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# Finite dimensional backstepping controller design

Varga K. Kalantarova, Türker Özsarıb,∗ and Kemal Cem Yılmazc aDepartment of
Mathematics, Koç University
Sarıyer, İstanbul, 34450 Turkey  bDepartment of Mathematics, Bilkent
University
Çankaya, Ankara, 06800 Turkey  cDepartment of Mathematics, İzmir Institute of
Technology
Urla, Izmir, 35430 Turkey

###### Abstract.

We introduce a finite dimensional version of backstepping controller design
for stabilizing solutions of PDEs from boundary. Our controller uses only a
finite number of Fourier modes of the state of solution, as opposed to the
classical backstepping controller which uses all (infinitely many) modes. We
apply our method to the reaction-diffusion equation, which serves only as a
canonical example but the method is applicable also to other PDEs whose
solutions can be decomposed into a slow finite-dimensional part and a fast
tail, where the former dominates the evolution in large time. One of the main
goals is to estimate the sufficient number of modes needed to stabilize the
plant at a prescribed rate. In addition, we find the minimal number of modes
that guarantee the stabilization at a certain (unprescribed) decay rate.
Theoretical findings are supported with numerical solutions.

###### Key words and phrases:

backstepping, boundary feedback, reaction-diffusion equation, stabilization.

###### 2020 Mathematics Subject Classification:

35B40, 35K57, 93C20, 93D15, 93D20, 93D23

This research was funded by TÜBİTAK 1001 Grant #122F084.

T. Özsarı’s research was partially supported by Science Academy’s Young
Scientist Award (BAGEP 2020)

*Corresponding author: Türker Özsarı<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. Problem statements

The goal of this paper is to introduce a finite dimensional analogue of the
classical backstepping control algorithm for stabilizing solutions of PDEs
from boundary. As a canonical example, we will use the reaction diffusion
equation posed on an interval with Dirichlet boundary conditions. However, the
method is applicable also to other evolutionary equations whose solutions
decompose into a slow finite-dimensional part and a fast tail, where the
former dominates the dynamics in long time. To this end, let us consider the
following plant:

$\displaystyle\begin{cases}u_{t}-\nu u_{xx}-\alpha u+u^{3}=0,\quad
x\in(0,L),t>0,\\\ u(0,t)=0,u(L,t)=g(t),\\\ u(x,0)=u_{0}(x).\end{cases}$ (1.1)

Here $\nu,\alpha>0$ are given constant coefficients and $g(t)$ is a
soughtafter feedback controller that involves only finitely many Fourier sine
modes of $u$. In the absence of control and for certain values of $\nu$,
$\alpha$ and $L$, zero equilibrium solution may be either asymptotically
stable or unstable. To be more precise, if $\nu\lambda_{1}-\alpha>0$, where
$\lambda_{1}$ is the first eigenvalue of the operator $-\frac{d^{2}}{dx^{2}}$
subject to Dirichlet boundary conditions, (1.1) has a unique equilibrium
solution $u\equiv 0$. For this case, it is asymptotically stable. Conversely,
if $\nu\lambda_{1}-\alpha<0$, then there will be at least two nontrivial
stationary solutions, exactly two of which are asymptotically stable, and all
solutions bifurcate from the zero equilibrium (see, e.g. [9, 18, 35] for a
more detailed discussion on this topic). So for the latter case, $u\equiv 0$
is no longer stable. Our aim is to construct a feedback law of the form

$\displaystyle u(L,t)=\int_{0}^{L}\xi(y)\Gamma[P_{N}u](y,t)dy,$ (1.2)

such that all solutions are steered asymptotically to zero. Here, $\Gamma$ is
a linear bounded operator on a certain $L^{2}-$based functional space, $\xi$
is a suitable smooth function to be constructed, and $P_{N}$ is the projection
operator

$P_{N}\varphi(x)=\sum_{j=1}^{N}e_{j}(x)\left(e_{j}(\cdot),\varphi(\cdot)\right)_{2},\quad
e_{j}(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{j\pi x}{L}\right).$

We will refer to boundary feedbacks of above form as a finite dimensional
backstepping controller. By abusing notation, we denote the feedback
controller defined by the integral at the right hand side of (1.2) (whose more
precise version is (1.7) below) as $g(t)$, $g(u(t))$, and $g(P_{N}u(t))$.

We are interested in two different stabilization problems.

######  Problem 1.1.

_Let $\nu,\alpha>0$ be such that $\nu\lambda_{1}-\alpha<0$ and $u_{0}\in
H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$. For given $\gamma>0$, can you find
$N>0$ and construct a feedback control law of the form $u(L,t)=g(P_{N}u(t))$
such that the solution of (1.1) satisfies
$\|u(t)\|_{H^{\ell}(0,L)}=\mathcal{O}(e^{-\gamma t})$ for $t\geq 0$?_

In contrast with the above _rapid stabilization_ problem, if one is only
interested in decay at a certain non-prescribed rate, then the problem takes
the following form:

######  Problem 1.2.

_Let $\nu,\alpha>0$ be such that $\nu\lambda_{1}-\alpha<0$ and $u_{0}\in
H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$. What is the minimum value of $N$
for which the solution of (1.1) satisfies
$\|u(t)\|_{H^{\ell}(0,L)}=\mathcal{O}(e^{-\gamma t})$, $t\geq 0$ for some
$\gamma>0$ with a boundary feedback of the form $u(L,t)=g(P_{N}u(t))$?_

######  Remark 1.3.

In Problem 1.1 and Problem 1.2, we are not only interested in establishing the
existence of a sufficiently large $N$ that fulfills the goal, in addition we
want to provide a precise estimate for $N$ in terms of problem parameters such
as the decay rate and the coefficients in the main equation.

Problem 1.1 and Problem 1.2 will be treated by combining ideas from the theory
of asymptotic dynamics of dissipative systems and boundary control of PDEs via
backstepping method, see e.g., [23]. We give an affirmative answer to Problem
1.1 in Theorems 1.4 and 1.5. Regarding Problem 1.2, we find that $N$ is
exactly equal to the instability level of (1.1). More precisely, if the number
of positive eigenvalues of the differential operator
$-\nu\frac{d^{2}}{dx^{2}}-\alpha I$ with domain $\\{\varphi\in
H^{2}(0,L)\,|\,\varphi(0)=\varphi(L)=0\\}$ is $N$, then it suffices to design
a boundary controller involving only the first $N$ Fourier sine modes of $u$.
Corresponding results are stated in Theorem 1.7 and Theorem 1.9.

###  1.2. Discussion of previous work and motivation

Control and stabilization of evolutionary partial differential equations is an
important topic and became an area of interest for many researchers in recent
decades. Several different approaches exist for designing control systems. In
some of these systems, control inputs act from the interior of the medium,
perhaps only from a local part. Other systems may use boundary controllers,
especially when access to medium is restricted. The control mechanism is of
feedback type if the controller depends on the state of the model. A
backstepping based boundary feedback controller is a classical example. See
[1, 24, 28, 36, 37, 38] for its application to second order PDEs and [7, 8,
11, 31, 32, 33] for its application to third order PDEs.

Stabilization of a nonlinear diffusion equation (with a quadratic
nonlinearity) via the standard backstepping transform was previously achieved
by [41]. Designing a finite dimensional controller for a similar problem as in
the current paper is more challenging. For instance, the usual methods for
proving bounded invertibility of the standard backstepping transformation
fails in the current finite dimensional case and one needs to give a new proof
of such result. Moreover, proofs of decay rate estimates through multipliers
are more delicate in the present case as projections are involved both in
equations and in the estimates. Designing a control system where the
controller involves only finitely many parameters of the solution rather than
the full state of the solution also offers an important practical use.

The idea of stabilization of solutions of nonlinear parabolic equations by
finite-dimensional controllers goes back to the pioneering works on 2D Navier-
Stokes equations (see [6], [16]). The results obtained in these works and the
pioneering results on infinite-dimensional disspative systems that generate 2D
Navier-Stokes equations, nonlinear parabolic equations, nonlinear damped wave
equations and related systems of PDE’s (see, e.g., [4], [25], [15]) inspired
further investigations on stabilization of these equations by internal and
boundary finite-dimensional controllers (see [3], [10], [19], [20], [27], [29]
and references therein). A number of papers are devoted to the boundary
stabilization of nonlinear parabolic equations by finite-dimensional
controllers ([5], [29], [30]). In particular, Barbu [5] studied the problem of
stabilization of the stationary solution of the system

$\begin{cases}u_{t}=\Delta u+f(x,u),\ \ \mbox{in}\ \
(0,\infty)\times\Omega,\\\ u=v,\ \ \mbox{on}\ \ (0,\infty)\times\Gamma_{1},\ \
\frac{\partial u}{\partial n}=0\ \ \mbox{on}\ \ (0,\infty)\times\Gamma_{2},\\\
u(x,0)=u_{0}(x)\ \ \mbox{in}\ \ \Omega,\end{cases}$

with feedback boundary controller depending on finitely many parameters. Here
$\Omega\subset\mathbb{R}^{d}$ is a bounded domain with the boundary
$\partial\Omega=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_{1},\Gamma_{2}$ are
disjoint connected components of $\partial\Omega.$ They assume that the
system, $\\{(\partial e_{j}/\partial n)\,|\,1\leq j\leq N\\}$, of unstable
Fourier modes is linearly independent on $\Gamma_{1}$. Their strategy works
successfuly for $d\geq 2$. However, in one spatial dimension, this assumption
is true only if the number of unstable Fourier modes is one. This situation
results in a restriction $\alpha<\lambda_{2}$ on the coefficient of the zero
order term of the main equation, where $\lambda_{2}$ is the second eigenvalue
of the differential operator $-\frac{d^{2}}{dx^{2}}$ with domain
$\\{\varphi\in H^{2}(0,L)\,|\,\varphi^{\prime}(0)=\varphi(L)=0\\}$. The
strategy that we design for Problems 1.1 and Problem 1.2 will work for
arbitrary choice of $\alpha$, i.e. arbitrary levels of instability.

Very recently, stabilization of the 1D linear heat equation by observer-based
finite dimensional controllers using in-domain point measurement and boundary
measurement was studied in [21] and [26] respectively. There are also similar
results for the semilinear heat equation, see [22]. Their strategy relies on
homogenizing the boundary conditions by changing the variables to transfer the
boundary control input into the domain, and then decomposing the main equation
with respect to the Fourier modes. Utilizing the fact that the eigenvalues
$\lambda_{j},j\in\mathbb{Z}^{+}$, of the differential operator are countable
and ordered, and therefore unstable Fourier modes are finitely many, say the
first $N_{0}$, they control only first $N_{0}$ modes by considering
$N_{0}-$dimensional system of ODEs. Another recent study on the stabilization
of linear heat equation in higher dimensions is [13] (see also [12] for 1D
case), which is again based on stabilizing the model via spectral truncation
stabilizer.

These studies mentioned in the previous paragraph are based on stabilizing the
infinite dimensional dynamical systems by controlling a corresponding finite
dimensional system, which eventually amounts to use finite dimensional
feedback controllers. In our present study, we will follow a different
approach that allows us to stabilize the zero equilibrium by controlling
directly the infinite dimensional system via finite dimensional boundary
feedback controllers. Our approach is inspired by the fact that dissipative
dynamical systems are known to possess finite-dimensional determining
parameters (see, e.g., [4, 14, 17, 34, 39]). We combine this theory with the
backstepping method by choosing the so called _target_ model in a novel way.
However, the advantage of our strategy is that, it allows us to gain
exponential stabilization of zero equilibrium with a desired decay rate, while
the instability level of the model is allowed to be arbitrarily large. This in
turn amounts to say that the decay rate can be chosen arbitrarily large, also
known as rapid stabilization. We would also like to note that the method in
our present study is applicable for other second order dissipative nonlinear
PDEs such as Ginzburg-Landau equation, Fitzhugh–Nagumo equation, viscous
Burgers’ equation and so on.

It should be remarked that recently a nice strategy, called late-lumping, that
combines the method of backstepping with a finite dimensional controller was
introduced, see for instance [2] and [40] for its application to the heat
equation. There are certain important differences when the methodology of
these papers are compared with ours. The first one of these is that the
aforementioned works use the standard backstepping transformation as in [23].
Therefore, it does not involve the projection operator inside the integral
term and this contrasts with our transformation (see equation (1.5) below),
which is a priori given with the projection $P_{N}$ in its definition. The
motivation for this is to obtain a target model with homogeneous boundary
conditions ameanable to a Lyapunov analysis that yields a precise information
on the sufficient number of Fourier modes for the desired purpose
(stabilization with either a prescribed or an unprescribed rate). Such
estimate is not provided in the late-lumping approach.

In addition, the late-lumping approach is based on a crucial assumption that
the finite dimensional controller uniformly converges to the standard
controller that uses the full state of the solution. Such assumption was
verified in [2, Lemma 5], however its proof relies on the embedding
$H^{1}(0,L)\hookrightarrow L^{\infty}(0,L)$. This forces solutions and in turn
the initial state to be taken from $H^{1}(0,L)$, too, for proving decay
estimates at $L^{2}$-level. On the other hand, our method does not rely on the
verification of such property and we establish stabilization at $L^{2}$-level
with initial datum taken from $L^{2}(0,L)$. Furthermore, we also provide decay
rate estimates at $H^{1}-$level with data from $H^{1}(0,L)$ \- the space used
for $L^{2}$-level decay with late-late lumping approach.

###  1.3. Methodology

Let us now describe our strategy. We first consider the following linearized
model associated with (1.1):

$\begin{cases}u_{t}-\nu u_{xx}-\alpha u=0,\quad x\in(0,L),t>0,\\\
u(0,t)=0,u(L,t)=g(t),\\\ u(x,0)=u_{0}(x).\end{cases}$ (1.3)

The steps of our strategy are given below:
Step i. We first design the following target model:

$\begin{cases}w_{t}-\nu w_{xx}-\alpha w+\mu P_{N}w=0,\quad x\in(0,L),t>0,\\\
w(0,t)=w(L,t)=0\\\ w(x,0)=w_{0}(x),\end{cases}$ (1.4)

where $w_{0}$ to be computed in step (iii) below. This target model is chosen
to convert the feedback action on the boundary to an interior stabilizing
effect (see, e.g. [28]). In our case, the feedback involves a finite number of
Fourier modes, therefore, we consider a target model with a stabilizing term
given by $\mu P_{N}w$. Indeed, dissipative dynamical systems possess finite
dimensional asymptotic dynamics (see e.g., [17, 34, 39]) since such systems
possess a finite number of determining modes [14]. Therefore, the term $\mu
P_{N}w$ is capable of preventing large fluctuations or uncontrolled growth of
the solution for large $N$ due to the term $-\alpha w$. We will show that
solutions of (1.4) can be steered to zero with any prescribed exponential
decay rate provided $\mu>\nu\lambda_{1}-\alpha$ and $N$ fulfills certain
criteria.
Step ii. Next, we introduce our backstepping transformation:

$\begin{split}u(x,t)&=w(x,t)+\int_{0}^{x}k(x,y)P_{N}w(y,t)dy\\\
&\doteq[(I+\Upsilon_{k}P_{N})w](x,t)\\\ &\doteq[T_{N}w](x,t).\end{split}$
(1.5)

In Section 2 we show that if $k$ is a sufficiently smooth solution of a
certain boundary value problem (see (2.1)) posed on the triangular region
$\Delta_{x,y}\doteq\left\\{(x,y)\in\mathbb{R}^{2}:x\in(0,L),y\in(0,x)\right\\}$,
then (1.5) successfully maps the target model (1.4) to the linearized model
(1.3). Finding the kernel $k$ allows us to write the control input as

$\displaystyle g(t)=\int_{0}^{L}k(L,y)P_{N}w(y,t)dy$ (1.6)

(1.6) will be rewritten in the form of a feedback using the inverse operator
discussed in the next step.
Step iii. We prove that the transformation $I+\Upsilon_{k}P_{N}$ is bounded
invertible if $(\mu,N)$ is an admissible decay rate-mode pair (see Definition
2.1 in Section 2). Bounded invertibility and step (ii) imply that the decay
properties of the target model (1.4) are inherited by the linear plant (1.3).
Moreover, the inverse operator can be expressed as $I-\Phi_{N}$, where
$\Phi_{N}:L^{2}(0,L)\to H^{\ell}(0,L)$ is a bounded linear operator. Thus, the
boundary controller (1.6) can be expressed as

$\displaystyle g(t)=\int_{0}^{L}k(L,y)\Gamma[P_{N}u](y,t)dy,$ (1.7)

where $\Gamma\doteq(I-P_{N}\Phi_{N}):H^{\ell}(0,L)\to H^{\ell}(0,L)$. The
initial state of the target model takes the form
$w_{0}\doteq(I-\Phi_{N})u_{0}$.

We use the same backstepping transformation to transform the nonlinear model
(1.1) into the corresponding nonlinear target model below.

$\begin{cases}w_{t}-\nu w_{xx}-\alpha w+\mu P_{N}w+f(w)=0,\quad
x\in(0,L),t>0,\\\ w(0,t)=w(L,t)=0,\\\ w(x,0)=w_{0}(x),\end{cases}$

where $f(w)=(I-\Phi_{N})\left[((I+\Upsilon_{k}P_{N})w)^{3}\right].$ We will
see in Section 4 that choosing $\mu>\nu\lambda_{1}-\alpha$ and $N$
sufficiently large, solutions of the above nonlinear target model tend to zero
at a prescribed exponential decay rate, provided that
$\|w_{0}\|_{H^{\ell}(0,L)}$ is sufficiently small. Now thanks to steps
(ii)-(iii) above, under a smallness assumption on $u_{0}$, the same decay
result will also hold for the nonlinear plant (1.1).

###  1.4. Preliminaries

$L^{p}(0,L)$, $1\leq p\leq\infty$ is the usual Lebesgue space and given
$\varphi\in L^{p}(0,L)$ we denote its $L^{p}-$norm by
$\|\varphi\|_{L^{p}(0,L)}$. For $p=2$, we write $\|\varphi\|$ instead of
$\|\varphi\|_{L^{2}(0,L)}$. Given $\ell>0$, we represent the $L^{2}-$based
Sobolev space by $H^{\ell}(0,L)$. $H_{0}^{1}(0,L)$ is the closed subspace of
$H^{1}(0,L)$ that involves those functions in $H_{0}^{1}(0,L)$ which vanish
(have zero trace) on the boundary of $(0,L)$. If $A:H^{\ell}(0,L)\to
H^{\ell}(0,L)$ is a linear bounded operator, $\ell=0,1$, we denote its
operator norm as $\|A\|_{H^{\ell}(0,L)\to H^{\ell}(0,L)}$. We write $a\lesssim
b$ to denote an inequality $a\leq cb$, where $c>0$ may depend on only some
fixed parameters. With $\lambda_{j}$, $j=1,2,\dotsc$, where
$\lambda_{j}=j^{2}\lambda_{1}=\left(\frac{j\pi}{L}\right)^{2}$, we denote the
$j-$th eigenvalue of the operator $-\frac{d^{2}}{dx^{2}}$ subject to Dirichlet
boundary conditions, and $e_{j}=\sqrt{\frac{2}{L}}\sin(\frac{j\pi x}{L})$ is
the corresponding $L^{2}-$normalized eigenfunction. We introduce the
(solution) spaces

$X_{T}^{\ell}\equiv L^{\infty}(0,T;H^{\ell}(0,L))\cap
L^{2}(0,T;H^{1+\ell}(0,L)),\quad\ell=0,1.$

We equip these spaces with the norms

$\displaystyle\|\psi\|_{X_{T}^{\ell}}^{2}\equiv\operatorname*{ess\,sup}_{t\in[0,T]}\|\psi(\cdot,t)\|_{H^{\ell}(0,L)}^{2}+\int_{0}^{T}\|\psi(\cdot,t)\|_{H^{1+\ell}(0,L)}^{2}dt.$

The following inequalities are useful while carrying out some of the norm
estimates.

  * •

Cauchy’s inequality with $\epsilon$: $ab\leq\frac{\epsilon
a^{2}}{2}+\frac{b^{2}}{2\epsilon}$, $\epsilon>0$.

  * •

Cauchy-Schwarz inequality: For $\varphi,\psi\in L^{2}(0,L)$,
$\left|\int_{0}^{L}\varphi(x)\psi(x)dx\right|\leq\|\varphi\|\|\psi\|.$

  * •

Poincaré inequality: For $\varphi\in H_{0}^{1}(0,L)$ or for $\varphi\in
H^{1}(0,L)$ with $\int_{0}^{L}\varphi dx=0$, it follows
$\|\varphi\|^{2}\leq\lambda_{1}^{-1}\|\varphi^{\prime}\|^{2}.$ In particular,
$\|\varphi^{\prime}\|_{2}^{2}\leq\lambda_{1}^{-1}\|\varphi^{\prime\prime}\|_{2}^{2}$
for $\varphi\in H_{0}^{1}(0,L)\cap H^{2}(0,L)$.

  * •

Poincaré type inequality: For $u\in H_{0}^{1}(0,L)$,

$\|u-P_{N}u\|^{2}\leq\lambda_{N+1}^{-1}\|u^{\prime}\|^{2},\quad\lambda_{N+1}=(N+1)^{2}\lambda_{1},$

The proof of this inequality is easy, indeed with $\hat{u}_{j}=(u,e_{j})_{2}$,
one has

$\|u^{\prime}\|^{2}=\sum\limits_{j=1}^{\infty}\lambda_{j}\hat{u}_{j}^{2}\geq\sum\limits_{j=N+1}^{\infty}\lambda_{j}\hat{u}_{j}^{2}\geq\lambda_{N+1}\sum\limits_{j=N+1}^{\infty}\hat{u}_{j}^{2}=\lambda_{N+1}\|u-P_{N}u\|^{2}.$

  * •

Gagliardo-Nirenberg’s inequality: Let $u\in H_{0}^{1}(0,L)$, $p\geq 2$,
$\alpha=\frac{1}{2}-\frac{1}{p}$. Then $\|u\|_{L^{p}(0,L)}\leq
c^{*}\|u^{\prime}\|^{\alpha}\|u\|^{1-\alpha},$ where $c^{*}>0$ (is say best
GN-constant) depends on $p,\alpha$, and $L$.

###  1.5. Main results

We first solve the rapid stabilization problem for the linearized model and
prove the theorem below. $(\mu,N)$ is assumed to be an admissible decay rate-
mode pair throughout (see Definition 2.1 in Section 2).

######  Theorem 1.4.

Let $\nu,\alpha>0$, $\mu>\alpha-\nu\lambda_{1}\geq 0$, and $u_{0}\in
H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$. Then the weak solution $u$ of the
linear problem (1.3), which satisfies the feedback control law
$u(L,t)=g(P_{N}u(t))$, belongs to $X_{T}^{\ell}$. Moreover, if

$N>\max\left\\{\frac{\mu}{2\nu\lambda_{1}}-1,\frac{\mu}{\mu+\nu\lambda_{1}-\alpha}-1\right\\},$

then $u$ satisfies the decay estimate

$\|u(t)\|_{H^{\ell}(0,L)}\leq c_{k}e^{-\gamma
t}\|u_{0}\|_{H^{\ell}(0,L)},\quad\text{ for a.e. }t\geq 0,$

where $\gamma=\nu\lambda_{1}-\alpha+\mu(1-\frac{1}{N+1})$ and $c_{k}$ is a
nonnegative constant dependent on the kernel $k$ and independent of $u_{0}$.

Secondly, we treat the corresponding nonlinear model.

######  Theorem 1.5.

Let $\nu,\alpha>0$, $\mu>\alpha-\nu\lambda_{1}\geq 0$, $d\in(0,1)$, and
$u_{0}\in H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$. Let $\epsilon>0$ be any
number satisfying
$\epsilon\lambda_{1}\leq\max\\{\nu-\frac{\mu}{N+1},2\gamma\\}$ and assume that
$\|u_{0}\|_{H^{\ell}}$ is sufficiently small in the sense that
$\|u_{0}^{(\ell)}\|_{2}^{2}\leq
d\frac{\sqrt{\epsilon(2\gamma-\epsilon\lambda_{1})}\lambda_{1}^{\ell}}{c_{0}c_{1}}{\|T_{N}^{-1}\|_{H^{\ell}(0,L)\rightarrow
H^{\ell}(0,L)}^{-2}}$, where $\gamma$ is given by (3.7), $c_{0},c_{1}$ are
defined in (4.10) and (4.11), respectively. If

$N>\max\left\\{\frac{\mu}{2\nu\lambda_{1}}-1,\frac{\mu}{\mu+\nu\lambda_{1}-\alpha}-1\right\\},$

then the nonlinear problem (1.1) has a global solution $u\in X_{T}^{\ell}$,
which satisfies the feedback control law $u(L,t)=g(P_{N}u(t))$, and has the
decay estimate

$\|u(t)\|_{H^{\ell}(0,L)}\leq
c_{k}e^{-\gamma^{\prime}t}\|u_{0}\|_{H^{\ell}(0,L)}\text{ for a.e. }t\geq 0,$

for any $\gamma^{\prime}$ with
$0<\gamma^{\prime}<\gamma=\nu\lambda_{1}-\alpha+\mu\left(1-\frac{1}{N+1}\right)$,
where $c_{k}$ is a non-negative constant dependent on the kernel $k$, $d$ and
independent of $u_{0}$.

######  Remark 1.6.

The smallness assumption on initial datum is used only for stabilization and
it is not a necessary assumption for local existence of solutions. Smallness
condition for stabilization of nonlinear PDEs via backstepping controllers is
common, see for instance [8]. This is because the backstepping transformation
turns the original nonlinear plant into another plant (target model) in which
the monotone structure of the nonlinear term is disrupted.

Next, we consider the stabilization problem with a certain unprescribed not
necessarily large decay rate for the linearized model. We show that if $N$ is
the number of unstable modes, it suffices to employ a controller that involves
only the first $N$ modes to achieve exponential stabilization.

######  Theorem 1.7.

Let $\nu,\alpha>0$, $u_{0}\in H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$, $N$
be such that $\lambda_{N}<\frac{\alpha}{\nu}<\lambda_{N+1}$, and $\mu$ satisfy

$2(\alpha-\nu\lambda_{1})\left(1-\frac{1}{(N+1)^{2}}\right)^{-1}<\mu<2\nu\lambda_{N+1}.$
(1.8)

Then the solution $u$ of linear problem (1.3), which satisfies the feedback
control law $u(L,t)=g(P_{N}u(t))$, belongs to $X_{T}^{\ell}$ and has the decay
estimate

$\|u(t)\|_{H^{\ell}(0,L)}\leq c_{k}e^{-\rho
t}\|u_{0}\|_{H^{\ell}(0,L)},\quad\text{ for a.e. }t\geq 0,$

where $\rho=\nu\lambda_{1}-\alpha+\frac{\mu}{2}(1-\frac{1}{(N+1)^{2}})$ and,
$c_{k}$ is a nonnegative constant depending on $k$ and independent of $u_{0}$.

######  Remark 1.8.

The set of $\mu$ that satisfies (1.8) is nonempty as shown in Section 5.
Moreover, $\rho$ is strictly positive by construction. Similar remarks also
apply to Theorem 1.9 below, where we extend the above result to the nonlinear
model.

######  Theorem 1.9.

Let $\nu,\alpha>0$, $d\in(0,1)$ and $u_{0}\in H^{\ell}(0,L)$, where $\ell=0$
or $\ell=1$, $N$ be such that $\lambda_{N}<\frac{\alpha}{\nu}<\lambda_{N+1}$.
Assume that $\|u_{0}\|_{H^{\ell}}$ is sufficiently small in the sense of
Theorem 1.5 and $\mu$ satisfies

$2(\alpha-\nu\lambda_{1})\left(1-\frac{1}{(N+1)^{2}}\right)^{-1}<\mu<2\nu\lambda_{N+1}.$

Then for any $\rho^{\prime}$ with
$0<\rho^{\prime}<\rho=\nu\lambda_{1}-\alpha+\frac{\mu}{2}\left(1-\frac{1}{(N+1)^{2}}\right)$,
the nonlinear problem (1.1) has a solution $u\in X_{T}^{\ell}$, which
satisfies the feedback control law $u(L,t)=g(P_{N}u(t))$, and has the decay
estimate

$\|u(t)\|_{H^{\ell}(0,L)}\leq
c_{k}e^{-\rho^{\prime}t}\|u_{0}\|_{H^{\ell}(0,L)}\quad\text{ for a.e. }t\geq
0,$

where $c_{k}$ is a nonnegative constant depending on $k$, $d$, and independent
of $u_{0}$.

###  1.6. Orientation

This paper consists of six sections. In Section 2 we find sufficient
conditions for the backstepping kernel so that the corresponding backstepping
transformation maps the linearized plant to a desired target model. Using
these conditions, we obtain an explicit representation of the kernel. Then, we
prove the invertibility of the backstepping transformation with a bounded
inverse. In Sections 3 and 4 we answer Problem 1.1, and in Section 5 we answer
Problem 1.2. Finally, in Section 6, we give some numerical simulations
verifying our theoretical results.

##  2\. Backstepping transformation: existence and bounded invertibility

###  2.1. Kernel

Here, we state the sufficient conditions for the kernel so that the
backstepping transformation maps the target model to the linearized plant.
Adapting arguments given in [23, Section 4] to the operator in (1.5) , it
turns out that it is enough if $k$ solves the boundary value problem

$\begin{cases}\nu(k_{xx}-k_{yy})+\mu k=0,\quad(x,y)\in\Delta_{x,y},\\\
k(x,0)=0,\quad k(x,x)=-\frac{\mu x}{2\nu},\quad x\in(0,L),\end{cases}$ (2.1)

on $\Delta_{x,y}$. Moreover, the solution can be expressed explicitly as

$k(x,y)=-\frac{\mu
y}{2\nu}\sum\limits_{m=0}^{\infty}\left(-\frac{\mu}{4\nu}\right)^{m}\frac{(x^{2}-y^{2})^{m}}{m!(m+1)!},$
(2.2)

which is a uniformly and absolutely convergent series on
$\overline{\Delta_{x,y}}$. It is important that $P_{N}$ and
$\frac{d^{2}}{dy^{2}}$ commutes in derivation of (2.1).

###  2.2. Invertibility

In this section, we discuss the bounded invertibility of the backstepping
transformation on $H^{\ell}(0,L)$, $\ell=0,1,2$.

At first, we argue that the invertibility does not necessarily hold for every
choice of $\mu>0$ in (2.1). To infer this, let $\Upsilon_{k}:H^{\ell}(0,L)\to
H^{\ell}(0,L)$ be the integral operator defined by
$(\Upsilon_{k}\psi)(x)\doteq\int_{0}^{x}k(x,y)\psi(y)dy$. Let $N=1$ as an
example and consider the backstepping transformation
$I+\Upsilon_{k}P_{N}=I+\Upsilon_{k}P_{1}$, which uses a single Fourier mode.
Let $\mu>0$ be such that $(e_{1},\Upsilon_{k}e_{1})_{2}=-1$ (such $\mu$ exists
by intermediate value theorem). Let $\varphi$ be a function in $H^{\ell}(0,L)$
with the property $\hat{\varphi}_{1}=(\varphi,e_{1})_{2}\neq 0$. We claim that
there is no $\psi$ with $\varphi=(I+\Upsilon_{k}P_{1})\psi$ because if it is
the case then setting $v=\Upsilon_{k}P_{1}\psi$, we see that $v$ must satisfy

$v=\Upsilon_{k}P_{1}(\varphi-v)=(\hat{\varphi}_{1}-\hat{v}_{1})\Upsilon_{k}e_{1}.$

Taking $L^{2}-$inner product of both sides with $e_{1}$ and using
$(e_{1},\Upsilon_{k}e_{1})_{2}=-1$, it follows that $0=-\hat{\varphi}_{1}\neq
0$, which is a contradiction. Hence, $(I+\Upsilon_{k}P_{1})$ cannot be
invertible if $\mu$ (equivalently the associated kernel $k=k(x,y;\mu)$) is
such that $(e_{1},\Upsilon_{k}e_{1})_{2}=-1$.

Note however, if $\mu>0$ is such that $(e_{1},\Upsilon_{k}e_{1})_{2}\neq-1$,
then given $\varphi$, the function $\psi=\varphi-v$, where
$v=\hat{v}_{1}\Upsilon_{k}e_{1}$ with

$\hat{v}_{1}=\frac{\hat{\varphi}_{1}}{1+(e_{1},\Upsilon_{k}e_{1})_{2}}$

satisfies $(I+\Upsilon_{k}P_{1})\psi=\varphi$. The preimage

$\displaystyle\varphi-v=\varphi-\frac{\hat{\varphi}_{1}}{1+(e_{1},\Upsilon_{k}e_{1})_{2}}\Upsilon_{k}e_{1}$

leads to a well-defined inverse operator in the form $I-\Phi_{1}$, where

$\Phi_{1}\varphi=\frac{\hat{\varphi}_{1}}{1+(e_{1},\Upsilon_{k}e_{1})_{2}}\Upsilon_{k}e_{1}.$

Now, let us consider the case of two modes ($N=2$) and the transformation
$I+\Upsilon_{k}P_{2}$. Suppose $\mu>0$ is such that
$(e_{1},\Upsilon_{k}e_{1})_{2}\neq-1$ so that the operator $\Phi_{1}$ given in
above paragraph is well-defined. Let us write
$\varphi=(I+\Upsilon_{k}P_{2})\psi$ and set $v=\Upsilon_{k}P_{2}\psi$. Then,
we have

$v=\Upsilon_{k}P_{2}(\varphi-v)=\Upsilon_{k}P_{2}\varphi-\Upsilon_{k}P_{1}v-\hat{v}_{2}\Upsilon_{k}e_{2},$

from which it follows that
$(I+\Upsilon_{k}P_{1})v=\Upsilon_{k}P_{2}\varphi-\hat{v}_{2}\Upsilon_{k}e_{2}$.
Now, applying $I-\Phi_{1}$ to both sides, we get

$v=(I-\Phi_{1})[\Upsilon_{k}P_{2}\varphi]-\hat{v}_{2}(I-\Phi_{1})[\Upsilon_{k}e_{2}].$

Taking inner product of above expression with $e_{2}$ and rearranging the
terms, we obtain

$\hat{v}_{2}=\frac{\left((I-\Phi_{1})[\Upsilon_{k}P_{2}\varphi],e_{2}\right)_{2}}{1+\left((I-\Phi_{1})[\Upsilon_{k}e_{2}],e_{2}\right)_{2}}$

provided $\mu>0$ is also such that
$\left((I-\Phi_{1})[\Upsilon_{k}e_{2}],e_{2}\right)_{2}\neq-1.$ Under the
aforementioned two conditions on $\mu$, the inverse takes the form
$I-\Phi_{2}$, where

$\Phi_{2}\varphi=(I-\Phi_{1})[\Upsilon_{k}P_{2}\varphi]-\frac{\left((I-\Phi_{1})[\Upsilon_{k}P_{2}\varphi],e_{2}\right)_{2}}{1+\left((I-\Phi_{1})[\Upsilon_{k}e_{2}],e_{2}\right)_{2}}\times(I-\Phi_{1})[\Upsilon_{k}e_{2}].$

The above argument can be extended to any $N>2$ via recursion (whose details
are given in Step 3 in the proof of Lemma 2.2 below). The following definition
plays a crucial role in order for such recursive argument to work:

######  Definition 2.1 (Admissible decay-mode pair).

Let $\mu>0$ and $N\in\mathbb{Z}_{+}$. Then, $(\mu,N)$ is said to be an
admissible finite dimensional backstepping decay rate-mode pair if

$\left((I-\Phi_{j-1})[\Upsilon_{k}e_{j}],e_{j}\right)_{2}\neq-1$

for $1\leq j\leq N$, where $\Phi_{0}\doteq 0$ and

$\Phi_{j}\varphi=(I-\Phi_{j-1})[\Upsilon_{k}P_{j}\varphi]-\frac{\left((I-\Phi_{j-1})[\Upsilon_{k}P_{j}\varphi],e_{j}\right)_{2}}{1+\left((I-\Phi_{j-1})[\Upsilon_{k}e_{j}],e_{j}\right)_{2}}\times(I-\Phi_{j-1})[\Upsilon_{k}e_{j}]$

for $1\leq j\leq N$.

######  Lemma 2.2.

Let $\ell\in\\{0,1,2\\}$ and $(\mu,N)$ be an admissible decay-mode pair. Then
$T_{N}\doteq I+\Upsilon_{k}P_{N}:H^{\ell}(0,L)\to H^{\ell}(0,L)$, is bounded
invertible. Moreover $T_{N}^{-1}$ can be written as $I-\Phi_{N}$, where
$\Phi_{N}:L^{2}(0,L)\to H^{\ell}(0,L)$ is linear bounded.

###### Proof.

We write the backstepping transformation in operator form as
$\varphi=(I+\Upsilon_{k}P_{N})\psi$. Set $v=\Upsilon_{k}P_{N}\psi$. Then, we
have $\psi=\varphi-v$ and we get

$v=\Upsilon_{k}P_{N}(\varphi-v).$ (2.3)

Given $\psi\in H^{\ell}(0,L)$, we have $(I+\Upsilon_{k}P_{N})\psi\in
H^{\ell}(0,L).$ Therefore, we have the inclusion
$R(I+\Upsilon_{k}P_{N})\subset H^{\ell}(0,L).$ We will prove the invertibility
with induction on $N$.

  1. Step 1:

We first consider the case $N=1$. Using (2.3) with $N=1$, we write
$v=\Upsilon_{k}P_{1}(\varphi-v)$. Note that
$P_{1}\varphi=\hat{\varphi}_{1}e_{1}$ and $P_{1}v=\hat{v}_{1}e_{1}$, where
$\hat{\varphi}_{1}=(\varphi,e_{1})_{2}$ and $\hat{v}_{1}=(v,e_{1})_{2}$. Using
linearity of $\Upsilon_{k}$, we get

$v=\Upsilon_{k}(\hat{\varphi}_{1}-\hat{v}_{1})e_{1}=(\hat{\varphi}_{1}-\hat{v}_{1})\Upsilon_{k}e_{1}.$
(2.4)

Now, we look for a solution in the form $v=\alpha_{1}\Upsilon_{k}e_{1}$. Note
that then
$\hat{v}_{1}=(e_{1},\alpha_{1}\Upsilon_{k}e_{1})_{2}=\alpha_{1}\beta_{1}$,
where $\beta_{1}=\int_{0}^{L}e_{1}(s)[\Upsilon_{k}e_{1}](s)ds$. Using this
equality in (2.4), we find that $\alpha_{1}$ should satisfy
$\alpha_{1}\Upsilon_{k}e_{1}=(\hat{\varphi}_{1}-\alpha_{1}\beta_{1})\Upsilon_{k}e_{1}$.
This equation will hold with the choice
$\alpha_{1}(1+\beta_{1})=\hat{\varphi}_{1}$. Since $(\mu,N)$ is assumed to be
an admissible decay rate-mode pair, we have $\beta_{1}\neq-1$, so that we can
take $\alpha_{1}=\frac{\hat{\varphi}_{1}}{1+\beta_{1}}.$ This shows
surjectivity of $I+\Upsilon_{k}P_{1}$. Note that we in particular have $v\in
H^{\ell}(0,L)$ (indeed better than this) so that $\psi=\varphi-v\in
H^{\ell}(0,L)$. Note we have

$\psi=\varphi-v=\varphi-\frac{\hat{\varphi}_{1}}{1+\beta_{1}}\Upsilon_{k}e_{1}=\varphi-\frac{1}{1+\beta_{1}}\Upsilon_{k}P_{1}\varphi.$

Now we set
$\Phi_{1}\varphi\equiv\frac{1}{1+\beta_{1}}\Upsilon_{k}P_{1}\varphi$. It is
easy to verify that $I-\Phi_{1}$ is both a right inverse and a left inverse
for $I+\Upsilon_{k}P_{1}$ and moreover $\Phi_{1}:L^{2}(0,L)\to H^{\ell}(0,L)$
is a linear bounded operator. Therefore, $(I+\Upsilon_{k}P_{1})^{-1}$ exists
and is given by $(I+\Upsilon_{k}P_{1})^{-1}{\varphi}=(I-\Phi_{1})\varphi$.

  2. Step 2:

We assume that there exists some $K\geq 1$ such that the statement of the
lemma holds true for $N=K$.

  3. Step 3:

Now, we claim that the statement of the lemma must also be true for $N=K+1$.
Replacing $N$ by $K+1$ we have

$\begin{split}v&=\Upsilon_{k}P_{K+1}(\varphi-v)\\\
&=\Upsilon_{k}P_{K+1}\varphi-\Upsilon_{k}P_{K}v-\Upsilon_{k}E_{K+1}v,\end{split}$
(2.5)

where we used $P_{K+1}=P_{K}+E_{K+1}$ with $E_{K+1}$ being the projection onto
the $(K+1)-$th Fourier sine mode. Rearranging the terms, we get
$(I+\Upsilon_{k}P_{K})v=\Upsilon_{k}P_{K+1}\varphi-\hat{v}_{K+1}\Upsilon_{k}e_{K+1}$,
where $\hat{v}_{K+1}$ is the $(K+1)-$th Fourier sine mode of $v$. By using the
induction assumption in Step 2, we obtain

$v=(I-\Phi_{K})[\Upsilon_{k}P_{K+1}\varphi]-\hat{v}_{K+1}(I-\Phi_{K})[\Upsilon_{k}e_{K+1}].$

Taking inner product in both sides of the above equation with $e_{K+1}$, we
get

$\hat{v}_{K+1}=\frac{\left((I-\Phi_{K})[\Upsilon_{k}P_{K+1}\varphi],e_{K+1}\right)_{2}}{1+\left((I-\Phi_{K})[\Upsilon_{k}e_{K+1}],e_{K+1}\right)_{2}}.$

Note that $\hat{v}_{K+1}$ is well-defined since the denominator does not
vanish above due to the assumption that $(\mu,N)$ is an admissible decay rate-
mode pair. Therefore, we find

$\begin{split}v=&(I-\Phi_{K})[\Upsilon_{k}P_{K+1}\varphi]\\\
&-\frac{\left((I-\Phi_{K})[\Upsilon_{k}P_{K+1}\varphi],e_{K+1}\right)_{2}}{1+\left((I-\Phi_{K})[\Upsilon_{k}e_{K+1}],e_{K+1}\right)_{2}}\times(I-\Phi_{K})[\Upsilon_{k}e_{K+1}]\equiv\Phi_{K+1}\varphi.\end{split}$
(2.6)

This shows that $I+\Upsilon_{k}P_{K+1}$ is surjective and it has a right
inverse given by $I-\Phi_{K+1}$, where $\Phi_{K+1}$ is given in (2.6).
Moreover, given $\varphi$, we see from above that $v$ is uniquely determined,
and therefore the right inverse is unique. This implies the right inverse is
also a left inverse and $I+\Upsilon_{k}P_{K+1}$ is invertible and we have

$\psi=(I+\Upsilon_{k}P_{K+1})^{-1}{\varphi}=\varphi-v=(I-\Phi_{K+1})\varphi.$

It is easy to check that $\Phi_{K+1}:L^{2}(0,L)\to H^{\ell}(0,L)$ is a linear
bounded operator. This follows from the definition of $\Phi_{K+1}$ together
with the induction assumption in Step 2.

∎

######  Remark 2.3.

Replacing $w$ in the control input $g(t)=\int_{0}^{L}k(x,L)P_{N}w(y,t)dy$ with
$w=(I-\Phi_{N})u$, and noting $\Phi_{N}u=\Phi_{N}P_{N}u$ we see that $g$ can
be expressed as

$g(t)=\int_{0}^{L}k(L,y)\Gamma_{N}[P_{N}u](y,t)dy,$

where $\Gamma_{N}\doteq I-P_{N}\Phi_{N}:H^{\ell}(0,L)\to H^{\ell}(0,L)$ is a
bounded operator. Consequently, (i) our control input is indeed feedback type
and (ii) it is calculated by using only finitely many Fourier sine modes of
the state of solution.

##  3\. Linear theory (Proof of Theorem 1.4)

In this section, we prove well-posedness of the linearized model (1.3) and
establish uniform decay rate estimates for its solutions. Our analysis will be
first carried out through the target model (1.4) and then, we will use
boundedness of the backstepping transformation and its inverse to deduce that
the wellposedness and decay results that we obtain for (1.4) are inherited by
(1.3).

Existence of a local solution of (1.4) can be obtained rigorously by using the
standard Galerkin method. We will only give a proof showing the uniform
exponential decay of solutions of (1.4) at topological levels $H^{\ell}(0,L)$
with $\ell=0$ and $\ell=1$, which will also imply global existence and
uniqueness of solutions. This will be done formally by using the multiplier
method. Indeed, one first construct approximate regular solutions in the form
$w^{m}(t)=\sum_{k=1}^{m}d_{m}^{k}(t)e_{k}\in C([0,T];H_{0}^{\ell}(0,L))\cap
L^{2}(0,T;H^{\ell+1}(0,L))$ via Galerkin method, apply the multiplier
technique with these solutions and then use a limiting argument to obtain the
same type of final estimates for the soughtafter weak solution. This procedure
is rather standard and will be omitted.

Let us first consider the case $u_{0}\in L^{2}(0,L)$. This implies
$w_{0}=(I-\Upsilon_{k}P_{N})^{-1}u_{0}\in L^{2}(0,L)$ thanks to the bounded
invertibility of the backstepping transformation on $L^{2}(0,L)$ (see Lemma
2.2). Now, taking the $L^{2}-$inner product of the main equation in (1.4) by
$2w$, we get

$\frac{d}{dt}\|w(t)\|^{2}+2\nu\|w_{x}(t)\|^{2}-2\alpha\|w(t)\|^{2}+2\mu\int_{0}^{L}w(x,t)P_{N}w(x,t)dx=0.$
(3.1)

Using the Cauchy-Schwarz inequality and Cauchy’s inequality with
$\epsilon_{1}>0$, we see that the last term at the left hand side of (3.1) is
bounded from below as

$\begin{split}2\mu\int_{0}^{L}w(x,t)P_{N}w(x,t)dx&=-2\mu\int_{0}^{L}w(x,t)\left(w-P_{N}w\right)(x,t)dx+2\mu\|w(\cdot,t)\|^{2}\\\
&\geq(2\mu-\epsilon_{1}\mu)\|w(\cdot,t)\|^{2}-\frac{\mu}{\epsilon_{1}}\|\left(w-P_{N}w\right)(\cdot,t)\|^{2}.\end{split}$

Using the above bound in (3.1), we get

$\frac{d}{dt}\|w(t)\|^{2}+2\nu\|w_{x}(t)\|^{2}-2\alpha\|w(t)\|^{2}+(2\mu-\mu\epsilon_{1})\|w(t)\|^{2}-\frac{\mu}{\epsilon_{1}}\|\left(w-P_{N}w\right)(t)\|^{2}\leq
0.$

Employing the Poincaré type inequality, we obtain

$\frac{d}{dt}\|w(t)\|^{2}+2\left(\nu-\frac{\mu}{2\epsilon_{1}\lambda_{1}(N+1)^{2}}\right)\|w_{x}(t)\|^{2}+(2\mu-\mu\epsilon_{1}-2\alpha)\|w(t)\|^{2}\leq
0.$ (3.2)

For $\epsilon_{1}>0$ satisfying

$\nu-\frac{\mu}{2\epsilon_{1}\lambda_{1}(N+1)^{2}}>0,$ (3.3)

we apply the Poincaré inequality to the second term at the left hand side of
in (3.2):

$\frac{d}{dt}\|w(t)\|^{2}+2\left[\nu\lambda_{1}-\alpha+\mu\left(1-\frac{\epsilon_{1}}{2}-\frac{1}{2\epsilon_{1}(N+1)^{2}}\right)\right]\|w(t)\|^{2}\leq
0.$ (3.4)

To maximize the stabilizing effect one must choose
$\epsilon_{1}=\frac{1}{N+1}$. Note that (3.3) with
$\epsilon_{1}=\frac{1}{N+1}$ will hold provided

$N>\frac{\mu}{2\nu\lambda_{1}}-1.$ (3.5)

Therefore, under condition (3.5) on $N$, we get

$\displaystyle\frac{d}{dt}\|w(t)\|^{2}+2\gamma\|w(t)\|^{2}\leq 0,$ (3.6)

where

$\gamma=\nu\lambda_{1}-\alpha+\mu\left(1-\frac{1}{(N+1)}\right).$ (3.7)

In order to have decay, we must have $\gamma>0$ which imposes

$N>\frac{\mu}{\mu+\nu\lambda_{1}-\alpha}-1.$ (3.8)

Finally, integrating (3.6), we deduce

$\|w(t)\|\leq e^{-\gamma t}\|w_{0}\|,\quad\text{ for a.e. }t\geq 0,$ (3.9)

under the assumption that

$N>\max\left\\{\frac{\mu}{2\nu\lambda_{1}}-1,\frac{\mu}{\mu+\nu\lambda_{1}-\alpha}-1\right\\}.$
(3.10)

Next, we obtain uniform estimates for solutions of (1.4) in the case $u_{0}\in
H^{1}(0,L)$. We take $L^{2}-$inner product of the main equation in (1.4) by
$-2w_{xx}$ and deduce

$\frac{d}{dt}\|w_{x}(t)\|^{2}+2\nu\|w_{xx}(t)\|^{2}-2\alpha\|w_{x}(t)\|^{2}-2\mu\int_{0}^{L}w_{xx}(x,t)P_{N}w(x,t)dx=0.$
(3.11)

Applying the Cauchy-Schwarz inequality and then Cauchy’s inequality with
$\epsilon_{2}>0$, the last term at the left hand side of the above identity
can be bounded from below as

$\begin{split}-2\mu\int_{0}^{L}w_{xx}(x,t)P_{N}w(x,t)dx=&2\mu\int_{0}^{L}w_{xx}(x,t)(w-P_{N}w)(x,t)dx\\\
&-2\mu\int_{0}^{L}w_{xx}(x,t)w(x,t)dx\\\
\geq&-\mu\epsilon_{2}\|w_{xx}(t)\|^{2}-\frac{\mu}{\epsilon_{2}}\|(w-P_{N}w)(t)\|^{2}+2\mu\|w_{x}(t)\|^{2}.\end{split}$
(3.12)

Next we use the Poincaré type inequality to get

$-2\mu\int_{0}^{L}w_{xx}(x,t)P_{N}w(x,t)dx\geq-\mu\epsilon_{2}\|w_{xx}(t)\|^{2}+\left(2\mu-\frac{\mu}{\epsilon_{2}\lambda_{1}(N+1)^{2}}\right)\|w_{x}(t)\|^{2}.$

Combining the above estimate with (3.11), we obtain

$\frac{d}{dt}\|w_{x}(t)\|^{2}+2\left(\nu-\frac{\mu\epsilon_{2}}{2}\right)\|w_{xx}(t)\|^{2}+2\left(\mu-\alpha-\frac{\mu}{2\epsilon_{2}\lambda_{1}(N+1)^{2}}\right)\|w_{x}(t)\|^{2}\leq
0.$ (3.13)

Now for a given $\mu$, we can find $\epsilon_{2}>0$ such that

$\nu-\frac{\mu\epsilon_{2}}{2}>0.$ (3.14)

For this choice of $\epsilon_{2}$, we apply the Poincaré inequality to the
second term at the left hand side of (3.13) and get

$\frac{d}{dt}\|w_{x}(t)\|^{2}+2\left[\nu\lambda_{1}-\alpha+\mu\left(1-\frac{\epsilon_{2}\lambda_{1}}{2}-\frac{1}{2\epsilon_{2}\lambda_{1}(N+1)^{2}}\right)\right]\|w_{x}(t)\|^{2}\leq
0.$ (3.15)

Observe that the inner parenthesis is maximized for the choice
$\epsilon_{2}=\frac{1}{\lambda_{1}(N+1)}$. Using this choice we rewrite (3.15)
and get

$\frac{d}{dt}\|w_{x}(t)\|^{2}+2\gamma\|w_{x}(t)\|^{2}\leq 0,$ (3.16)

where $\gamma$ is given by (3.7). Note that letting
$\epsilon_{2}=\frac{1}{\lambda_{1}(N+1)}$ in (3.14) yields the same condition
(3.5) on $N$ as we obtained in the $L^{2}-$decay case. Finally, combining
(3.16) with (3.6), we obtain the following decay result

$\|w(t)\|_{H^{1}(0,L)}\leq e^{-\gamma t}\|w_{0}\|_{H^{1}(0,L)},\quad\text{ for
a.e. }t\geq 0,$

under condition (3.10) on $N$. Finally, we deduce $w\in X_{T}^{\ell}$ by using
the definition of norm of $X_{T}^{\ell}$ once we integrate (3.6) and (3.16)
over $(0,T)$ for the cases $\ell=0$ and $\ell=1$, respectively.

The following proposition follows.

######  Proposition 3.1.

_Let $\nu,\alpha>0$,$w_{0}\in H^{\ell}(0,L)$, where $\ell=0$ or $\ell=1$,
$\mu>\alpha-\nu\lambda_{1}\geq 0$, $N$ satisfy (3.10), and let $(\mu,N)$ be
admissible (Definition 2.1). Then, the solution $w$ of (1.4), which belongs to
$X_{T}^{\ell}$, satisfies_

$\|w(t)\|_{H^{\ell}(0,L)}\leq e^{-\gamma
t}\|w_{0}\|_{H^{\ell}(0,L)},\quad\forall t\geq 0.$

_where $\gamma$ is given by (3.7)._

######  Remark 3.2.

If one uses all Fourier sine modes ($N=\infty$), then the main equation of the
target model takes the form

$w_{t}-\nu w_{xx}-\alpha w+\mu w=0.$

Using standard multipliers, one can see that
$\|w(t)\|_{H^{\ell}(0,L)}=\mathcal{O}(e^{-(\nu\lambda_{1}-\alpha+\mu)t})$,
$t\geq 0$. Hence, the condition $\mu>\alpha-\nu\lambda_{1}$ is necessary for
solutions to decay. Therefore, this condition, which also appears in the above
proposition is a natural assumption.

Using boundedness of the backstepping transformation, we deduce that $u$, the
solution of (1.3), also belongs to $X_{T}^{\ell}$. Moreoever, from Proposition
3.1, we have

$\begin{split}\|u(t)\|_{H^{\ell}(0,L)}&\leq\left(1+\|k\|_{H^{\ell}(\Delta_{x,y})}\right)\|w(t)\|_{H^{\ell}(0,L)}\\\
&\leq\left(1+\|k\|_{H^{\ell}(\Delta_{x,y})}\right)e^{-\gamma
t}\|w_{0}\|_{H^{\ell}(0,L)}.\end{split}$ (3.17)

Due to the invertibility of the backstepping transformation with a bounded
inverse, we get

$\|w_{0}\|_{H^{\ell}(0,L)}\leq\|T_{N}^{-1}\|_{H^{\ell}(0,L)\to
H^{\ell}(0,L)}\|u_{0}\|_{H^{\ell}(0,L)}.$ (3.18)

Combining (3.17) and (3.18), we conclude that

$\|u(t)\|_{H^{\ell}(0,L)}\leq c_{k}e^{-\gamma
t}\|u_{0}\|_{H^{\ell}(0,L)},\quad\forall t\geq 0$ (3.19)

where
$c_{k}=(1+\|k\|_{H^{\ell}(\Delta_{x,y})})\|T_{N}^{-1}\|_{H^{\ell}(0,L)\to
H^{\ell}(0,L)}$ is a nonnegative constant independent of the initial datum.
(3.19) also provides continuous dependence of solutions on the data. Hence,
complete the proof of Theorem 1.4.

##  4\. Nonlinear theory (Proof of Theorem 1.5)

In this section, we study the well-posedness and stabilization of the
nonlinear plant (1.1). The same backstepping transformation that was used in
Section 2 relates the following target system posed on $(0,L)\times(0,T)$ to
the original nonlinear plant, where
$Fw=(I-\Phi_{N})\left[((I+\Upsilon_{k}P_{N})w)^{3}\right],$ with $\Phi_{N}$
being the operator defined in Lemma 2.2:

$\begin{cases}w_{t}-\nu w_{xx}-\alpha w+\mu P_{N}w=-Fw,\quad x\in(0,L),t>0,\\\
w(0,t)=0,w(L,t)=0,\\\ w(x,0)=w_{0}(x).\end{cases}$ (4.1)

We first prove a linear nonhomogeneous estimate in $X_{T}^{0}$ assuming
$q_{0}\in L^{2}(0,L)$. Taking $L^{2}-$inner product of the main equation by
$2q$ and integrating by parts, we get

$\frac{d}{dt}\|q(t)\|^{2}+2\nu\|q_{x}(t)\|^{2}-2\alpha\|q(t)\|^{2}+2\mu\int_{0}^{L}q(t)P_{N}q(t)dx\leq
2\int_{0}^{L}|f(t)||q(t)|dx.$ (4.2)

Integrating above inequality over $[0,t]$ and using Cauchy-Schwarz inequality
in $x$, we get

$\begin{split}\min&\\{1,2\nu\\}\cdot\left(\|q(t)\|^{2}+\int_{0}^{t}\|q_{x}(s)\|^{2}ds\right)\\\
&\leq
2\alpha\int_{0}^{t}\|q(s)\|^{2}ds+2\mu\int_{0}^{t}\|q(s)\|\|P_{N}q(s)\|ds+\|q_{0}\|^{2}+2\int_{0}^{t}\|q(s)\|\|f(s)\|ds\\\
&\leq\|q_{0}\|^{2}+2(\alpha+\mu)\int_{0}^{T}\|q(s)\|^{2}ds+2\left(\operatorname*{ess\,sup}_{t\in[0,T]}\|q(t)\|\right)\cdot\|f\|_{L^{1}(0,T;L^{2}(0,T))}.\end{split}$

It follows that for a.e. $t\in[0,T]$, in view of Cauchy’s inequality, for
$\epsilon>0$, we get

$\begin{split}\|&q(t)\|^{2}+\int_{0}^{t}\|q_{x}(s)\|^{2}ds\\\ &\leq
C_{1}\|q_{0}\|^{2}+C_{2}T\operatorname*{ess\,sup}_{t\in[0,T]}\|q(t)\|^{2}+{\epsilon}\left(\operatorname*{ess\,sup}_{t\in[0,T]}\|q(t)\|^{2}\right)+C_{\epsilon}\|f\|_{L^{1}(0,T;L^{2}(0,T))}^{2},\end{split}$

where the constants at the right hand side are given by
$C_{1}=\frac{1}{\min\\{1,2\nu\\}}$,
$C_{2}=\frac{2(\alpha+\mu)}{\min\\{1,2\nu\\}}$, and
$C_{\epsilon}=\frac{1}{\epsilon\cdot\min\\{1,2\nu\\}^{2}}$. Now, taking
$\operatorname*{ess\,sup}$ with respect to $t$ on $[0,T]$, we get

$\displaystyle\|q\|_{X_{T}^{0}}\leq
c_{T}(\|q_{0}\|+\|f\|_{L^{1}(0,T;L^{2}(0,L))}),$ (4.3)

where $c_{T}=\frac{\max\\{C_{1},C_{\epsilon}\\}}{1-\epsilon-C_{2}T}>0$ with
$\epsilon$ and $T$ small enough that $\epsilon+C_{2}T<1$.

Next, we prove a linear non-homogeneous estimate in $X_{T}^{1}$ assuming
$q_{0}\in H_{0}^{1}(0,L)$. Taking the $L^{2}-$inner product of the main
equation by $-2q_{xx}$,

$\frac{d}{dt}\|q_{x}(t)\|^{2}+2\nu\|q_{xx}(t)\|^{2}-2\alpha\|q_{x}(t)\|^{2}-2\mu\int_{0}^{L}q_{xx}(x,t)P_{N}q(x,t)dx\leq
2\int_{0}^{L}|f(x,t)||q_{xx}(x,t)|dx.$

Using arguments similar to those used for (4.3), we obtain

$\|q\|_{X_{T}^{1}}\leq
c_{T}(\|q_{0}\|_{H_{0}^{1}(0,L)}+\|f\|_{L^{1}(0,T;L^{2}(0,L))}).$ (4.4)

####  4.0.1. Contraction

We assume $u_{0}\in H^{\ell}(0,L)$ where either $\ell=0$ or $\ell=1$.
Therefore, by bounded invertibilility of the backstepping transformation, we
have $w_{0}\in H^{\ell}(0,L)$ for the same value of $\ell$. We set the
solution map

$w=\Psi z(t)\doteq S(t)w_{0}+\int_{0}^{t}S(t-\tau)Fz(\tau)d\tau,$ (4.5)

where $S(t)w_{0}$ is the solution of the corresponding linear problem (1.4).
Using estimates of Section LABEL:apriori and boundedness of
$T_{N}^{-1}=I-\Phi_{N}$ on $L^{2}(0,L)$, we obtain

$\begin{split}\|\Psi
z\|_{X_{T}^{\ell}}&\leq\left\|S(t)w_{0}+\int_{0}^{t}S(t-\tau)Fz(\tau)d\tau\right\|_{X_{T}^{\ell}}\\\
&\leq
c_{T}\|w_{0}\|_{H^{\ell}(0,L)}+c_{T,k}\int_{0}^{T}\left\|(T_{N}z)^{3}(t)\right\|dt.\end{split}$

Applying the Gagliardo-Nirenberg inequality and using boundedness of the
backstepping transformation, we get

$\begin{split}\left\|\left(T_{N}z\right)^{3}(t)\right\|&=\|T_{N}z\|_{L^{6}(0,L)}^{3}\\\
&\leq c_{L}\|\partial_{x}((T_{N}z)(t)\|\|T_{N}z(t)\|^{2}\\\ &\leq
c_{k,L}\|z_{x}(t)\|\|z(t)\|^{2}\end{split}$ (4.6)

from which it follows that

$\begin{split}\int_{0}^{T}\left\|((I+\Upsilon_{k}P_{N})z)^{3}(t)\right\|dt&\leq\int_{0}^{T}\left(c_{k,L}\|z_{x}(t)\|\|z(t)\|^{2}\right)dt\\\
&\leq
c_{k,L}\sqrt{T}\|z\|_{L^{\infty}(0,T;L^{2}(0,L))}^{2}\|z_{x}\|_{L^{2}(0,T;L^{2}(0,L))}\\\
&\leq c_{k,L}\sqrt{T}\|z\|_{X_{T}^{\ell}}^{3}.\end{split}$ (4.7)

Combining (4.6)-(4.7), we get

$\|\Psi z\|_{X_{T}^{\ell}}\leq
c_{T}\left(\|w_{0}\|_{H^{\ell}(0,L)}+\sqrt{T}\|z\|_{X_{T}^{\ell}}^{3}\right),$
(4.8)

where $c_{T}$ is uniformly bounded in $T$, and it also depends on parameters
$\nu,\alpha,\mu,k,L$, which are fixed throughout.

To prove local existence, it is enough to show that the mapping (4.5) has a
fixed point for some $T>0$. Let $B_{R}^{T}\doteq\left\\{z\in
X_{T}^{\ell}\,|\,\|z\|_{X_{T}^{\ell}}\leq R\right\\}$, where $R\doteq
2c_{T}\|w_{0}\|_{H^{\ell}(0,L)}$. Then,

$\|\Psi z\|_{X_{T}^{\ell}}\leq\frac{R}{2}+c_{T}\sqrt{T}R^{3}.$

Let us choose $T>0$ small enough that $2c_{T}\sqrt{T}R^{2}<1$ holds. This
yields $\|\Psi z\|_{X_{T}^{\ell}}<R$, i.e. $\Psi$ maps $B_{R}^{T}$ onto itself
for sufficiently small $T>0$.

Next, we show that $\Psi$ is a contraction on $B_{R}^{T}$ for sufficiently
small $T$. Let $z_{1},z_{2}\in B_{R}^{T}$. Then

$\begin{split}\left\|\Psi z_{1}-\Psi
z_{2}\right\|_{X_{T}^{\ell}}=&\left\|\int_{0}^{t}S(\cdot-s)\left(Fz_{1}-Fz_{2}\right)ds\right\|_{X_{T}^{\ell}}\\\
\leq&c_{T}\int_{0}^{T}\left\|Fz_{1}-Fz_{2}\right\|dt\\\
=&c_{T}\int_{0}^{T}\left\|T_{N}^{-1}\left[\left(T_{N}z_{1}\right)^{3}-\left(T_{N}z_{2}\right)^{3}\right]\right\|dt\\\
\leq&c_{T}\int_{0}^{T}\left\|\left(T_{N}z_{1}\right)^{3}-\left(T_{N}z_{2}\right)^{3}\right\|dt\\\
\leq&c_{T}\int_{0}^{T}\left\|\left(T_{N}(z_{1}-z_{2})\right)\left(\left(T_{N}z_{1}\right)^{2}+\left(T_{N}z_{2}\right)^{2}\right)\right\|dt\\\
\leq&c_{T}\int_{0}^{T}\left(\|T_{B}z_{1}\|_{\infty}^{2}+\|T_{N}z_{2}\|_{\infty}^{2}\right)\|T_{N}(z_{1}-z_{2})\|dt\\\
\leq&c_{T}\int_{0}^{T}\left(\|\partial_{x}z_{1}\|\|z_{1}\|+\|\partial_{x}z_{2}\|\|z_{2}\|\right)\|z_{1}-z_{2}\|dt\\\
\leq&c_{T}\sqrt{T}(\|z_{1}\|_{X_{T}^{\ell}}^{2}+\|z_{2}\|_{X_{T}^{\ell}}^{2})\|z_{1}-z_{2}\|_{X_{T}^{\ell}}.\end{split}$

where we used the boundedness of the backstepping transformation as well as
its inverse. For sufficiently small $T>0$, we guarantee that
$c_{T}T^{\frac{1}{2}}R^{2}<\frac{1}{2}$. Hence $\Psi$ becomes a contraction on
a closed ball $B_{R}^{T}$. This yields a unique solution in $B_{R}^{T}\subset
X_{T}^{\ell}$.

###  4.1. Stabilization

In this section, we prove the exponential decay for solutions of (4.1) in
$H^{\ell}(0,L)$ for $\ell=0$ and $\ell=1$.

####  4.1.1. $L^{2}-$decay

Let $u_{0}\in L^{2}(0,L)$ so that $w_{0}=T_{N}^{-1}u_{0}\in L^{2}(0,L)$.
Taking $L^{2}-$inner product of (4.1) by $2w$:

$\begin{split}\frac{d}{dt}\|w(t)\|^{2}&+2\nu\|w_{x}(t)\|^{2}-2\alpha\|w(t)\|^{2}+2\mu\int_{0}^{L}w(x,t)P_{N}w(x,t)dx\\\
&=-2\int_{0}^{L}T_{N}^{-1}\left[(T_{N}w)^{3}\right]w(x,t)dx.\end{split}$ (4.9)

Applying Cauchy-Schwarz inequality and using boundedness of $T_{N}^{-1}$ on
$L^{2}(0,L)$, the term at the right hand side of (4.9) is dominated by
$2c_{0}\left\|(T_{N}w)^{3}(t)\right\|\|w(t)\|,$ where

$c_{0}=\|T_{N}^{-1}\|_{2\rightarrow 2}.$ (4.10)

Applying arguments similar to those in (4.6) and using Cauchy’s inequality
with $\epsilon$, we get

$2c_{0}\left\|\left(T_{N}w\right)^{3}(t)\right\|\|w(t)\|\leq
2c_{0}c_{1}\|w_{x}(t)\|\|w(t)\|^{3}\leq\epsilon\|w_{x}(t)\|^{2}+\frac{c_{0}^{2}c_{1}^{2}}{\epsilon}\|w(t)\|^{6},$

where

$c_{1}=c^{*}\|T_{N}\|_{H^{1}(0,L)\rightarrow
H^{1}(0,L)}\|T_{N}\|_{2\rightarrow 2}^{2}$ (4.11)

with $c^{*}$ being the Gagliardo-Nirenberg constant. Using

$2\mu\int_{0}^{L}w(x,t)P_{N}w(x,t)dx\geq(2\mu-\epsilon_{1}\mu)\|w(t)\|^{2}-\frac{\mu}{\epsilon_{1}}\|(w-P_{N}w)(t)\|^{2}$

with $\epsilon_{1}=1/(N+1)$, we get

$\begin{split}\frac{d}{dt}&\|w(t)\|^{2}+(2\nu-\epsilon)\|w_{x}(t)\|^{2}+\left(2\mu-\frac{\mu}{N+1}-2\alpha\right)\|w(t)\|^{2}\\\
&-\mu(N+1)\|\left(w-P_{N}w\right)(t)\|^{2}\leq\frac{c_{0}^{2}c_{1}^{2}}{\epsilon}\|w(t)\|^{6}.\end{split}$
(4.12)

Thanks to the Poincaré type inequality, we have

$\begin{split}\frac{d}{dt}&\|w(t)\|^{2}+2(\nu-\frac{\epsilon}{2}-\frac{\mu}{2\lambda_{1}(N+1)})\|w_{x}(t)\|^{2}\\\
&+\left(2\mu-\frac{\mu}{N+1}-2\alpha\right)\|w(t)\|^{2}\leq\frac{c_{0}^{2}c_{1}^{2}}{\epsilon}\|w(t)\|^{6}.\end{split}$
(4.13)

One can find $\epsilon$ such that

$\nu-\frac{\mu}{2\lambda_{1}(N+1)}\geq\frac{\epsilon}{2}>0$

provided the same condition (3.5) on $N$ given in the linear theory holds. For
such $N$, using Poincaré inequality above, we get

$\frac{d}{dt}\|w(t)\|^{2}+2\left(\gamma-\frac{\epsilon\lambda_{1}}{2}\right)\|w(t)\|^{2}\leq\frac{c_{0}^{2}c_{1}^{2}}{\epsilon}\|w(t)\|^{6},$
(4.14)

where $\gamma$ is as in (3.7). Now if $\mu>\alpha-\nu\lambda_{1}\geq 0$ and
$N$ in addition satisfies (3.8), then (4.14) implies $L^{2}-$decay provided
$\|w_{0}\|$ is small (see Proposition 4.2 below). This follows from Lemma 4.1
by taking $y(t)=\|w(t)\|^{2}$.

####  4.1.2. $H^{1}-$decay

Here, we consider the case $u_{0}\in H^{1}(0,L)$ so that
$w_{0}=T_{N}^{-1}u_{0}\in H^{1}(0,L)$. We take $L^{2}-$inner product of the
main equation in (4.1) by $-2w_{xx}$, proceed as in (3.12)-(3.13) with
$\epsilon_{2}=\frac{1}{\lambda_{1}(N+1)}$, and thereby obtain

$\begin{split}\frac{d}{dt}\|w_{x}(t)\|^{2}&+2\left(\nu-\frac{\mu}{2\lambda_{1}(N+1)}\right)\|w_{xx}(t)\|^{2}+2\left(\mu-\alpha-\frac{\mu}{2(N+1)}\right)\|w_{x}(t)\|^{2}\\\
&\leq
2\int_{0}^{L}T_{N}^{-1}\left[(T_{N}w)^{3}\right](t)w_{xx}(t)dx.\end{split}$
(4.15)

Applying Cauchy-Schwarz inequality and using boundedness of $T_{N}^{-1}$ on
$L^{2}(0,L)$, the right hand side of (4.15) can be estimated as

$2\int_{0}^{L}T_{N}^{-1}\left[(T_{N}w)^{3}\right](x,t)w_{xx}(x,t)dx\\\ \leq
2c_{0}\left\|(T_{N}w)^{3}(t)\right\|\|w_{xx}(t)\|.$ (4.16)

Proceeding as in (4.6), applying Poincaré inequality and Cauchy’s inequality
with $\epsilon>0$, we get

$\begin{split}2c_{0}\left\|(T_{N}w)^{3}(t)\right\|\|w_{xx}(t)\|&\leq\left(2c_{0}c_{1}\|w_{x}(t)\|\|w(t)\|^{2}\right)\|w_{xx}(t)\|\\\
&\leq 2c_{0}c_{1}\lambda_{1}^{-1}\|w_{x}(t)\|^{3}\|w_{xx}(t)\|\\\
&\leq\frac{c_{0}^{2}c_{1}^{2}\lambda_{1}^{-2}}{\epsilon}\|w_{x}(t)\|^{6}+\epsilon\|w_{xx}(t)\|^{2}.\end{split}$
(4.17)

Using (4.16)-(4.17) in (4.15), we get

$\begin{split}\frac{d}{dt}&\|w_{x}(t)\|^{2}+2\left(\nu-\frac{\epsilon}{2}-\frac{\mu}{2\lambda_{1}(N+1)}\right)\|w_{xx}(t)\|^{2}\\\
&+2\left(\mu-\alpha-\frac{\mu}{2(N+1)}\right)\|w_{x}(t)\|^{2}\leq\frac{c_{0}^{2}c_{1}^{2}\lambda_{1}^{-2}}{\epsilon}\|w_{x}(t)\|^{6}.\end{split}$

Again, one can find $\epsilon>0$ such that

$\nu-\frac{\mu}{2\lambda_{1}(N+1)}\geq\frac{\epsilon}{2}>0$

provided the same condition (3.5) on $N$ given in the linear theory holds. So
assuming (3.5), we can apply the Poincaré inequality and get

$\displaystyle\frac{d}{dt}\|w_{x}(t)\|^{2}+2\left(\gamma-\dfrac{\epsilon\lambda_{1}}{2}\right)\|w_{x}(t)\|^{2}\leq\dfrac{c_{0}^{2}c_{1}^{2}\lambda_{1}^{-2}}{\epsilon}\|w_{x}(t)\|^{6},$

where $\gamma>0$ is given by (3.7). If $\mu>\nu\lambda_{1}-\alpha$, then
(4.1.2) yields $H^{1}-$decay of solutions. This can be obtained from the
following lemma by setting $y(t)=\|w_{x}(t)\|^{2}$.

######  Lemma 4.1.

Let $a,b>0$, $d\in(0,1)$, and $y=y(t)>0$ satisfy

$\displaystyle y^{\prime}(t)+ay(t)-by^{3}(t)\leq 0,\quad t\geq 0.$ (4.18)

If $y(0)\leq d\sqrt{\frac{a}{b}}$, then
$y(t)\leq\frac{1}{\sqrt{1-d^{2}}}y(0)e^{-at}$, $t\geq 0$.

###### Proof.

(4.18) is a Bernoulli type differential inequality. Solving this inequality,
we obtain

$\displaystyle
y^{2}(t)\leq\left(\left(\dfrac{1}{y^{2}(0)}-\dfrac{b}{a}\right)e^{2at}+\dfrac{b}{a}\right)^{-1}.$

The result follows from the assumption $y(0)\leq d\sqrt{\frac{a}{b}}$. ∎

Applying the above lemma to the $H^{\ell}-$estimates, we establish the
proposition below for the target system.

######  Proposition 4.2.

Let $\nu,\alpha>0$, $\mu>\alpha-\nu\lambda_{1}\geq 0$, $d\in(0,1)$, $w_{0}\in
H^{\ell}(0,L)$, $\ell=0$ or $\ell=1$. Let $\epsilon>0$ be any number
satisfying
$\epsilon\lambda_{1}\leq\max\left\\{\nu-\frac{\mu}{N+1},2\gamma\right\\}$,
suppose $\|w_{0}^{(\ell)}\|_{2}^{2}\leq
d\frac{\sqrt{\epsilon(2\gamma-\epsilon\lambda_{1})}\lambda_{1}^{\ell}}{c_{0}c_{1}}$
and $N$ satisfies (3.10). Then, the solution of (4.1), which belongs to
$X_{T}^{\ell}$, has the decay estimate

$\|w(t)\|_{H^{\ell}(0,L)}\lesssim
e^{-\left(\gamma-\frac{\epsilon\lambda_{1}}{2}\right)t}\|w_{0}\|_{H^{\ell}(0,L)},\text{
for a.e. }t\geq 0,$

where $\gamma$ is given by (3.7), $c_{0},c_{1}$ are defined in (4.10) and
(4.11), respectively.

Proceeding as in (3.17)-(3.19), we conclude that well-posedness and
exponential decay results for the target model are also true for the original
plant. Hence, we have Theorem 1.5.

######  Remark 4.3.

Recall that $w_{0}=T_{N}^{-1}u_{0}=(I-\Phi_{N})u_{0}$, therefore

$\|w_{0}\|_{H^{\ell}(0,L)}\leq\|T_{N}^{-1}\|_{H^{\ell}(0,L)\rightarrow
H^{\ell}(0,L)}\|u_{0}\|_{H^{\ell}(0,L)}$

Hence, the smallness condition of Proposition 4.2 will hold provided

$\|u_{0}^{(\ell)}\|_{2}^{2}\leq
d\frac{\sqrt{\epsilon(2\gamma-\epsilon\lambda_{1})}\lambda_{1}^{\ell}}{c_{0}c_{1}}{\|T_{N}^{-1}\|_{H^{\ell}(0,L)\rightarrow
H^{\ell}(0,L)}^{-2}},$

which characterizes the domain of attraction for the original plant. Note that
the above estimate is still implicit, however with the help of the definition
of $\Phi_{N}$, one may be able to find an upper bound for the quantities
$c_{0}$ and $\|T_{N}^{-1}\|_{H^{\ell}(0,L)\to H^{\ell}(0,L)}$, which would
yield an explicit estimate for the domain of attraction.

##  5\. Minimal number of Fourier modes for decay (Proofs of Theorem 1.7 and
Theorem 1.9 )

Section 3 and Section 4 focused on the rapid stabilization problem. In this
section, we are interested in determining the minimal number of Fourier sine
modes of $u$ to gain exponential stabilization of zero equilibrium for some
unprescribed and not necessarily large decay rate. To this end, we will assume
that we want to gain exponential stabilization with $N$ Fourier sine modes,
and we will extract the conditions that the problem parameters must satisfy.
Recall from (3.3) and (3.4) that the conditions

$\mu<2\epsilon_{1}\nu\lambda_{1}(N+1)^{2}\quad\text{and}\quad\mu>\frac{\alpha-\nu\lambda_{1}}{1-\frac{\epsilon_{1}}{2}-\frac{1}{2\epsilon_{1}(N+1)^{2}}}$

must be satisfied by $\mu$. To guarantee that such $\mu$ exists, we must have

$\frac{\alpha-\nu\lambda_{1}}{1-\frac{\epsilon_{1}}{2}-\frac{1}{2\epsilon_{1}(N+1)^{2}}}<2\epsilon_{1}\nu\lambda_{1}(N+1)^{2},$
(5.1)

which holds provided

$\frac{\alpha}{\nu}<\lambda_{1}(N+1)^{2}(2\epsilon_{1}-\epsilon_{1}^{2})=\lambda_{N+1}(2\epsilon_{1}-\epsilon_{1}^{2}).$
(5.2)

Note that for $\lambda_{j+1}>\frac{\alpha}{\nu}>\lambda_{j}$, where
$j\in\\{1,2,\dotsc\\}$, the first $j$ Fourier modes are unstable while the
$(j+1)-$th Fourier mode is stable. So let us maximize the right hand side of
(5.2) in order to maximize the range selection for $\frac{\alpha}{\nu}$. This
is done with the choice $\epsilon_{1}=1$, which yields

$\frac{\alpha}{\nu}<\lambda_{N+1}.$ (5.3)

Similarly, from (3.14) and (3.15), we need

$\mu<\frac{2\nu}{\epsilon_{2}}\quad\text{and}\quad\mu>\frac{\alpha-\nu\lambda_{1}}{1-\frac{\epsilon_{2}\lambda_{1}}{2}-\frac{1}{2\epsilon_{2}\lambda_{1}(N+1)^{2}}}.$
(5.4)

Combining these inequalities, we deduce that we must have

$\frac{\alpha}{\nu}<\frac{2}{\epsilon_{2}}-\frac{1}{\epsilon_{2}^{2}\lambda_{1}(N+1)^{2}}$

in order to guarantee that $\mu$ satisfying (5.4) exists. We again maximize
the right hand side with $\epsilon_{2}=\frac{1}{\lambda_{1}(N+1)^{2}}$, and
therefore get the same inequality given by (5.3). Consequently, as long as the
$(N+1)-$th Fourier mode is stable, we can achieve exponential stabilization of
zero equilibrium by using a boundary feedback controller involving only the
first $N$ Fourier modes. Now using the above choices of $\epsilon_{1}$ and
$\epsilon_{2}$ in (3.4) and (3.15), respectively, one can see that the decay
rate becomes

$\rho=\nu\lambda_{1}-\alpha+\frac{\mu}{2}\left(1-\frac{1}{(N+1)^{2}}\right)>0,$

where $\mu$ obeys

$\displaystyle
2(\alpha-\nu\lambda_{1})\left(1-\frac{1}{(N+1)^{2}}\right)^{-1}<\mu<2\nu\lambda_{N+1}.$
(5.5)

Since conditions (5.1) and (5.4) above are required for the stabilization of
both linear and nonlinear target models, the conclusion we obtain here is
valid for both. The resuts are stated in Theorem 1.7 and Theorem 1.9.

##  6\. Numerical algorithm and simulations

Notice that our control input involves the integral operator $\Upsilon_{k}$
and the inverse backstepping operator $I-\Phi_{N}$. So, before deriving a
numerical scheme for our continuous models, we need to express discrete
counterparts of these operators.

To this end, let $N_{x}>1$ be a positive integer, $1\leq j\leq i\leq N_{x}$,
and $(x_{i},y_{j})$ be distinct points of the triangular region
$\Delta_{x,y}$, where

$\delta_{x}=\frac{L}{N_{x}-1},\quad x_{i}=(i-1)\delta_{x},\quad
y_{j}=(j-1)\delta_{x}.$

We derive $k$ approximately by setting

$k^{M}(x_{i},y_{j})=-\frac{\mu
y_{i}}{2\nu}\sum_{m=0}^{M}\left(-\frac{\mu}{4\nu}\right)^{m}\frac{(x_{i}^{2}-y_{j}^{2})^{m}}{m!(m+1)!},$

where $M$ is an appropriate value in the sense that $\underset{1\leq j\leq
i\leq N_{x}}{\max}|k^{M+1}(x_{i},y_{j})-k^{M}(x_{i},y_{j})|$ is less than a
tolerance value. Using $k^{M}$ and applying composite trapezoidal rule for the
integration, discrete counterpart, $\mathbf{\Upsilon}_{k}^{h}$, of
$\Upsilon_{k}$ can be expressed by an $N_{x}$ dimensional square matrix with
elements

$(\mathbf{\Upsilon}_{k}^{h})_{i,j}=\begin{cases}0,&\text{if }j>i,\\\
\frac{\delta_{x}}{2}k(x_{i},x_{j}),&\text{if }j=i,\\\
\delta_{x}k(x_{i},x_{j}),&\text{else}.\end{cases}$

Discrete counterpart of the operator $P_{N}$, say $\mathbf{P}_{N}^{h}$, can
also be expressed by an ${N_{x}}$ dimensional square matrix
$\mathbf{P}_{N}^{h}=\delta_{x}\mathbf{W}\mathbf{W}^{T}$, where $\mathbf{W}$ is
an $N_{x}\times N$ dimensional matrix with elements

$(\mathbf{W})_{j,n}=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi
x_{j}}{L}\right),\quad n=1,\dotsc,N.$

Next, we obtain an approximation to $\Phi_{N}u$ using the iteration

$\displaystyle\Phi_{N}u$ $\displaystyle\equiv$
$\displaystyle(I-\Phi_{N-1})[\Upsilon_{k}P_{N}u]-\dfrac{\left((I-\Phi_{N-1})[\Upsilon_{k}P_{N}u],e_{N}\right)_{2}}{1+\left((I-\Phi_{N-1})[\Upsilon_{k}e_{N}],e_{N}\right)_{2}}(I-\Phi_{N-1})[\Upsilon_{k}e_{N}],$
(6.1) $\displaystyle\Phi_{0}u$ $\displaystyle=$ $\displaystyle 0.$

Iteration (6.1) can be viewed as a mapping, say $\Psi$, from $p-$th level to
$(p+1)-$th level, $p=1,2,\dotsc,N-1$, since derivation of $\Phi_{N}u$ via
(6.1) requires information related with $\Phi_{N-1}$. More precisely one
explicitly needs to know $\Phi_{N-1}[\Upsilon_{k}P_{N}u]$ and
$\Phi_{N-1}[\Upsilon_{k}e_{N}]$ to derive $\Phi_{N}u$. Let us illustrate this
situation schematically as follows:

$\left.\begin{aligned} &\Phi_{N-1}[\Upsilon_{k}P_{N}u]\\\
&\Phi_{N-1}[\Upsilon_{k}e_{N}]\end{aligned}\right\\}\xlongrightarrow{\Psi}\Phi_{N}u.$
(6.2)

However, $\Phi_{N-1}[\Upsilon_{k}P_{N}u]$ and $\Phi_{N-1}[\Upsilon_{k}e_{N}]$
are unknowns as well, since they depend on $\Phi_{N-2}$ due to iteration
(6.1). In other words, considering (6.1) for $N-1$ and replacing $u$ with
$\Upsilon_{k}P_{N}u$ or $\Upsilon_{k}e_{N}$, we see that we need information
about $\Phi_{N-2}[(\Upsilon_{k}P_{N-1})(\Upsilon_{k}P_{N})u]$ and
$\Phi_{N-2}[\Upsilon_{k}e_{N-1}]$ or
$\Phi_{N-2}[(\Upsilon_{k}P_{N-1})(\Upsilon_{k}e_{N})]$ and
$\Phi_{N-2}[\Upsilon_{k}e_{N-1}]$, respectively. Together with the iteration
(6.2), let us illustrate this as the following scheme:

$\left.\begin{aligned} \left.\begin{aligned}
&\Phi_{N-2}[(\Upsilon_{k}P_{N-1})(\Upsilon_{k}P_{N})u]\\\
&\Phi_{N-2}[\Upsilon_{k}e_{N-1}]\end{aligned}\right\\}\xlongrightarrow{\Psi}&\Phi_{N-1}[\Upsilon_{k}P_{N}u]\\\
\left.\begin{aligned} &\Phi_{N-2}[(\Upsilon_{k}P_{N-1})(\Upsilon e_{N})]\\\
&\Phi_{N-2}[\Upsilon_{k}e_{N-1}]\end{aligned}\right\\}\xlongrightarrow{\Psi}&\Phi_{N-1}[\Upsilon_{k}e_{N}]\end{aligned}\right\\}\xlongrightarrow{\Psi}\Phi_{N}u.$
(6.3)

The functions lying at the most left end of scheme (6.3) are unknown since
they depend on $\Phi_{N-3}$. Iteratively, each backward step to determine
$\Phi_{j}$ gives rise to a new unknown including $\Phi_{j-1}$, $j=3,\dotsc,N$.
Therefore, in order to understand what total information is required to derive
$\Phi_{N}u$, we need to go backward step by step until we reach $\Phi_{2}$,
since it depends on $\Phi_{1}$, which is already known since
$\Phi_{0}\varphi=0$ implies

$\Phi_{1}\varphi=\frac{1}{1+\beta_{1}}\Upsilon_{k}P_{1}\varphi,$

where $\beta_{1}=\int_{0}^{L}e_{1}(s)[\Upsilon_{k}e_{1}](s)ds$. For example,
if one needs to find $\Phi_{3}u$, then one can see by taking $N=3$ on (6.3)
that $(\Upsilon_{k}P_{1})[\Upsilon_{k}e_{2}]$,
$(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})[\Upsilon_{k}e_{3}]$ and
$(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})(\Upsilon_{k}P_{3})u$ are required
inputs. In general, to derive $\Phi_{N}u$ explicitly, we need

$(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})\dotsm(\Upsilon_{k}P_{N-1})(\Upsilon_{k}P_{N})u$
(6.4)

and

$\displaystyle(\Upsilon_{k}P_{1})[\Upsilon_{k}e_{2}]$ (6.4–2)
$\displaystyle(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})[\Upsilon_{k}e_{3}]$
(6.4–3) $\displaystyle\quad\vdots$
$\displaystyle(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})\dotsc(\Upsilon_{k}P_{j-1})[\Upsilon_{k}e_{j}]$
(6.4–j) $\displaystyle\quad\vdots$
$\displaystyle(\Upsilon_{k}P_{1})(\Upsilon_{k}P_{2})\dotsc(\Upsilon_{k}P_{j-1})\dotsc(\Upsilon_{k}P_{N-1})[\Upsilon_{k}e_{N}]$
(6.4–N)

as an input.

Algorithm 1 gives a numerical construction of this.

0: $\varphi$

1: if $N==1$ then

2:
$\mathbf{\Phi_{N}^{h}}u\leftarrow\frac{1}{1+\beta_{1}}\mathbf{\Upsilon_{k}^{h}}\mathbf{P_{N}^{h}}\varphi;$

3: else if $N>1$ then

4: $K_{\text{old}}(1)\leftarrow\text{expression }\eqref{inviter_input1};$

5: for $j=2\to N$ do

6: $K_{\text{old}}(j)\leftarrow\text{expression }\eqref{inviter_inputj};$

7: end for

8: $i\leftarrow N;$

9: for $p=2\to i$ do

10: $K_{\text{new}}\leftarrow\mathbf{0}_{i-1}$

11: $K_{\text{new}}(1)\leftarrow\Psi(K_{\text{old}}(1),K_{\text{old}}(2));$

12: for $j=3\to i$ do

13:
$K_{\text{new}}(j-1)\leftarrow\Psi\left(K_{\text{old}}(2),K_{\text{old}}(j)\right);$

14: end for

15: clear $K_{\text{old}};$

16: $K_{\text{old}}\leftarrow K_{\text{new}};$

17: $i\leftarrow i-1;$

18: end for

19: $\mathbf{\Phi_{N}^{h}}\varphi\leftarrow K_{\text{old}}(1);$

20: end if

Algorithm 1 Numerical algorithm for derivation of
$\mathbf{\Phi_{N}^{h}}\varphi$.

In view of the above discretization procedure, control input is of the form

$g(u(:,t))=\int_{0}^{L}k(L,y)\mathbf{P}_{N}^{h}[(\mathbf{I}_{N_{x}}-\mathbf{\Phi}_{N}^{h})u(y,t)]dy,$
(6.5)

where $\mathbf{\Phi}_{N}^{h}$ is derived by using the above iterative scheme
involving finite dimensional operators $\mathbf{\Upsilon}_{k}^{h}$,
$\mathbf{P}_{N}^{h}$ and $\mathbf{I}_{N_{x}}$ is the $N_{x}$ dimensional
identity matrix. Note that we evaluate the integral (6.5) by the composite
trapezoidal rule.

Next, we derive discrete schemes for our linear and nonlinear continuous
models.

##### Linear case.

Define the finite dimensional space
$\mathrm{X}^{h}\doteq\left\\{\mathbf{\varphi}\,|\,\mathbf{\varphi}=[\varphi_{1}\,\dotsm\,\varphi_{N_{x}}]^{T}\in\mathbb{R}^{N_{x}}\right\\}$
with the property that $\varphi_{1}=0$ and $\varphi_{N_{x}}=g(\varphi)$.
Define the operator $\mathbf{A}:\mathrm{X}^{h}\to\mathrm{X}^{h}$,

$\mathbf{A}\doteq-\nu\mathbf{\Delta}-\alpha\mathbf{I}_{N_{x}}+\mu\mathbf{P}_{N}^{h},$

where $\mathbf{\Delta}$ is the $N_{x}$ dimensional tridiagonal matrix obtained
by applying central difference approximation to the second order derivative.

Let $N_{t}$ denotes the number of time steps, $T_{\text{max}}$ is the final
time and $\delta_{t}=\frac{T_{\text{max}}}{N_{t}-1}$. Let
$\mathbf{u}^{n}=[u_{1}^{n}\,\dotsm\,u_{N_{x}}^{n}]^{T}\in\mathrm{X}^{h}$ be an
approximation of the solution $u$ at $n-$th time level. Using Crank-Nicolson
time stepping scheme, which yields unconditionally numerical stable results in
time, we end up with the following fully discrete problem for the linear
target model (1.4):

$\begin{cases}\text{Given }\mathbf{u}^{n}\in\mathrm{X}^{h},\text{ find
}\mathbf{u}^{n+1}\in\mathrm{X}^{h}\text{ such that}\\\
\left(\mathbf{I}_{N_{x}}+\frac{\delta_{t}}{2}\mathbf{A}\right)\mathbf{u}^{n+1}=\mathbf{B}_{l}^{n},\quad
n=0,1,\dotsc,N_{t},\\\ u^{n+1}_{N_{x}}=g(\mathbf{u}^{n}),\end{cases}$ (6.6)

where

$\mathbf{B}_{l}^{n}\doteq\left(\mathbf{I}_{N_{x}}-\frac{\delta_{t}}{2}\mathbf{A}\right)\mathbf{u}^{n}.$
(6.7)

Now (6.6)-(6.7) is linear in $\mathbf{u}^{n+1}$, hence can be solved directly.

See Algorithm 2 for the algorithm to get simulations for linear model.

0: Initial state $u_{0}$.

1: $\mathbf{u}^{1}\leftarrow\mathbf{u_{0}}$;

2: for $n=1\to N_{t}-1$ do

3: $u^{n+1}(1)\leftarrow 0$;

4:
$u^{n+1}(N_{x})\leftarrow\text{trapz}(y,k(L,y_{j})\mathbf{P}_{N}^{h}[(\mathbf{I}_{N_{x}}-\mathbf{\Phi}_{N}^{h})u^{n}(y_{j})])$;

5: Solve (6.6)-(6.7) for $\mathbf{u}^{n+1}$;

6: end for

Algorithm 2 Numerical algorithm for linear model.

##### Nonlinear case.

Applying the same discretization procedure as in the linear case, we obtain
the following fully discrete system of equations for nonlinear model (1.1)

$\left(\mathbf{I}_{N_{x}}+\frac{\delta_{t}}{2}\mathbf{A}\right)\mathbf{u}^{n+1}+\frac{\delta_{t}}{2}\left(\mathbf{u}^{n+1}\right)^{3}=\mathbf{B}_{l}^{n}+\mathbf{B}_{nl}^{n},$
(6.8)

where $\mathbf{B}_{l}^{n}$ is given by (6.7) and
$\mathbf{B}_{nl}^{n}=-\frac{\delta_{t}}{2}\left(\mathbf{u}^{n}\right)^{3}$.
Observe that (6.8) is nonlinear in the unknown $\mathbf{u}^{n+1}$. To solve
(6.8) for $\mathbf{u}^{n+1}$, we apply the following linearization scheme: Let
$\mathbf{u}^{n,p}$ be an approximation to the unknown $\mathbf{u}^{n+1}$ and
consider performing the iteration

$\begin{cases}\mathbf{u}^{n,p+1}=\mathbf{u}^{n,p}+\mathbf{du},\quad
p=0,1,2\dotsc,\\\ \mathbf{u}^{n,0}=\mathbf{u}^{n},\end{cases}$ (6.9)

together with

$u^{n,p+1}(N_{x})=g(\mathbf{u}^{n,p}),$ (6.10)

to obtain a better approximation, $\mathbf{u}^{n,p+1}$, until $\underset{1\leq
i\leq N_{x}}{\max}|\mathbf{du}|$ is small enough. To determine $\mathbf{du}$
in each iteration, we replace $\mathbf{u}^{n+1}$ by
$\mathbf{u}^{n,p}+\mathbf{du}$ for the linear term and approximate the
nonlinear term $\left(\mathbf{u}^{n+1}\right)^{3}$ linearly in $\mathbf{du}$
by
$\left(\mathbf{u}^{n,p}\right)^{3}+3\mathbf{du}\left(\mathbf{u}^{n,p}\right)^{2}$.
Then, we get

$\left(\mathbf{I}_{N_{x}}+\frac{\delta_{t}}{2}\mathbf{A}\right)(\mathbf{u}^{n,p}+\mathbf{du})\frac{\delta_{t}}{2}\left(\left(\mathbf{u}^{n,p}\right)^{3}+3\mathbf{du}\left(\mathbf{u}^{n,p}\right)^{2}\right)=\mathbf{B}_{l}^{n}+\mathbf{B}_{nl}^{n}.$
(6.11)

Now we reexpress (6.11) in terms of $\mathbf{du}$ and write

$\left(\mathbf{I}_{N_{x}}+\frac{\delta_{t}}{2}\mathbf{A}+\frac{3\delta_{t}}{2}\text{diag}\left(\mathbf{u}^{n,p}\right)^{2}\right)\mathbf{du}=\mathbf{B}_{l}^{n}+\mathbf{B}_{nl}^{n}-\left(\mathbf{I}_{N_{x}}+\frac{\delta_{t}}{2}\mathbf{A}\right)\mathbf{u}^{n,p}-\frac{\delta_{t}}{2}\left(\mathbf{u}^{n,p}\right)^{3}.$
(6.12)

Here $\text{diag}\left(\mathbf{u}^{n,p}\right)^{2}$ represents $N_{x}$
dimensional diagonal matrix, where the diagonal elements are
$\left(u^{n,p}(x_{i})\right)^{2}$, $i=1,\dotsc,N_{x}$. Observe that (6.12) is
linear in $\mathbf{du}$. Consequently for given $\mathbf{u}^{n,p}$, (6.12) can
be solved directly to obtain $\mathbf{du}$. Once $\underset{1\leq i\leq
N_{x}}{\max}|\mathbf{du}|$ is small enough, we stop the iteration (6.9)-(6.10)
and set $\mathbf{u}^{n+1}=\mathbf{u}^{n,p}+\mathbf{du}$.

See Algorithm 3 for the algorithm to get simulations for nonlinear model.

0: Initial state $u_{0}$.

1: $\text{TOL}\leftarrow 10^{-12}$;

2: $\mathbf{u}^{1}\leftarrow\mathbf{u_{0}}$;

3: for $n=1\to N_{t}-1$ do

4: $u^{n+1}(1)\leftarrow 0$;

5: $p\leftarrow 0$;

6: $\mathbf{u}^{n,p}\leftarrow\mathbf{u}^{n}$;

7: $\mathbf{du}=\mathbf{1}_{N_{x}}$;

8: while $\text{max}|\mathbf{du}|>\text{TOL}$ do

9: Solve (6.12) for $\mathbf{du}$;

10: $\mathbf{u}^{n,p+1}\leftarrow\mathbf{u}^{n,p}+\mathbf{du}$;

11:
$u^{n,p+1}(N_{x})\leftarrow\text{trapz}(y,k(L,y_{j})\mathbf{P}_{N}^{h}[(\mathbf{I}_{N_{x}}-\mathbf{\Phi}_{N}^{h})u^{n,p}(y_{j})])$;

12: $p\leftarrow p+1$;

13: end while

14: $\mathbf{u}^{n+1}=\mathbf{u}^{n,p+1}$;

15: end for

Algorithm 3 Numerical algorithm for nonlinear model.

In the next part of this section, we present two experiments which verify our
theoretical stabilization results both for linear and nonlinear models.
Results are obtained by taking $N_{x}=1000$ spatial nodes and $N_{t}=1000$
time steps. Value $M$ for the partial sum of the series representation of
backstepping kernel $k$ is chosen such that $\underset{1\leq j\leq i\leq
N_{x}}{\max}|k^{M+1}(x_{i},y_{j})-k^{M}(x_{i},y_{j})|\sim 10^{-12}$.

##### Experiment 1 - Stabilization with a single Fourier mode.

Consider the linear model

$\begin{cases}u_{t}-u_{xx}-12u=0,\quad x\in(0,1),t>0,\\\
u(0,t)=0,u(1,t)=g(t),\quad t>0,\\\
u(x,0)=10x\left(x-\frac{1}{2}\right)(x-1)^{2},\quad x\in(0,1).\end{cases}$
(6.13)

We have $\nu=1,\alpha=12$, and $\lambda_{1}=\pi^{2}<12<4\pi^{2}=\lambda_{2}$.
So, in the absence of control, zero equilibrium is instable. As the
instability level is one, we should be able to stabilize the zero equilibrium
by a controller involving a single Fourier mode. So let us take $N=1$. Then,
in order to fulfill the condition (5.5), $\mu$ should be such that

$\frac{8(12-\pi^{2})}{3}<\mu<8\pi^{2}.$

We take $\mu=6$. With this choice $1+\beta_{1}=0.632\neq 0$, so
$(\mu,N)=(6,1)$ is an admissible decay rate-mode pair. Thus, with these
problem parameters, we ensure that Lemma 2.2 holds.

See Figure 1 for numerical simulations of the stabilized linear model (6.13).

(a) Time evolution of the solution.

(b) Time evolution of $L^{2}-$norm of the solution.

Figure 1. Simulations for the stabilized linear model (6.13).

##### Experiment 2 - Rapid stabilization of a nonlinear model

Now we consider the nonlinear model

$\begin{cases}u_{t}-u_{xx}-15u+u^{3}=0,\quad x\in(0,1),t>0,\\\
u(0,t)=0,u(1,t)=g(t),\quad t>0,\\\ u(x,0)=\sin(2\pi x)-\frac{1}{2}\sin(3\pi
x),\quad x\in(0,1).\end{cases}$ (6.14)

We have $\alpha-\nu\lambda_{1}=15-\pi^{2}>0$. Therefore, if there is no
control input acting on the model, then zero equilibrium is unstable and the
given initial state will evolve into a nontrivial equilibrium in time (see
Figure 2).

Figure 2. Time evolution of the solution of uncontrolled nonlinear model
(6.14).

Let us take $\mu=15>\alpha-\nu\lambda_{1}=15-\pi^{2}$. Then choosing $N=2$
fulfills the condition

$N=2>\max\left\\{\frac{\mu}{2\nu\lambda_{1}}-1,\frac{\mu}{\mu+\nu\lambda_{1}-\alpha}-1\right\\}=\max\left\\{\frac{15}{2\pi^{2}}-1,\frac{15}{\pi^{2}}-1\right\\}.$

With this choice of $\mu$ and $N$, we have $1+\beta_{1}=1.746\neq 0$ and
$1+\left((I-\Phi_{1})[\Upsilon_{k}e_{2}],e_{2}\right)_{2}=0.845\neq 0$. Hence
$(\mu,N)=(15,2)$ is an admissible decay rate-mode pair and we ensure that
Lemma 2.2 holds.

See Figure 3 for the numerical simulation of the stabilized linear model
(6.13).

(a) Time evolution of the solution.

(b) Time evolution of $L^{2}-$norm of the solution.

Figure 3. Simulations for the stabilized nonlinear model (6.14).

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