}.$ (44)

Finally, if there is no $T$ such that (43) is satisfied, then

$\displaystyle\hat{\theta}_{1}(t)\leq\theta_{1}+\frac{\sigma\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{1}\overline{T}_{1}}{\underline{T}_{1}}\qquad\forall
t\geq 0.$ (45)

Using the same argument for $\hat{\theta}_{2}$, we conclude that (42) is
verified with

$\displaystyle M_{1}$
$\displaystyle:=\theta_{1}+\frac{\sigma\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{1}\overline{T}_{1}}{\underline{T}_{1}}+2\Delta_{1}\overline{T}_{1},$
(46) $\displaystyle M_{2}$
$\displaystyle:=\theta_{2}+\frac{\sigma\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{2}\overline{T}_{2}}{\underline{T}_{2}}+2\Delta_{2}\overline{T}_{2}.$
(47)

Step 3: To analyze $V_{1}+V_{2}$, we define the sequences
$\\{T_{i}\\}_{i=0}^{\infty}$ and $\\{T^{\prime}_{i}\\}_{i=1}^{\infty}$ such
that $T_{i}:=t_{2i+1}$ and $T^{\prime}_{i}:=t_{2i}$. In particular, we note
that, for all $i\in\mathbb{N}$, we have

$\displaystyle\underline{T}_{1}+\underline{T}_{2}\leq
T_{i+1}-T_{i}\leq\overline{T}_{1}+\overline{T}_{2},$ (48)
$\displaystyle\underline{T}_{1}+\underline{T}_{2}\leq
T^{\prime}_{i+2}-T^{\prime}_{i+1}\leq\overline{T}_{1}+\overline{T}_{2}.$ (49)

Since $(\hat{\theta}_{1},\hat{\theta}_{2})$ are nondecreasing and become
constant after some finite time, we conclude on the existence of at most a
finite number of intervals $[t_{2k-1},t_{2k})\subset I_{1}$, on which
$\hat{\theta}_{1}$ may increase. On the latter intervals, we know that $V_{1}$
is governed by $\dot{V}_{1}\leq\theta_{1}V_{1}$, while on the remaining
intervals $V_{1}$ does not verify the inequality in R1), implying its
exponential decay. The same reasoning applies to $\hat{\theta}_{2}$ and
$V_{2}$. More precisely, for each $(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$,
we can find $N_{1}^{*},N_{2}^{*}\in\mathbb{N}$ such that, for each locally
absolutely continuous solution $(V_{1},V_{2})$ to (13), there exists two
finite increasing sub-sequences
$\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}\subset\mathbb{N}$ and
$\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}\subset\mathbb{N}^{*}$ such that

  * •

For each $i\in\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have

$V_{1}(T_{i+1})\leq V_{1}(T_{i})\exp^{\theta_{1}(T_{i+1}-T_{i})}.$ (50)

  * •

For each $j\in\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have

$V_{2}(T_{j+1}^{\prime})\leq
V_{2}(T_{j}^{\prime})\exp^{\theta_{2}(T_{j+1}^{\prime}-T_{j}^{\prime})}.$ (51)

  * •

For each $i\in\mathbb{N}/\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have

$V_{1}(T_{i+1})\leq V_{1}(T_{i})\exp^{-\sigma(T_{i+1}-T_{i})}.$ (52)

  * •

For each $j\in\mathbb{N^{*}}/\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have

$V_{2}(T_{j+1}^{\prime})\leq
V_{2}(T_{j}^{\prime})\exp^{-\sigma(T_{j+1}^{\prime}-T_{j}^{\prime})}.$ (53)

Using Lemma 6.6 in the appendix, while replacing $(V,M,\psi,N^{*})$ therein by
$(V_{1},0,\theta_{1},N^{*}_{1})$, we conclude that, for each $i\geq 0$,

$\displaystyle V_{1}(T_{i})$
$\displaystyle\leq\exp^{(\theta_{1}+\sigma)N^{*}_{1}\left(\overline{T}_{1}+\overline{T}_{2}\right)}V_{1}(0)\exp^{-\sigma
T_{i}}.$ (54)

Similarly, using Lemma 6.6 in the appendix, while replacing
$(V,M,\psi,\\{T_{i}\\}^{\infty}_{i=1},N^{*})$ therein by
$(V_{2},0,\theta_{2},\\{T^{\prime}_{i}\\}^{\infty}_{i=1},N^{*}_{2})$, we
conclude that, for each $i\geq 0$,

$\displaystyle V_{2}(T_{i}^{\prime})$
$\displaystyle\leq\exp^{(\theta_{2}+\sigma)N^{*}_{2}\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{2}\overline{T}_{1}}V_{2}(0)\exp^{-\sigma
T_{i}^{\prime}}.$ (55)

Let $t\in[T_{i},T_{i+1}]$, we have

$V_{1}(t)\leq
V_{1}(T_{i})\exp^{\theta_{1}\left(\overline{T}_{1}+\overline{T}_{2}\right)}.$
(56)

By combining (54) and (56), and using the fact that
$t-T_{i}\leq\overline{T}_{1}+\overline{T}_{2}$, we obtain

$V_{1}(t)\leq\exp^{(\theta_{1}+\sigma)(N^{*}_{1}+1)\left(\overline{T}_{1}+\overline{T}_{2}\right)}V_{1}(0)\exp^{-\sigma
t}.$ (57)

Similarly, for $t\in[T_{i}^{\prime},T_{i+1}^{\prime}]$, we have

$\displaystyle
V_{2}(t)\leq\exp^{(\theta_{2}+\sigma)(N^{*}_{2}+1)\left(\overline{T}_{1}+\overline{T}_{2}\right)+\theta_{2}\overline{T}_{1}}V_{2}(0)\exp^{-\sigma
t}.$ (58)

Hence, (8) follows by summing up (57) and (58).

###  4.2 Proof of Theorem 3.4

We first show that $\hat{\theta}$ admits an upperbound that does not depend on
$V(0)$. As a result, since $\hat{\theta}$ is nondecreasing and, when it
increases, it does so according to (20), we conclude that $\hat{\theta}$
becomes constant after some finite time $T\geq 0$. Next, we analyze the
Lyapunov function candidate $V$ in (15) and show that, after a finite time,
$V$ starts decaying exponentially towards a neighborhood of the origin, whose
size is proportional to $\epsilon$. Hence, we conclude $L^{2}$-GpA. Finally,
using boundedness of $(V,\hat{\theta})$ and the structure of the feedback law
$\kappa$, boundedness of the control input $u_{1}$ follows.

Let us show that

$\displaystyle\hat{\theta}(t)\leq\max\\{\theta+C^{2}/\epsilon+\sigma+\Delta,\hat{\theta}(0)\\}\quad\forall
t\geq 0.$ (59)

To prove (59), we first suppose that
$\hat{\theta}(0)\leq\theta+C^{2}/\epsilon+\sigma$. As a result, either

$\hat{\theta}(t)\leq\theta+C^{2}/\epsilon+\sigma\quad\forall t\geq 0.$

Otherwise, in view of R1), there exist $k^{*}\geq 1$ and
$t\in[(k^{*}-1)\tau,k^{*}\tau)$ such that

$\theta+C^{2}/\epsilon+\sigma<\hat{\theta}(t)\leq\theta+C^{2}/\epsilon+\sigma+\Delta.$

Using (19), we then conclude that

$\displaystyle V(s)\leq
V(t)\exp^{-\sigma(s-t)}+\frac{\epsilon}{\sigma}\quad\forall
s\in[t,k^{*}\tau].$

This implies that $\hat{\theta}$ is constant on $[t,k^{*}\tau]$. Since
$\hat{\theta}$ is nondecreasing, it follows that (19) is verified for all
$s\geq t$ and thus

$\displaystyle V(s)\leq
V(t)\exp^{-\sigma(s-t)}+\frac{\epsilon}{\sigma}\qquad\forall s\geq t,$

which in turn implies that

$\hat{\theta}(s)\leq\hat{\theta}(t)\leq\theta+C^{2}/\epsilon+\sigma+\Delta\qquad\forall
s\geq t.$

If $\hat{\theta}(0)>\theta+C^{2}/\epsilon+\sigma,$ we use the fact that
$\hat{\theta}$ is nondecreasing to conclude that the inequality in (19) is
verified for all $s\geq 0$ and thus

$\displaystyle V(s)\leq
V(0)\exp^{-\sigma(s-t)}+\frac{\epsilon}{\sigma}\qquad\forall s\geq t,$

which in turn implies that

$\hat{\theta}(s)\leq\hat{\theta}(0)\qquad\forall s\geq t.$

To analyze the function $V$, we let $k\geq 0$ such that $\hat{\theta}$ is
constant on $[k\tau,+\infty)$. As a result, for each $t\in[k\tau,+\infty)$,
there exists $n\geq 0$ such that $t\in[(k+n)\tau,(k+1+n)\tau)$. Now, according
to R1), we have

$\displaystyle V(t)$ $\displaystyle\leq
V((k+n)\tau)\exp^{-\sigma(t-(k+n)\tau)}+\frac{\epsilon}{\sigma}$ (60)
$\displaystyle\qquad\qquad\qquad\forall t\in[(k+n)\tau,(k+1+n)\tau].$

Moreover, by continuity of $V$ and according to R1), we conclude that, for
each $i\in\\{1,...,n\\}$, we have

$\displaystyle V((k+i)\tau)\leq
V((k+i-1)\tau)\exp^{-\sigma\tau}+\frac{\epsilon}{\sigma}.$ (61)

We show next that

$\displaystyle V((k+n)\tau)\leq V(k\tau)\exp^{-\sigma
n\tau}+\left(\frac{1}{1-\exp^{-\sigma\tau}}\right)\frac{\epsilon}{\sigma}.$
(62)

Indeed, the latter inequality combined with (60) allows us to conclude
$L^{2}$-GpA.

To prove (62), we note that

$\displaystyle V$ $\displaystyle((k+n)\tau)\leq
V((k+n-1)\tau)\exp^{-\sigma\tau}+\frac{\epsilon}{\sigma}$ $\displaystyle\leq
V((k+n-2)\tau)\exp^{-2\sigma\tau}+\frac{\epsilon}{\sigma}\left(1+\exp^{-\sigma\tau}\right)$
$\displaystyle\leq V(k\tau)\exp^{-\sigma
n\tau}+\frac{\epsilon}{\sigma}\left(\sum_{j=0}^{n}\exp^{-j\sigma\tau}\right).$

Finally, (62) follows using the fact that

$\displaystyle\sum_{j=0}^{n}\exp^{-j\sigma\tau}\leq\sum_{j=0}^{\infty}\exp^{-j\sigma\tau}\leq\frac{1}{1-\exp^{-\sigma\tau}}.$

###  4.3 Proof of Theorem 3.7

The proof follows in four steps. In Step 1, by setting $(u_{1},u_{2})$ as in
(6), we show that the Lyapunov functions $(V_{1},V_{2})$ in (5), along the
closed-loop system, satisfy the inequalities in (27). In Step 2, we prove that
$(\hat{\theta}_{1},\hat{\theta}_{2})$ become constant after some $T>0$.
Furthermore, $(\hat{\theta}_{1},\hat{\theta}_{2})$ never exceed
$\max\\{\hat{\theta}_{1}(0),M_{1}\\}$ and
$\max\\{\hat{\theta}_{2}(0),M_{2}\\}$, respectively, where $M_{1},M_{2}>0$
depend on
$(\sigma,\theta_{1},\theta_{2},C_{1},C_{2},\overline{T}_{1},\overline{T}_{2},\underline{T}_{1},\underline{T}_{2},\Delta_{1},\Delta_{2})$.
In Step 3, we analyze the function $V_{1}+V_{2}$ and show that the $L^{2}$-ISS
property with respect to $f$ is verified. Finally, using the structure of
$\kappa$ and the fact that $(\hat{\theta}_{1},\hat{\theta}_{2},V_{1},V_{2})$
are bounded to conclude that $(u_{1},u_{2})$ remain bounded.

Step 1: According to Lemma 6.2 in the Appendix, $(V_{1},V_{2})$ verify almost
everywhere the inequalities

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+C_{1}\sqrt{V}_{1}+\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0),$
(63)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}+C_{2}\sqrt{V}_{2}-\frac{u_{2}^{3}}{3}-u_{2}v_{xxx}(1).$
(64)

By setting $u_{1}=0$ on $I_{2}$, and $u_{2}=0$ on $I_{1}$, we therefore obtain

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+C_{1}\sqrt{V_{1}}\quad\text{a.e.
on}~{}I_{2},$ (65)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}+C_{2}\sqrt{V_{2}}\quad\text{a.e.
on}~{}I_{1}.$ (66)

Next, due to our choice of $\kappa$, we obtain

$\displaystyle u_{1}^{3}+3u_{1}w_{xxx}(0)$
$\displaystyle\leq-3\hat{\theta}_{1}V_{1}\quad\text{on}~{}I_{1},$ (67)
$\displaystyle-u_{2}^{3}-3u_{2}v_{xxx}(1)$
$\displaystyle\leq-3\hat{\theta}_{2}V_{2}\quad\text{on}~{}I_{2}.$ (68)

Using the inequalities $\sqrt{V_{1}}\leq V_{1}+1$ and $\sqrt{V_{2}}\leq
V_{2}+1$, we find

$\displaystyle\dot{V}_{1}\leq(\theta_{1}+C_{1}-\hat{\theta}_{1})V_{1}+C_{1}\quad\text{a.e.
on}~{}I_{1},$ (69)
$\displaystyle\dot{V}_{2}\leq(\theta_{2}+C_{2}-\hat{\theta}_{2})V_{2}+C_{2}\quad\text{a.e.
on}~{}I_{2}.$ (70)

Step 2: We show that $(\hat{\theta}_{1},\hat{\theta}_{2})$ is constant after
some $T\geq 0$ using contradiction. Assume that there is not such a finite
time $T\geq 0$ such that

$\dot{\hat{\theta}}_{1}(t)=\dot{\hat{\theta}}_{2}(t)=0\qquad\forall t\geq T.$

As a result, according to R1)-R3) in Algorithm 3,

$\lim_{t\to\infty}\hat{\theta}_{1}(t)=\lim_{t\to\infty}\hat{\theta}_{2}(t)=\infty.$

Therefore, there exists $T\geq 0$ such that, for all $t\geq T$, we have

$\displaystyle\hat{\theta}_{1}(t)$
$\displaystyle\geq\theta_{1}+C_{1}+\frac{(\theta_{1}+1)\overline{T}_{2}+\sigma(\overline{T}_{1}+\overline{T}_{2})}{\underline{T}_{1}}+1,$
(71) $\displaystyle\hat{\theta}_{2}(t)$
$\displaystyle\geq\theta_{2}+C_{2}+\frac{(\theta_{2}+1)\overline{T}_{1}+\sigma(\overline{T}_{1}+\overline{T}_{2})}{\underline{T}_{2}}+1.$
(72)

Let $k>1$ be such that $t_{2k-3}\geq T$. Using (71), we obtain

$V_{1}(t_{2k-2})\leq V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-2}-t_{2k-3})}+C_{1}.$
(73)

Moreover, using Lemma 6.4, we find

$\displaystyle V_{1}(t_{2k-1})\leq$
$\displaystyle~{}V_{1}(t_{2k-2})\exp^{(\theta_{1}+1)(t_{2k-1}-t_{2k-2})}$
$\displaystyle~{}+(C_{1}^{2}/4)\exp^{(\hat{\theta}_{1}(t_{2k-3})+1)(t_{2k-1}-t_{2k-2})}.$

By combining the latter inequality to (73), we obtain

$\displaystyle V_{1}(t_{2k-1})$ $\displaystyle\leq
V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})}+$
$\displaystyle\left(C_{1}+C_{1}^{2}/4\right)\exp^{(\hat{\theta}_{1}(t_{2k-3})+1)\left(t_{2k-1}-t_{2k-3}\right)},$

which implies, according to R1) in Algorithm 3, that
$\dot{\hat{\theta}}_{1}(t)=0$ for all $t\geq t_{2k-1}$. We show in a similar
way that $\dot{\hat{\theta}}_{2}(t)=0$ for all $t\geq t_{2k}$, which leads to
a contradiction.

To show (42), we suppose the existence of $T\geq 0$ such that

$\hat{\theta}_{1}(T)=\theta_{1}+C_{1}+\frac{(\theta_{1}+1)\overline{T}_{2}+\sigma(\overline{T}_{1}+\overline{T}_{2})}{\underline{T}_{1}}+1.$

Then we follow the exact same steps as in the proof of Theorem 3.1 to conclude
that we can take

$\displaystyle M_{1}$
$\displaystyle:=\theta_{1}+C_{1}+\frac{(\theta_{1}+1)\overline{T}_{2}+\sigma(\overline{T}_{1}+\overline{T}_{2})}{\underline{T}_{1}}+1+2\Delta_{1}\overline{T}_{1},$
(74) $\displaystyle M_{2}$
$\displaystyle:=\theta_{2}+C_{2}+\frac{(\theta_{2}+1)\overline{T}_{1}+\sigma(\overline{T}_{1}+\overline{T}_{2})}{\underline{T}_{2}}+1+2\Delta_{2}\overline{T}_{2}.$
(75)

Step 3: To study the function $V_{1}+V_{2}$, we introduce the sequences
$\\{T_{i}\\}_{i=0}^{\infty}$ and $\\{T_{i}^{\prime}\\}_{i=1}^{\infty}$, such
that $T_{i}:=t_{2i+1}$ and $T_{i}^{\prime}:=t_{2i}$.

Since $(\hat{\theta}_{1},\hat{\theta}_{2})$ are nondecreasing and become
constant after some finite time, we conclude the existence of at most a finite
number of intervals $[t_{2k-1},t_{2k})\subset I_{1}$, on which,
$\hat{\theta}_{1}$ may increase. On the latter intervals, we know that $V_{1}$
is governed by the inequality in Lemma 6.4, while on the remaining intervals
$V_{1}$ does not verify the inequality in R1). The same reasoning applies to
$\hat{\theta}_{2}$ and $V_{2}$. More precisely, for each initial conditions
$(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exist two integers
$N^{*}_{1},N_{2}^{*}\in\mathbb{N}$ such that, for each locally absolutely
continuous solution $(V_{1},V_{2})$ to (27), there exist two finite increasing
subsequences $\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}\subset\mathbb{N}$ and
$\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}\subset\mathbb{N}^{*}$ such that

  * •

For each $i\in\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have

$\displaystyle V_{1}(T_{i+1})$
$\displaystyle\leq\left(V_{1}(T_{i})+\frac{C_{1}^{2}}{4}\right)\exp^{(M_{1}+1)(T_{i+1}-T_{i})}.$

  * •

For each $j\in\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have

$\displaystyle V_{2}(T_{j+1}^{\prime})$
$\displaystyle\leq\left(V_{2}(T_{j}^{\prime})+\frac{C_{2}^{2}}{4}\right)\exp^{(M_{2}+1)(T_{j+1}^{\prime}-T_{j}^{\prime})}.$

  * •

For each $i\in\mathbb{N}/\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have

$\displaystyle V_{1}(T_{i+1})$ $\displaystyle\leq
V_{1}(T_{i})\exp^{-\sigma(T_{i+1}-T_{i})}$
$\displaystyle+\left(C_{1}+\frac{C_{1}^{2}}{4}\right)\exp^{(M_{1}+1)(T_{i+1}-T_{i})}.$

  * •

For each $j\in\mathbb{N}^{*}/\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have

$\displaystyle V_{2}(T_{j+1}^{\prime})$ $\displaystyle\leq
V_{2}(T_{j}^{\prime})\exp^{-\sigma(T_{j+1}^{\prime}-T_{i}^{\prime})}$
$\displaystyle+\left(C_{2}+\frac{C_{2}^{2}}{4}\right)\exp^{(M_{2}+1)(T_{j+1}^{\prime}-T_{j}^{\prime})}.$

Using Lemma 6.6 in the appendix, while replacing $(V,M,\psi,N^{*})$ therein by
$(V_{1},C_{1},M_{1}+1,N^{*}_{1})$, we obtain the inequality

$\displaystyle V_{1}(T_{i})$ $\displaystyle\leq\gamma_{1}V_{1}(0)\exp^{-\sigma
T_{i}}+\Phi_{1}(C_{1})\quad\forall i\in\mathbb{N},$ (76)

for some $\gamma_{1}>0$ and $\Phi_{1}\in\mathcal{K}$. Similarly, using Lemma
6.6 in the appendix, while replacing
$(V,M,\psi,N^{*},\\{T_{i}\\}^{\infty}_{i=1})$ therein by
$(V_{2},C_{2},M_{2}+1,N^{*}_{2},\\{T^{\prime}_{i}\\}^{\infty}_{i=1})$, we
obtain

$\displaystyle V_{2}(T_{i}^{\prime})$
$\displaystyle\leq\gamma_{2}V_{2}(0)\exp^{-\sigma
T_{i}^{\prime}}+\Phi_{2}(C_{2})\quad\forall i\in\mathbb{N}^{*},$ (77)

for some $\gamma_{2}>0$ and $\Phi_{2}\in\mathcal{K}$. As a consequence, for
each $t\in[T_{i},T_{i+1}]$, we have

$\displaystyle V_{1}(t)$
$\displaystyle\leq\gamma_{1}\exp^{(\theta_{1}+1+\sigma)(\overline{T}_{1}+\overline{T}_{2})}V_{1}(0)\exp^{-\sigma
t}$
$\displaystyle+\left[\Phi_{1}(C_{1})+\frac{C_{1}^{2}}{4(\theta_{1}+1)}\right]\exp^{(\theta_{1}+1)(\overline{T}_{1}+\overline{T}_{2})}.$
(78)

Similarly, for each $t\in[T_{i}^{\prime},T_{i+1}^{\prime}]$, we have

$\displaystyle V_{2}(t)$
$\displaystyle\leq\gamma_{2}\exp^{(\theta_{2}+2+\sigma)(\overline{T}_{1}+\overline{T}_{2})}V_{2}(0)\exp^{-\sigma
t}$
$\displaystyle+\left[\Phi_{2}(C_{2})+\frac{C_{2}^{2}}{4(\theta_{2}+1)}\right]\exp^{(\theta_{2}+1)(\overline{T}_{1}+\overline{T}_{2})}.$
(79)

Defining

$\displaystyle\gamma$
$\displaystyle:=\max\left\\{\gamma_{1}\exp^{(\theta_{1}+1+\sigma)(\overline{T}_{1}+\overline{T}_{2})},\right.$
$\displaystyle\qquad\left.\gamma_{2}\exp^{(\theta_{2}+2+\sigma)(\overline{T}_{1}+\overline{T}_{2})}\right\\}$
$\displaystyle\Phi(|f|_{\infty})$
$\displaystyle:=\left[\Phi_{1}(C_{1})+\frac{C_{1}^{2}}{4(\theta_{1}+1)}\right]\exp^{(\theta_{1}+1)(\overline{T}_{1}+\overline{T}_{2})}$
$\displaystyle~{}+\left[\Phi_{2}(C_{2})+\frac{C_{2}^{2}}{4(\theta_{2}+1)}\right]\exp^{(\theta_{2}+1)(\overline{T}_{1}+\overline{T}_{2})},$

and combining (78) and (79), we obtain, for all $t\geq 0$,

$\displaystyle V_{1}(t)+V_{2}(t)\leq\gamma(V_{1}(0)+V_{2}(0))\exp^{-\sigma
t}+\Phi(|f|_{\infty}).$ (80)

###  4.4 Proof of Theorem 3.9

According to the first step in the proof of Theorem 3.7, along the closed-loop
solutions of (3) with $(u_{1},u_{2})$ given by (6) with (7), $(V_{1},V_{2})$
verify (27). The rest of the proof follows in three steps. In Step 1, we prove
that $(\hat{\theta}_{1},\hat{\theta}_{2})$ become constant after some $T>0$.
Furthermore, $(\hat{\theta}_{1},\hat{\theta}_{2})$ never exceed
$\max\\{\hat{\theta}_{1}(0),M_{1}\\}$ and
$\max\\{\hat{\theta}_{2}(0),M_{2}\\}$, respectively, where $M_{1},M_{2}>0$
depend on
$(\sigma,\theta_{1},\theta_{2},C_{1},C_{2},\overline{T}_{1},\overline{T}_{2},\underline{T}_{1},\underline{T}_{2},\Delta_{1},\Delta_{2})$.
In Step 2, we analyze the function $V_{1}+V_{2}$ and show the $L^{2}$-GUUB
property. Finally, as a last step, we use the structure of $\kappa$ and the
fact that $(\hat{\theta}_{1},\hat{\theta}_{2},V_{1},V_{2})$ are bounded to
conclude that $(u_{1},u_{2})$ remain bounded.

Step 1: We first show that $(\hat{\theta}_{1},\hat{\theta}_{2})$ become
constant after some $T\geq 0$ using contradiction. Namely, we assume that
there is not such a $T\geq 0$ such that, for all $t\geq T$,
$\dot{\hat{\theta}}_{1}(t)=\dot{\hat{\theta}}_{2}(t)=0$. As a result,
according to R1)-R3) in Algorithm 4, we conclude that
$\lim_{t\to\infty}\hat{\theta}_{1}(t)=\lim_{t\to\infty}\hat{\theta}_{2}(t)=\infty$.
Therefore, there exists $T\geq 0$ such that, for all $t\geq T$, (71) and (72)
hold. Let $k>1$ be such that $t_{2k-3}\geq T$. We have

$V_{1}(t_{2k-2})\leq
V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-2}-t_{2k-3})}+\hat{\theta}_{1}(t_{2k-3}).$
(81)

Moreover, using Lemma 6.4, we obtain

$\displaystyle V_{1}(t_{2k-1})$ $\displaystyle\leq
V_{1}(t_{2k-2})\exp^{(\theta_{1}+1)(t_{2k-1}-t_{2k-2})}$
$\displaystyle~{}+\frac{\hat{\theta}_{1}(t_{2k-3})^{2}}{4}\exp^{(\hat{\theta}_{1}(t_{2k-3})+1)(t_{2k-1}-t_{2k-2})}.$

By combining the latter inequality to (81), we obtain

$\displaystyle V_{1}(t_{2k-1})\leq
V_{1}(t_{2k-3})\exp^{-\sigma(t_{2k-1}-t_{2k-3})}+$
$\displaystyle\left(\hat{\theta}_{1}(t_{2k-3})+\frac{\hat{\theta}_{1}(t_{2k-3})^{2}}{4}\right)\exp^{(\hat{\theta}_{1}(t_{2k-3})+1)\left(t_{2k-1}-t_{2k-3}\right)},$

which implies, according to R1) in Algorithm 4, that
$\dot{\hat{\theta}}_{1}(t)=0$ for all $t\geq t_{2k-1}$. We show in a similar
way that $\dot{\hat{\theta}}_{2}(t)=0$ for all $t\geq t_{2k}$, which leads to
a contradiction.

To find the constants $M_{1}$ and $M_{2}$ verifying (42), we follow the same
steps as in the proof of Theorem 3.1. That is, we can show that (42) holds
with $M_{1},M_{2}$ as in (74) and (75), respectively.

Step 2: To study the function $V_{1}+V_{2}$, we first define the sequences
$\\{T_{i}\\}_{i=0}^{\infty}$ and $\\{T_{i}^{\prime}\\}_{i=1}^{\infty}$, such
that $T_{i}:=t_{2i+1}$ and $T_{i}^{\prime}:=t_{2i}$. As in the proof of
Theorem 3.7, for each initial condition
$(\hat{\theta}_{1}(0),\hat{\theta}_{2}(0))$, there exist two integers
$N^{*}_{1},N_{2}^{*}\in\mathbb{N}$ such that, for each locally absolutely
continuous solution $(V_{1},V_{2})$ to (27), there exist two finite increasing
subsequences $\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}\subset\mathbb{N}$ and
$\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}\subset\mathbb{N}^{*}$ such that

  * •

For each $i\in\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have
$V_{1}(T_{i+1})\leq\left(V_{1}(T_{i})+\frac{M_{1}^{2}}{4}\right)\exp^{(M_{1}+1)(T_{i+1}-T_{i})}.$

  * •

For each $j\in\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have
$V_{2}(T_{j+1}^{\prime})\leq\left(V_{2}(T_{j}^{\prime})+\frac{M_{2}^{2}}{4}\right)\exp^{(M_{2}+1)(T_{j+1}^{\prime}-T_{j}^{\prime})}$.

  * •

For each $i\in\mathbb{N}/\\{i_{1},i_{2},...,i_{N_{1}^{*}}\\}$, we have
$V_{1}(T_{i+1})\leq V_{1}(T_{i})\exp^{-\sigma(T_{i+1}-T_{i})}\\\
+\left(M_{1}+\frac{M_{1}^{2}}{4}\right)\exp^{(M_{1}+1)(T_{i+1}-T_{i})}.$

  * •

For each $j\in\mathbb{N}^{*}/\\{j_{1},j_{2},...,j_{N_{2}^{*}}\\}$, we have
$V_{2}(T_{j+1}^{\prime})\leq
V_{2}(T_{j}^{\prime})\exp^{-\sigma(T_{j+1}^{\prime}-T_{j}^{\prime})}\\\
+\left(M_{2}+\frac{M_{2}^{2}}{4}\right)\exp^{(M_{2}+1)(T_{j+1}^{\prime}-T_{j}^{\prime})}.$

Using Lemma 6.6 in the appendix, while replacing $(V,M,\psi,N^{*})$ therein by
$(V_{1},M_{1},M_{1}+1,N^{*}_{1})$, we obtain the inequality

$\displaystyle V_{1}(T_{i})$ $\displaystyle\leq\gamma_{1}V_{1}(0)\exp^{-\sigma
T_{i}}+\Phi_{1}(M_{1})\quad\forall i\in\mathbb{N},$ (82)

for some $\gamma_{1}>0$ and $\Phi_{1}\in\mathcal{K}$. Similarly, using Lemma
6.6 in the appendix, while replacing
$(V,M,\psi,N^{*},\\{T_{i}\\}^{\infty}_{i=1})$ therein by
$(V_{2},M_{2},M_{2}+1,N^{*}_{2},\\{T^{\prime}_{i}\\}^{\infty}_{i=1})$, we
obtain

$\displaystyle V_{2}(T_{i}^{\prime})$
$\displaystyle\leq\gamma_{2}V_{2}(0)\exp^{-\sigma
T_{i}^{\prime}}+\Phi_{2}(M_{2})\quad\forall i\in\mathbb{N}^{*},$ (83)

for some $\gamma_{2}>0$ and $\Phi_{2}\in\mathcal{K}$.

As a consequence, for each $t\in[T_{i},T_{i+1}]$,

$\displaystyle V_{1}(t)$
$\displaystyle\leq\gamma_{1}\exp^{(\theta_{1}+1+\sigma)(\overline{T}_{1}+\overline{T}_{2})}V_{1}(0)\exp^{-\sigma
t}$
$\displaystyle+\left[\Phi_{1}(M_{1})+\frac{M_{1}^{2}}{4(\theta_{1}+1)}\right]\exp^{(\theta_{1}+1)(\overline{T}_{1}+\overline{T}_{2})}.$
(84)

Moreover, for each $t\in[T_{i}^{\prime},T_{i+1}^{\prime}]$,

$\displaystyle V_{2}(t)$
$\displaystyle\leq\gamma_{2}\exp^{(\theta_{2}+2+\sigma)(\overline{T}_{1}+\overline{T}_{2})}V_{2}(0)\exp^{-\sigma
t}$
$\displaystyle+\left[\Phi_{2}(M_{2})+\frac{M_{2}^{2}}{4(\theta_{2}+1)}\right]\exp^{(\theta_{2}+1)(\overline{T}_{1}+\overline{T}_{2})}.$
(85)

Defining

$\displaystyle\gamma:=\max\left\\{\gamma_{1}\exp^{(\theta_{1}+1+\sigma)(\overline{T}_{1}+\overline{T}_{2})},\gamma_{2}\exp^{(\theta_{2}+2+\sigma)(\overline{T}_{1}+\overline{T}_{2})}\right\\},$

and

$\displaystyle\Phi$
$\displaystyle:=\left[\Phi_{1}(M_{1})+\frac{M_{1}^{2}}{4(\theta_{1}+1)}\right]\exp^{(\theta_{1}+1)(\overline{T}_{1}+\overline{T}_{2})}$
$\displaystyle+\left[\Phi_{2}(M_{2})+\frac{M_{2}^{2}}{4(\theta_{2}+1)}\right]\exp^{(\theta_{2}+1)(\overline{T}_{1}+\overline{T}_{2})},$

and summing (84) and (85), which are valid for all $t\geq 0$, we obtain

$\displaystyle V_{1}(t)+V_{2}(t)\leq\gamma(V_{1}(0)+V_{2}(0))\exp^{-\sigma
t}+\Phi.$ (86)

Let $r:=\Phi+\epsilon$, where $\epsilon$ is any positive constant, and suppose
that $V_{1}(0)+V_{2}(0)\leq R$. We conclude that $V_{1}(t)+V_{2}(t)\leq\gamma
R\exp^{-\sigma t}+\Phi$ for all $t\geq 0$. Hence, to guarantee that $\gamma
R\exp^{-\sigma t}+\Phi\leq r$, it is sufficient to have $t\geq
T(R):=\frac{1}{\sigma}\log\left(\frac{\gamma R}{\epsilon}\right)$.

##  5 Numerical Results

In this section, we propose to illustrate our results in Theorems 3.1 and 3.9
via simulations.

###  5.1 Numerical Scheme

The used numerical scheme is based on the mesh-free collocation method using
radial basis functions (RBF)s [30]. We estimate $w_{xxx}(0)$ and $w_{x}(0)$
using the Euler forward scheme and $v_{xxx}(1)$ and $v_{x}(1)$ using the Euler
backward scheme. The Lyapunov function candidates $(V_{1},V_{2})$ are
approximated using Riemannian sums. Furthermore, we select multiquadric RBFs,
which depend on a shape parameter $c>0$. We set $c:=0.4$ in all the
simulations.

###  5.2 The Different Parameters

We select the initial time to be $t_{o}:=0$, the final time to be
$t_{f}:=8\times 10^{-3}$, and the time step $\Delta t:=10^{-7}$. We select
$10$ uniformly separated collocation points on the interval $[0,Y]$ (with
$Y:=0.5$) ranging from $x_{o}:=0$ to $x_{9}:=Y$. We select the same number of
collocation points on the interval $[Y,1]$. We select the anti-diffusion
parameter $\lambda:=4\pi^{2}/0.25+50$, for which, the linearization of (1)
under $u(0)=u(1)=u_{x}(0)=u_{x}(1)=0$ is unstable [19]. The initial condition
is $u(x,0):=u_{o}:=-A(\cos(4\pi x)-1)$ for all $x\in(0,1)$, where $A>0$ is the
amplitude to be selected. The sequences $I_{1}$ and $I_{2}$ are given by

$\displaystyle I_{1}$
$\displaystyle=[0,1)\cup[2,2.8)\cup[3.9,5)\cup[5.5,6.5)\cup[7,7.6)\times
10^{-3},$ $\displaystyle I_{2}$
$\displaystyle=[1,2)\cup[2.8,3.9)\cup[5,5.5)\cup[6.5,7)\cup[7.6,8)\times
10^{-3}.$

###  5.3 Closed-Loop Response When $f=0$

Starting from the initial condition $u_{o}$ with $A:=3$, we simulate the
solution to (1) under (4), $(u_{1},u_{2})$ as in (6), and the adaptation
parameters starting from $\hat{\theta}_{1}(0)=\hat{\theta}_{2}(0)=0$ and
updated according to Algorithm 1 with $\Delta_{1}=\Delta_{2}=0.01$ and
$\sigma=100$. The obtained solution is depicted in Figure 1. Along such a
solution, the Lyapunov functions $(V_{1},V_{2})$ are depicted in Figure 2. The
obtained simulations confirm the stability statement in Theorem 3.1. In
particular, on the time intervals where $u_{1}=0$, we observe an exponential
growth in $V_{1}$. This growth is compensated on the intervals where
$u_{1}\neq 0$. Similarly, on the time intervals where $u_{2}=0$, we notice an
exponential growth in $V_{2}$, which is compensated on the intervals where
$u_{2}\neq 0$. In Figure 3, we plot the evolution of the adaptation parameters
$(\hat{\theta}_{1},\hat{\theta}_{2})$. We observe the existence of an
adaptation phase. Indeed, according to Figure 2, $V_{1}$ does not decrease on
the time interval $[0,2)$. Its decrease starts on the time interval $[2,2.8)$
and corresponds, according to Figure 3, to an increase in the value of
$\hat{\theta}_{1}$. Note also that, as expected, $\hat{\theta}_{1}$ and
$\hat{\theta}_{2}$ increase until they become constant, which corresponds to a
decay of $V_{1}$ and $V_{2}$ at the prescribed rate of $\sigma=100$. It is
interesting to observe that $\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ do not
exceed $\theta_{1}$ and $\theta_{2}$. Indeed, according to Figure 3, we have
$|\hat{\theta}_{1}|_{\infty}=80$ and $|\hat{\theta}_{2}|_{\infty}=50$ while,
according to Lemma 6.2, the nominal parameters are
$\theta_{1}=\theta_{2}=106880$. This illustrates some conservatism of our
Lyapunov-based approach.

Figure 1: The KS response to (4), (6), and Algorithm 1 Figure 2: Lyapunov
functions along the KS response under (4), (6), and Algorithm 1 Figure 3: The
adaptation parameters $(\hat{\theta}_{1},\hat{\theta}_{2})$ constructed
according to Algorithm 1

###  5.4 Closed-Loop Response When $f\neq 0$

Next, we set $f(t):=12\times 10^{3}\sin(20\times 10^{3}t)$. We simulate the
solution to (2) under (4), $(u_{1},u_{2})$ as in (6), and the adaptation
parameters starting from $\hat{\theta}_{1}(0)=\hat{\theta}_{2}(0)=0$ and
updated according to Algorithm 4 with $\Delta_{1}=\Delta_{2}=0.01$ and
$\sigma=100$. In Figure 4, we plot $V_{1}+V_{2}$ for $A\in\\{3,5,7\\}$. The
conclusions of Theorem 3.9 are in agreement with Figure 4. Indeed, we observe
the finite-time convergence of $V_{1}+V_{2}$ to a bound that is uniform
despite the different initial conditions.

Figure 4: $V_{1}+V_{2}$ along the NKS response to (4), (6), and Algorithm 4,
for different initial conditions

The evolution of the adaptation parameters
$(\hat{\theta}_{1},\hat{\theta}_{2})$ is the same as in Figure 3 (i.e. we
obtain the same plot). In particular, $|\hat{\theta}_{1}|_{\infty}=80$, and
$|\hat{\theta}_{2}|_{\infty}=50$, which means that $\hat{\theta}_{1}$ and
$\hat{\theta}_{2}$ do not exceed the nominal parameters
$\theta_{1}=\theta_{2}=106880$.

##  6 Conclusion

We proposed an adaptive boundary-control approach to stabilize the origin of
the NKS equation under different assumptions on the perturbation $f$ and the
measurements available. In particular, under the considered sensing scenario,
we guaranteed $L^{2}$-GES when $f=0$, $L^{2}$-ISS with respect to $f$ when
$|f|_{\infty}\neq 0$ is known, and $L^{2}$-GUUB when $|f|_{\infty}\neq 0$ is
unknown. Note that we do not constrain $\lambda$ to be constant or in a
specific region and we only require $|\lambda|_{\infty}$ and
$|\lambda^{\prime}|_{\infty}$ to be bounded. In the future, it is of interest
to study well-posedness of the obtained closed-loop system in each case. This
may require different tools than those in existing PDE control literature.
Furthermore, it would be desirable to propose an alternative to our
discontinuous feedback law that is continuous and guaranteed to remain bounded
along the closed-loop solutions. Finally, it would be of interest to consider
a scanning sensing scenario, in which, we suppose that $Y$ evolves with time
and that we measure the state, and/or control the PDE, only around $Y$.

## References

  * [1] L. Baudouin, E. Cerpa, E. Crépeau, and A. Mercado.  Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation.  Applicable Analysis, 92(10):2084–2102, 2013.
  * [2] M. C Belhadjoudja, M. Maghenem, E. Witrant, and C. Prieur.  Adaptive stabilization of the Kuramoto-Sivashinsky equation subject to intermittent sensing.  In Proceedings of the 2023 American Control Conference, 1608-1613, San Diego, CA, USA.
  * [3] D-J. Benny.  Long waves in liquid film.  J. Math. and Phys., 45:150–155, 1966.
  * [4] H. Cao, D. Lu, L. Tian, and Y. Cheng.  Boundary control of the Kuramoto-Sivashinsky equation with an external excitation.  International Journal of Nonlinear Science, 1(2):67–81, 2006.
  * [5] J-M. Coron and Q. Lü.  Fredholm transform and local rapid stabilization for a Kuramoto–Sivashinsky equation.  Journal of Differential Equations, 259(8):3683–3729, 2015.
  * [6] R. Cuerno, H. A. Makse, S. Tomassone, S. T. Harrington, and H. E. Stanley.  Stochastic model for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology.  Phys. Rev. Lett., 75:4464–4467, 1995.
  * [7] D. Gajardo, A. Mercado, and J. C. Muñoz.  Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation.  Journal of Mathematical Analysis and Applications, 495(2):124747, 2021.
  * [8] D. Gilbarg, N. S. Trudinger, D. Gilbarg, and N. S. Trudinger.  Elliptic partial differential equations of second order, volume 224.  Springer, 1977.
  * [9] I. Karafyllis, M. Kontorinaki, and M. Krstić.  Adaptive control by regulation-triggered batch least squares.  IEEE Trans. on Automatic Control, 65(7):2842–2855, 2019.
  * [10] T. Kobayashi.  Adaptive stabilization of the Kuramoto-Sivashinsky equation.  Internat. J. Systems Sci., 33:3:175–180, 2002.
  * [11] L. Kocarev, Z. Tasev, and U. Parlitz.  Synchronizing spatiotemporal chaos of partial differential equations.  Physical Review Letters, 79(1):51, 1997.
  * [12] M. Krstić.  On global stabilization of Burgers’ equation by boundary control.  Systems & Control Letters, 37:123–141, 1999.
  * [13] Y. Kuramoto.  Diffusion-induced chaos in reaction systems.  Progress of Theoretical Physics Supplement, 64:346–367, 1978.
  * [14] R. LaQuey, S. Mahajan, P. Rutherford, and W. Tang.  Nonlinear saturation of the trapped-ion mode.  Physical Review Letters, 33:391–394, 1975.
  * [15] K. B. Lauritsen, R. Cuerno, and H. A. Makse.  Noisy Kuramoto-Sivashinsky equation for an erosion model.  Physical Review E, 54:3577–3580, 1996.
  * [16] H. Lhachemi.  Local output feedback stabilization of a nonlinear Kuramoto-Sivashinsky equation.  IEEE Trans. on Automatic Control, pages 1–6, 2023.
  * [17] T. Liard, I. Balogoun, S. Marx, and F. Plestan.  Boundary sliding mode control of a system of linear hyperbolic equations: a Lyapunov approach.  Automatica, 135:109964, 2022.
  * [18] S-P. Lin.  Finite amplitude side-band stability of a viscous fluid.  J. Fluid Mech., 63:417–429, 1974.
  * [19] W-J. Liu and M. Krstić.  Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation.  Nonlinear Analysis: Theory, Methods & Applications, 43(4):485–507, 2001.
  * [20] M. Maghenem, C. Prieur, and E. Witrant.  Boundary control of the Kuramoto-Sivashinsky equation under intermittent data availability.  In Proceedings of the 2022 American Control Conference, 2227-2232, Atlanta, GA, USA.
  * [21] D. E. Miller.  Near optimal LQR performance for a compact set of plants.  IEEE Trans. on Automatic Control, 51(9):1423–1439, 2006.
  * [22] D. E. Miller and J. R. Vale.  Pole placement adaptive control with persistent jumps in the plant parameters.  Mathematics of Control, Signals, and Systems, 26(2):177–214, 2014\.
  * [23] D. E. Miller and E. J. Davison.  An adaptive controller which provides Lyapunov stability.  IEEE Trans. on Automatic Control, 34(6):599–609, 1989.
  * [24] D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink.  Inequalities involving functions and their integrals and derivatives.  Kluwer Academic Publishers, Dordrecht/Boston/London, 1991.
  * [25] P. Parmananda.  Generalized synchronization of spatiotemporal chemical chaos.  Phys. Rev. E, 56:1595–1598, 1997.
  * [26] G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations.  Acta Astronautica, 4(11-12):1177–1206, 1977.
  * [27] G. Sivashinsky.  On flame propagation under conditions of stoichiometry.  SIAM Journal on Applied Mathematics, 39(1):67–82, 1980.
  * [28] Y. Song, K. Zhao, and M. Krstić.  Adaptive control with exponential regulation in the absence of persistent excitation.  IEEE Trans. on Automatic Control, 62(5):2589–2596, 2016.
  * [29] Z. Tasev, L. Kocarev, L. Junge, and U. Parlitz.  Synchronization of Kuramoto–Sivashinsky equations using spatially local coupling.  International Journal of Bifurcation and Chaos, 10(04):869–873, 2000.
  * [30] M. Uddin, S. Haq, and S. ul Islam.  A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations.  Applied Mathematics and Computations. 212, pages 458–469, 2009\.
  * [31] J-M. Wang, J-J. Liu, B. Ren, and J. Chen.  Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance.  Automatica, 52:23–34, 2015.

## Appendix

We recall a key inequality that links the $L^{2}$-norm of a function to the
$L^{2}$-norms of its first- and second-order derivatives.

######  Lemma 6.1 ([24], page 84).

Given $u\in C^{2}(a,b)$ and $\epsilon>0$, we have

$\displaystyle\int_{a}^{b}(u^{\prime})^{2}dx\leq$
$\displaystyle~{}\left[\frac{1}{\epsilon}+\frac{12}{(b-a)^{2}}\right]\int_{a}^{b}u^{2}dx+\epsilon\int_{a}^{b}(u^{\prime\prime})^{2}dx.$
(87)

By virtue of Lemma 6.1, we are able to establish the following intermediate
result.

######  Lemma 6.2.

Consider system (3) such that Assumption 2 holds. Then, along (3), the
Lyapunov functions $(V_{1},V_{2})$ in (5) satisfy

$\displaystyle\dot{V}_{1}\leq\theta_{1}V_{1}+C_{1}\sqrt{V_{1}}+\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0),$
(88)
$\displaystyle\dot{V}_{2}\leq\theta_{2}V_{2}+C_{2}\sqrt{V_{2}}-\frac{u_{2}^{3}}{3}-u_{2}v_{xxx}(1),$
(89)

where

$\displaystyle
C_{1}:=\sqrt{2Y}|f|_{\infty},~{}C_{2}:=\sqrt{2(1-Y)}|f|_{\infty},$ (90)

and $(\theta_{1},\theta_{2})$ are given by

$\displaystyle\theta_{1}:=|\lambda^{\prime}|_{\infty}+2\left(|\lambda|_{\infty}+\frac{1}{2}\right)\left(\left(|\lambda|_{\infty}+\frac{1}{2}\right)+\frac{12}{Y^{2}}\right),$
$\displaystyle\theta_{2}:=|\lambda^{\prime}|_{\infty}+2\left(|\lambda|_{\infty}+\frac{1}{2}\right)\left(\left(|\lambda|_{\infty}+\frac{1}{2}\right)+\frac{12}{(1-Y)^{2}}\right).$

######  Proof 6.3.

By differentiating $V_{1}$ along (3), we obtain

$\displaystyle\dot{V}_{1}$ $\displaystyle=\int_{0}^{Y}w(x)w_{t}(x)dx$ (91)
$\displaystyle=\int_{0}^{Y}w(x)\left[-w(x)w_{x}(x)-\lambda(x)w_{xx}(x)\right.$
$\displaystyle\left.-w_{xxxx}(x)+f(x)\right]dx.$

Note that $-3\int_{0}^{Y}w(x)^{2}w_{x}(x)dx=-w(Y)^{3}+w(0)^{3}$. Furthermore,
using integration by part, we obtain

$\displaystyle-\int^{Y}_{0}w(x)w_{xxxx}(x)dx$
$\displaystyle=-\left[w(x)w_{xxx}(x)\right]^{Y}_{0}+\int^{Y}_{0}w_{x}(x)w_{xxx}(x)dx$
$\displaystyle=-\left[w(x)w_{xxx}(x)\right]^{Y}_{0}+\left[w_{x}(x)w_{xx}(x)\right]^{Y}_{0}-\int^{Y}_{0}w_{xx}(x)^{2}dx.$

Using the boundary conditions $w_{x}(0)=w_{x}(Y)=0$, we obtain

$\displaystyle\int^{Y}_{0}w(x)w_{xxxx}(x)dx$
$\displaystyle={\left[w(x)w_{xxx}(x)\right]}^{Y}_{0}-\int^{Y}_{0}w_{xx}(x)^{2}dx$
$\displaystyle=-u_{1}w_{xxx}(0)+\int^{Y}_{0}w_{xx}(x)^{2}dx.$

Next, we note that

$\displaystyle-\int_{0}^{Y}\lambda(x)w(x)w_{xx}(x)dx=-[\lambda(x)w(x)w_{x}(x)]_{x=0}^{x=1}$
$\displaystyle+\int_{0}^{Y}\lambda(x)w_{x}(x)^{2}dx+\int_{0}^{Y}\lambda^{\prime}(x)w(x)w_{x}(x)dx.$

Using the boundary conditions $w_{x}(0)=w_{x}(Y)=0$ and Young inequality, we
find

$\displaystyle-\int_{0}^{Y}\lambda(x)w(x)w_{xx}(x)dx\leq$
$\displaystyle~{}\left(|\lambda|_{\infty}+\frac{1}{2}\right)\int_{0}^{Y}w_{x}(x)^{2}dx$
$\displaystyle~{}+|\lambda^{\prime}|_{\infty}^{2}V_{1}.$

Finally, using Cauchy-Schwarz inequality, we obtain

$\displaystyle\int_{0}^{Y}w(x)f(x)dx\leq|f|_{\infty}\int_{0}^{Y}|w(x)|dx\leq
C_{1}\sqrt{V_{1}},$ (92)

where $C_{1}$ is defined in (90). As a consequence, we obtain

$\displaystyle\dot{V}_{1}$
$\displaystyle\leq|\lambda^{\prime}|_{\infty}^{2}V_{1}+\left(|\lambda|_{\infty}+\frac{1}{2}\right)\int_{0}^{Y}w_{x}(x)^{2}dx$
$\displaystyle-\int_{0}^{Y}w_{xx}(x)^{2}dx+C_{1}\sqrt{V_{1}}+\frac{u_{1}^{3}}{3}+u_{1}w_{xxx}(0).$

Invoking Lemma 6.1 with $\epsilon:=1/(|\lambda|_{\infty}+\frac{1}{2})$, we
find

$\displaystyle\left(|\lambda|_{\infty}+\frac{1}{2}\right)\int_{0}^{Y}w_{x}^{2}dx-\int_{0}^{Y}w_{xx}^{2}dx\leq(\theta_{1}-|\lambda^{\prime}|_{\infty})V_{1},$

which proves inequality (88). We show inequality (89) in a similar way.

######  Lemma 6.4.

Let $V:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ be a locally
absolutely continuous solution to the differential inequality

$\dot{V}\leq\theta V+C\sqrt{V}\quad\text{a.e. on}\
[0,T]\subset\mathbb{R}_{\geq 0},$ (93)

where $\theta,C\geq 0$ are constants. Then, for any constant $\delta>0$ we
have, for all $t\in[0,T]$,

$V(t)\leq
V(0)\exp^{(\theta+\delta)t}+\frac{C^{2}/4\delta}{(\theta+\delta)}\left(\exp^{(\theta+\delta)t}-1\right).$
(94)

######  Proof 6.5.

Let $\delta>0$ and consider the function $f(V):=\sqrt{V}-\frac{\delta
V}{C}-\frac{C}{4\delta}$. By differentiating $f$, we find for all $V>0$,
$f^{\prime}(V)=\frac{1}{2\sqrt{V}}-\frac{\delta}{C}$. As a result, the
function $f$ is strictly increasing on $[0,C^{2}/(4\delta^{2})]$ and strictly
decreasing on $[C^{2}/(4\delta^{2}),\infty)$. Moreover, $f(0)=-C/4\delta$, and
$f(C^{2}/(4\delta^{2}))=0$. Therefore, for all $V\geq 0$, we have $f(V)\leq
0$. As a consequence, we can rewrite (93) as $\dot{V}\leq\theta
V+C\sqrt{V}\leq(\theta+\delta)V+\frac{C^{2}}{4\delta}$. By integrating this
inequality from $0$ to $t$, (94) follows.

######  Lemma 6.6.

Let the function $V:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be locally
absolutely continuous, let a sequence $\\{T_{i}\\}_{i=0}^{\infty}$ and
$\underline{T},\overline{T}>0$ such that
$T_{0}=0~{}~{}\text{and}~{}~{}\overline{T}\geq
T_{i+1}-T_{i}\geq\underline{T}\quad\forall i\in\mathbb{N}$. Let
$\\{i_{1},i_{2},...,i_{N^{*}}\\}\subset\mathbb{N}$, with $N^{*}\in\mathbb{N}$,
and let $(M,\psi,\sigma)$ be nonnegative constants. Assume that

  * •

For each $i\in\\{i_{1},i_{2},...,i_{N^{*}}\\}$,

$\displaystyle
V(T_{i+1})\leq\left(V(T_{i})+\frac{M^{2}}{4}\right)\exp^{\psi(T_{i+1}-T_{i})}.$
(95)

  * •

For each $i\in\mathbb{N}/\\{i_{1},i_{2},...,i_{N^{*}}\\}$,

$\displaystyle V(T_{i+1})$ $\displaystyle\leq
V(T_{i})\exp^{-\sigma(T_{i+1}-T_{i})}$ (96)
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi(T_{i+1}-T_{i})}.$

Then, there exists $\gamma\geq 1$ and $\Phi\in\mathcal{K}$ such that

$\displaystyle V(T_{i})$ $\displaystyle\leq\gamma V(0)\exp^{-\sigma
T_{i}}+\Phi(M)\qquad\forall i\in\mathbb{N}.$

######  Proof 6.7.

To prove the Lemma, it is enough to show that

$\displaystyle V(T_{i})$
$\displaystyle\leq\exp^{(\sigma+\psi)N^{*}\overline{T}}\left(V(0)\exp^{-\sigma
T_{i}}+\eta(i)\right)~{}~{}\forall i\in\mathbb{N},$ (97)

where, for each $i\in\mathbb{N}$,

$\displaystyle\eta(i)$
$\displaystyle:=\left(\sum_{k=0}^{i}\exp^{-k\sigma\underline{T}}\right)\frac{4M+M^{2}}{4}\exp^{\psi\overline{T}}$
$\displaystyle\leq\left(\frac{1}{1-\exp^{-\sigma\underline{T}}}\right)\frac{4M+M^{2}}{4}\exp^{\psi\overline{T}}.$

To prove (97), it is sufficient to show that

$\displaystyle V(T_{i})$
$\displaystyle\leq\exp^{(\sigma+\psi)N(i)\overline{T}}\left(V(0)\exp^{-\sigma
T_{i}}+\eta(i)\right)~{}~{}~{}\forall i\in\mathbb{N},$ (98)

where $N(i):=\text{card}\\{[T_{j},T_{j+1}]:j+1\leq
i,~{}j\in\\{i_{1},i_{2},...,i_{N^{*}}\\}\\}$ is the number of time intervals
$[T_{j},T_{j+1}]$, $j\in\\{i_{1},i_{2},...,i_{N^{*}}\\}$, prior to $T_{i}$,
which satisfies $N(i)\leq N^{*}$ for all $i\in\mathbb{N}$. To show (98), we
proceed by recurrence. Indeed, for $i=0$, the inequality in (98) is trivially
satisfied. Suppose now that the inequality in (98) is verified for
$i\in\mathbb{N}$ and let us show that it is also verified for $i+1$.

Note that either $N(i+1)=N(i)$ or $N(i+1)=N(i)+1$. If $N(i+1)=N(i)$ then,
using (96), we obtain

$\displaystyle V(T_{i+1})$ $\displaystyle\leq
V(T_{i})\exp^{-\sigma(T_{i+1}-T_{i})}$
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{(\hat{\theta}+1)(T_{i+1}-T_{i})}$
$\displaystyle\leq\exp^{(\sigma+\psi)N(i)\overline{T}}V(0)\exp^{-\sigma
T_{i+1}}$
$\displaystyle+\left(\eta(i)\exp^{-\sigma(T_{i+1}-T_{i})}\right)\exp^{(\sigma+\psi)N(i)\overline{T}}$
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi(T_{i+1}-T_{i})}.$

Using the fact that

$\displaystyle\eta(i)$
$\displaystyle\exp^{-\sigma(T_{i+1}-T_{i})}\leq\bigg{[}\sum_{k=1}^{i+1}\exp^{-k\sigma\underline{T}}\bigg{]}\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}},$

we obtain

$\displaystyle V$
$\displaystyle(T_{i+1})\leq\exp^{(\sigma+\psi)N(i)\overline{T}}V(0)\exp^{-\sigma
T_{i+1}}$
$\displaystyle+\bigg{[}\sum_{k=1}^{i+1}\exp^{-k\sigma\underline{T}}\bigg{]}\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}\exp^{(\sigma+\psi)N(i)\overline{T}}$
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi(T_{i+1}-T_{i})}.$

It implies that

$\displaystyle
V(T_{i+1})\leq\exp^{(\sigma+\psi)N(i)\overline{T}}V(0)\exp^{-\sigma T_{i+1}}$
$\displaystyle+\bigg{[}\sum_{k=1}^{i+1}\exp^{-k\sigma\underline{T}}\bigg{]}\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}\exp^{(\sigma+\psi)N(i)\overline{T}}$
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\bar{T}}\exp^{(\sigma+\psi)N(i)\overline{T}}.$

Combining the latter two terms, we obtain

$\displaystyle
V(T_{i+1})\leq\exp^{(\sigma+\psi)N(i)\overline{T}}V(0)\exp^{-\sigma T_{i+1}}$
$\displaystyle+\bigg{[}\sum_{k=0}^{i+1}\exp^{-k\sigma\underline{T}}\bigg{]}\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}\exp^{(\sigma+\psi)N(i)\overline{T}}.$

Finally, since $N(i)=N(i+1)\leq N^{*}$ and
$\eta(i+1)=\bigg{[}\sum_{k=0}^{i+1}\exp^{-k\sigma\underline{T}}\bigg{]}\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}},$
we obtain

$\displaystyle V(T_{i+1})$
$\displaystyle\leq\exp^{(\sigma+\psi)N(i+1)\overline{T}}V(0)\exp^{-\sigma
T_{i+1}}$ $\displaystyle+\eta(i+1)\exp^{(\sigma+\psi)N(i)\overline{T}}.$

If $N(i+1)=N(i)+1$, we use (95) to conclude that

$\displaystyle V(T_{i+1})$
$\displaystyle\leq\left(V(T_{i})+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}$
$\displaystyle\leq\exp^{(\sigma+\psi)N(i)\overline{T}}\exp^{\psi\overline{T}}$
$\displaystyle\times\left(V(0)\exp^{-\sigma
T_{i}}+\eta(i)+\exp^{-(\sigma+\psi)N(i)\overline{T}}\frac{M^{2}}{4}\right).$

Now, using the fact that $N(i+1)=N(i)+1$, we obtain

$\displaystyle
V(T_{i+1})\leq\exp^{(\sigma+\psi)N(i+1)\overline{T}}\exp^{\psi\overline{T}}\exp^{-(\sigma+\psi)\overline{T}}$
$\displaystyle\times\left(V(0)\exp^{-\sigma
T_{i}}+\eta(i)+\exp^{-(\sigma+\psi)N(i)\overline{T}}\frac{M^{2}}{4}\right)$
$\displaystyle\leq\exp^{(\sigma+\psi)N(i+1)\overline{T}}\exp^{\psi\overline{T}}$
$\displaystyle\times\left(V(0)\exp^{-\sigma
T_{i}}+\eta(i)\exp^{-(\sigma+\psi)\overline{T}}+\frac{M^{2}}{4}\right).$

We finish by showing
$\eta(i)\exp^{-(\sigma+\psi)\overline{T}}+M^{2}/4\leq\eta(i+1)$. Indeed, we
note that

$\displaystyle\eta(i)\exp^{-(\sigma+\psi)\overline{T}}+\frac{M^{2}}{4}$
$\displaystyle\leq\eta(i)\exp^{-\sigma\overline{T}}\exp^{-\psi\overline{T}}+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}$
$\displaystyle\leq\left(\sum_{k=1}^{i+1}\exp^{-k\sigma\underline{T}}\right)\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}$
$\displaystyle+\left(M+\frac{M^{2}}{4}\right)\exp^{\psi\overline{T}}$
$\displaystyle=\eta(i+1).$

Mohamed Camil Belhadjoudja received a Control-Engineer and MS degrees from the
National Polytechnic School, Algiers, Algeria, in 2021, and a MS degree in
Control Theory from the Université Grenoble Alpes, Grenoble, France, in 2022.
He is currently a Ph.D. student at the Grenoble Image Parole Signal
Automatique Laboratory (GIPSA-lab), Grenoble, France. His research interests
include nonlinear partial differential equations, model-free control, and
time-delay systems.

Mohamed Maghenem received his Control-Engineer degree from the Polytechnical
School of Algiers, Algeria, in 2013, his PhD degree on Automatic Control from
the University of Paris-Saclay, France, in 2017. He was a Postdoctoral Fellow
at the Electrical and Computer Engineering Department at the University of
California at Santa Cruz from 2018 through 2021. M. Maghenem holds a research
position at the French National Centre of Scientific Research (CNRS) since
January 2021. His research interests include control systems theory (linear,
non-linear, and hybrid) to ensure (stability, safety, reachability,
synchronisation, and robustness); with applications to mechanical systems,
power systems, cyber-physical systems, and some partial differential
equations.

Emmanuel Witrant obtained a B.Sc. in Aerospace Eng. from Georgia Tech in 2001
and a Ph.D. in Automatic Control from Grenoble University in 2005. He joined
the Physics department of Univ. Grenoble Alpes and GIPSA-lab as an Associate
Professor in 2007 and became Professor in 2020. His research interest is
focused on finding new methods for modeling and control of inhomogeneous
transport phenomena (of information, energy, gases…), with real-time and/or
optimization constraints. Such methods provide new results for automatic
control (time-delay and distributed systems), controlled thermonuclear fusion
(temperature/magnetic flux profiles in tokamak plasma), environmental sciences
(atmospheric history of trace gas from ice cores measurements) and
Poiseuille’s flows (ventilation control in mines, car engines and intelligent
buildings).

Christophe Prieur graduated in Mathematics from the Ecole Normale Supérieure
de Cachan, France in 2000. He received the Ph.D. degree in 2001 in Applied
Mathematics from Université Paris-Sud, France. From 2002 he was a researcher
at CNRS at the laboratory SATIE, Cachan, France, and at the LAAS, Toulouse,
France (2004–2010). In 2010 he joined the Gipsa-lab, Grenoble, France where he
is currently a senior researcher at CNRS (since 2011). His current research
interests include nonlinear control theory, hybrid systems, and control of
partial differential equations, with applications including navigation and
object tracking, fluid dynamics, and fusion control. He is an IMA Fellow, and
an IEEE Fellow.