# Input-to-state stabilization of $1$-D parabolic equations with Dirichlet
boundary disturbances under boundary fixed-time control

Jun Zheng1,2,⋆<EMAIL_ADDRESS>Guchuan Zhu2<EMAIL_ADDRESS>1School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan
611756, P. R. of China
2Department of Electrical Engineering, Polytechnique Montréal, P.O. Box 6079,
Station Centre-Ville, Montreal, QC, Canada H3T 1J4

###### Abstract

This paper addresses the problem of stabilization of $1$-D parabolic equations
with destabilizing terms and Dirichlet boundary disturbances. By using the
method of backstepping and the technique of splitting, a boundary feedback
controller is designed to ensure the input-to-state stability (ISS) of the
closed-loop system with Dirichlet boundary disturbances, while preserving
fixed-time stability (FTS) of the corresponding disturbance-free system, for
which the fixed time is either determined by the Riemann zeta function or
freely prescribed. To overcome the difficulty brought by Dirichlet boundary
disturbances, the ISS and FTS properties of the involved systems are assessed
by applying the generalized Lyapunov method. Numerical simulations are
conducted to illustrate the effectiveness of the proposed scheme of control
design.

###### keywords:

Finite-time stability fixed-time stability prescribed-time stability input-to-
state stability Dirichlet boundary disturbance parabolic equation

††thanks: Corresponding author

,

##  1 Introduction

It is well-known that the backstepping method (see [19]) is effective for
designing boundary controllers to stabilize partial differential equations
(PDEs) with destabilizing terms; see, e.g., [2, 4, 9, 15, 22, 25, 32, 33] for
PDEs with different types of coefficients, and [1, 6, 11, 17, 18, 10, 27] for
coupled PDEs, respectively. In addition, under the framework of input-to-state
stability (ISS) theory of PDEs, in [16, 26, 41], the method of backstepping
was used to design boundary controllers to ensure the ISS of parabolic PDEs
with destabilizing terms when boundary disturbances are involved.

It is worth mentioning that a notable feature in the problem of input-to-state
stabilization is that the disturbance-free system in closed loop is
asymptotically stable under the designed controller. However, many control
systems exhibit a property stronger than the asymptotic stability, that is the
finite-time stability, which requires the state of the disturbance-free system
to reach equilibrium within a finite time [23, 12, 14, 20, 21, 24, 32, 34,
40].

In the past few years, the problem of finite-time stabilization of PDEs has
attracted much attention and significant results have been obtained [3, 5, 7,
13, 28, 29, 31, 30, 37, 38, 36, 35, 39]. Notably, by using backstepping, the
authors of [5] designed boundary controllers to achieve finite-time stability
of parabolic PDEs in the absence of external disturbances, while the authors
of [13] designed state-dependent switching control laws to ensure the so-
called dual properties of the heat equation in closed loop, namely, the heat
equation is finite-time stable in the absence of external disturbances, while
preserving the ISS in the presence of distributed in-domain disturbances.
Nevertheless, for parabolic PDEs with destabilizing terms and boundary
disturbances, the so-called dual properties of the closed-loop system have not
been studied under the backstepping control.

In this paper, we study the stabilization problem of $1$-D linear parabolic
equations involving not only distributed in-domain but also Dirichlet boundary
disturbances. Moreover, unlike [13], the open-loop system considered in this
paper is allowed to be unstable. Based on the technique of splitting and the
method applied in [5], a boundary controller is designed by means of
backstepping to simultaneously ensure the ISS of the closed-loop system and
the finite-time stability of the disturbance-free closed-loop system. In
particular, the finite time is independent of initial data.

It should be noted that, compared to [13], the control design scheme proposed
in this paper is much easier while Dirichlet boundary disturbances present
more challenges in the stability analysis due to the fact that the classical
Lyapunov method is not suitable in the appearance of Dirichlet boundary
disturbances. To avoid using state-dependent switching control laws presented
in [13], we apply the technique of splitting in the control design via
backstepping, while, to overcome the difficulty in the stability analysis, we
handle the terms of Dirichlet boundary disturbances by using the generalized
Lyapunov method, which was proposed in [42, Remark 4.1] recently. More
specifically, we split first the considered system into two sub-systems, among
which one is a disturbed sub-system with zero initial datum while another one
is a disturbance-free sub-system with non-zero initial datum. Then, for the
disturbed sub-system, by using backstepping, we design a boundary controller
via a time-invariant kernel function. By using the generalized Lyapunov method
proposed in [42], we prove that the boundedness of the solution to this
subsystem is determined only by the external disturbances. For the
disturbance-free sub-system, following [5], we design fixed-time boundary
controllers over different time intervals, and derive sufficient conditions
for time segments to ensure the fixed-time stability of the sub-system. In
particular, the fixed time is either chosen based on the Riemann zeta function
or freely prescribed. Finally, by using controllers of the two sub-systems, we
design a boundary controller for the original system with distributed in-
domain and Dirichlet boundary disturbances to simultaneously ensure the ISS of
the closed-loop system and the fixed-time stability of the corresponding
disturbance-free system.

The main contribution of this paper lies in the design of a boundary
controller, which ensures that the parabolic equation in closed loop is ISS
with respect to (w.r.t.) Dirichlet boundary disturbances and fixed-time stable
in the absence of external disturbances.

In the rest of the paper, we present first some basic notations. In Section 2,
we introduce the problem formulation and provide some preliminaries on kernel
functions. In Section 3, we design boundary controllers and state main results
in two cases, where the finite time is determined by the Riemann zeta function
and freely prescribed, respectively. In Section 4, we assess stability for the
closed-loop system under the designed boundary control. Numerical simulations
are provided in Section 5 to illustrate the effectiveness of the proposed
scheme. Some concluding remarks are given in Section 6.

#### Notation.

Let $\mathbb{N}$ and $\mathbb{N}_{+}$ be the sets of non-negative and positive
integers, respectively. Let $\mathbb{R}:=(-\infty,+\infty)$, $\mathbb{R}_{\geq
0}:=[0,+\infty)$, $\mathbb{R}_{>0}:=(0,+\infty)$, and $\mathbb{R}_{\leq
0}:=(-\infty,0]$.

For $\tau\in\mathbb{R}_{>0}$, let $Q_{\tau}:=(0,1)\times(0,\tau)$. Let
$Q_{\infty}:=(0,1)\times(0,+\infty)$ and
$\overline{Q}_{\infty}:=[0,1]\times[0,+\infty)$. For
$v:D\rightarrow\mathbb{R}$ with $D\subset\overline{Q}_{\infty}$, the notation
$v[t]$ (or $v[y]$) denotes the profile at certain $t\in{\mathbb{R}_{\geq 0}}$,
i.e., $v[t](y)=v(y,t)$.

Throughout this paper, all the functional spaces, as well as the norms in
normed linear spaces, are defined in the standard way as in, e.g., [8].

For classes of comparison functions, let

$\displaystyle\mathcal{P}:=$ $\displaystyle\\{\gamma\in C(\mathbb{R}_{\geq
0};\mathbb{R}_{\geq 0})|\gamma(0)=0,\gamma(s)>0,\forall
s\in\mathbb{R}_{>0}\\},$ $\displaystyle\mathcal{K}:=$
$\displaystyle\\{\gamma\in\mathcal{P}|\gamma\ \text{is\ strictly\
increasing}\\},$ $\displaystyle\mathcal{L}:=$ $\displaystyle\\{\gamma\in
C(\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})|\gamma\ \text{is\ strictly\
decreasing\ with}\ \lim\limits_{s\to+\infty}\gamma(s)=0\\},$
$\displaystyle\mathcal{K}\mathcal{L}:=$ $\displaystyle\\{\beta\in
C(\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0};\mathbb{R}_{\geq
0})|\beta(\cdot,t)\in\mathcal{K},\forall t\in\mathbb{R}_{\geq 0};\
\beta(s,\cdot)\in\mathcal{L},\forall s\in{{\mathbb{R}_{>0}}}\\}.$

##  2 Problem setting and preliminaries

In this section, we present the problem formulation and provide some
preliminaries.

###  2.1 Problem setting

For certain fixed time $T_{0}>0$, we consider the problem of input-to-state
stabilization for the following $1$-D parabolic equation:

$\displaystyle u_{t}(x,t)=$
$\displaystyle(a(x)u_{x}(x,t))_{x}+c(x)u(x,t)+f(x,t),(x,t)\in Q_{T_{0}},$ (1a)
$\displaystyle u(0,t)=$ $\displaystyle{d_{0}(t)},t\in(0,T_{0}),$ (1b)
$\displaystyle u(1,t)=$ $\displaystyle U(t)+d_{1}(t),t\in(0,T_{0}),$ (1c)
$\displaystyle u(x,0)=$ $\displaystyle u_{0}(x),x\in(0,1),$ (1d)

where $u_{0}$ is the initial datum, $f$ represents the in-domain disturbance,
$d_{0},d_{1}$ represent Dirichlet boundary disturbances, and $U(t)$ is the
control input, which will be designed later.

Throughout this paper, we assume that $a\in H^{2}(0,1)$ and there exists a
constant $\Lambda\geq 1$ such that

$\displaystyle\frac{1}{\Lambda}\leq a(x)\leq\Lambda,\forall x\in(0,1).$

We assume that $c\in H^{1}(0,1)$, $f\in C((0,1)\times\mathbb{R}_{\geq 0})$,
$d_{0},d_{1}\in C(\mathbb{R}_{\geq 0})$, and the initial datum $u_{0}\in
L^{2}(0,1)$.

The goal of this paper is to design a boundary control law $U(t)$ by using
backstepping such that the following so-called dual properties, which was
proposed in [13], hold true simultaneously, i.e.,

  1. (i)

the disturbed system (1), i.e., system (1) with
$f^{2}+d_{0}^{2}+d_{1}^{2}\not\equiv 0$, is input-to-state stable in the
spatial $L^{2}$-norm w.r.t. in-domain and boundary disturbances, namely, there
exist functions $\beta\in\mathcal{K}\mathcal{L}$ and
$\gamma,\gamma_{0},\gamma_{1}\in\mathcal{K}$ such that

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$
$\displaystyle\beta\left(\|u_{0}\|_{L^{2}(0,1)},t\right)+\gamma\left(\|f\|_{L^{\infty}({Q_{t}})}\right)+\gamma_{0}\left(\|d_{0}\|_{{L^{\infty}(0,t)}}\right)+\gamma_{1}\left(\|d_{1}\|_{{L^{\infty}(0,t)}}\right),\forall
t\in(0,T_{0});$

  2. (ii)

the disturbance-free system (1), i.e., system (1) with $f\equiv d_{0}\equiv
d_{1}\equiv 0$, is fixed-time stable in the spatial $L^{2}$-norm in time
$T_{0}>0$, i.e., $\|u[t]\|_{L^{2}(0,1)}\to 0$ as $t\to T_{0}^{-}$, where the
time $T_{0}$ is determined by the Riemann zeta function or freely prescribed.

###  2.2 Preliminaries

Throughout this paper, let $\lambda_{0}>0$ be a fixed constant, and define
$D:=\\{(x,y)\in[0,1]\times[0,1]|y\leq x\\}$ and

$\displaystyle\frac{\operatorname{d}\\!{}}{\operatorname{d}\\!{x}}g(x,x):=g_{x}(x,x)+g_{y}(x,x),\forall
g\in H^{1}(D).$

We present some results on the existence and _a priori_ estimates of kernel
functions, which will be used in the control design.

######  Lemma 1 ([5, Corollary 1]).

For every $\lambda\geq\lambda_{0}$, the kernel equation

$\displaystyle(a(x)k_{x}(x,y))_{x}-(a(y)k_{y}(x,y))_{y}=$
$\displaystyle(\lambda+c(y))k(x,y),\forall(x,y)\in D,$ (2a) $\displaystyle
2a(x)\frac{\operatorname{d}\\!{}}{\operatorname{d}\\!{x}}k(x,x)+a_{x}(x)k(x,x)=$
$\displaystyle-(\lambda+c(x)),\forall x\in[0,1],$ (2b) $\displaystyle k(x,0)=$
$\displaystyle 0,\forall x\in[0,1],$ (2c)

admits a unique solution $k\in H^{1}(D)$ having the estimate

$\displaystyle\|k\|_{H^{1}(D)}\leq e^{C\sqrt{\lambda}}$

for some positive constant $C$ independent of
$\lambda\in[\lambda_{0},+\infty)$.

######  Lemma 2 ([5, Corollary 2]).

For every $\lambda\geq\lambda_{0}$, the kernel equation

$\displaystyle(a(x)l_{x}(x,y))_{x}-(a(y)l_{y}(x,y))_{y}=$
$\displaystyle-(\lambda+c(y))l(x,y),\forall(x,y)\in D,$ (3a) $\displaystyle
2a(x)\frac{\operatorname{d}\\!{}}{\operatorname{d}\\!{x}}l(x,x)+a_{x}(x)l(x,x)=$
$\displaystyle-(\lambda+c(x)),\forall x\in[0,1],$ (3b) $\displaystyle l(x,0)=$
$\displaystyle 0,\forall x\in[0,1],$ (3c)

admits a unique solution $l\in H^{1}(D)$ having the estimate

$\displaystyle\|l\|_{H^{1}(D)}\leq C\lambda^{2}$

for some positive constant $C$ independent of
$\lambda\in[\lambda_{0},+\infty)$.

######  Lemma 3 ([5, Lemma 4]).

Let $k\in H^{1}(D)$ and $l\in H^{1}(D)$ be the unique solution to (2) and (3)
with $\lambda\geq\lambda_{0}$, respectively. Let $h\in L^{2}(0,1)$, and for
almost every $x\in[0,1]$ define

$\displaystyle\tilde{h}(x):=h(x)-\int_{0}^{x}k(x,y)h(y)\text{d}y.$

Then, it holds for almost every $x\in[0,1]$ that

$\displaystyle h(x):=\tilde{h}(x)+\int_{0}^{x}l(x,y)\tilde{h}(y)\text{d}y.$

######  Remark 1.

As indicated in [19, Chapter 4], for general functions $a$ and $c$ depending
on the spatial variable, there is no a closed-form solution $k$ (or $l$) to
the kernel equation (2) (or (3)). However, for positive constants $a$ and $c$,
it can be proved that (see [19, Chapter 4])

$\displaystyle k(x,y)=$
$\displaystyle-\frac{\lambda+c}{a}y\frac{I_{1}\left(\sqrt{\frac{\lambda+c}{a}(x^{2}-y^{2})}\right)}{\sqrt{\frac{\lambda+c}{a}(x^{2}-y^{2})}},$
$\displaystyle l(x,y)=$
$\displaystyle-\frac{\lambda+c}{a}y\frac{J_{1}\left(\sqrt{\frac{\lambda+c}{a}(x^{2}-y^{2})}\right)}{\sqrt{\frac{\lambda+c}{a}(x^{2}-y^{2})}},$

where $I_{1}$ is the first-order modified Bessel function given by

$\displaystyle
I_{1}(x):=\sum_{m=0}^{\infty}\frac{\left(\frac{x}{2}\right)^{2m+1}}{m!(m+1)!},$

and $J_{1}$ is the first-order Bessel function given by

$\displaystyle
J_{1}(x):=\sum_{m=0}^{\infty}\frac{(-1)^{m}\left(\frac{x}{2}\right)^{2m+1}}{m!(m+1)!}.$

##  3 Control design

In this section, we design boundary controllers in two cases:

Case I the fixed time $T_{0}$ is determined by the Riemann zeta function;

Case II the fixed time $T_{0}$ is freely prescribed.

The main method adopted in the control design is backstepping applied in [5]
and the technique of splitting, for which we consider $u$ as $u:=v+w$ with $v$
and $w$ being the solutions of a disturbance-free system and a disturbed
system under different boundary controls, respectively.

###  3.1 A lemma on fixed-time stability of disturbance-free system

Throughout this paper, let $\\{t_{n}\\}_{n\in\mathbb{N}}$ be a strictly
increasing sequence of real numbers such that $t_{0}=0$. For given constants
$\lambda_{n}\geq\lambda_{0}$, let $k_{n}\in H^{1}(D)$ be the unique solution
to the kernel equation corresponding to $n\in\mathbb{N}$:

$\displaystyle(a(x)k_{n,x}(x,y))_{x}-(a(y)k_{n,y}(x,y))_{y}=$
$\displaystyle(\lambda_{n}+c(y))k_{n}(x,y),\forall(x,y)\in D,$ $\displaystyle
2a(x)\frac{\operatorname{d}\\!{}}{\operatorname{d}\\!{x}}k_{n}(x,x)+a_{x}(x)k_{n}(x,x)=$
$\displaystyle-(\lambda_{n}+c(x)),\forall x\in[0,1],$ $\displaystyle
k_{n}(x,0)=$ $\displaystyle 0,\forall x\in[0,1].$

Motivated by [5], we consider the disturbance-free system:

$\displaystyle v_{t}(x,t)=$
$\displaystyle(a(x)v_{x}(x,t))_{x}+c(x)v(x,t),(x,t)\in[0,1]\times(t_{n},t_{n+1}),$
(4a) $\displaystyle v(0,t)=$ $\displaystyle 0,t\in(t_{n},t_{n+1}),$ (4b)
$\displaystyle v(1,t)=$ $\displaystyle V(t),t\in(t_{n},t_{n+1}),$ (4c)

with boundary control law

$\displaystyle
V(t):=\int_{0}^{1}k_{n}(1,y)v(y,t)\operatorname{d}\\!{y},\forall
t\in(t_{n},t_{n+1}),$ (5)

and initial condition

$\displaystyle v(x,t_{0}):=u_{0}(x),\forall x\in(0,1).$

Set $s_{0}:=0$ and

$\displaystyle s_{n}:=$
$\displaystyle{\sum\limits_{j=0}^{n-1}}\lambda_{j}(t_{j+1}-t_{j}),n\in\mathbb{N}_{+}.$
(6)

In [5], it was proved the following stability result, which indicates rapid
convergence of solutions to disturbance-free systems, as well as their fixed-
time stability when $n\rightarrow+\infty$.

######  Lemma 4 ([5, Proposition 1]).

For system (4) under the control law (5), there exists a positive constant
$\gamma_{0}$ depending only on $a$ and $c$, such that if, for large $n$,

$\displaystyle(t_{n+1}-t_{n})\lambda_{n}\geq\gamma_{0}\sqrt{\lambda_{n+1}},$

then, for $t\in(t_{n},t_{n+1})$,

$\displaystyle\|v[t]\|_{L^{2}(0,1)}\leq$ $\displaystyle
Ce^{-\frac{s_{n-1}}{4}+C(n-1)}\|u_{0}\|_{L^{2}(0,1)},$ (7a)
$\displaystyle|V(t)|\leq$ $\displaystyle
Ce^{-\frac{s_{n-1}}{4}+C(n-1)+C\sqrt{\lambda_{n}}}\|u_{0}\|_{L^{2}(0,1)},$
(7b)

where $C$ is a positive constant independent of $n$ and $u_{0}$.

In particular, if, in addition,

$\displaystyle\lim_{n\to+\infty}\frac{s_{n}}{n+\sqrt{\lambda_{n+1}}}=+\infty,$
(8)

and $T:=\lim_{n\to\infty}t_{n}$ exists, then, it holds that

$\displaystyle\lim_{t\to
T^{-}}\|v[t]\|_{L^{2}(0,1)}=0\quad\text{and}\quad\lim_{t\to T^{-}}V(t)=0,$ (9)

which give the fixed-time stability of system (4) in the fixed time $T$.

It is worth noting that, under the transformation

$\displaystyle\widetilde{v}(x,t):=v(x,t)-\int_{0}^{x}k_{n}(x,y)v(y,t)\operatorname{d}\\!{y},$

system (4) is converted into (see [5])

$\displaystyle\widetilde{v}_{t}(x,t)=$
$\displaystyle(a(x)\widetilde{v}_{x}(x,t))_{x}-\lambda_{n}\widetilde{v}(x,t),(x,t)\in[0,1]\times(t_{n},t_{n+1}),$
(10a) $\displaystyle\widetilde{v}(0,t)=$ $\displaystyle
0,t\in(t_{n},t_{n+1}),$ (10b) $\displaystyle\widetilde{v}(1,t)=$
$\displaystyle 0,t\in(t_{n},t_{n+1}),$ (10c)

with
$\widetilde{v}(x,t_{0}):=u_{0}(x)-\int_{0}^{x}k_{n}(x,y)u_{0}(y)\operatorname{d}\\!{y}.$
Since system (10) admits a unique (weak) solution, which is defined in the
sense of [5, (4.3) and (4.4)] and belongs to
$C([t_{n},t_{n+1});L^{2}(0,1))\cap C^{1}((t_{n},t_{n+1});L^{2}(0,1))$, by
virtue of Lemma 3, system (4) also admits a unique (weak) solution belonging
to $C([t_{n},t_{n+1});L^{2}(0,1))\cap C^{1}((t_{n},t_{n+1});L^{2}(0,1))$.

###  3.2 Control design in Case I

First, recall that the Riemann zeta function $\zeta(s)$ is given by (in the
field of real numbers)

$\displaystyle\zeta(s):=\sum_{i=1}^{\infty}\frac{1}{i^{s}},\forall s>1,$

which converges for any fixed $s>1$.

By virtue of Lemma 4 and the definition of Riemann zeta function, we set

$\displaystyle p>$ $\displaystyle 1,$ (11a) $\displaystyle t_{0}:=$
$\displaystyle 0,$ (11b) $\displaystyle t_{n+1}-{t_{n}}:=$
$\displaystyle\frac{1}{(n+1)^{p}},n\in\mathbb{N},$ (11c)
$\displaystyle\lambda_{n}:=$ $\displaystyle
n^{2(p+1)}+\lambda_{0},n\in\mathbb{N}.$ (11d)

It is clear that

$\displaystyle\frac{s_{n}}{n+\sqrt{\lambda_{n+1}}}\geq$
$\displaystyle\frac{\lambda_{n-1}(t_{n}-{t_{n-1}})}{n+\sqrt{\lambda_{n+1}}}$
$\displaystyle\geq$
$\displaystyle\frac{\frac{(n-1)^{2(p+1)}}{n^{p}}}{n+\sqrt{2}(n+1)^{p+1}}$
$\displaystyle\geq$
$\displaystyle\frac{(n-1)^{2(p+1)}}{2\sqrt{2}(n+1)^{2p+1}},\forall
n\geq\max\\{1,\lambda_{0}\\},$

which implies that the condition (8) is fulfilled.

For $p>1$, it holds that

$\displaystyle
T_{0}:=t_{0}+\sum_{i=0}^{\infty}(t_{i+1}-{t_{i}})=\sum_{i=0}^{\infty}\frac{1}{(i+1)^{p}}=\zeta(p).$

For any constant $\sigma>0$, let $k$ be the unique solution to the kernel
equation (2) with

$\displaystyle\lambda:=\sigma.$

Consider the disturbed system with zero initial datum

$\displaystyle w_{t}(x,t)=$
$\displaystyle(a(x)w_{x}(x,t))_{x}+c(x)w(x,t)+f(x,t),(x,t)\in Q_{T_{0}},$
(12a) $\displaystyle w(0,t)=$ $\displaystyle d_{0}(t),t\in(0,T_{0}),$ (12b)
$\displaystyle w(1,t)=$ $\displaystyle W(t)+d_{1}(t),t\in(0,T_{0}),$ (12c)
$\displaystyle w(x,0)=$ $\displaystyle 0,x\in(0,1),$ (12d)

where

$\displaystyle W(t):=\int_{0}^{1}k(1,y)w(y,t)\operatorname{d}\\!{y},\forall
t\in(0,T_{0}).$ (13)

To stabilize system (1) in the fixed time $T_{0}$, we define the control law:

$\displaystyle U(t):=V(t)+W(t),\forall t\in(t_{n},t_{n+1}),\forall
n\in\mathbb{N},$ (14)

where $V(t)$ and $W(t)$ are determined by (5) and (13), respectively.

We state the following theorem, which is the first main result obtained in
this paper, and its proof is provided in Section 4.

######  Theorem 1.

Let $p$, $\\{t_{n}\\}$, and $\\{\lambda_{n}\\}$ be determined by (11). Under
the control law (14), system (1) admits the following properties:

  1. (i)

the disturbance-free system (1), i.e., system (1) with $f\equiv d_{0}\equiv
d_{1}\equiv 0$, is fixed-time stable in the time $T_{0}:=\zeta(p)$, i.e.,

$\displaystyle\lim_{t\to T_{0}^{-}}\|u[t]\|_{L^{2}(0,1)}=$ $\displaystyle 0;$

  2. (ii)

the disturbed system (1), i.e., system (1) with
$f^{2}+d_{0}^{2}+d_{1}^{2}\not\equiv 0$, is ISS, having the following
estimate:

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$ $\displaystyle
C_{0}e^{-t}\|u_{0}\|_{L^{2}(0,1)}+C_{1}\left(\|f\|_{L^{\infty}{(Q_{t})}}+\|d_{0}\|_{{L^{\infty}(0,t)}}+\|d_{1}\|_{{L^{\infty}(0,t)}}\right),\forall
t\in(0,T_{0}),$

where $C_{0}$ and $C_{1}$ are positive constants independent of $n$, $u_{0}$,
$f$, $d_{0}$, and $d_{1}$.

It is worth noting that, under the transformation

$\displaystyle\widetilde{w}(x,t):=w(x,t)-\int_{0}^{x}k(x,y)w(y,t)\operatorname{d}\\!{y},$

system (12) is converted into

$\displaystyle\widetilde{w}_{t}(x,t)=$
$\displaystyle(a(x)\widetilde{w}_{x}(x,t))_{x}-\sigma\widetilde{w}(x,t)+\widetilde{f}(x,t),(x,t)\in
Q_{T_{0}},$ (15a) $\displaystyle\widetilde{w}(0,t)=$ $\displaystyle
d_{0}(t),t\in(0,T_{0}),$ (15b) $\displaystyle\widetilde{w}(1,t)=$
$\displaystyle d_{1}(t),t\in(0,T_{0}),$ (15c)
$\displaystyle\widetilde{w}(x,0)=$ $\displaystyle 0,x\in(0,1),$ (15d)

where
$\widetilde{f}(x,t):=f(x,t)-\int_{0}^{x}k(x,y)f(y,t)\operatorname{d}\\!{y}.$
Since system (15) admits a unique (weak) solution, which is defined in the
sense of [5, (4.3) and (4.4)] and belongs to $C([0,T_{0});L^{2}(0,1))\cap
C^{1}((0,T_{0});L^{2}(0,1))$, by virtue of Lemma 3, system (12) also admits a
unique (weak) solution belonging to $C([0,T_{0});L^{2}(0,1))\cap
C^{1}((0,T_{0});L^{2}(0,1))$. Therefore, the disturbed system (1) in closed
loop admits a unique (weak) solution belonging to $C([0,T_{0});L^{2}(0,1))$
with $u_{t}[t]\in L^{2}(0,1)$ for any fixed $t\in(t_{n},t_{n+1})$,
$n\in\mathbb{N}$.

It is also worth mentioning that the regularity (in the spatial variable) of
the solution to system (1) can not be improved due to the fact that the
initial datum of system (10) (or system (4)) is only in $L^{2}$-space.
However, the regularity of the solution to system (12) (or system (15)) can be
improved due to the fact that its initial datum and external disturbances are
all continuous. For instance, by taking spatial $H^{1}$-approximations of
external disturbances, it can be shown that the solutions to system (15) and
system (12) belong to $C([0,T_{0});L^{\infty}(0,1))$.

###  3.3 Control design in Case II

In this section, we design a boundary controller to stabilize system (1) in
arbitrarily given time. For this, we let $T_{0}>0$ be freely prescribed and
independent of initial conditions. By virtue of Lemma 4, we set

$\displaystyle{t_{n}}:=$ $\displaystyle
T_{0}-\frac{T_{0}}{n+1},n\in\mathbb{N},$ (16a) $\displaystyle\lambda_{n}:=$
$\displaystyle n^{6}+\lambda_{0},n\in\mathbb{N}.$ (16b)

Consider (4) and (12) with $V(t)$ and $W(t)$ being determined by (5) and (13),
respectively. To stabilize system (1) in the freely prescribed finite time
$T_{0}$, we define the control law by (14).

Note that for $n\geq\max\\{1,\lambda_{0}\\}$ we have

$\displaystyle\frac{s_{n}}{n+\sqrt{\lambda_{n+1}}}\geq$
$\displaystyle\frac{\lambda_{n-1}(t_{n}-{t_{n-1}})}{n+\sqrt{\lambda_{n+1}}}\geq\frac{T(n-1)^{6}}{2\sqrt{2}(n+1)^{5}}.$

Thus, the condition (8) is fulfilled and hence (9) holds true.

Analogous to Theorem 1, we have the following result, which can be proved in
the same way as Theorem 1, and its proof is given in Section 4.

######  Theorem 2.

Let $\\{t_{n}\\}$ and $\\{\lambda_{n}\\}$ be determined by (16). Under the
control law (14), system (1) admits the following properties:

  1. (i)

the disturbance-free system (1), i.e., system (1) with $f\equiv d_{0}\equiv
d_{1}\equiv 0$, is fixed-time stable in the freely prescribed finite time
$T_{0}$;

  2. (ii)

the disturbed system (1), i.e., system (1) with
$f^{2}+d_{0}^{2}+d_{1}^{2}\not\equiv 0$, is ISS.

######  Remark 2.

Although Theorem 1 and Theorem 2 theoretically guarantee the fixed-time
stability of the disturbance-free system, as well as the ISS of the disturbed
system, it is not feasible for practical problems to stabilize the system by
designing controllers over infinite time intervals. Nevertheless, within an
allowable error of fixed-time decay estimate, the convergence estimates
presented in (7) can provide a numeric computation scheme to implement the
boundary controller over finite time intervals.

##  4 Stability assessment

In order to prove Theorem 1 and Theorem 2, we present a result on the
$L^{\infty}$-estimate of the solution to system (15), which plays a crucial
role in establishing the ISS w.r.t. Dirichlet boundary disturbances for the
original system (1) in closed loop.

######  Lemma 5.

The solution of system (15) admits the following estimate:

$\displaystyle\|\widetilde{w}[t]\|_{L^{\infty}(0,1)}\leq$
$\displaystyle\frac{1}{\sigma}\|\widetilde{f}\|_{L^{\infty}({Q_{t})}}+\|d_{0}\|_{{L^{\infty}(0,t)}}+\|d_{1}\|_{{L^{\infty}(0,t)}},\forall
t\in(0,T_{0}).$ (17)

Proof. To deal with boundary terms in the Lyapunov arguments, following [42],
we use the generalized Lyapunov method to prove the $L^{\infty}$-estimate of
the solution. We first define Stampacchia’s truncation functions:

$\displaystyle g(s):=$
$\displaystyle\begin{cases}\ln(1+s^{2}),&s\in\mathbb{R}_{>0},\\\
0,&s\in\mathbb{R}_{\leq 0},\end{cases}$ $\displaystyle G(s):=$
$\displaystyle\int_{0}^{s}g(\tau)\mathrm{d}\tau,\forall s\in\mathbb{R}.$

It is easy to see that

$\displaystyle 0\leq$ $\displaystyle g(s)\leq 2|s|,\forall s\in\mathbb{R},$
(18a) $\displaystyle g^{\prime}(s)\geq$ $\displaystyle 0,G(s)\geq 0,\forall
s\in\mathbb{R},$ (18b) $\displaystyle g(s)=$ $\displaystyle G(s)=0,\forall
s\in\mathbb{R}_{\leq 0}.$ (18c)

For any $T\in(0,T_{0})$, let

$\displaystyle\Omega:=\max\left\\{\frac{1}{\sigma}\|\widetilde{f}\|_{L^{\infty}({Q_{T})}},\|d_{0}\|_{L^{\infty}(0,T)},\|d_{1}\|_{L^{\infty}(0,T)}\right\\}.$

In the generalized Lyapunov arguments, we shall derive
$\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(\widetilde{w}-\Omega\right)\text{d}x$
and
$\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(-\widetilde{w}-\Omega\right)\text{d}x$,
respectively.

Indeed, the boundary and initial conditions, the definition of $\Omega$, and
(18c) imply that

$\displaystyle
g\left(\widetilde{w}(0,t)-\Omega\right)=g\left(\widetilde{w}(1,t)-\Omega\right)=0,\forall
t\in{[0,T)}.$ (19)

Note that the following implication holds true for all $t\in(0,T)$:

$\displaystyle\widetilde{w}(\cdot,t)-\Omega\geq
0\Rightarrow{\sigma}\widetilde{w}(\cdot,t)\geq\|\widetilde{f}\|_{L^{\infty}(Q_{T})}\geq\widetilde{f}(\cdot,t).$

Therefore, it holds that

$\displaystyle\int_{0}^{1}g\left(\widetilde{w}-\Omega\right)\left(\widetilde{f}-{\sigma}\widetilde{w}\right)\text{d}x\leq
0,\forall t\in(0,T).$ (20)

Then, we deduce by (18b), (19), and (20) that

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(\widetilde{w}-\Omega\right)\text{d}x=$
$\displaystyle\int_{0}^{1}g\left(\widetilde{w}-\Omega\right)\widetilde{w}_{t}\text{d}x$
$\displaystyle=$
$\displaystyle\int_{0}^{1}g\left(\widetilde{w}-\Omega\right)\left(\left(a\widetilde{w}_{x}\right)_{x}-\sigma\widetilde{w}+\widetilde{f}\right)\text{d}x$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}ag^{\prime}\left(\widetilde{w}-\Omega\right)\widetilde{w}_{x}^{2}\text{d}x+\int_{0}^{1}g\left(\widetilde{w}-\Omega\right)\left(\widetilde{f}-{\sigma}\widetilde{w}\right)\text{d}x$
$\displaystyle\leq$ $\displaystyle 0,\forall t\in(0,T),$

which implies that

$\displaystyle\int_{0}^{1}G\left(\widetilde{w}(x,t)-\Omega\right)\text{d}x\leq\int_{0}^{1}G\left(\widetilde{w}(x,0)-\Omega\right)\text{d}x\leq
0,\forall t\in(0,T).$

Therefore, $G\left(\widetilde{w}-\Omega\right)\equiv 0$ in $Q_{T}$ and thus,
for every $t\in(0,T)$, we have

$\displaystyle\widetilde{w}(\cdot,t)\leq\Omega\ \ \text{a.e.\ in}\ (0,1).$

Applying similar arguments, we obtain

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(-\widetilde{w}-\Omega\right)\text{d}x\leq
0,\forall t\in(0,T).$

Furthermore, for every $t\in(0,T)$, we have

$\displaystyle-\widetilde{w}(\cdot,t)\leq\Omega\ \ \text{a.e.\ in}\ (0,1).$

Finally, for every $t\in(0,T)$, we conclude that

$\displaystyle|\widetilde{w}(\cdot,t)|\leq\Omega\ \ \text{a.e.\ in}\ (0,1),$

which along with the arbitrariness of $t$ and $T$ implies (17).
$\hfill\blacksquare$

Proof of Theorem 1 and Theorem 2. The statements of Theorem 1(i) and Theorem
2(i) are guaranteed by Lemma 4. Therefore, it suffices to prove the statements
of Theorem 1(ii) and Theorem 2(ii), which can be proceeded in a unified way.

Let $T_{0}$ be determined by the Riemann zeta function in Theorem 1, or freely
prescribed in Theorem 2. In the sequel, denoted by $C$ (or $C^{\prime}$) a
positive constant, which may be different from each other when appearing in
different places. In view of Lemma 5 and Lemma 1, we have

$\displaystyle\|\widetilde{w}[t]\|_{L^{2}(0,1)}\leq$
$\displaystyle\|\widetilde{w}[t]\|_{L^{\infty}(0,1)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\sigma}\|\widetilde{f}\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sigma}\left(1+e^{C\sqrt{\sigma}}\right)\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)},\forall
t\in(0,T_{0}).$ (21)

By (21), and applying Lemma 2 and Lemma 3 with $\lambda:=\sigma$, we deduce
that

$\displaystyle\|w[t]\|_{L^{2}(0,1)}\leq$
$\displaystyle\left(1+\|l\|_{L^{\infty}(D)}\right)\|\widetilde{w}[t]\|_{L^{2}(0,1)}$
$\displaystyle\leq$
$\displaystyle\left(1+\|l\|_{H^{1}(D)}\right)\|\widetilde{w}[t]\|_{L^{2}(0,1)}$
$\displaystyle\leq$
$\displaystyle\left(1+C\sigma^{2}\right)\|\widetilde{w}[t]\|_{L^{2}(0,1)}$
$\displaystyle\leq$ $\displaystyle
C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right),\forall
t\in(0,T_{0}),$ (22)

where $C$ and $C^{\prime}$ are positive constants independent of $n$, $u_{0}$,
$f$, $d_{0}$, and $d_{1}$.

We infer from $u=v+w$, (22), and Lemma 4 that

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$
$\displaystyle\|v[t]\|_{L^{2}(0,1)}+\|w[t]\|_{L^{2}(0,1)}$ $\displaystyle\leq$
$\displaystyle
Ce^{-\frac{s_{n-1}}{4}+C(n-1)}\|u_{0}\|_{L^{2}(0,1)}+C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right),\forall
t\in(t_{n},t_{n+1}),$ (23)

where $\\{s_{n}\\}$ is defined by (6), and $C,C^{\prime}$ are positive
constants independent of $n$, $u_{0}$, $f$, $d_{0}$, and $d_{1}$.

In view of (8), there exists a sufficiently large integer
$N_{0}\in\mathbb{N}_{+}$ such that

$\displaystyle\frac{s_{n-1}}{4}-C(n-1)>T_{0},\forall n>N_{0}.$ (24)

Moreover, for any $t\in(0,T_{0})$, there exists a positive integer $n_{0}$
such that $t\in(t_{n_{0}},t_{n_{0}+1})$ with either $n_{0}>N_{0}$ or
$n_{0}+1\leq N_{0}$.

If $n_{0}>N_{0}$, we deduce by (23) and (24) that

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$ $\displaystyle
Ce^{-T}\|u_{0}\|_{L^{2}(0,1)}+C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right)$
$\displaystyle\leq$ $\displaystyle
Ce^{-t}\|u_{0}\|_{L^{2}(0,1)}+C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right),$
(25)

where $C$ and $C^{\prime}$ are positive constants independent of $n$, $u_{0}$,
$f$, $d_{0}$, and $d_{1}$.

If $n_{0}+1\leq N_{0}$, we infer from (23) that there exists a positive
constant $C^{\prime\prime}$ depending only on $N_{0}$ such that

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$ $\displaystyle
C^{\prime\prime}\|u_{0}\|_{L^{2}(0,1)}+C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right)$
$\displaystyle\leq$ $\displaystyle
C^{\prime\prime}e^{t_{N_{0}+1}}e^{-t}\|u_{0}\|_{L^{2}(0,1)}+C^{\prime}\left(\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right),$
(26)

where $C^{\prime}$ is positive constant independent of $n$, $u_{0}$, $f$,
$d_{0}$, and $d_{1}$.

Combining (25) and (26), we conclude that disturbed system in closed loop is
ISS, having the estimate for all $t\in(0,T_{0})$:

$\displaystyle\|u[t]\|_{L^{2}(0,1)}\leq$ $\displaystyle
C\left(e^{-t}\|u_{0}\|_{L^{2}(0,1)}+\|f\|_{L^{\infty}(Q_{t})}+\|d_{0}\|_{L^{\infty}(0,t)}+\|d_{1}\|_{L^{\infty}(0,t)}\right),$

where $C$ is a positive constant independent of $n$, $u_{0}$, $f$, $d_{0}$,
and $d_{1}$. $\hfill\blacksquare$

##  5 Numerical example

We consider the following system

$\displaystyle u_{t}(x,t)=$ $\displaystyle u_{xx}(x,t)+cu(x,t),(x,t)\in
Q_{T_{0}},$ (27a) $\displaystyle u(0,t)=$ $\displaystyle 0,t\in(0,T_{0}),$
(27b) $\displaystyle u(1,t)=$ $\displaystyle U(t)+d_{1}(t),t\in(0,T_{0}),$
(27c)

with $c=24$, $d_{1}(t)=A\sin(30t)$, and $A\in\\{0,1,2\\}$, where $A$ is used
to describe the amplitude of different Dirichlet boundary disturbances.

The initial condition is given by

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

or,

$\displaystyle u(x,0)=$ $\displaystyle 10u_{0}(x),x\in(0,1),$

where $u_{0}(x):=-4\sin(15(x-0.5))$.

It is known that the disturbance-free system (27) in open loop is unstable due
to the fact that $c>\pi^{2}$; see also Fig. 1

Figure 1: $L^{2}$-norm of solution to the open-loop system (27) with
$d_{1}\equiv 0$ and initial datum $u_{0}(x)$

In order to input-to-state stabilize system (27), we split it into two
subsystems:

$\displaystyle v_{t}(x,t)=$ $\displaystyle
v_{xx}(x,t)+cv(x,t),(x,t)\in(0,1)\times(t_{n},t_{n+1}),$ $\displaystyle
v(0,t)=$ $\displaystyle 0,t\in(t_{n},t_{n+1}),$ $\displaystyle v(1,t)=$
$\displaystyle V(t),t\in(t_{n},t_{n+1}),$ $\displaystyle v(x,t_{0})=$
$\displaystyle u(x,0),x\in(0,1),$

and

$\displaystyle w_{t}(x,t)=$ $\displaystyle w_{xx}(x,t)+cw(x,t),(x,t)\in
Q_{T_{0}},$ $\displaystyle w(0,t)=$ $\displaystyle 0,t\in(0,T_{0}),$
$\displaystyle w(1,t)=$ $\displaystyle W(t)+d_{1}(t),t\in(0,T_{0}),$
$\displaystyle w(x,0)=$ $\displaystyle 0,x\in(0,1),$

where

$\displaystyle V(t)=$
$\displaystyle\int_{0}^{1}k_{n}(1,y)v(y,t)\operatorname{d}\\!{y},\forall
t\in(t_{n},t_{n+1}),$ $\displaystyle W(t)=$
$\displaystyle\int_{0}^{1}k(1,y)w(y,t)\operatorname{d}\\!{y},\forall
t\in(0,T_{0}).$

According to Remark 1, the kernel functions are given by

$\displaystyle k_{n}(1,y)=$
$\displaystyle-(\lambda_{n}+c)y\frac{I_{1}\left(\sqrt{(\lambda_{n}+c)(1-y^{2})}\right)}{\sqrt{(\lambda_{n}+c)(1-y^{2})}},$
$\displaystyle k(1,y)=$
$\displaystyle-(\sigma+c)y\frac{I_{1}\left(\sqrt{(\sigma+c)(1-y^{2})}\right)}{\sqrt{(\sigma+c)(1-y^{2})}}.$

According to Theorem 1 and Theorem 2, the following control law

$\displaystyle U(t):=V(t)+W(t),\forall t\in(t_{n},t_{n+1})$ (28)

can be used to input-to-state stabilize system (27) after choosing appropriate
$t_{n},\lambda_{n}\geq\lambda_{0}$, and $\sigma$.

In simulations, we always choose

$\displaystyle\lambda_{0}:=3.5\quad\text{and}\quad\sigma={1}.$

Numerical results in Case I. In this case, we choose $p=1.9$, then the fixed
time is given by

$\displaystyle T_{0}=\zeta(p)=1.7497,$

and $t_{n},\lambda_{n}$ are given by

$\displaystyle t_{0}:=$ $\displaystyle 0,$ $\displaystyle t_{n+1}=$
$\displaystyle{t_{n}}+\frac{1}{(n+1)^{p}},{n\in\mathbb{N}_{+}},$
$\displaystyle\lambda_{n}=$ $\displaystyle
n^{2(p+1)}+\lambda_{0},n\in\mathbb{N}.$

By virtue of Remark 2, for numeric computations, we consider $n=0,1,2$. Thus,
the time interval $(0,T_{0})$ is divided into

$\displaystyle(t_{0},t_{1})=$ $\displaystyle(0,1),$
$\displaystyle(t_{1},t_{2})=$ $\displaystyle(1,1.2679),$
$\displaystyle(t_{2},t_{3})=$ $\displaystyle(1.2679,1.7497),$

for which $\\{\lambda_{n}\\}$ is given by

$\displaystyle{\lambda_{0}=}{3.5},\quad\lambda_{1}=4.5,\quad\text{and}\quad\lambda_{2}=58.2152,$

respectively.

Figure 2 (a) shows that system (27) with initial datum $u_{0}$ is fixed-time
stable in $L^{2}$-norm in the time $T_{0}=1.7497$, while Fig. 2 (b) shows that
the system is still fixed-time stable in the same time $T_{0}$ when the
initial datum is increased to be $10u_{0}$.

Under the same initial condition, Fig. 2 (a) and Fig. 3 show that the
$L^{2}$-norms of the solutions to system (27) with different boundary
disturbances remains bounded due to the facts that the disturbances are
bounded. However, the norms are more influenced by disturbances with large
amplitudes. This reflects well the ISS property of the system in the closed
loop.

(a) $u(x,0)=u_{0}(x),A=0$

(b) $u(x,0)=10u_{0}(x),A=0$

Figure 2: $L^{2}$-norms of the solutions to the disturbance-free closed-loop
system with different initial data under the control law (28) in Case I

(a) $u(x,0)=u_{0}(x),A=1$

(b) $u(x,0)=u_{0}(x),A=2$

Figure 3: $L^{2}$-norms of the solutions to the closed-loop system (27) with
different disturbances under the control law (28) in Case I

Numerical results in Case II. In this case, we freely prescribe the finite
time to be

$\displaystyle T_{0}=1.5.$

Then, $t_{n},\lambda_{n}$ are given by

$\displaystyle t_{0}=$ $\displaystyle 0,$ $\displaystyle{t_{n}}=$
$\displaystyle T_{0}-\frac{T_{0}}{n+1},n\in\mathbb{N}_{+},$
$\displaystyle\lambda_{n}=$ $\displaystyle n^{6}+\lambda_{0},n\in\mathbb{N}.$

By virtue of Remark 2, for numeric computations, we consider $n=0,1,2$. Thus,
the time interval $(0,T_{0})$ is divided into

$\displaystyle(t_{0},t_{1})=$ $\displaystyle(0,0.75),$
$\displaystyle(t_{1},t_{2})=$ $\displaystyle(0.75,1),$
$\displaystyle(t_{2},t_{3})=$ $\displaystyle(1,1.5),$

associated with

$\displaystyle\lambda_{0}=3.5,\quad\lambda_{1}=4.5,\quad\text{and}\quad\lambda_{2}=67.5,$

respectively.

Figure 4 (a) shows that system (27) with initial datum $u_{0}$ is fixed-time
stable in $L^{2}$-norm in the time $T_{0}=1.5$, while fig. 4 (b) shows that
the system is still fixed-time stable in the same time $T_{0}$ when the
initial datum is increased to be $10u_{0}$. As in Fig. 2 (a) and Fig. 3, under
the same initial condition, Fig. 4 (a) and Fig. 5 also reflect well the ISS
property of the system in the closed loop.

(a) $u(x,0)=u_{0}(x),A=0$

(b)$u(x,0)=10u_{0}(x),A=0$

Figure 4: $L^{2}$-norms of the solutions to the disturbance-free closed-loop
system (27) with different initial data under the control law (28) in Case II

(a) $u(x,0)=u_{0}(x),A=1$

(b) $u(x,0)=u_{0}(x),A=2$

Figure 5: $L^{2}$-norms of the solutions to the closed-loop system (27) with
different disturbances under the control law (28) in Case II

##  6 Conclusion

This paper addressed the problem of input-to-state stabilization for $1$-D
parabolic equations in the presence of distributed in-domain and Dirichlet
boundary disturbances while ensuring the fixed-time stability of the
corresponding disturbance-free system. A boundary controller was designed by
using backstepping based on the technique of splitting, and the stability was
assessed by using the generalized Lyapunov method, which can be used to deal
with boundary terms easily. Numerical simulations were conducted, and the
results confirmed the ISS properties of the considered system with in-domain
and Dirichlet boundary disturbances, as well as the fixed-time stability of
the associated disturbance-free system.

It is worth mentioning that the proposed scheme is suitable for input-to-state
stabilization of $1$-D linear parabolic equations with Robin or Neumann
boundary disturbances, as well as for fixed-time stabilization of the
corresponding disturbance-free system, for which the classical Lyapunov method
remains valid for stability analysis. However, it is still challenging to
directly apply the proposed method to fixed-time input-to-state stabilization
of the system with Dirichlet boundary disturbances due to the fact that the
fixed-time ISS is a much stronger property than the ISS. This problem will be
addressed in our future work.

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# Constructing multi-cusped hyperbolic manifolds that are isospectral and not
isometric

Benjamin Linowitz  Department of Mathematics
10 North Professor Street
Oberlin, OH 44074<EMAIL_ADDRESS>
###### Abstract.

In a recent paper Garoufalidis and Reid constructed pairs of $1$-cusped
hyperbolic $3$-manifolds which are isospectral but not isometric. In this
paper we extend this work to the multi-cusped setting by constructing
isospectral but not isometric hyperbolic $3$-manifolds with arbitrarily many
cusps. The manifolds we construct have the same Eisenstein series, the same
infinite discrete spectrum and the same complex length spectrum. Our
construction makes crucial use of Sunada’s method and the Strong Approximation
Theorem of Nori and Weisfeiler.

##  1\. Introduction

In 1966 Kac [10] famously asked “Can one hear the shape of a drum?” In other
words, can one deduce the shape of a planar domain given knowledge of the
frequencies at which it resonates? Long before Kac had posed his question
mathematicians had considered analogous problems in more general settings and
sought to determine the extent to which the geometry and topology of a
Riemannian manifold is determined by its Laplace eigenvalue spectrum.

Early constructions of isospectral non-isometric manifolds include
$16$-dimensional flat tori (Milnor [13]), compact Riemann surfaces (Vignéras
[22]) and lens spaces (Ikeda [9]). For an excellent survey of the long history
of the construction of isospectral non-isometric manifolds we refer the reader
to [7].

In this paper we consider a problem posed by Gordon, Perry and Schueth [8,
Problem 1.2]: to construct complete, non-compact manifolds that are
isospectral and non-isometric. This problem has received a great deal of
attention in the case of surfaces. For example, Brooks and Davidovich [1] were
able to use Sunada’s method [18] in order to construct a number of examples of
isospectral non-isometric hyperbolic $2$-orbifolds. For more examples, see
[8].

In a recent paper Garoufalidis and Reid [4] constructed the first known
examples of isospectral non-isometric $1$-cusped hyperbolic $3$-manifolds. The
main result of this paper extends the work of Garoufalidis and Reid to the
multi-cusped setting.

######  Theorem 1.1.

There exist finite volume orientable $n$-cusped hyperbolic $3$-manifolds that
are isopectral and not isometric for arbitrarily large positive integers $n$.

Moreover, the manifolds we construct will be shown to have the same Eisenstein
series, the same infinite discrete spectrum and the same complex length
spectrum.

The author would like to thank Dubi Kelmer, Emilio Lauret, Ben McReynolds,
Djordje Milićević, Alan Reid and Ralf Spatzier for useful conversations
concerning the material in this paper. The author is especially indebted to
Jeff Meyer for his close reading of this paper and his many suggestions and
comments. The work of the author is partially supported by NSF Grant Number
DMS-1905437.

##  2\. Preliminaries

Given a positive integer $d\geq 2$ we define ${\bf H}^{d}$ to be
$d$-dimensional hyperbolic space, that is, the connected and simply connected
Riemannian manifold of dimension $d$ having constant curvature $-1$. Let
$\Gamma$ be a torsion-free discrete group of orientation preserving isometries
of ${\bf H}^{d}$ such that the quotient space ${\bf H}^{d}/\Gamma$ has finite
hyperbolic volume. Thus $M={\bf H}^{d}/\Gamma$ is a finite volume orientable
hyperbolic $d$-manifold.

There exists a compact hyperbolic $d$-manifold $M^{\prime}$ with boundary
(possibly empty) such that the complement $M-M^{\prime}$ consists of at most
finitely many disjoint unbounded ends of finite volume, the cusps of $M$. Each
cusp is homeomorphic to $N\times(0,\infty)$ where $N$ is a compact Euclidean
$(d-1)$-manifold.

Let $\Lambda$ denote the limit set of $\Gamma$ (i.e., the set of limit points
of all the orbits of the action of $\Gamma$ on ${\bf H}^{d}$). A point
$c\in\Lambda$ is called a parabolic limit point if it is the fixed point of
some parabolic isometry $\gamma\in\Gamma$. The stabilizer $\Gamma_{c}<\Gamma$
of such a $c$ is called a maximal parabolic subgroup of $\Gamma$. A cusp of
$\Gamma$ is a $\Gamma$-equivalence class of parabolic limit points and will be
denoted by $[c]_{\Gamma}$. We will omit the subscript when the group is clear
from context. The correspondence between cusps of $M$ and cusps of $\Gamma$ is
given by the fact if $C$ is a cusp of $M$ then $C$ may be identified as
$C=V_{c}/\Gamma_{c}$ where $V_{c}\subset{\bf H}^{d}$ is a precisely invariant
horoball based at $c$ for some cusp $[c]$ of $\Gamma$.

##  3\. Spectrum of the Laplacian

It is known that the space $L^{2}(M)$ has a decomposition

$L^{2}(M)=L^{2}_{disc}(M)\oplus L^{2}_{cont}(M)$

where $L^{2}_{disc}(M)$ corresponds to the discrete spectrum of the Laplacian
on $M$ and $L^{2}_{cont}(M)$ corresponds to the continuous spectrum of $M$.
The discrete spectrum of $M$ is a collection of eigenvalues
$0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots$ where each $\lambda_{j}$ occurs
with a finite multiplicity. The continuous spectrum of $M$ is empty when $M$
is compact and otherwise is a union of finitely many intervals (one for each
cusp of $M$) of the form

$\left[\frac{(d-1)^{2}}{4},\infty\right).$

When $M$ is compact it is known that the discrete spectrum is infinite and
obeys Weyl’s Asymptotic Law. The precise analogue of Weyl’s Asymptotic Law is
in general not available when $M$ is not compact, though it is known in the
case that $\Gamma$ is an arithmetic congruence group [17, 19, 20, 21].

The following elementary lemma will be useful in proving that certain
manifolds have infinite discrete spectrum.

######  Lemma 3.1.

Let $M={\bf H}^{d}/\Gamma$ be a non-compact hyperbolic $d$-manifold and
$M^{\prime}={\bf H}^{d}/\Gamma^{\prime}$ be a finite cover of $M$. If $M$ has
an infinite discrete Laplace spectrum then so does $M^{\prime}$.

###### Proof.

The eigenfunctions associated to the discrete Laplace spectrum of $M$ are the
set of eigenfunctions of the Laplacian that are invariant under $\Gamma$ and
which are $L^{2}$-integrable over some (and hence any) fundamental domain for
$\Gamma$. Any such function is also invariant under $\Gamma^{\prime}$, and
since the fundamental domain of $\Gamma^{\prime}$ is a finite union of
fundamental domains of $\Gamma$, the function will also be $L^{2}$ integrable
over a fundamental domain for $\Gamma^{\prime}$. It follows that $M^{\prime}$
has an infinite discrete Laplace spectrum if $M$ does.∎

In order to discuss the spectrum of $M$ further we need to make clear the
contribution of Eisenstein series. Let $[c]$ be a cusp of $\Gamma$ with
stabilizer $\Gamma_{c}$. The Eisenstein series on $M$ associated to $[c]$ is
defined to be the convergent series

$E_{M,c}(w,s)=\sum_{\gamma\in\Gamma_{c}\backslash\Gamma}y(\sigma^{-1}\gamma
w)^{s},\qquad w\in{\bf H}^{d},s\in\mathbf{C},\mathrm{Re}(s)>d-1,$

where $\gamma\in\Gamma$ represents a non-identity coset $\Gamma_{c}\gamma$ of
$\Gamma_{c}$ in $\Gamma$ and $\sigma$ is the orientation preserving isometry
of hyperbolic space taking the point at infinity to the cusp point $c$. Here
we use the coordinates $z=(x,y)\in{\bf H}^{d}={\bf R}^{d-1}\times{\bf R}^{+}$
for the upper half-space.

Let $c_{1},\dots,c_{\kappa}$ be representatives of a full set of inequivalent
cusps of $\Gamma$. To ease notation we will temporarily refer to the
Eisenstein series associated to the $i$-th cusp by $E_{i}(w,s)$. The constant
term of $E_{i}(w,s)$ with respect to $c_{j}$ is denoted $E_{ij}(w,s)$ and
satisfies

$E_{ij}(w,s)=\delta_{ij}y(\sigma_{j}^{-1}w)^{s}+\phi_{ij}(s)y(\sigma_{j}^{-1}w)^{d-1-s},$

where $\sigma_{j}$ is the orientation preserving isometry of hyperbolic space
taking the point at infinity to the cusp point $c_{j}$ and where the
coefficients $\phi_{ij}(s)$ define the scattering matrix
$\Phi(s)=(\phi_{ij})$. We define the scattering determinant to be the function
$\varphi(s)=\det\Phi(s)$. The Eisenstein series $E_{j}(w,s)$, the scattering
matrix $\Phi(s)$ and the scattering determinant $\phi(s)$ have meromorphic
extensions to the complex plane. The poles of $\varphi(s)$ are poles of the
Eisenstein series and all lie in the half-plane
$\mathrm{Re}(s)<\frac{d-1}{2}$, except for at most finitely many poles in the
interval $(\frac{d-1}{2},d-1]$. The latter poles are related to the discrete
spectrum as follows. Taking the residue of $E_{j}(w,s)$ at one of the latter
poles yields an eigenfunction of the Laplacian with eigenvalue $s(d-1-s)$. The
subset of the discrete spectrum arising from residues of poles of Eisenstein
series (equivalently, of $\varphi(s)$) is called the residual spectrum. If $t$
is such a pole then we define the multiplicity at $t$ to be the order of the
pole at $t$, plus the dimension of the eigenspace in the case when $t$
contributes to the residual spectrum as described above. This discussion
motivates the following definition.

######  Definition 3.2.

Let $M_{1},M_{2}$ be $n$-cusped hyperbolic $d$-manifolds (for some positive
integer $n$) of finite volume with scattering determinants
$\varphi_{1}(s),\varphi_{2}(s)$. We say that $M_{1}$ and $M_{2}$ are
isospectral if

  * •

$M_{1}$ and $M_{2}$ have the same discrete spectrum, counting multiplicities;

  * •

$\varphi_{1}(s)$ and $\varphi_{2}(s)$ have the same set of poles and
multiplicities.

The scattering determinant is in general very difficult to compute explicitly,
although it has been worked out in several special case. For example, the
scattering determinants associated to Hilbert modular groups over number
fields have been computed in terms of Dedekind zeta functions by Efrat and
Sarnak [3] and Masri [12].

##  4\. Cusps of finite covers of hyperbolic manifolds

We begin with a group theoretic lemma. Let $G$ be a group, $g$ be an element
of $G$, and $H,K$ be subgroups of $G$. We define the double coset $HgK$ by

$HgK=\\{hgk:h\in H,k\in K\\}.$

######  Lemma 4.1.

There is a bijection between the cosets of $H$ in $HgK$ and the cosets of
$gKg^{-1}\cap H$ in $gKg^{-1}$.

###### Proof.

Recall that $HgK$ is the union of the cosets $Hgk$ as $k$ varies over the
elements of $K$. As right cosets of $H$ in $G$, two cosets $Hgk_{1}$ and
$Hgk_{2}$ intersect if and only if they are equal. Observe that
$Hgk_{1}=Hgk_{2}$ if and only if there is an element $h\in H$ such that
$gk_{1}=hgk_{2}$, or equivalently, if and only if $k_{1}k_{2}^{-1}\in
g^{-1}Hg$ (and thus is an element of $K\cap g^{-1}Hg$). This shows that
$Hgk_{1}=Hgk_{2}$ if and only if $(K\cap g^{-1}Hg)k_{1}=(K\cap
g^{-1}Hg)k_{2}$. We have therefore shown that the map $f$ given by
$f(Hgk)=(K\cap g^{-1}Hg)k$ is a bijection between the cosets of $H$ in $HgK$
and of $K\cap g^{-1}Hg$ in $K$. We can now conjugate by $g$ to obtain a
bijection between the cosets of $H$ in $HgK$ and the cosets of $(gKg^{-1}\cap
H)$ in $gKg^{-1}$. ∎

Let $\Gamma$ be a discrete subgroup of $\mathrm{Isom}^{+}({\bf H}^{d})$ and
$x,y\in\partial{\bf H}^{d}$ be $\Gamma$-equivalent. Let $G$ be a subgroup of
$\Gamma$ of finite index. We now define the set

$\Gamma_{x,y}=\\{\gamma\in\Gamma:\gamma x\in G\cdot y\\}.$

######  Lemma 4.2.

There is an equality of sets $\Gamma_{x,y}=G\gamma P_{x}$, where
$P_{x}=\mathrm{Stab}_{\Gamma}(x)$ and $\gamma$ is any element of $\Gamma$ such
that $\gamma x=y$.

###### Proof.

That any element of $G\gamma P_{x}$ lies in $\Gamma_{x,y}$ is clear. Suppose
therefore that $\delta\in\Gamma_{x,y}$ and that $\delta x=gy=g(\gamma x)$.
Then $(g\gamma)^{-1}\delta x=x$, hence $\gamma^{-1}g^{-1}\delta\in P_{x}$ and
there exists $p\in P_{x}$ such that $\gamma^{-1}g^{-1}\delta=p$. This implies
that $\delta=g\gamma p\in G\gamma P_{x}$ and completes the proof of the lemma.
∎

Let $M={\bf H}^{d}/\Gamma$ and $N={\bf H}^{d}/G$ be non-compact hyperbolic
$d$-manifolds of finite volume and

$\pi:N\longrightarrow M$

be a covering. Let $c$ represent a cusp of $\Gamma$ and
$P=\mathrm{Stab}_{\Gamma}(c)$.

######  Definition.

The preimage of a cusp of $M$ is always a union of cusps of $N$. We say a cusp
of $M$ remains a cusp of $N$ relative to $\pi$ when the preimage of that cusp
has precisely one cusp of $N$. Algebraically, this is equivalent to
$[c]_{\Gamma}=[c]_{G}$.

######  Lemma 4.3.

Suppose $c$ is a cusp representative of both $\Gamma$ and $G$ and that
$[c]_{\Gamma}=[c]_{G}$. Then there is an equality of sets $\Gamma=GP$.

###### Proof.

That $GP\subseteq\Gamma$ is clear as both $G$ and $P$ are subgroups of
$\Gamma$. Now let $\gamma\in\Gamma$. Since $\Gamma c=Gc$ there exists an
element $g\in G$ such that $\gamma c=gc$. It follows that $(g^{-1}\gamma)c=c$,
hence $g^{-1}\gamma\in P$ and there exists $p\in P$ such that
$g^{-1}\gamma=p$. This implies that $\gamma=gp$, concluding the proof. ∎

######  Theorem 4.4.

Let $\\{d_{1},\dots,d_{m}\\}$ represent the $G$-orbits on the elements of
$\partial{\bf H}^{d}$ belonging to the cusp $[c]$ of $\Gamma$. Then

$[\Gamma:G]=\sum_{i=1}^{m}[\mathrm{Stab}_{\Gamma}(d_{i}):\mathrm{Stab}_{\Gamma}(d_{i})\cap
G].$

###### Proof.

Write $\Gamma$ as a disjoint union of cosets $G\gamma_{i}$:

$\Gamma=\bigcup_{i=1}^{r}G\gamma_{i}.$

Since $\Gamma$ acts transitively on $[c]$, every element of $[c]$ is in the
$G$ orbit of $\gamma_{i}d_{1}$ for some $i$. For each $j\in\\{1,\dots,m\\}$,
fix $\delta_{j}\in\Gamma$ such that $\delta_{j}d_{1}=d_{j}$. By Lemma 4.2,
$\Gamma_{d_{1},d_{j}}=G\delta_{j}\mathrm{Stab}_{\Gamma}(d_{1})$. Lemma 4.1
shows that $\Gamma_{d_{1},d_{j}}$ is the union of $n$ cosets of $G$, where $n$
is the index of $\delta_{j}\mathrm{Stab}_{\Gamma}(d_{1})\delta_{j}^{-1}\cap G$
in $\delta_{j}\mathrm{Stab}_{\Gamma}(d_{1})\delta_{j}^{-1}$. As
$\delta_{j}\mathrm{Stab}_{\Gamma}(d_{1})\delta_{j}^{-1}=\mathrm{Stab}_{\Gamma}(\delta_{j}d_{1})=\mathrm{Stab}_{\Gamma}(d_{j})$,
we see that
$n=[\mathrm{Stab}_{\Gamma}(d_{j}):\mathrm{Stab}_{\Gamma}(d_{j})\cap G]$.

Putting all of this together, we see that $\Gamma$ is the disjoint union of
$\Gamma_{d_{1},d_{j}}$ as $j$ varies over $\\{1,\dots,m\\}$. Since each of
these is the disjoint union of
$[\mathrm{Stab}_{\Gamma}(d_{j}):\mathrm{Stab}_{\Gamma}(d_{j})\cap G]$ cosets
of $G$, we conclude that

$[\Gamma:G]=\sum_{i=1}^{m}[\mathrm{Stab}_{\Gamma}(d_{i}):\mathrm{Stab}_{\Gamma}(d_{i})\cap
G],$

which completes our proof.∎

######  Corollary 4.5.

We have an equality of indices
$[\Gamma:G]=[\mathrm{Stab}_{\Gamma}(d):\mathrm{Stab}_{\Gamma}(d)\cap G]$ for
all cusps $[d]$ of $G$ if and only if every cusp of $M$ remains a cusp of $N$.

###### Proof.

We first prove that if every cusp of $M$ remains a cusp of $N$ then
$[\Gamma:G]=[\mathrm{Stab}_{\Gamma}(d):\mathrm{Stab}_{G}(d)]$ for all cusps
$[d]$ of $G$. Fix a cusp $[d]$ of $G$ and define
$P=\mathrm{Stab}_{\Gamma}(d)$. We must show that $[P:P\cap G]=[\Gamma:G]$. To
that end, suppose that $p_{1},p_{2}\in P$. Then

$\displaystyle Gp_{1}\cap Gp_{2}\neq\emptyset$ $\displaystyle\iff
Gp_{1}=Gp_{2}$ $\displaystyle\iff p_{1}=gp_{2}\text{ for some }g\in G$
$\displaystyle\iff p_{1}p_{2}^{-1}=g$ $\displaystyle\iff p_{1}p_{2}^{-1}\in
P\cap G$ $\displaystyle\iff(P\cap G)p_{1}=(P\cap G)p_{2}.$

We have therefore exhibited a bijection between the cosets of $G$ in
$GP=\Gamma$ (the equality follows from Lemma 4.3) and the cosets of $(P\cap
G)$ in $P$, hence $[\Gamma:G]=[P:P\cap G]$.

As the reverse direction is an immediate consequence of Theorem 4.4, our proof
is complete. ∎

######  Corollary 4.6.

Suppose that $N$ is a normal cover of $M$. Let $[c]$ be a cusp of $\Gamma$ and
$[d]$ be a cusp of $G$ contained in $[c]$. The number of cusps of $G$
contained in $[c]$ is

$\frac{[\Gamma:G]}{[\mathrm{Stab}_{\Gamma}(d):\mathrm{Stab}_{\Gamma}(d)\cap
G]}.$

###### Proof.

In light of Theorem 4.4 it suffices to prove that if $[d_{i}],[d_{j}]$ are
cusps of $G$ contained in the cusp $[c]$ of $\Gamma$ then
$[\mathrm{Stab}_{\Gamma}(d_{i}):\mathrm{Stab}_{\Gamma}(d_{i})\cap
G]=[\mathrm{Stab}_{\Gamma}(d_{j}):\mathrm{Stab}_{\Gamma}(d_{j})\cap G]$. To
that end, let $\gamma\in\Gamma$ be such that $\gamma d_{i}=d_{j}$. Then

$\mathrm{Stab}_{\Gamma}(d_{j})=\mathrm{Stab}_{\Gamma}(\gamma
d_{i})=\gamma\mathrm{Stab}_{\Gamma}(d_{i})\gamma^{-1},$

hence, as $G=\gamma G\gamma^{-1}$, we have

$[\mathrm{Stab}_{\Gamma}(d_{j}):\mathrm{Stab}_{\Gamma}(d_{j})\cap
G]=[\gamma\mathrm{Stab}_{\Gamma}(d_{i})\gamma^{-1}:\gamma\mathrm{Stab}_{\Gamma}(d_{i})\gamma^{-1}\cap\gamma
G\gamma^{-1}]=[\mathrm{Stab}_{\Gamma}(d_{i}):\mathrm{Stab}_{\Gamma}(d_{i})\cap
G],$

which completes the proof. ∎

##  5\. Eisenstein series

######  Theorem 5.1.

Let $M={\bf H}^{d}/\Gamma$ be a non-compact hyperbolic $d$-manifold and
$N={\bf H}^{d}/G$ be a finite cover of $M$ with covering degree $n$. If a cusp
$[c]$ of $\Gamma$ is also a cusp of $G$ (i.e., the preimage in $N$ of the
corresponding cusp of $M$ is a single cusp) then $E_{M,c}(w,s)=E_{N,c}(w,s)$.

###### Proof.

Let $c$ represent a fixed cusp of $\Gamma$ and $P=\mathrm{Stab}_{\Gamma}(c)$.
We begin our proof by noting that Theorem 4.4 shows that $[\Gamma:G]=[P:P\cap
G]$, hence we may select a collection of coset representatives for $P\cap G$
in $P$ which is also a collection of coset representatives for $G$ in
$\Gamma$. Let $\\{\delta_{1},\dots,\delta_{n}\\}\subset P$ be such a
collection.

An arbitrary term of $E_{M,c}(w,s)$ is of the form $y(\sigma^{-1}\gamma
w)^{s}$ where $\gamma\in\Gamma$ represents a non-identity coset $P\gamma$ of
$P$ in $\Gamma$ and $\sigma$ is the orientation preserving isometry of
hyperbolic space taking the point at infinity to the cusp point $c$. Here we
use the coordinates $z=(x,y)\in{\bf H}^{d}={\bf R}^{d-1}\times{\bf R}^{+}$ for
the upper half-space. Using our decomposition of $\Gamma$ into cosets of $G$
we see that there exists $\delta_{j}$ and $g\in G$ such that
$\gamma=\delta_{j}g$. Because $\delta_{j}\in P$, the coset
$P\gamma=P\delta_{j}g$ is equal to the coset $Pg$ as cosets of
$P\backslash\Gamma$. In particular this implies that we may choose
representatives for the cosets $P\backslash\Gamma$ to all lie in $G$. Note
that for all $g_{1},g_{2}\in G$ we have

$\displaystyle Pg_{1}=Pg_{2}$ $\displaystyle\iff g_{1}g_{2}^{-1}\in P$
$\displaystyle\iff g_{1}g_{2}^{-1}\in P\cap G$ $\displaystyle\iff(P\cap
G)g_{1}=(P\cap G)g_{2}.$

It follows that

$E_{M,c}(w,s)=\sum_{\gamma\in P\backslash\Gamma}y(\sigma^{-1}\gamma
w)^{s}=\sum_{g\in P\cap G\backslash G}y(\sigma^{-1}gw)^{s}=E_{N,c}(w,s).$

∎

The following is an immediate consequence of Theorem 5.1.

######  Corollary 5.2.

Suppose that $M$ is a cusped orientable finite volume hyperbolic $d$-manifold
and that $M_{1},M_{2}$ are finite covers of $M$ with the same covering degree
and having the property that every cusp of $M$ remains a cusp of $M_{i}$
($i=1,2$). Then all of the Eisenstein series of $M_{1}$ and $M_{2}$ are equal.

##  6\. Congruence covers and $p$-reps

Let $M$ be a non-compact finite volume orientable hyperbolic $3$-manifold. Let
$c_{1},\dots,c_{\kappa}$ represent a complete set of inequivalent cusps of
$\pi_{1}(M)$ and $P_{i}$ be the subgroup of $\pi_{1}(M)$ that fixes $c_{i}$.

######  Definition 6.1.

A surjective homomorphism $\rho:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,p)$
is called a $p$-rep if, for all $i$, $\rho(P_{i})$ is non-trivial and all non-
trivial elements of $\rho(P_{i})$ are parabolic elements of
$\operatorname{PSL}(2,p)$.

We remark that if $\rho:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,p)$ is a
$p$-rep then $\rho(P_{i})$ must be a subgroup of $\operatorname{PSL}(2,p)$ of
order $p$.

######  Theorem 6.2.

Let $M$ be a $1$-cusped, non-arithmetic, finite volume orientable hyperbolic
$3$-manifold with $p$-reps $\rho:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,7)$
and $\rho^{\prime}:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,11)$. Let $k$ be
a number field with ring of integers $\mathcal{O}_{k}$ and degree not
divisible by $3$. Assume that the faithful discrete representation of
$\pi_{1}(M)$ can be conjugated to lie in
$\operatorname{PSL}(2,\mathcal{O}_{k})$. There exist infinitely many prime
powers $q$ and covers $M_{q}$ of $M$ such that:

  1. (i)

the composite homomorphism

$\rho_{q}:=\rho\circ\iota:\pi_{1}(M_{q})\hookrightarrow\pi_{1}(M)\rightarrow\operatorname{PSL}(2,7)$

is a $p$-rep,

  2. (ii)

the degree over $M$ of the cover $M_{q}$ is $\frac{11}{2}(q^{3}-q)$,

  3. (iii)

the number of cusps of $M_{q}$ is at least $q+1$, and

  4. (iv)

$M_{q}$ has an infinite discrete spectrum.

###### Proof.

We begin by constructing a finite cover $\widetilde{M}$ of $M$ which has an
infinite discrete spectrum. The manifold $M_{q}$ will arise as a finite cover
of $\widetilde{M}$ and will therefore have an infinite discrete spectrum by
virtue of Lemma 3.1. To that end, let $H$ be an index $11$ subgroup of
$\operatorname{PSL}(2,11)$. Such a subgroup is well-known to exist, and the
cover of $M$ associated to the pullback subgroup of $H$ by $\rho^{\prime}$ is
a degree $11$ cover of $M$. Denote this cover by $\widetilde{M}$. We claim
that $\widetilde{M}$ has one cusp. Let $P$ be the subgroup of $\pi_{1}(M)$
stabilizing the cusp of $M$. As was commented above, $\rho^{\prime}(P)$ must
be a cyclic subgroup of $\operatorname{PSL}(2,11)$ of order $11$. Since $H$
has index $11$ in $\operatorname{PSL}(2,11)$ and
$|\operatorname{PSL}(2,11)|=660=2^{2}\cdot 3\cdot 5\cdot 11$ it must be the
case that $\rho^{\prime}(P)\cap H$ is trivial. It follows that
$[P:P\cap\pi_{1}(\widetilde{M})]=11=[\pi_{1}(M):\pi_{1}(\widetilde{M})]$,
hence $\widetilde{M}$ has one cusp by Corollary 4.5. It now follows from [4,
Theorem 2.4] that $\widetilde{M}$ has an infinite discrete spectrum. We note
that [4, Theorem 2.4] has two hypotheses: that $\widetilde{M}$ be non-
arithmetic and that $\widetilde{M}$ not be the minimal element in its
commensurability class. That $\widetilde{M}$ is non-arithmetic is clear, since
it is a finite cover of $M$, which is non-arithmetic. It is equally clear that
$\widetilde{M}$ is not the minimal element of its commensurability class,
since such an element cannot be a finite cover of another hyperbolic
$3$-manifold.

We claim that $\pi_{1}(\widetilde{M})$ also admits a $p$-rep to
$\operatorname{PSL}(2,7)$. In particular, we will show the homomorphism to
$\operatorname{PSL}(2,7)$ obtained by composing the inclusion map
$\pi_{1}(\widetilde{M})\hookrightarrow\pi_{1}(M)$ with
$\rho:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,7)$ is a $p$-rep. To see this,
note that because $\gcd(11,|\operatorname{PSL}(2,7)|)=1$, the map $g\mapsto
g^{11}$ is a bijection from $\operatorname{PSL}(2,7)$ to itself, hence our
claim follows from the fact that for every $\gamma\in\pi_{1}(M)$ the element
$\gamma^{11}$ lies in $\pi_{1}(\widetilde{M})$.

Given a proper, non-zero ideal $I$ of $\mathcal{O}_{k}$ we have a composite
homomorphism

$\phi_{I}:\pi_{1}(\widetilde{M})\longrightarrow\operatorname{PSL}(2,\mathcal{O}_{k})\longrightarrow\operatorname{PSL}(2,\mathcal{O}_{k}/I)$

called the level $I$ congruence homomorphism. It follows from the Strong
Approximation Theorem of Nori [14] and Weisfeiler [23] that for all but
finitely many prime ideals $\mathfrak{p}$ of $\mathcal{O}_{k}$ the level
$\mathfrak{p}$ congruence homomorphism $\phi_{\mathfrak{p}}$ is surjective.

By Dirichlet’s Theorem on Primes in Arithmetic Progressions we may choose a
prime $p$ satisfying $p\equiv 5\pmod{168}$ which does not divide the
discriminant of $k$. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_{k}$
lying above $p$ which has inertia degree $f$ satisfying $\gcd(f,3)=1$. Note
that the existence of such a prime ideal $\mathfrak{p}$ follows from the well-
known equality in algebraic number theory

$[k:\mathbf{Q}]=\sum_{i=1}^{g}e(\mathfrak{p}_{i}/p)f(\mathfrak{p}_{i}/p),$

where $p\mathcal{O}_{k}=\mathfrak{p}_{1}\cdots\mathfrak{p}_{g}$,
$e(\mathfrak{p}_{i}/p)$ denotes the ramification degree of $\mathfrak{p}_{i}$
over $p$ and $f(\mathfrak{p}_{i}/p)$ denotes the inertia degree of
$\mathfrak{p}_{i}$ over $p$. In particular our assertion follows from the
hypothesis that $[k:\mathbf{Q}]$ not be divisible by $3$ and the fact that all
of the ramification degrees $e(\mathfrak{p}_{i}/p)$ are equal to one (since
$p$ doesn’t divide the discriminant of $k$ and thus does not ramify in $k$).

We observed above that it follows from the Strong Approximation Theorem that
for all but finitely many primes the associated congruence homomorphism is
surjective. In light of our use of Dirichlet’s Theorem on Primes in Arithmetic
Progressions in the previous paragraph we may assume that $\mathfrak{p}$ was
selected so that $\phi_{\mathfrak{p}}$ is surjective. Let $M_{q}$ be the cover
of $\widetilde{M}$ associated to the kernel of $\phi_{\mathfrak{p}}$. The
cover $M_{q}$ of $\widetilde{M}$ is normal of degree

$|\operatorname{PSL}(2,\mathcal{O}_{k}/\mathfrak{p})|=|\operatorname{PSL}(2,p^{f})|=\frac{p^{3f}-p^{f}}{2},$

which proves (ii) upon setting $q=p^{f}$.

Assertion (iii) follows from assertion (ii) and Corollary 4.6 since the image
under $\phi_{\mathfrak{p}}$ of a cusp stabilizer $P_{i}$ will be an abelian
subgroup of $\operatorname{PSL}(2,p^{f})$ and thus will have order at most
$\frac{p^{f}(p^{f}-1)}{2}$ by the classification of subgroups of
$\operatorname{PSL}(2,q)$ (see [2]).

We now prove assertion (i). We will abuse notation and denote by $\rho$ the
$p$-rep from $\pi_{1}(\widetilde{M})$ onto $\operatorname{PSL}(2,7)$. Because
this $p$-rep was obtained by composing the inclusion of
$\pi_{1}(\widetilde{M})$ into $\pi_{1}(M)$ with the $p$-rep from $\pi_{1}(M)$
onto $\operatorname{PSL}(2,7)$ (which was also denoted $\rho$), it suffices to
prove assertion (i) with $\widetilde{M}$ in place of $M$. Let
$N=\frac{p^{3f}-p^{f}}{2}=[\pi_{1}(\widetilde{M}):\pi_{1}(M_{q})]$. As
$\rho_{q}(\pi_{1}(M_{q}))$ contains
$\rho_{q}(\gamma^{N})=\rho(\gamma^{N})=\rho(\gamma)^{N}$ for all
$\gamma\in\pi_{1}(\widetilde{M})$ and
$\rho:\pi_{1}(M)\rightarrow\operatorname{PSL}(2,7)$ is surjective, the
surjectivity of $\rho_{q}$ follows from the fact (easily verifiable in SAGE
[16]) that $\operatorname{PSL}(2,7)$ is generated by the $N$th powers of its
elements whenever $p\equiv 5\pmod{168}$ and $\gcd(f,3)=1$.

Let $P_{0}$ be the subgroup of $\pi_{1}(M_{q})$ which fixes some cusp of
$M_{q}$ and $P$ be the subgroup of $\pi_{1}(\widetilde{M})$ fixing the
corresponding cusp of $\widetilde{M}$. Because
$\rho:\pi_{1}(\widetilde{M})\rightarrow\operatorname{PSL}(2,7)$ is a $p$-rep,
$\rho(P)$ consists entirely of parabolic elements and therefore is a subgroup
of $\operatorname{PSL}(2,7)$ of order $7$. Note that $[P:P_{0}]=d$ for some
divisor $d$ of $N$. We will show that $N$, and thus $d$, is not divisible by
$7$. Because $p$ was chosen so that $p\equiv 5\pmod{168}$, we also have
$p\equiv 5\pmod{7}$ (since $168=2^{3}\cdot 3\cdot 7$). It is now an easy
exercise in elementary number theory to show that $N=\frac{p^{3f}-p^{f}}{2}$
is not divisible by $7$ whenever $\gcd(f,3)=1$. Having shown that
$\gcd(d,7)=1$, we observe that if $\gamma\in P$ has non-trivial image in
$\operatorname{PSL}(2,7)$ then $\gamma^{d}\in P_{0}$ and thus
$\rho_{q}(\gamma^{d})=\rho(\gamma)^{d}$ is non-trivial in
$\operatorname{PSL}(2,7)$. Since $\rho_{q}(P_{0})$ is a subgroup of $\rho(P)$
and thus also consists entirely of parabolic elements, this proves assertion
(i). ∎

##  7\. Sunada’s Method for constructing isospectral manifolds

We begin this section by recalling the statement of Sunada’s theorem [18].

Given a finite group $G$ with subgroups $H_{1}$ and $H_{2}$ we say that
$H_{1}$ and $H_{2}$ are almost conjugate if, for all $g\in G$,

$\\#(H_{1}\cap[g])=\\#(H_{2}\cap[g])$

where $[g]$ denotes the conjugacy class of $g$ in $G$.

######  Theorem 7.1 (Sunada).

Let $M$ be a Riemannian manifold and $\rho:\pi_{1}(M)\rightarrow G$ be a
surjective homomorphism. The coverings $M^{H_{1}}$ and $M^{H_{2}}$ of $M$ with
fundamental groups $\rho^{-1}(H_{1})$ and $\rho^{-1}(H_{2})$ are isospectral.

The following is a group theoretic lemma of Prasad and Rajan [15, Lemma 1]
which they used to reprove Sunada’s theorem. In what follows, if $G$ is a
group and $V$ is a $G$-module then $V^{G}$ is the submodule of invariants of
$G$.

######  Lemma 7.2.

Suppose that $G$ is a finite group with almost conjugate subgroups $H_{1}$ and
$H_{2}$. Assume that $V$ is a representation space of $G$ over a field $k$ of
characteristic zero. Then there exists an isomorphism $i:V^{H_{1}}\rightarrow
V^{H_{2}}$, commuting with the action of any endomorphism $\Delta$ of $V$
which commutes with the action of $G$ on $V$; i.e. the following diagram
commutes:

$V^{H_{1}}$$V^{H_{2}}$$V^{H_{1}}$$V^{H_{2}}$$\Delta$$i$$\Delta$$i$

######  Theorem 7.3.

Let $M={\bf H}^{3}/\Gamma$ be a cusped finite volume orientable hyperbolic
$3$-manifold that is non-arithmetic and that is the minimal element in its
commensurability class (i.e., $\Gamma=\mathrm{Comm}(\Gamma)$ where
$\mathrm{Comm}(\cdot)$ denotes the commensurator). Let $M_{0}={\bf
H}^{3}/\Gamma_{0}$ be a finite cover of $M$, $G$ be a finite group and
$H_{1},H_{2}$ be non-conjugate almost conjugate subgroups of $G$. Suppose that
$\Gamma$ admits a homomorphism onto $G$ such that the induced composite
homomorphism $\Gamma_{0}\hookrightarrow\Gamma\rightarrow G$ is also onto. Let
$M_{1},M_{2}$ be the finite covers of $M_{0}$ associated to the pullback
subgroups of $H_{1}$ and $H_{2}$ and assume that $M_{1}$ and $M_{2}$ both have
the same number of cusps as $M_{0}$. Then $M_{1}$ and $M_{2}$ are are
isospectral, have the same complex length spectra, are non-isometric and have
infinite discrete spectra.

###### Proof.

Our proof will largely follow the proof of the analogous result of
Garoufalidis and Reid [4, Theorem 3.1].

We begin by proving that the manifolds $M_{1}$ and $M_{2}$ are non-isometric.
Let $\Gamma_{1},\Gamma_{2}$ be such that $M_{1}={\bf H}^{3}/\Gamma_{1}$ and
$M_{2}={\bf H}^{3}/\Gamma_{2}$. If $M_{1}$ and $M_{2}$ are isometric then
there exists $g\in\mathrm{Isom}({\bf H}^{3})$ such that
$g\Gamma_{1}g^{-1}=\Gamma_{2}$. Such an element $g$ necessarily lies in the
commensurator $\mathrm{Comm}(\Gamma)$ of $\Gamma$, and since
$\Gamma=\mathrm{Comm}(\Gamma)$ we see that $g\in\Gamma$. By hypothesis there
exists a surjective homomorphism $\rho:\Gamma\rightarrow G$. Projecting onto
$G$ we see that $\rho(g)H_{1}\rho(g)^{-1}=H_{2}$, which contradicts our
hypothesis that $H_{1}$ and $H_{2}$ be non-conjugate.

To prove that $M_{1}$ and $M_{2}$ are isospectral we must show that their
scattering determinants have the same poles with multiplicities and that they
have the same discrete spectrum. Since $M_{1}$ and $M_{2}$ have the same
covering degree over $M_{0}$, that their scattering determinants have the same
poles with multiplicities follows immediately from Theorem 5.1, which in fact
shows that all of their Eisenstein series coincide. That $M_{1}$ and $M_{2}$
have the same discrete spectrum follows from Lemma 7.2 with $k=\mathbf{C}$,
$V=L^{2}_{disc}(M_{0})$ and $\Delta$ the Laplacian.

That $M_{1}$ and $M_{2}$ have the same complex length spectra follows from the
proof given by Sunada [18, Section 4].

That $M_{1}$ and $M_{2}$ have infinite discrete spectra follows from [4,
Theorem 2.4].

∎

##  8\. Proof of Theorem 1.1

In light of Theorems 6.2 and 7.3 it suffices to exhibit a non-arithmetic,
$1$-cusped finite volume hyperbolic $3$-manifold $M$ which is the minimal
element in its commensurability class and which admits $p$-reps onto
$\operatorname{PSL}(2,7)$ and $\operatorname{PSL}(2,11)$.

To prove this assertion, let $M$ be a hyperbolic $3$-manifold as in the
previous paragraph and assume that $\pi_{1}(M)$ can be conjugated to lie in
$\operatorname{PSL}(2,\mathcal{O}_{k})$ for some number field $k$ whose degree
is not divisible by $3$. (We will construct such a manifold below.) It follows
from Theorem 6.2 that there exist infinitely many prime powers $q$ and covers
$M_{q}$ of $M$ such that composing the inclusion
$\pi_{1}(M_{q})\hookrightarrow\pi_{1}(M)$ with the $p$-rep
$\pi_{1}(M)\rightarrow\operatorname{PSL}(2,7)$ yields a $p$-rep and such that
$M_{q}$ has at least $q+1$ cusps.

We have seen that there is a surjective homomorphism
$\rho:\pi_{1}(M_{q})\rightarrow\operatorname{PSL}(2,7)$. It is well known that
$\operatorname{PSL}(2,7)$ contains a pair of non-conjugate, almost conjugate
subgroups of index $7$. Call these subgroups $H_{1}$ and $H_{2}$ and observe
that since $|PSL(2,7)|=168$, it must be that $H_{1}$ and $H_{2}$ have order
$24$. Let $M_{i}={\bf H}^{3}/\Gamma_{i}$ ($i=1,2$) be the manifold covers of
$M_{q}$ associated to $H_{1}$ and $H_{2}$.

Fix $i\in\\{1,2\\}$ and let $[d]$ be a cusp of $\Gamma_{i}$. Let
$P_{i}=\mathrm{Stab}_{\Gamma_{i}}(d)$ and
$P=\mathrm{Stab}_{\pi_{1}(M_{q})}(d)$. Because the homomorphism
$\rho:\pi_{1}(M_{q})\rightarrow\operatorname{PSL}(2,7)$ is a $p$-rep,
$\rho(P)$ is a cyclic subgroup of $\operatorname{PSL}(2,7)$ of order $7$.
Since $H_{i}$ has order $24$ it must be that $\rho(P)\cap H_{i}$ is trivial.
In particular it follows that $\rho(P_{i})=1$ and consequently that
$[\pi_{1}(M_{q}):\Gamma_{i}]=7=[P:P_{i}]$. Corollary 4.5 now implies that
every cusp of $M_{q}$ remains a cusp of $M_{i}$. In particular this shows that
$M_{1}$ and $M_{2}$ both have the same number of cusps as $M_{q}$, and this
number can be made arbitrarily large by taking the prime power $q$ (from
Theorem 6.2) to be arbitrarily large. Theorem 1.1 now follows from Theorem
7.3.

We now construct a non-arithmetic, $1$-cusped finite volume hyperbolic
$3$-manifold $M$ which is the minimal element in its commensurability class
and which admits $p$-reps onto $\operatorname{PSL}(2,7)$ and
$\operatorname{PSL}(2,11)$. We will additionally show that $\pi_{1}(M)$ can be
conjugated to lie in $\operatorname{PSL}(2,\mathcal{O}_{k})$ where $k$ is a
number field of degree $8$.

Figure 1. The knot K11n116.

To that end, let $K$ be the knot K11n116 of the Hoste-Thistlethwaite table
shown in Figure 1. The manifold $M=S^{3}\setminus K={\bf H}^{3}/\Gamma$ has
$1$ cusp, volume $7.7544537602\cdots$ and invariant trace field
$k=\mathbf{Q}(t)$ where $t=0.00106+0.9101192i$ is a root of the polynomial
$x^{8}-2x^{7}-x^{6}+4x^{5}-3x^{3}+x+1$. It was proven in [6] that $M$ is the
minimal element in its commensurability class (i.e., that
$\Gamma=\mathrm{Comm}(\Gamma)$ where $\mathrm{Comm}(\Gamma)$ denotes the
commensurator of $\Gamma$). The work of Margulis [11] shows that this implies
$M$ must be non-arithmetic. Moreover, a computation in Snap [5] shows that
$\Gamma$ has presentation

$\Gamma=\langle a,b,c\text{ }|\text{ }aaCbAccBB,\text{ }aacbCbAAB\rangle,$

and peripheral structure

$\mu=CbAcb,\qquad\lambda=AAbCCbacb.$

Here $A=a^{-1},B=b^{-1},C=c^{-1}$. In terms of matrices, we may represent
$\Gamma$ as a subgroup of $\operatorname{PSL}(2,\mathcal{O}_{k})$ via

$a=\begin{pmatrix}-t^{2}+t-1&t^{7}-3t^{6}+4t^{5}-t^{4}+t^{2}-t\\\
-t^{2}+t-1&0\end{pmatrix},$

$b=\begin{pmatrix}-t^{7}+2t^{6}-2t^{5}-3t^{3}+2t^{2}-3t-1&t^{6}-2t^{5}+t^{4}+3t^{3}-2t^{2}+3t+2\\\
-t^{7}+3t^{6}-5t^{5}+4t^{4}-4t^{3}+2t^{2}-2t-1&t^{7}-3t^{6}+5t^{5}-4t^{4}+4t^{3}-t^{2}+t+2\end{pmatrix},$

and

$c=\begin{pmatrix}-t^{6}+4t^{5}-8t^{4}+7t^{3}-5t^{2}-t&-2t^{7}+7t^{6}-14t^{5}+15t^{4}-12t^{3}+t^{2}+3t-1\\\
t^{5}-3t^{4}+4t^{3}-3t^{2}+t&-t^{7}+4t^{6}-9t^{5}+11t^{4}-9t^{3}+3t^{2}+t-2\end{pmatrix}.$

We now show that $\Gamma$ admits $p$-reps onto $\operatorname{PSL}(2,7)$ and
$\operatorname{PSL}(2,11)$. We begin by exhibiting the $p$-rep onto
$\operatorname{PSL}(2,7)$. As the discriminant of $k$ is $156166337$, which is
not divisible by $7$, we see that $7$ is unramified in $k/\mathbf{Q}$. Using
SAGE [16] we find that
$7\mathcal{O}_{k}=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3},$ where the
inertia degrees of the $\mathfrak{p}_{i}$ are $1,2,5$. We note that the prime
$\mathfrak{p}_{1}$ of norm $7$ is equal to the principal ideal $(t-1)$. Upon
identifying $\mathcal{O}_{k}/\mathfrak{p}_{1}$ with ${\bf F}_{7}$ we obtain a
homomorphism from $\Gamma$ to $\operatorname{PSL}(2,7)$ by reducing the matrix
entries of $a,b,c$ modulo $\mathfrak{p}_{1}$. The images of $a,b,c$ in
$\operatorname{PSL}(2,7)$ are represented by

$a=\begin{pmatrix}6&1\\\ 6&0\end{pmatrix},\qquad b=\begin{pmatrix}1&6\\\
3&5\end{pmatrix},\qquad c=\begin{pmatrix}3&4\\\ 0&5\end{pmatrix},$

while the images of $\mu,\lambda$ in $\operatorname{PSL}(2,7)$ are represented
by the parabolic matrices

$\mu=\begin{pmatrix}0&4\\\
5&5\end{pmatrix},\qquad\lambda=\begin{pmatrix}2&5\\\ 1&3\end{pmatrix}.$

It remains only to show that the homomorphism we have defined, call it
$\rho_{7}$, is surjective. Our proof of this will make use of the following
easy lemma.

######  Lemma 8.1.

Let $p$ be a prime. The group $\operatorname{SL}(2,p)$ is generated by the
matrices

$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\qquad U=\begin{pmatrix}1&0\\\
1&1\end{pmatrix}.$

###### Proof.

The lemma follows from the fact that $\operatorname{SL}(2,\mathbf{Z})$ is
generated by the matrices in the lemma’s statement. To see this, note that the
usual generators of $\operatorname{SL}(2,\mathbf{Z})$ are

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\qquad T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},$

and $S=T^{-1}UT^{-1}$.∎

Surjectivity of our homomorphism
$\rho_{7}:\Gamma\rightarrow\operatorname{PSL}(2,7)$ now follows from the fact
that

$\begin{pmatrix}1&1\\\
0&1\end{pmatrix}=\rho_{7}(b)^{-1}\rho_{7}(a)^{-2}\rho_{7}(b)^{-1}\rho_{7}(a)\rho_{7}(b)^{-1}$

and

$\begin{pmatrix}1&0\\\
1&1\end{pmatrix}=\rho_{7}(c)\rho_{7}(a)^{-1}\rho_{7}(b)\rho_{7}(c)^{2}.$

We have just shown that $\Gamma$ admits a $p$-rep onto
$\operatorname{PSL}(2,7)$. We now show that $\Gamma$ admits a $p$-rep onto
$\operatorname{PSL}(2,11)$ as well. In $k$ we have the factorization
$11\mathcal{O}_{k}=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}$ where the
inertia degrees of the $\mathfrak{p}_{i}$ are $1,1,6$. We may assume without
loss of generality that $\mathfrak{p}_{1}=(t-4)$. Identifying
$\mathcal{O}_{k}/\mathfrak{p}_{1}$ with ${\bf F}_{11}$ we see that the images
in $\operatorname{PSL}(2,11)$ of $a,b,c$ are represented by the matrices

$a=\begin{pmatrix}9&6\\\ 9&0\end{pmatrix},\qquad b=\begin{pmatrix}4&3\\\
1&1\end{pmatrix},\qquad c=\begin{pmatrix}10&1\\\ 6&4\end{pmatrix},$

while the images of $\mu,\lambda$ in $\operatorname{PSL}(2,11)$ are
represented by the parabolic matrices

$\mu=\begin{pmatrix}10&10\\\
10&10\end{pmatrix},\qquad\lambda=\begin{pmatrix}10&0\\\ 6&10\end{pmatrix}.$

Finally, we show that our homomorphism
$\rho_{11}:\Gamma\rightarrow\operatorname{PSL}(2,11)$ is surjective by
applying Lemma 8.1. To that end we simply note that

$\begin{pmatrix}1&1\\\
0&1\end{pmatrix}=\rho_{11}(a)^{-1}\rho_{11}(b)\rho_{11}(c)^{-1}$

and

$\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}=\rho_{11}(c)\rho_{11}(a)^{2}.$

This completes the proof of Theorem 1.1.

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