#  Diffusion and robustness of boundary feedback stabilization of hyperbolic
systems
À notre maitre et ami Eduardo

Georges Bastin , Jean-Michel Coron and Amaury Hayat Department of Mathematical
Engineering, ICTEAM, UCLouvain, Louvain-La-Neuve, Belgium.
<EMAIL_ADDRESS>Jacques-Louis Lions, Sorbonne
Université, Université de Paris, CNRS, INRIA, équipe Cage, Paris, France.
<EMAIL_ADDRESS>École des Ponts ParisTech, 6
- 8, Avenue Blaise Pascal, Cité Descartes - Champs sur Marne, 77455 Marne la
Vallée, France<EMAIL_ADDRESS>
###### Abstract

We consider the problem of boundary feedback control of single-input-single-
output (SISO) one-dimensional linear hyperbolic systems when sensing and
actuation are anti-located. The main issue of the output feedback
stabilization is that it requires dynamic control laws that include delayed
values of the output (directly or through state observers) which may not be
robust to infinitesimal uncertainties on the characteristic velocities. The
purpose of this paper is to highlight some features of this problem by
addressing the feedback stabilization of an unstable open-loop system which is
made up of two interconnected transport equations and provided with anti-
located boundary sensing and actuation. The main contribution is to show that
the robustness of the control against delay uncertainties is recovered as soon
as an arbitrary small diffusion is present in the system. Our analysis also
reveals that the effect of diffusion on stability is far from being an obvious
issue by exhibiting an alternative simple example where the presence of
diffusion has a destabilizing effect instead.

##  1 Introduction

The output feedback stabilization of single-input-single-output (SISO) one-
dimensional linear hyperbolic systems is a subject that has been widely
studied in the scientific literature since the nineties when both actuation
and sensing are located at the boundaries. In the case where the control input
and the measured output are co-located at the same boundary, the problem is
now relatively well understood and has given rise to numerous publications,
both in the linear case [1, 3] and in the nonlinear case [5, 13], in
particular in fluid mechanics for Saint-Venant equations [10, 5, 16] or Euler
equations [14], to name just a few of the many publications on the subject.

In contrast, when the actuator acts through one boundary, whereas the sensor
is placed at the other boundary, the output feedback stabilization problem can
become much more complicated and remains largely unexplored in the literature.
As Krstic et al. pointed out in [18], the difficulty arises from the fact that
“the input-output operator is no longer passive (…) which precludes the
application of simple controllers”. Anti-located sensing and actuation
requires to use dynamic compensators that include delayed values of the output
(directly or through state observers). A meaningful example is the output
feedback stabilization of a simple unstable wave equation addressed in [18]
using a separation principle that combines a state feedback control with a
state observer. This approach is extended to the adaptive stabilization of
more general linear hyperbolic systems with unknown parameters in [2] and [7].
Recently, an experimental application to the control of hydraulic waves is
reported in [23], where the actuation is provided by a moving boundary while
the water level is measured at the other boundary.

We can also mention references [4, 17] which deal with MIMO systems with anti-
located multivariable sensors and actuators for $n\times n$ linear hyperbolic
systems expressed in a characteristic form having a very specific input/output
structure.

It is important to note that the use of control laws containing delayed output
feedback or state observers can however prove to be problematic because it can
be strongly sensitive to uncertainties on the characteristic velocities of the
plant model [20, 12, 8]. This happens when the control design relies on a
(supposed) exact knowledge of some characteristic velocities that must be
exactly compensated in the control law such that the stability can be
destroyed by arbitrarily small modelling uncertainties.

In this paper, our purpose is to highlight some features of this problem by
addressing the feedback stabilization of an unstable open-loop system which is
made up of two interconnected transport equations and provided with anti-
located boundary sensing and actuation.

Our paper is organized as follows. The control problem is described in Section
2. It is first shown that the considered control system is open loop unstable
and cannot be stabilized by a simple proportional output feedback. Then, it is
shown that the system can be stabilized by a dynamic controller that involves
a delayed output feedback. However, this control turns out not to be robust
with respect to delay uncertainties precisely because the control requires a
(utopian) exact knowledge of the transport velocity.

The main contribution of this paper is to show that the robustness of the
control against delay uncertainties is recovered as soon as an arbitrary small
diffusion is present in the system. For that purpose, in Section 3, it is
first assumed that the considered plant is subject to a slight phenomenon of
diffusion, interpreted as a viscosity and the corresponding (unstable) input-
output transfer function is computed. Then in Sections 4 and 5, we show that
the dynamic output (non robust) feedback designed for the inviscid case also
stabilizes exponentially the viscous system when the (unknown) diffusion is
small, and that, in this case, the control proves to be perfectly robust, even
if the diffusion is almost negligible. Interestingly, an upper bound on the
decay rate appears when adding a small viscosity, and this upper bound is
uniform with respect to the diffusion parameter $\eta$ when it is small, while
for the unperturbed system with $\eta=0$ the decay rate is infinite (the
system is finite-time stable).

Our analysis in Sections 4 and 5 also reveals that the effect of diffusion on
stability is far from being an obvious issue, contrary to what one might
expect. It is indeed well known that, in hyperbolic systems, the presence of
diffusion (or friction) can have a destabilizing as well as a stabilizing
effect (see for instance the references [11, 22]). This issue is further
discussed in Section 6 where we present an example of another simple
hyperbolic system which simplifies the previous case and for which, however,
the same diffusion term destroys the stability instead of strengthening it.

Some final conclusions are given in Section 7.

##  2 Description of the control problem

We consider the open loop control system represented in Figure 1. The system
is made up of the positive feedback interconnection of two identical transport
systems. The system dynamics are described in the time domain by the following
equations:

$\displaystyle\partial_{t}y_{1}(t,x)+\upsilon\partial_{x}y_{1}(t,x)=0,$ (1a)
$\displaystyle\partial_{t}y_{2}(t,x)+\upsilon\partial_{x}y_{2}(t,x)=0,$ (1b)
$\displaystyle y_{1}(t,0)=y_{2}(t,1)+U(t),$ (1c) $\displaystyle
y_{2}(t,0)=y_{1}(t,1),$ (1d) $\displaystyle Y(t)=y_{1}(t,1).$ (1e)

where $U(t)$ is the control input and $Y(t)$ is the measurable output. In the
classical pure transport equations (1a) and (1b), the parameter $\upsilon>0$
denotes the transport velocity.

Figure 1: The structure of the open loop control system considered in this
paper.

In the frequency domain, it is well known that the transfer function of the
transport systems (1a) and (1b) is

$f_{o}(s)=e^{-s\tau}\hskip 14.22636pt\text{with the time delay}\hskip
14.22636pt\tau=\dfrac{1}{\upsilon},$ (2)

where $s$ denotes the Laplace complex variable.

It follows that the overall input-output transfer function of the open loop
system (1) is

$G(s)=\dfrac{\textrm{Y}(s)}{\textrm{U}(s)}=\dfrac{f_{o}(s)}{1-f_{o}^{2}(s)}=\dfrac{e^{s\tau}}{e^{2s\tau}-1},$
(3)

where $\textrm{Y}(s)$ and $\textrm{U}(s)$ denote the Laplace transforms of the
output $Y(t)$ and the input $U(t)$ respectively.

The poles of the system are the roots of the characteristic equation

$e^{2s\tau}-1=0.$ (4)

This open loop control system is clearly not asymptotically stable since all
the poles are located on the imaginary axis.

In order to illustrate the challenge that arises when actuation and sensing
are anti-located, we shall first show that, despite its apparent simplicity,
this unstable system cannot be stabilized with a simple proportional output
feedback, i.e. with a static proportional controller of the form

$U(t)=-2k_{p}Y(t)$ (5)

where $k_{p}\neq 0$ is a control tuning parameter.

In the frequency domain, for the system (3) with the control law (5) the
characteristic equation of the closed loop system is:

$e^{2s\tau}+2k_{p}e^{s\tau}-1=0.$ (6)

Solving this equation for $e^{s\tau}$, we get

$e^{s\tau}=-k_{p}\pm\sqrt{1+k_{p}^{2}}.$ (7)

Then for any $k_{p}\neq 0$ there is an infinity of system poles
$\sigma+i\omega$ lying on two vertical lines with real parts:

$\sigma=\upsilon\ln\left(\sqrt{1+k_{p}^{2}}+|k_{p}|\right)>0\hskip
14.22636pt\text{and}\hskip
14.22636pt\sigma=\upsilon\ln\left(\sqrt{1+k_{p}^{2}}-|k_{p}|\right)<0.$ (8)

It follows that the unstable system (3) cannot be stabilized with the static
controller (5).

Let us now show that the system can actually be stabilized by a dynamic
controller that involves a delayed output feedback.

From (3), it follows that the input-output dynamics of the system (1) in the
time domain can alternatively be represented by the delay-difference equation

$Y(t)-Y(t-2\tau)=U(t-\tau).$ (9)

A simple and natural candidate for a stabilizing feedback control law is then

$U(t)=-2k_{1}Y(t)-k_{2}Y(t-\tau)$ (10)

where $k_{1}$ and $k_{2}$ are control tuning parameters. With this control
law, the closed loop dynamics are

$Y(t)+2k_{1}Y(t-\tau)+(k_{2}-1)Y(t-2\tau)=0.$ (11)

Clearly, we can conclude that the controller (10) exponentially stabilizes the
system (1) if the tuning parameters $k_{1}$ and $k_{2}$ are selected such that
the roots of the polynomial

$w^{2}+2k_{1}w+(k_{2}-1)$ (12)

are located inside the unit circle. However, this should be considered with
caution because it is well known that boundary feedback stabilization of
hyperbolic systems with delayed control may be sensitive to delay modeling
errors (see e.g. [20]). To clarify this point we consider the time domain
representation of the dynamical control law (10) defined as

$\displaystyle\begin{split}&\partial_{t}\hat{y}_{2}(t,x)+\upsilon\partial_{x}\hat{y}_{2}(t,x)=0,\\\
&\hat{y}_{2}(t,0)=y_{1}(t,1),\end{split}$ (13) $\displaystyle
U(t)=-2k_{1}y_{1}(t,1)-k_{2}\hat{y}_{2}(t,1),$ (14)

where the transport equation (13) with transport velocity $\upsilon=1/\tau$ is
equivalent to the time delay $\tau$ of the control law (10). With this
definition, the closed loop system (1), (13), (14) is then represented as
follows:

$\displaystyle\partial_{t}y_{1}(t,x)+\upsilon\partial_{x}y_{1}(t,x)=0,$ (15a)
$\displaystyle\partial_{t}y_{2}(t,x)+\upsilon\partial_{x}y_{2}(t,x)=0,$ (15b)
$\displaystyle\partial_{t}\hat{y}_{2}(t,x)+\upsilon\partial_{x}\hat{y}_{2}(t,x)=0,$
(15c) $\displaystyle\begin{pmatrix}y_{1}(t,0)\\\\[5.0pt] y_{2}(t,0)\\\\[5.0pt]
\hat{y}_{2}(t,0)\end{pmatrix}=\underbrace{\begin{pmatrix}-2k_{1}&1&-k_{2}\\\\[5.0pt]
1&0&0\\\\[5.0pt] \
1&0&0\end{pmatrix}}_{\displaystyle\mathbf{K}}\begin{pmatrix}y_{1}(t,1)\\\\[5.0pt]
y_{2}(t,1)\\\\[5.0pt] \hat{y}_{2}(t,1)\end{pmatrix}.$ (15d)

Now, as proved in Appendix A, for the matrix $\mathbf{K}$ defined in (15d), it
can be shown that

$\bar{\rho}(\mathbf{K})=|k_{1}|+\sqrt{1+k_{1}^{2}+|k_{2}|}\;\geqslant 1\hskip
14.22636pt\text{for all }(k_{1},k_{2})\in\mathbb{R}^{2},$ (16)

where $\bar{\rho}(\mathbf{K})$ is defined as follows:

$\bar{\rho}(\mathbf{K}):=\max\\{\rho(\text{diag}\\!\left\\{e^{-i\theta_{1}},e^{-i\theta_{2}},e^{-i\theta_{3}}\right\\}\mathbf{K});(\theta_{1},\theta_{2},\theta_{3})^{\\!\mathsf{T}}\in\mathbb{R}^{3}\\},$
(17)

$\rho(M)$ denoting the spectral radius of the matrix $M$. By [21] (see also
[15, Chapter 9, Theorem 6.1] and [5, Chapter 3]), we know that
$\bar{\rho}(\mathbf{K})<1$ is a necessary (and sufficient) condition to have a
stability which is robust against small uncertainties in the characteristic
velocities. Although the ideal closed loop system (15) is exponentially stable
(with all the poles strictly located in the left half complex plane provided
$k_{1}$ and $k_{2}$ are chosen accordingly), the stability can be destroyed by
an arbitrarily small difference in characteristic velocities between the plant
equations (15a), (15b) and the controller equation (15c). More precisely, if
we assume that the physical transport velocities are
$\upsilon+\varepsilon_{1}$, $\upsilon+\varepsilon_{2}$ with
$\varepsilon_{1},\varepsilon_{2}$ representing uncertainties in the plant
equations (15a), (15b) rewritten as follows:

$\displaystyle\partial_{t}y_{1}(t,x)+(\upsilon+\varepsilon_{1})\partial_{x}y_{1}(t,x)=0,$
(18a)
$\displaystyle\partial_{t}y_{2}(t,x)+(\upsilon+\varepsilon_{2})\partial_{x}y_{2}(t,x)=0,$
(18b)

then the closed loop system (15) with (15a), (15b) replaced by (18a), (18b)
may become unstable, with poles moving to the right half complex plane even
for arbitrarily small $\varepsilon_{i}$ perturbations.

Should this necessarily mean that the control law (10) with delayed feedback
could not be applied in practice? Our objective, in this paper, is exactly to
prove the opposite! Indeed, our main contribution will be to show that, even
with the simple control law (10), the robustness of the output feedback
stabilization against delay uncertainties can be recovered as soon as an
arbitrary small diffusion is present in the system. Our analysis will also
reveal, however, that the effect on stability of adding an arbitrarily small
diffusion is far from obvious, contrary to what one might expect. Indeed,
while diffusion strengthens the robustness of the exponential stability for
the $2\times 2$ problem (18), it can also destroy the stability of similar
simpler systems as we will see in Section 6.

We shall consider the special case of a dead beat control where $k_{1}=0$ and
$k_{2}=1$. In that case the characteristic equation of the closed loop
inviscid system reduces to $e^{2s\tau}=0$, meaning that all the poles of the
system have negative real parts that are moved off to infinity. However, for
that system we have

$\bar{\rho}(\mathbf{K})=\sqrt{2}$ (19)

showing a strict lack of robustness of the control w.r.t. delay inaccuracy.

##  3 The open loop control system with diffusion

We consider again a control system as represented in Figure 1. However, we
assume here that the two transport systems are subject to a slight phenomenon
of diffusion. The system is therefore made up of the feedback interconnection
of two identical transport systems that are perturbed by a diffusion term
which can be interpreted for example as a viscosity in the case of a fluid.
For simplicity and without loss of generality, we assume a unit nominal
transport velocity $\upsilon=1$. The dynamics of the open loop control system
are therefore described in the time domain by the following equations:

$\displaystyle\partial_{t}y_{1}(t,x)+\partial_{x}y_{1}(t,x)-\eta\partial^{2}_{xx}y_{1}(t,x)=0,$
(20a)
$\displaystyle\partial_{t}y_{2}(t,x)+\partial_{x}y_{2}(t,x)-\eta\partial^{2}_{xx}y_{2}(t,x)=0,$
(20b) $\displaystyle y_{1}(t,0)=y_{2}(t,1)+U(t),$ (20c) $\displaystyle
y_{2}(t,0)=y_{1}(t,1),$ (20d)
$\displaystyle\partial_{x}y_{1}(t,1)=\partial_{x}y_{2}(t,1)=0$ (20e)
$\displaystyle Y(t)=y_{1}(t,1).$ (20f)

As above $U(t)$ is the control input and $Y(t)$ is the measurable output while
$\eta>0$ is the viscosity coefficient.

In the frequency domain, with $\textrm{y}_{\\!1}(s,x)$ and
$\textrm{y}_{2}(s,x)$ denoting the Laplace transforms of $y_{1}(t,x)$ and
$y_{2}(t,x)$, the system is written

$\displaystyle
s\textrm{y}_{\\!1}(s,x)+\partial_{x}\textrm{y}_{\\!1}(s,x)-\eta\partial^{2}_{xx}\textrm{y}_{\\!1}(s,x)=0,$
(21a) $\displaystyle
s\textrm{y}_{2}(s,x)+\partial_{x}\textrm{y}_{2}(s,x)-\eta\partial^{2}_{xx}\textrm{y}_{2}(s,x)=0,$
(21b) $\displaystyle\textrm{y}_{\\!1}(s,0)=\textrm{y}_{2}(s,1)+\textrm{U}(s),$
(21c) $\displaystyle\textrm{y}_{2}(s,0)=\textrm{y}_{\\!1}(s,1),$ (21d)
$\displaystyle\partial_{x}\textrm{y}_{\\!1}(s,1)=\partial_{x}\textrm{y}_{2}(s,1)=0$
(21e) $\displaystyle\textrm{Y}(s)=\textrm{y}_{\\!1}(s,1).$ (21f)

For any value of s, the solutions of the differential equations (21a), (21b)
are written

$\textrm{y}_{i}(s,x)=A_{i}(s)e^{\lambda_{1}(s)x}+B_{i}(s)e^{\lambda_{2}(s)x},\hskip
14.22636pti=1,2,$ (22)

where $\lambda_{1}(s)$, $\lambda_{2}(s)$ are the roots of the polynomial

$\eta\lambda^{2}-\lambda-s=0,$ (23)

which implies that

$\lambda_{1}(s)=\dfrac{1+\sqrt{1+4\eta s}}{2\eta},\hskip
14.22636pt\lambda_{2}(s)=\dfrac{1-\sqrt{1+4\eta s}}{2\eta}.$ (24)

In (24) and in the following $\sqrt{\cdot}$ denotes the principal value of the
square root, which is well defined except when $1+4\eta s\in\mathbb{R}_{-}$;
however this particular case implies that $s$ is real and $s\leq-1/4\eta$ and
therefore is negative and converges to $-\infty$ when $\eta\rightarrow 0^{+}$.
We will see later on that this case can be considered separately. Using the
solution (22) and the boundary condition (21e), we have for each $i=1,2$,

$\displaystyle A_{i}(s)+B_{i}(s)=\textrm{y}_{i}(s,0),$ (25)
$\displaystyle\lambda_{1}(s)A_{i}(s)e^{\lambda_{1}(s)}+\lambda_{2}(s)B_{i}(s)e^{\lambda_{2}(s)}=0,$
(26)
$\displaystyle\textrm{y}_{i}(s,1)=A_{i}(s)e^{\lambda_{1}(s)}+B_{i}(s)e^{\lambda_{2}(s)}.$
(27)

Eliminating $A_{i}(s)$ and $B_{i}(s)$ between these three equations, we get
the transfer function $f_{\eta}(s)$ of each viscous transport system:

$f_{\eta}(s)=\dfrac{\textrm{y}_{i}(s,1)}{\textrm{y}_{i}(s,0)}=\dfrac{\lambda_{1}(s)-\lambda_{2}(s)}{\lambda_{1}(s)e^{-\lambda_{2}(s)}-\lambda_{2}(s)e^{-\lambda_{1}(s)}}.$
(28)

Remark that in the notation $f_{\eta}$, we use a subscript to emphasize the
dependency on the viscosity parameter $\eta$. Remark also that in the limit,
in the absence of a viscosity term (i.e. $\eta=0$), we recover the transfer
function (2): $f_{o}(s)=e^{-s\tau}$.

It follows that the system (21) is equivalent to:

$\displaystyle\textrm{y}_{\\!1}(s,1)=f_{\eta}(s)\textrm{y}_{\\!1}(s,0),$ (29a)
$\displaystyle\textrm{y}_{2}(s,1)=f_{\eta}(s)\textrm{y}_{2}(s,0),$ (29b)
$\displaystyle\textrm{y}_{\\!1}(s,0)=\textrm{y}_{2}(s,1)+\textrm{U}(s),$ (29c)
$\displaystyle\textrm{y}_{2}(s,0)=\textrm{y}_{\\!1}(s,1),$ (29d)
$\displaystyle\textrm{Y}(s)=\textrm{y}_{\\!1}(s,1).$ (29e)

Eliminating $\textrm{y}_{i}(s,0),\textrm{y}_{i}(s,1),(i=1,2),$ between these
equations, we get that the overall input-output transfer function of the open
loop system (20) is

$G(s)=\dfrac{\textrm{Y}(s)}{\textrm{U}(s)}=\dfrac{f_{\eta}(s)}{1-f_{\eta}^{2}(s)}.$
(30)

The poles of the system are the roots of the characteristic equation

$f_{\eta}^{2}(s)-1=0.$ (31)

In particular, it can be checked that $f_{\eta}(0)=1$ for all $\eta\neq 0$,
which means that there is a pole at the origin. Therefore, as in the inviscid
case, the open loop system (20) is not asymptotically stable whatever the
value of the viscosity $\eta$. As a matter of illustration, in Figure 2, we
present the spectrum of the system for $\eta=0.1$.

Figure 2: The spectrum of the open loop system with viscosity $\eta=0.1$.

In the next section, we shall show that the system can be stabilized by output
feedback when the (unknown) viscosity is small and even almost negligible.

##  4 Output feedback stabilization of the viscous system

We now assume that the open loop control system (20) is closed with a dead
beat output feedback controller

$U(t)=-Y(t-1).$ (32)

Note that this corresponds to the special case of the controller (10) where
$k_{1}=0$, $k_{2}=1$ and $\tau=1/\upsilon=1$. We know from Section 2 that, in
this case, the inviscid system is exponentially stable but the stability is
not robust to small perturbations in the propagation speeds. We shall show
here that the exponential stability remains in the viscous system and we shall
check in Section 5 that, in addition, the exponential stability is robust with
small variations in the propagation speeds. In the frequency domain, the
closed loop system (20), (32) is then:

$\displaystyle\textrm{y}_{\\!1}(s,1)=f_{\eta}(s)\textrm{y}_{\\!1}(s,0),$ (33a)
$\displaystyle\textrm{y}_{2}(s,1)=f_{\eta}(s)\textrm{y}_{2}(s,0),$ (33b)
$\displaystyle\hat{\textrm{y}}_{2}(s,1)=e^{-s}\hat{\textrm{y}}_{2}(s,0)$ (33c)
$\displaystyle\textrm{y}_{\\!1}(s,0)=\textrm{y}_{2}(s,1)-\hat{\textrm{y}}_{2}(s,1),$
(33d) $\displaystyle\textrm{y}_{2}(s,0)=\textrm{y}_{\\!1}(s,1),$ (33e)
$\displaystyle\hat{\textrm{y}}_{2}(s,0)=\textrm{y}_{\\!1}(s,1),$ (33f)

From these equations, it follows that the characteristic equation of the
closed loop system is:

$f_{\eta}^{2}(s)-f_{\eta}(s)e^{-s}-1=0.$ (34)

Our purpose is now to address the stability of this closed loop system. For a
given value of the viscosity $\eta$, the spectrum $\mathcal{S}_{\eta}$ of the
closed loop system is the set of the poles which are the roots of the
characteristic equation (34):

$\mathcal{S}_{\eta}=\\{s:f_{\eta}^{2}(s)-f_{\eta}(s)e^{-s}-1=0\\}.$ (35)

Moreover, the maximal spectral abscissa is defined as the supremum of the real
parts of the spectrum and denoted as follows:

$\sigma_{\eta}=\sup\\{\text{Re}(s):s\in\mathcal{S}_{\eta}\\}.$ (36)

As a matter of illustration, we present in Figure 3 the spectrum of the closed
loop system for $\eta=0.1$.

Figure 3: The spectrum $\mathcal{S}_{\eta}$ of the closed loop system for
$\eta=0.1$.

The stability of the closed loop system (33) can be deduced using the spectral
mapping theorem. See [15, Chapter 9, Theorem 3.5] and [19] in the case
$\eta=0$, and reference [6] for the case $\eta\neq 0$. In particular the
system is exponentially stable if (and only if) $\sigma_{\eta}<0$.

Then, one of the main results of this paper is given in the following
stability theorem.

######  Theorem 1.

For any $\delta>0$, there exists $\eta_{1}>0$ such that, for all
$\eta\in(0,\eta_{1})$, the maximal spectral abscissa satisfies

$\sigma_{\eta}\leqslant-\ln(2)+\delta.$ (37)

∎

######  Conjecture 1.

The bound $\ln(2)$ on the decay rate is optimal. More precisely, for any
$\delta>0$ and $\eta_{1}>0$ there exists $\eta\in(0,\eta_{1})$ such that the
maximal abscissa satisfies

$-\ln(2)-\delta\leqslant\sigma_{\eta}\leqslant-\ln(2)+\delta.$ (38)

∎

######  Remark 1 (Loss of continuity of the spectral abscissa).

Conjecture 1 seems to be verified when looking at the spectrum numerically
(see Figure 4). This would imply a loss of continuity in the sense that any
decay rate is achievable when $\eta=0$. However there is a bound $\ln(2)$ as
soon as $\eta>0$, even if $\eta$ is arbitrarily small.

In order to prove Theorem 1, it is useful to define the complex variable

$z=\sqrt{1+4\eta s},$ (39)

where we recall that $\sqrt{\cdot}$ is the principal value of the square root.
The functions $\lambda_{1}$ and $\lambda_{2}$ become

$\lambda_{1}=\dfrac{(1+z)}{2\eta}\hskip 14.22636pt\text{and}\hskip
14.22636pt\lambda_{2}=\dfrac{(1-z)}{2\eta}.$ (40)

Then, defining the function

$X_{\eta}(z)=\dfrac{1+z}{2}e^{-(1-z)/2\eta}-\dfrac{1-z}{2}e^{-(1+z)/2\eta},$
(41)

we get, using (28), that the characteristic equation (34) is equivalent to

$X^{2}_{\eta}(z)+ze^{-(z^{2}-1)/4\eta}X_{\eta}(z)-z^{2}=0.$ (42)

Let us now observe that $X_{\eta}(0)=0$. Consequently $z=0$ is a root of
equation (42) for all $\eta$. This implies obviously that
$s=-1/4\eta\in\mathcal{S}_{\eta}$ is a pole of the system (i.e. a root of the
characteristic equation (34)). Note that this pole tends to $-\infty$ as
$\eta\rightarrow 0^{+}$ so this is a very stable pole for small (positive)
$\eta$.

Let $\mathcal{Z}_{\eta}$ denote the set of non-zero roots of equation (42).
Then, using definition (41) and solving equation (42) with respect to
$X_{\eta}$, we have that every $z\in\mathcal{Z}_{\eta}$ satisfies the equation

$\dfrac{(1+z)}{z}e^{-(1-z)/2\eta}-\dfrac{(1-z)}{z}e^{-(1+z)/2\eta}=-e^{-(z^{2}-1)/4\eta}\pm\sqrt{e^{-(z^{2}-1)/2\eta}+4}.$
(43)

We now consider a sequence

$(\eta_{n})_{n\in\mathbb{N}}\hskip 14.22636pt\text{with}\hskip
14.22636pt0<\eta_{n}\in\mathbb{R},\;\forall n\in\mathbb{N}\hskip
14.22636pt\text{and}\hskip
14.22636pt\lim_{n\rightarrow+\infty}\eta_{n}=0^{+},$ (44)

and an associated sequence

$(s_{n})_{n\in\mathbb{N}}\hskip 14.22636pt\text{such that}\hskip
14.22636pts_{n}\in\mathcal{S}_{\eta_{n}},\forall n\in\mathbb{N}.$ (45)

In order to prove Theorem 1, we will look to the adherent points of the
sequences $(s_{n})_{n\in\mathbb{N}}$ when $n\rightarrow+\infty$ (i.e. when
$\eta_{n}\rightarrow 0^{+}$). By definition, we know that $\bar{s}$ is an
adherent point of a sequence $(s_{n})_{n\in\mathbb{N}}$ if and only if there
exists a subsequence which converges to $\bar{s}$. With a slight abuse of
notation, we will write

$s_{n}\longrightarrow\bar{s}\hskip 14.22636pt\text{or}\hskip
14.22636pt\lim_{n\rightarrow+\infty}s_{n}=\bar{s}$ (46)

to signify that $\bar{s}$ is an adherent point of a sequence
$(s_{n})_{n\in\mathbb{N}}$ but it is implied that the convergence in fact only
relies on the adequate subsequence. This holds also for all other sequences
that are introduced later in this article.

The proof of Theorem 1 is built from the two following lemmas.

######  Lemma 1.

Let $(\eta_{n})_{n\in\mathbb{N}}$ be a sequence of the form (44) and
$(z_{n})_{n\in\mathbb{N}}$ be any associated sequence of elements of
$\mathcal{Z}_{\eta}$. Let $\bar{z}$ be an adherent point of the sequence
$(z_{n})_{n\in\mathbb{N}}$. Then

$\displaystyle\text{a)}\;\;\text{Re}(\bar{z}^{2})\leqslant 1\hskip
14.22636pt\big{(}\text{precisely:
}\text{Re}(\bar{z}^{2})\in[-\infty,1]\;\big{)};$ (47)
$\displaystyle\text{b)}\;\;\text{if }\text{Re}(\bar{z}^{2})=1\text{ then
}\left(\text{Re}(\bar{z})\right)^{\\!2}=1\text{ and }\text{Im}(\bar{z})=0.$
(48)

###### Proof.

The proof of this lemma is given in Appendix B. ∎

######  Lemma 2.

Let $(\eta_{n})_{n\in\mathbb{N}}$ be a sequence of the form (44) and
$(s_{n})_{n\in\mathbb{N}}$ be any associated sequence of the form (45). Let
$\bar{s}$ be an adherent point of $(s_{n})_{n\in\mathbb{N}}$. Then

$\text{Re}(\bar{s})\leqslant-\ln(2)\hskip 14.22636pt\big{(}\text{precisely:
}\text{Re}(\bar{s})\in[-\infty,-\ln(2)]\;\big{)}.$ (49)

###### Proof.

Let $\bar{s}$ be an adherence point of the considered sequence
$(s_{n})_{n\in\mathbb{N}}$. We restrict to a subsequence (still denoted
$(s_{n})_{n\in\mathbb{N}}$) such that $(s_{n})_{n\in\mathbb{N}}$ converges to
$\bar{s}$. Consider the sequence

$(z_{n})_{n\in\mathbb{N}}\hskip 14.22636pt\text{such that}\hskip
14.22636ptz_{n}=\sqrt{1+4\eta s_{n}},\forall n\in\mathbb{N}.$ (50)

By definition $z_{n}\in\mathcal{Z}_{\eta}$ for any $n\in\mathbb{N}$. Let
$\bar{z}$ be an adherence value of this sequence $(z_{n})_{n\in\mathbb{N}}$
and let us again restrict to a subsequence (still denoted
$(z_{n})_{n\in\mathbb{N}}$) that converges to $\bar{z}$. Note that since it is
a subsequence, we still have $s_{n}\rightarrow\bar{s}$. From Lemma 1, we know
that necessarily $\text{Re}(\bar{z}^{2})\in[-\infty,1]$.

Let us first assume that $\text{Re}(\bar{z}^{2})$ is strictly smaller than 1,
i.e. $\text{Re}(\bar{z}^{2})\in[-\infty,1)$. In that case, from the definition
of $s_{n}$ in (45), we have

$\text{Re}(\bar{s})=\lim_{n\rightarrow+\infty}\text{Re}(s_{n})=\lim_{n\rightarrow+\infty}\dfrac{\text{Re}(z_{n}^{2})-1}{4\eta_{n}}=\dfrac{\lim_{n\rightarrow+\infty}\text{Re}(z_{n}^{2})-1}{\lim_{n\rightarrow+\infty}4\eta_{n}}=\dfrac{\text{Re}(\bar{z}^{2})-1}{\lim_{n\rightarrow+\infty}4\eta_{n}}.$
(51)

Since, by the definition (44), we know that
$\lim_{n\rightarrow+\infty}4\eta_{n}=0^{+}$, we can conclude

$\text{Re}(\bar{s})=-\infty$ (52)

and the lemma is proved.

Let us now assume that $\text{Re}(\bar{z}^{2})=1$. From Lemma 1 we know that,
if $\text{Re}(\bar{z}^{2})=1$, then necessarily $\text{Re}(\bar{z})=\pm 1$ and
$\text{Im}(\bar{z})=0$; hence $\bar{z}=\pm 1$. We address the case where
$\bar{z}=1$ (the reader can easily handle the case $\bar{z}=-1$ by symmetry)
and we introduce the notations

$y_{n}=z_{n}-1,\hskip 14.22636pta_{n}=\text{Re}(y_{n}),\hskip
14.22636ptb_{n}=\text{Im}(y_{n}).$ (53)

With these notations we have

$\text{Re}(s_{n})=\dfrac{\text{Re}(z_{n}^{2})-1}{4\eta_{n}}=\dfrac{\text{Re}((y_{n}+1)^{2})-1}{4\eta_{n}}=\dfrac{a^{2}_{n}-b_{n}^{2}}{4\eta_{n}}+\dfrac{a_{n}}{2\eta_{n}}.$
(54)

Here the determination of adherent points of the sequence
$(\text{Re}(s_{n}))_{n\in\mathbb{N}}$, induced by the limit
$\text{Re}(z_{n})\rightarrow 1$ is clearly more delicate because it cannot be
directly derived from formula (54) and requires a development which is
detailed hereafter.

With the notation (53), the function $X_{\eta}$ defined in (41) becomes

$X_{\eta}=(1+\frac{y_{n}}{2})e^{y_{n}/2\eta_{n}}+\frac{y_{n}}{2}e^{-1/\eta_{n}-y_{n}/2\eta_{n}},$
(55)

such that the characteristic equation (42) is written

$\displaystyle((1+\frac{y_{n}}{2})e^{y_{n}/2\eta_{n}}+\frac{y_{n}}{2}e^{-1/\eta_{n}-y_{n}/2\eta_{n}})^{2}$
$\displaystyle\hskip 28.45274pt\hskip
28.45274pt+(1+y_{n})e^{-y_{n}/2\eta_{n}-y_{n}^{2}/4\eta_{n}}[(1+\frac{y_{n}}{2})e^{y_{n}/2\eta_{n}}+\frac{y_{n}}{2}e^{-1/4\eta_{n}-y_{n}/2\eta_{n}}]=(1+y_{n})^{2}.$
(56)

Now, because $\eta_{n}\rightarrow 0^{+}$ and $y_{n}\rightarrow 0$ (since
$z_{n}\rightarrow\bar{z}$), we have

$\displaystyle(1+\frac{y_{n}}{2})^{2}e^{y_{n}/\eta_{n}}+o(e^{-1/\eta_{n}})$
$\displaystyle\hskip 28.45274pt\hskip
28.45274pt+(1+y_{n})(1+\frac{y_{n}}{2})e^{-y_{n}^{2}/4\eta_{n}}+o(e^{-1/4\eta_{n}-y_{n}/\eta_{n}-y_{n}^{2}/4\eta_{n}})=(1+y_{n})^{2}.$
(57)

This implies that, for $n\rightarrow+\infty$,

$e^{y_{n}/\eta_{n}}+\frac{1+y_{n}}{1+\frac{y_{n}}{2}}e^{-y_{n}^{2}/4\eta_{n}}\longrightarrow
1.$ (58)

With (54), (58) becomes

$e^{a_{n}/\eta_{n}}e^{ib_{n}/\eta_{n}}+\frac{(1+a_{n}+ib_{n})(1+a_{n}/2-ib_{n}/2)}{|1+\frac{y_{n}}{2}|^{2}}e^{-a_{n}^{2}/4\eta_{n}}e^{b_{n}^{2}/4\eta_{n}}e^{-ia_{n}b_{n}/2\eta_{n}}\rightarrow
1,$ (59)

or, separating the real and imaginary parts,

$\displaystyle
e^{a_{n}/\eta_{n}}\cos({\textstyle\frac{b_{n}}{\eta_{n}}})+e^{-a_{n}^{2}/4\eta_{n}}e^{b_{n}^{2}/4\eta_{n}}\left[\left(1+{\textstyle\frac{3a_{n}+a_{n}^{2}+b_{n}^{2}}{2}}\right)\cos({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})+{\textstyle\frac{b_{n}}{2}}\sin({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})\right]\frac{1}{|1+\frac{y_{n}}{2}|^{2}}\rightarrow
1,$ (60) $\displaystyle
e^{a_{n}/\eta_{n}}\sin({\textstyle\frac{b_{n}}{\eta_{n}}})+e^{-a_{n}^{2}/4\eta_{n}}e^{b_{n}^{2}/4\eta_{n}}\left[-\left(1+{\textstyle\frac{3a_{n}+a_{n}^{2}+b_{n}^{2}}{2}}\right)\sin({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})+{\textstyle\frac{b_{n}}{2}}\cos({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})\right]\frac{1}{|1+\frac{y_{n}}{2}|^{2}}\rightarrow
0.$ (61)

Let us now denote $\bar{a}$ an adherence point of the sequence
$(a_{n}/2\eta_{n})_{n\in\mathbb{N}}$. We shall discuss successively the three
cases : $\bar{a}=-\infty$, $\bar{a}\in(-\infty,+\infty)$ and
$\bar{a}=+\infty$.

The case $\mathbf{\bar{a}=-\infty.}$
In this case the lemma is trivial because, since $a_{n}\rightarrow 0$ and
$a_{n}/2\eta_{n}\rightarrow\bar{a}=-\infty$, then
$a_{n}^{2}/4\eta_{n}+a_{n}/2\eta_{n}\rightarrow-\infty$ and therefore, from
(54), $\text{Re}(\bar{s})=-\infty$.

The case $\mathbf{\bar{a}\in(-\infty,+\infty).}$

In this case $a_{n}^{2}/2\eta_{n}\rightarrow 0$ and
$a_{n}b_{n}/2\eta_{n}\rightarrow 0$ because $a_{n}+ib_{n}=y_{n}\rightarrow 0$
while $a_{n}/2\eta_{n}\rightarrow\bar{a}\in\mathbb{R}$. Then from (60), we
have

$e^{a_{n}/\eta_{n}}\cos({\textstyle\frac{b_{n}}{\eta_{n}}})+e^{b_{n}^{2}/4\eta_{n}}(1+o(1))\longrightarrow
1.$ (62)

Let us now denote $\bar{c}\in[0,+\infty]$ an adherence point of the sequence
$(b_{n}^{2}/4\eta_{n})_{n\in\mathbb{N}}$ and $\bar{\kappa}\in[-1,1]$ an
adherent point of the sequence
$(\cos({\textstyle\frac{b_{n}}{\eta_{n}}}))_{n\in\mathbb{N}}$. Then, taking
the limit111This can be done by restricting to a subsequence where both
$(b_{n}^{2}/4\eta_{n})_{n\in\mathbb{N}}$ and
$(\cos({\textstyle\frac{b_{n}}{\eta_{n}}}))_{n\in\mathbb{N}}$ converge, using
a diagonal argument., we have from (62)

$e^{2\bar{a}}\bar{\kappa}+e^{\bar{c}}=1.$ (63)

Because $e^{\bar{a}}$ is bounded and $e^{\bar{a}}>0$, and
$b_{n}^{2}/4\eta_{n}\geq 0$ for any $n\in\mathbb{N}$, this implies necessarily
that

$\bar{c}\in[0,+\infty)\hskip 14.22636pt\text{and}\hskip
14.22636pte^{\bar{c}}\geqslant 1\hskip 14.22636pt\text{and}\hskip
14.22636pt\bar{\kappa}\in[-1,0].$ (64)

Then it means in particular that $b_{n}^{2}/\eta_{n}$ is bounded when
$\eta_{n}\rightarrow 0^{+}$, and from (61) we deduce that

$e^{a_{n}/\eta_{n}}\sin({\textstyle\frac{b_{n}}{\eta_{n}}})+o(1)\longrightarrow
0$ (65)

which implies that $\sin({\textstyle\frac{b_{n}}{\eta_{n}}})\rightarrow 0$ and
therefore that the only possible adherence points for
$\cos({\textstyle\frac{b_{n}}{\eta_{n}}})$ are $\bar{\kappa}=\pm 1$. Since
$\bar{\kappa}\in[-1,0]$, we deduce that $\bar{\kappa}=-1$. Therefore, from
(63),

$e^{\bar{c}}=1+e^{2\bar{a}}.$ (66)

Let us now consider the function $e^{-\text{Re}(s_{n})}$ with
$\text{Re}(s_{n})$ given by (54):

$e^{-\text{Re}(s_{n})}=e^{(b_{n}^{2}-a_{n}^{2})/4\eta_{n}-a_{n}/2\eta_{n}}\longrightarrow
e^{\bar{c}}e^{-\bar{a}}=e^{-\bar{a}}+e^{\bar{a}}=2\cosh(\bar{a})\geqslant 2.$
(67)

It follows directly that

$\text{Re}(\bar{s})\leqslant-\ln(2)$ (68)

and the lemma is proved.

The case $\mathbf{\bar{a}=+\infty.}$

In this case we first observe that
$e^{(a_{n}^{2}-b_{n}^{2})/4\eta_{n}-a_{n}/2\eta_{n}}\rightarrow 0$. Then
multiplying both sides of (60) by this quantity, we get:

$\displaystyle
e^{(a_{n}^{2}-b_{n}^{2})/4\eta_{n}+a_{n}/2\eta_{n}}\cos({\textstyle\frac{b_{n}}{\eta_{n}}})+e^{-a_{n}/2\eta_{n}}\left[(1+{\textstyle\frac{3a_{n}+a_{n}^{2}+b_{n}^{2}}{2}})\cos({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})+{\textstyle\frac{b_{n}}{2}}\sin({\textstyle\frac{a_{n}b_{n}}{2\eta_{n}}})\right]\frac{1}{|1+\frac{y_{n}}{2}|^{2}}$
$\displaystyle\hskip
227.62204pt-e^{(a_{n}^{2}-b_{n}^{2})/4\eta_{n}-a_{n}/2\eta_{n}}\rightarrow 0,$
(69)

From this expression, we deduce that

$e^{(a_{n}^{2}-b_{n}^{2})/4\eta_{n}+a_{n}/2\eta_{n}}\cos({\textstyle\frac{b_{n}}{\eta_{n}}})\longrightarrow
0.$ (70)

A similar manipulation of (61) gives

$e^{(a_{n}^{2}-b_{n}^{2})/4\eta_{n}+a_{n}/2\eta_{n}}\sin({\textstyle\frac{b_{n}}{\eta_{n}}})\longrightarrow
0.$ (71)

Combining (70) and (71) we obtain

$e^{(a_{n}^{2}-b_{n}^{2})/2\eta_{n}+a_{n}/\eta_{n}}=e^{2\text{Re}(s_{n})}\longrightarrow
0$ (72)

which implies that $\text{Re}(s_{n})\rightarrow-\infty$ and the lemma is
proved. ∎

Based on Lemma 2, we can now give the following proof of Theorem 1.

Proof of Theorem 1.
We assume by contradiction that the theorem does not hold:

$\text{For any }\delta>0,\nexists\,\eta_{1}>0\text{ such that
}\sigma_{\eta}\leqslant-\ln(2)+\delta\;\;\forall\eta\in(0,\eta_{1}).$ (73)

This implies that

$\text{For all }\eta>0,\;\exists\eta_{1}\in(0,\eta)\text{ and }\exists
s\in\mathcal{S}_{\eta_{1}}\text{ such that }\text{Re}(s)>-\ln(2)+\delta.$ (74)

It follows that we can build a sequence $(\eta_{n})_{n\in\mathbb{N}}$ of the
form (44) and an associated sequence $(s_{n})_{n\in\mathbb{N}}$ with
$s_{n}\in\mathcal{S}_{\eta_{n}}$. For this sequence we deduce from (74) that
necessarily $\text{Re}(\bar{s})>-\ln(2)+\delta$, which is in contradiction
with Lemma 2. ∎

Figure 4: The spectrum $\mathcal{S}_{\eta}$ of the closed loop system for
$\eta=0.02$ • and $\eta=0.2$ •.

The theorem is illustrated in Figure 4 where the spectrum is represented for
the values $\eta$ = 0.02 and 0.2, and where the effectiveness of the stability
margin $-\ln(2)$ can be appreciated.

##  5 Robustness analysis

We consider again the control problem of the previous section. However,
because we want to explicitly account for the sensitivity to delay
uncertainties, we introduce an additional perturbation $\varepsilon$ to the
nominal transport velocity $\upsilon=1$. The open loop system is therefore
written in the time domain as follows:

$\displaystyle\partial_{t}y_{1}(t,x)+(1+\varepsilon)\partial_{x}y_{1}(t,x)-\eta\partial^{2}_{xx}y_{1}(t,x)=0,$
(75a)
$\displaystyle\partial_{t}y_{2}(t,x)+(1+\varepsilon)\partial_{x}y_{2}(t,x)-\eta\partial^{2}_{xx}y_{2}(t,x)=0,$
(75b) $\displaystyle y_{1}(t,0)=y_{2}(t,1)+U(t),$ (75c) $\displaystyle
y_{2}(t,0)=y_{1}(t,1),$ (75d)
$\displaystyle\partial_{x}y_{1}(t,1)=\partial_{x}y_{2}(t,1)=0$ (75e)
$\displaystyle Y(t)=y_{1}(t,1),$ (75f)

We suppose, as before, that the system is closed with the dead beat output
feedback controller

$U(t)=-Y(t-1).$ (76)

Remark that this control law depends on the theoretical delay ($\tau=1$),
ignoring the uncertainty represented by $\varepsilon$.

Remark also that for simplicity we assume here that we have the same
uncertainty $\varepsilon$ on both physical subsystems represented by transport
equations (75a) and (75b). This may seem like a simplification but, actually,
it can be shown that this single perturbation, even if it is arbitrarily
small, is sufficient to destroy the closed loop stability when $\eta=0$ (see
Figure 5 hereafter).

In this case the transfer functions of the two transport subsystems become

$f_{\eta,\varepsilon}(s)=\dfrac{\lambda_{1}(s)-\lambda_{2}(s)}{\lambda_{1}(s)e^{-\lambda_{2}(s)}-\lambda_{2}(s)e^{-\lambda_{1}(s)}}.$
(77)

with the function $f_{\eta,\varepsilon}$ now depending on both $\eta$ and
$\varepsilon$ because the functions $\lambda_{1}$ and $\lambda_{2}$ are
modified as follows:

$\lambda_{1}(s)=\dfrac{(1+\varepsilon)+\sqrt{(1+\varepsilon)^{2}+4\eta
s}}{2\eta},\hskip
14.22636pt\lambda_{2}(s)=\dfrac{(1+\varepsilon)-\sqrt{(1+\varepsilon)^{2}+4\eta
s}}{2\eta}.$ (78)

With this definition, the characteristic equation of the closed loop system is
now

$f_{\eta,\varepsilon}^{2}(s)-f_{\eta,\varepsilon}(s)e^{-s}-1=0.$ (79)

As in the previous section, we introduce the spectrum
$\mathcal{S}_{\eta,\varepsilon}$ and the maximal abscissa
$\sigma_{\eta,\varepsilon}$ defined by

$\displaystyle\mathcal{S}_{\eta,\varepsilon}=\\{s\in\mathbb{C}:s\text{ is
solution of }\eqref{charac4}\\},$ (80)
$\displaystyle\sigma_{\eta,\varepsilon}=\sup\\{\text{Re}(s):s\in\mathcal{S}_{\eta,\varepsilon}\\}.$
(81)

We remark that, by definition, we have

$\mathcal{S}_{\eta,0}=\mathcal{S}_{\eta}\hskip 14.22636pt\text{ and }\hskip
14.22636pt\sigma_{\eta,0}=\sigma_{\eta}.$ (82)

We then have the following robustness theorem.

######  Theorem 2.

Let $\delta>0$ and $\eta>0$ be such that Theorem 1 holds, i.e.

$\sigma_{\eta,0}\leqslant-\text{ln}(2)+\delta.$ (83)

Then there exists $\varepsilon_{1}>0$ such that for any
$\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$ the maximal spectral
abscissa $\sigma_{\eta,\varepsilon}$ satisfies

$\sigma_{\eta,\varepsilon}\leqslant-\text{ln}(2)+2\delta$ (84)

###### Proof.

We proceed by contradiction and we assume that Theorem 2 does not hold. This
implies that for any $\varepsilon_{1}>0$ there exists
$\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$ such that

$\sigma_{\eta,\varepsilon}>-\text{ln}(2)+2\delta,$ (85)

and therefore that there exists $s\in\mathcal{S}_{\eta,\varepsilon}$ such that

$\text{Re}(s)>-\text{ln}(2)+2\delta.$ (86)

Hence we can define a sequence

$(\varepsilon_{n})_{n\in\mathbb{N}}\hskip 14.22636pt\text{with}\hskip
14.22636pt\lim_{n\rightarrow+\infty}\varepsilon_{n}=0,$ (87)

and an associated sequence $(s_{n})_{n\in\mathbb{N}}$ such that

$\text{Re}(s_{n})>-\text{ln}(2)+2\delta,\;\;\text{with}\;\;s_{n}\in\mathcal{S}_{\eta,\varepsilon_{n}}\;\forall
n\in\mathbb{N}.$ (88)

  * •

If the sequence $(s_{n})_{n\in\mathbb{N}}$ is bounded, then we can extract a
subsequence that converges to a limit $\bar{s}\in\mathbb{C}$. We still denote
this subsequence by $(s_{n})_{n\in\mathbb{N}}$. As $\varepsilon_{n}\rightarrow
0$ we can pass to the limit in (79) and deduce that
$\bar{s}\in\mathcal{S}_{\eta,0}=\mathcal{S}_{\eta}$ and therefore, from the
assumption on $\delta$ and $\eta$, that

$\text{Re}(\bar{s})\leqslant-\text{ln}(2)+\delta.$ (89)

On the other hand, again passing to the limit, we deduce from (88) that

$\text{Re}(\bar{s})\geqslant-\text{ln}(2)+2\delta>-\text{ln}(2)+\delta,$ (90)

which is in contradiction with (89).

  * •

If the sequence $(s_{n})_{n\in\mathbb{N}}$ is unbounded, then we can extract a
subsequence such that $|s_{n}|\rightarrow+\infty$. We still denote this
subsequence by $(s_{n})_{n\in\mathbb{N}}$. We denote
$s_{n}=r_{n}e^{i\theta_{n}}$, with $r_{n}\in\mathbb{R}_{+}$ and
$\theta_{n}\in[-\pi,\pi)$, for $n\in\mathbb{N}$. Since
$r_{n}=|s_{n}|\rightarrow+\infty$ and (88) holds, we deduce that for $n$
sufficiently large $\theta_{n}\in(-5\pi/8,5\pi/8)$. Denoting
$(1+\varepsilon_{n})+4\eta s_{n}=r_{n}^{1}e^{i\theta_{n}^{1}}$ with
$r_{n}^{1}\in\mathbb{R}_{+}$ and $\theta^{1}_{n}\in[-\pi,\pi)$ we deduce that
for $n$ sufficiently large (depending on $\eta$),
$r_{n}^{1}\rightarrow+\infty$ and $\theta_{n}^{1}\in(-3\pi/4,3\pi/4)$. Thus

$\sqrt{(1+\varepsilon_{n})+4\eta
s_{n}}=\sqrt{r_{n}^{1}}e^{i\theta_{n}^{1}/2},$ (91)

where $\theta_{n}^{1}/2\in(-3\pi/8,3\pi/8)$, which implies that
$\text{Re}(\sqrt{1+4\eta s_{n}})\rightarrow+\infty$. From the definition of
$\lambda_{1}$ and $\lambda_{2}$, we deduce that

$\text{Re}(\lambda_{1}(s_{n}))\rightarrow+\infty,\;\;\text{Re}(\lambda_{2}(s_{n}))\rightarrow-\infty,$
(92)

and

$f_{\eta,\varepsilon_{n}}=2\left[(\frac{(1+\varepsilon_{n})}{\sqrt{(1+\varepsilon_{n})+4\eta
s_{n}}}+1)e^{-\lambda_{2}(s_{n})}+(\frac{(1+\varepsilon_{n})}{\sqrt{(1+\varepsilon_{n})+4\eta
s_{n}}}-1)e^{-\lambda_{1}(s_{n})}\right]^{-1}.$ (93)

Using (92) and observing that
$(\frac{(1+\varepsilon_{n})}{\sqrt{(1+\varepsilon_{n})+4\eta
s_{n}}}+1)\rightarrow 1$, we obtain that $f_{\eta,\varepsilon_{n}}\rightarrow
0$. Moreover, from (88), $e^{-s_{n}}$ is bounded. Hence, we deduce that

$f_{\eta,\varepsilon_{n}}^{2}(s_{n})-f_{\eta,\varepsilon_{n}}(s_{n})e^{-s_{n}}-1\longrightarrow-1$
(94)

and, from (79), we obtain again a contradiction.

In both cases we obtain a contradiction, which means that there exists
$\varepsilon_{1}>0$ such that (84) holds for any
$\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$. This concludes the proof
of Theorem 2. ∎

Figure 5: The spectrum $\mathcal{S}_{\eta,\varepsilon}$ of the closed loop
system: influence of viscosity on stability in case of delay uncertainty.

The theorem is illustrated in Figure 5. In this figure, we can see what
happens in the situation where there is no diffusion ($\eta=0$) but a slight
uncertainty ($\varepsilon=0.1$) of the transport velocity:
$\upsilon=1+\varepsilon=1.1$ instead of $\upsilon=1$. Although the ideal
system (without modelling uncertainty) should be exponentially stable, it
appears that it becomes unstable with poles (represented by green dots in
Figure 5) moving to the right-half complex plane.

In contrast, when there is some diffusion ($\eta=0.1$) and no uncertainty
($\varepsilon=0$), we know from Theorem 1 that the closed loop system must be
stable as it can be seen with the spectrum of blue dots (actually reprinted
from Figure 3) which is entirely strictly located in the left half plane.

Then, illustrating Theorem 2, the robustness of the control in presence of
diffusion is clearly evidenced by the spectrum made up of red dots which
results from a small shift of the initial blue spectrum but remains entirely
in the left half plane.

##  6 Comparison with a simpler system

In this section, we consider the case of a system which is very close to the
previous one but slightly simpler. Somewhat surprisingly, we will see that in
this case the addition of a diffusion term does not seem to strengthen the
system stability but instead destroys the stability.

We start with an ideal system without diffusion nor modelling uncertainty
which is clearly a simplified form of the physical system (15) that we have
considered in Section 2 and is written as follows:

$\displaystyle\partial_{t}y(t,x)+\partial_{x}y(t,x)=0,$ (95a)
$\displaystyle\partial_{t}\hat{y}(t,x)+\partial_{x}\hat{y}(t,x)=0,$ (95b)
$\displaystyle\begin{pmatrix}y(t,0)\\\\[5.0pt]
\hat{y}(t,0)\end{pmatrix}=\underbrace{\begin{pmatrix}1&-1\\\\[5.0pt]
1&-1\end{pmatrix}}_{\displaystyle\mathbf{K}}\begin{pmatrix}y(t,1)\\\\[5.0pt]
\hat{y}(t,1)\end{pmatrix}.$ (95c)

In the frequency domain, the characteristic equation reduces to $e^{2s}=0$,
meaning that the system is exponentially stable (for any decay rate). In fact
one can observe easily that the system is finite time stable: for any time
$t\geq 1$, $\hat{y}(t,\cdot)=y(t,\cdot)$ on $[0,1]$, thus from the boundary
condition at $x=0$, $y(t,0)=0$, which means that $y(t,\cdot)\equiv 0$ for
$t\geq 2$. However, in this case also, the exponential stability is not robust
w.r.t. to delay inaccuracy because $\bar{\rho}(\mathbf{K})=\sqrt{2}$.

On the basis of our previous results in this paper, it seems natural to
conjecture that the addition of a diffusion term in equation (95a) should
allow to strengthen the system stability. To address this issue, the system
dynamics (95) are modified with an additional diffusion parameter $\eta$ as
follows:

$\displaystyle\partial_{t}y(t,x)+\partial_{x}y(t,x)-\eta\partial_{xx}y(t,x)=0,$
(96a) $\displaystyle\partial_{t}\hat{y}(t,x)+\partial_{x}\hat{y}(t,x)=0,$
(96b) $\displaystyle y(t,0)=y(t,1)-\hat{y}(t,1),$ (96c)
$\displaystyle\hat{y}(t,0)=y(t,0),$ (96d) $\displaystyle\partial_{x}y(t,1)=0.$
(96e)

For this system in the frequency domain, after calculations similar to those
in Sections 3 and 4, it can be shown that the characteristic equation is

$F_{\eta}(s)-1=0\hskip 14.22636pt\text{with}\hskip
14.22636ptF_{\eta}(s)=\dfrac{\left(\lambda_{1}(s)e^{\lambda_{1}(s)}-\lambda_{2}(s)e^{\lambda_{2}(s)}\right)(1+e^{-s})}{(\lambda_{1}(s)-\lambda_{2}(s))e^{\lambda_{1}(s)+\lambda_{2}(s)}},$
(97)

where $\lambda_{1}(s)$ and $\lambda_{2}(s)$ are given by (24) and repeated
here for convenience:

$\lambda_{1}(s)=\dfrac{1+\sqrt{1+4\eta s}}{2\eta},\hskip
14.22636pt\lambda_{2}(s)=\dfrac{1-\sqrt{1+4\eta s}}{2\eta}.$ (98)

For a given value of $\eta$, as in Section 4, we use the following notations
for the spectrum and the maximal spectral abscissa:

$\displaystyle\mathcal{S}_{\eta}=\\{s\in\mathbb{C}:F_{\eta}(s)-1=0\\},$ (99)
$\displaystyle\sigma_{\eta}=\sup\\{\text{Re}(s):s\in\mathcal{S}_{\eta}\\}.$
(100)

We then have the surprising observation that a slight diffusion in the system
has, in this case, a clear destabilizing effect. This is graphically
illustrated in Figure 6 and leads to the following conjecture.

######  Conjecture 2.

For all $\epsilon>0$, there exists $\eta_{1}>0$ such that for all
$\eta\in(0,\eta_{1})$ the maximal spectral abscissa satisfies the inequality
$\sigma_{\eta}>-\epsilon$. ∎

Figure 6: The spectrum $\mathcal{S}_{\eta}$ of the system (96) with
$\eta=0.1$.

##  7 Conclusion

We have discussed the output feedback stabilization of an unstable open loop
system which is made up of two interconnected transport equations and provided
with anti-located boundary sensing and actuation. We have shown that the
system can be stabilized by a dynamic controller that involves a delayed
output feedback which turns out to be non-robust with respect to delay
uncertainties. Then we have shown that the designed control law can however
stabilize the system in a robust way when there is a small unknown diffusion
in the plant.

Our work in progress on this topic [6] will be focused on the output feedback
stabilization of the motion of a viscous fluid represented by $2\times 2$
hyperbolic PDEs when the control input is the flow rate at one boundary while
the measurable output is the fluid density at the other boundary. There is, in
this case, an important difference which lies in how viscosity affects the
model, inducing a distributed internal coupling between the two partial
differential equations. This implies that the system can no longer be
considered as a feedback interconnection of two independent scalar transport
equations which may lead to additional difficulties for the stability
analysis.

## Acknowledgement

The authors would like to thank the Bernoulli center for fundamental studies
(CIB) and Joachim Krieger for their hospitality and their support, as well as
the projet PEPS JCJC 2022 of CNRS - INSMI.

## References

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##  Appendix A Computation of $\bar{\rho}(\mathbf{K})\geqslant 1$.

In this appendix we consider the matrix $\mathbf{K}$ defined in equation (15d)
as follows:

$\mathbf{K}=\begin{pmatrix}-2k_{1}&1&-k_{2}\\\\[5.0pt] 1&0&0\\\\[5.0pt] \
1&0&0\end{pmatrix}.$ (101)

From [9, Proposition 3.7] and [5, Theorem 3.12], we have

$\bar{\rho}(K)=\rho_{2}(K)$ (102)

with

$\displaystyle\rho_{2}(\mathbf{K})$
$\displaystyle:=\inf\\{\|D\mathbf{K}D^{-1}\|_{2};D\text{ is a positive
diagonal matrix}\\}$ (103)
$\displaystyle=\inf\left\\{\sqrt{\lambda_{\max}\big{[}\big{(}D\mathbf{K}D^{-1}\big{)}^{T}\big{(}D\mathbf{K}D^{-1}\big{)}\big{]}};D\text{
is a positive diagonal matrix}\right\\}.$ (104)

With the (normalized) matrix

$D=\begin{pmatrix}1&0&0\\\ 0&\theta_{2}&0\\\
0&0&\theta_{3}\end{pmatrix},\hskip 14.22636pt\theta_{2}>0,\;\theta_{3}>0,$
(105)

we have

$\mathcal{M}=\big{(}D\mathbf{K}D^{-1}\big{)}^{T}\big{(}D\mathbf{K}D^{-1}\big{)}=\begin{pmatrix}4k_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}&-2k_{1}\theta_{2}^{-1}&2k_{1}k_{2}\theta_{3}^{-1}\\\\[5.0pt]
-2k_{1}\theta_{2}^{-1}&\theta_{2}^{-2}&-k_{2}\theta_{2}^{-1}\theta_{3}^{-1}\\\\[5.0pt]
2k_{1}k_{2}\theta_{3}^{-1}&-k_{2}\theta_{2}^{-1}\theta_{3}^{-1}&k_{2}^{2}\theta_{3}^{-2}\end{pmatrix}.$
(106)

Then we have

$\det\big{(}\lambda\mathbf{I}_{3}-\mathcal{M}\big{)}=\lambda(\lambda^{2}-\beta\lambda+\gamma)$
(107)

with

$\displaystyle\beta=4k_{1}^{2}+(\theta_{2}^{2}+\theta_{2}^{-2})+(\theta_{3}^{2}+k_{2}^{2}\theta_{3}^{-2}),$
(108)
$\displaystyle\gamma=1+\theta_{2}^{-2}\theta_{3}^{2}+k_{2}^{2}+k_{2}^{2}\theta_{2}^{2}\theta_{3}^{-2}.$
(109)

From (107) we see that the eigenvalues of the matrix $\mathcal{M}$ are

$\lambda=0\hskip 14.22636pt\text{and}\hskip
14.22636pt\lambda=\dfrac{\beta\pm\sqrt{\beta^{2}-4\gamma}}{2}.$ (110)

From (108) we know that $\beta\geqslant 0$. Moreover,
$\beta^{2}-4\gamma\geqslant 0$ because the matrix $\mathcal{M}$ is symmetric.
It follows that

$\displaystyle\rho_{2}(\mathbf{K})=\inf_{\theta_{2},\theta_{3}}\sqrt{\dfrac{\beta+\sqrt{\beta^{2}-4\gamma}}{2}}.$
(111)

From (108) and (109), after some computations, we get

$\beta^{2}-4\gamma=16\,k_{1}^{4}+8k_{1}^{2}(\theta_{2}^{2}+\theta_{2}^{-2})+8k_{1}^{2}(\theta_{3}^{2}+k_{2}^{2}\theta_{3}^{-2})+(\theta_{2}^{4}+\theta_{2}^{-4})+(\theta_{3}^{4}+k_{2}^{4}\theta_{3}^{-4})\\\\[5.0pt]
+2(\theta_{2}^{2}\theta_{3}^{2}+k_{2}^{2}\theta_{2}^{-2}\theta_{3}^{-2})-2-2k_{2}^{2}-2(\theta_{2}^{-2}\theta_{3}^{2}+k_{2}^{2}\theta_{2}^{2}\theta_{3}^{-2}).$
(112)

From (108) and this latter expression it can be verified that
$\beta+\sqrt{\beta^{2}-4\gamma}$ is minimal if and only if

$\theta_{2}^{2}=1\hskip 14.22636pt\text{and}\hskip
14.22636pt\theta_{3}^{2}=|k_{2}|.$ (113)

With these values we then have

$\displaystyle\rho_{2}(\mathbf{K})$
$\displaystyle=\inf_{\theta_{2},\theta_{3}}\sqrt{\dfrac{\beta+\sqrt{\beta^{2}-4\gamma}}{2}}$
(114) $\displaystyle=|k_{1}|+\sqrt{1+k_{1}^{2}+|k_{2}|}\;\;\geqslant 1\hskip
14.22636pt\text{for all}\;(k_{1},k_{2}).$ (115)

##  Appendix B Proof of Lemma 1.

Part a)

Assume by contradiction that $\text{Re}(\bar{z}^{2})\in(1,+\infty]$. Then

$e^{-(\text{Re}(z_{n}^{2})-1)/4\eta_{n}}\longrightarrow 0$ (116)

and it follows that the right-hand side of (43) converges to $\pm 2$. Then,
denoting $a_{n}=\text{Re}(z_{n})$ and $b_{n}=\text{Im}(z_{n})$, (43) implies

$\frac{1}{2}\left(\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{-(1-a_{n}-ib_{n})/2\eta_{n}}-\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{-(1+a_{n}+ib_{n})/2\eta_{n}}\right)\rightarrow\pm
1.$ (117)

Since $\text{Re}(\bar{z}^{2})\in(1,+\infty]$ by assumption, it follows that
there exists a positive constant $c$ such that $a_{n}^{2}>c+1+b_{n}^{2}$ for
$n$ sufficiently large, and in particular that $|\text{Re}(\bar{z})|>1$. Let
us consider successively the two possibilities $\text{Re}(\bar{z})>1$ and
$\text{Re}(\bar{z})<-1$.

The case $\mathbf{\text{Re}(\bar{z})>1.}$

In this case, if $n$ is sufficiently large, we have

$\text{Re}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)\geq 1\hskip
14.22636pt\text{and}\hskip
14.22636pte^{(a_{n}-1)/2\eta_{n}}\longrightarrow+\infty.$ (118)

Thus

$\left|\frac{1}{2}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{(a_{n}-1)/2\eta_{n}}\right|\longrightarrow+\infty,$
(119)

while

$\frac{1}{2}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{(-a_{n}-1)/2\eta_{n}}e^{-ib_{n}/2\eta_{n}}\longrightarrow
0.$ (120)

This implies

$\left|\frac{1}{2}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{(a_{n}-1)/2\eta_{n}}e^{ib_{n}/2\eta_{n}}-\frac{1}{2}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{-(1+a_{n}+ib_{n})/2\eta_{n}}\right|\longrightarrow+\infty,$
(121)

which is in contradiction with (117).

The case $\mathbf{\text{Re}(\bar{z})<-1.}$

Similarly in this case we have

$e^{(-a_{n}-1)/2\eta_{n}}\longrightarrow+\infty\hskip
14.22636pt\text{and}\hskip
14.22636pt\left|\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{(-a_{n}-1)/2\eta_{n}}e^{-ib_{n}/2\eta_{n}}\right|\longrightarrow+\infty,$
(122)

while

$\left|\frac{1}{2}\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{(a_{n}-1)/2\eta_{n}}e^{ib_{n}/2\eta_{n}}\right|\longrightarrow
0.$ (123)

So we get again (121) and a contradiction with (117). This concludes the proof
of Lemma 1, Part a).

Part b)

We assume that

$\text{Re}(\bar{z}^{2})=1.$ (124)

We have

$\text{Re}(\bar{z}^{2})=\big{(}\text{Re}(\bar{z})\big{)}^{\\!2}-\big{(}\text{Im}(\bar{z})\big{)}^{\\!2}.$
(125)

From this expression and (124), either $|\text{Re}(\bar{z})|=1$ and
$\text{Im}(\bar{z})=0$, or $|\text{Re}(\bar{z})|>1$.

We shall show by contradiction that $|\text{Re}(\bar{z})|>1$ is actually not
possible. Let us thus assume that $\text{Re}(\bar{z})>1$ (the case
$\text{Re}(\bar{z})<-1$ can be easily handled by symmetry). In that case,
using again the notations $a_{n}=\text{Re}(z_{n})$ and
$b_{n}=\text{Im}(z_{n})$ and multiplying both sides of (43) by
$e^{(1-a_{n})/4\eta_{n}}$, we obtain:

$\begin{split}&\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{(a_{n}-1)/4\eta_{n}+ib_{n}/2\eta_{n}}-\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{-(1+3a_{n}+2ib_{n})/4\eta_{n}}\\\\[5.0pt]
&=-e^{(2-a_{n}^{2}+b_{n}^{2}-a_{n}-2ia_{n}b_{n})/4\eta_{n}}\pm\sqrt{e^{(2-a_{n}^{2}+b_{n}^{2}-a_{n}-2ia_{n}b_{n})/2\eta_{n}}+4e^{(1-a_{n})/2\eta_{n}}}.\end{split}$
(126)

Since

$1-a_{n}\rightarrow 1-\text{Re}(\bar{z})<0$ (127)

and, by (124), $2-a_{n}^{2}+b_{n}^{2}-a_{n}\rightarrow
1-\text{Re}(\bar{z})<0$, the right-hand side of (126) converges to 0 when
$\eta_{n}\rightarrow 0^{+}$. This implies that the left-hand side of (126)
also converges to 0, namely

$\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)e^{(a_{n}-1)/4\eta_{n}+ib_{n}/2\eta_{n}}-\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}-1\right)e^{-(1+3a_{n}+2ib_{n})/4\eta_{n}}\rightarrow
0.$ (128)

Then, since the $e^{-(1+3a_{n}+ib_{n})/2\eta_{n}}\rightarrow 0$, we should
have

$\left|\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)\right|e^{(a_{n}-1)/4\eta_{n}}\rightarrow
0,$ (129)

but it can be shown that this is impossible. Indeed, by (127), we know that
$a_{n}>0$ if $n$ is large enough, which implies that

$\left|\left(\frac{a_{n}-ib_{n}}{a_{n}^{2}+b_{n}^{2}}+1\right)\right|e^{(a_{n}-1)/4\eta_{n}}\geqslant
e^{(a_{n}-1)/4\eta_{n}}.$ (130)

From (127), $e^{\frac{(a_{n}-1)}{4\eta}}\rightarrow+\infty$ which leads to a
contradiction with (129) and (130).

This concludes the proof of Lemma 1. ∎


11institutetext: Elsevier B.V., Radarweg 29a, 1043 NX Amsterdam, Netherlands
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Depart of statistics Tarbiat
Modares University, Jalal Al Ahmad Street 7, Tehran, Iran
22email<EMAIL_ADDRESS>33institutetext: University of Amsterdam,
1012 WX Amsterdam, Netherlands
33email<EMAIL_ADDRESS>44institutetext: Elsevier Information Systems
GmbH, Franklinstr. 63, 60486, Frankfurt am Main, Germany
44email<EMAIL_ADDRESS>
# Learning Section Weights for Multi-Label Document Classification

Maziar Moradi Fard Paula Sorrolla Bayod 1111 Kiomars Motarjem 22 Mohammad
Alian Nejadi 33 Saber Akhondi 11 Camilo Thorne 44

###### Abstract

Multi-label document classification is a traditional task in NLP. Compared to
single-label classification, each document can be assigned multiple classes.
This problem is crucially important in various domains, such as tagging
scientific articles. Documents are often structured into several sections such
as abstract and title. Current approaches treat different sections equally for
multi-label classification. We argue that this is not a realistic assumption,
leading to sub-optimal results. Instead, we propose a new method called
Learning Section Weights (LSW), leveraging the contribution of each distinct
section for multi-label classification. Via multiple feed-forward layers, LSW
learns to assign weights to each section of, and incorporate the weights in
the prediction. We demonstrate our approach on scientific articles.
Experimental results on public (arXiv) and private (Elsevier) datasets confirm
the superiority of LSW, compared to state-of-the-art multi-label document
classification methods. In particular, LSW achieves a 1.3% improvement in
terms of Macro F-1 while it achieves 1.3% in terms of Macro recall on publicly
available arXiv dataset.

###### Keywords:

Classification Explainability Deep Neural Networks

##  1 Introduction

The structure of documents can vary from one domain and dataset to another,
but typically, they tend to be divided into several _sections_ , viz.,
thematically connected spans of text, such as e.g., introduction, methods,
results and conclusion sections for experimental science articles.
Furthermore, they can be ordered under potentially _multiple_ classes, giving
rise to a multi-labelling classification task.

One might be tempted to ignore concatenate all the sections and apply a model
[16]. However, ignoring sections and assuming that they have equal importance
is a weak assumption for document multi-label classification: in reality
different sections contain different information, and do not contribute
equally to the task, if at all. In our proposed approach, highlighting the
important sections is performed through two layers of feed forward neural
networks, which generate one weight per section. We additionally leverage
these weights to explain and quantify the importance of each section.

Our expectation is that the sections which include more substantial
information, are assigned higher weights. However, the weights will change per
document meaning that for one document, section A might be more important,
while to classify another document, section B might be more important. Our
work originates from the attention mechanism [1, 16] where the goal is to
assign higher weights to more important words. Our goal is to assign instead
higher weights to more important sections (multi-label classification task).
These learned weights can help researchers and content experts to better
analyse the performance of the algorithm.

The attention mechanism originates in neural machine translation [1] where,
given a sentence in a source language, the corresponding sentence in the
target language is predicted. As opposed to traditional neural machine
translation approaches [12] which treat words in the source sentence equally,
attention-based models assign weights to each word in the source sentence to
predict the next word in the target sentence. These weights can be interpreted
as the importance of each word in the source language.

Different types of attention mechanisms have been proposed so far. Sparse
Attention [6, 5, 17] has been proposed to reduce the complexity and memory
consumption of the original attention model. This type of attention focuses on
a limited number of pairs for which attention weights should be computed,
which yields a sparse matrix. Since there is a strong connection between
matrix sparsity and graph neural networks [18], Graph Attention Networks [14,
3, 11] have been proposed. These approaches, that try to maximize the utility
of sparse matrices, suffer from a lack of strong theoretical background but
can be used in specific models such as transformers [9] and generative
approaches [4]. Performer [7] was proposed to address these issues and
decrease the run time of the attention-based models. Self-attention and multi-
head attention [13] were proposed to 1) capture the relationship between words
in the source sentence and 2) obtain different possible relationships through
multi-head attention.

All these approaches try to compute the attention weights at a word or
sentence level [15]. As opposed to the approaches mentioned so far, we try to
propose a new Learning Section Weights (LSW) approach. Our proposed LSW
approach can measure the contribution of each section of the given article in
the downstream task (e.g., classification). In this paper, the downstream task
we have considered is multi-label text classification.

Indeed, as opposed to traditional BERT-based models [8] which treat the text
from different sections equally, we propose LSW network to learn the
importance of different sections. The LSW network is trained jointly with the
model used for classification, where back-propagation [10] will propagate the
classification error through the LSW and classifier parameters.

Also, our proposed approach helps to have a deeper understanding of the
different sections of a given article. Indeed, it brings additional
information which helps data analysts and data scientists to draw conclusions
based on the learned section weights.

##  2 Methodology

We focus our experiments on two datasets of scientific article snapshots,
containing a) abstracts, b) titles and c) keywords, and base our multi-label
classifiers on the SciBERT [2] BERT model. As core baseline, we concatenate
all sections and feed them to a BERT-base model to do classification.
Thereafter we improve on this baseline by adding feed-forward layers on top of
the BERT model.

Figure 1: LSW network architecture.

The drawback with both approaches is that different sections are considered
equally. To address this limitation, we propose to learn section weights to
highlight more important sections and allow them to contribute to the multi-
label classification task based on their importances.

Let $S=\\{s_{k}\\}_{k=1}^{K}$ represent all sections and
$W=\\{w_{k}\\}_{k=1}^{K}$ represent the corresponding weights computed per
section (representing sections importances). In the remainder, $\mathcal{X}$
will denote the set of documents to be classified, while $x\in\mathcal{X}$
represents a single document. $\text{CLS}_{k}\in\mathbb{R}^{d}$ ($d$ = 768)
denotes the representation of section $k$ obtained from the BERT model for
document $x$. $f_{\theta}$ : $\mathbb{R}^{d}\to\mathbb{R}^{p}$ ($p$ = 256) is
a first linear layer (of parameters $\theta$). $g_{\eta}$ :
$\mathbb{R}^{p}\to\mathbb{R}^{1}$ represents the second linear layer to
compute section weights (of parameters $\eta$) which is also a feed-forward
neural network. The output dimension of the second layer ($g_{\eta}$) equals
to one as each section weight is a scalar representing the importance and
contribution of the underlying section. Each layer is followed by a rectified
linear unit (relu) activation followed by a (first) softmax layer to estimate
section weights. Formally:

$\displaystyle y_{k}^{1}=\text{relu}(f_{\theta}(x)),\qquad
y_{k}^{2}=\text{relu}(g_{\theta}(x)),$ (1) $\displaystyle
w_{k}=\dfrac{\exp(y_{k}^{2})}{\sum_{k=1}^{K}{\exp(y_{k}^{2})}}.$ (2)

After computing Eq. 2, the section weights can be used for weighting each
section and then classifying the input document as follows:

$\displaystyle
y=\sum_{k=1}^{K}{w_{k}\cdot\text{CLS}_{k}},\qquad\text{output}=\text{softmax}(j_{\omega}(y))$
(3)

where $y$ is the summarization of all sections (instead of simply
concatenating them), and its dimension is equal to $\text{CLS}_{k}$ dimension
(768) as $w_{k}$ is a scalar, while
$j_{\omega}:\mathbb{R}^{d}\to\mathbb{R}^{p}\to\mathbb{R}^{m}$ ($m$ denotes the
number of classes) represents a final stack of feed-forward linear layers (of
parameters $\omega$) followed by a second softmax layer that performs the
classification. Binary Cross-Entropy has been used as the loss function in our
proposed LSW network. The proposed network has the following properties:

  * •

Since $\sum_{k=1}^{K}{w_{K}}$ = 1 (Eq. 2), further analysis per document
classification can be done through section weights analysis.

  * •

The contribution of each section in classification is determined by their
corresponding section weights meaning that noisy (less useful) sections will
contribute less to the multi-label classification. This improves
classification results.

The architecture of the proposed LSW network is shown in Fig. 1. Please note
that in our proposed approach: 1) all sections share the same BERT model, and
2) the BERT model parameters are not frozen, meaning that its trainable
parameters are updated through backpropagation.

Table 1: Tuned hyperparameters used for training on our private (Elsevier) dataset. Dataset | Optimizer | Learning Rate | #Epochs | Minibatch
---|---|---|---|---
LSW | Adam | $e^{-5}$ | 5 | 32
Baseline #1 | Adam | $e^{-5}$ | 10 | 32
Baseline #2 | Adam | $e^{-5}$ | 5 | 32
Baseline #3 | Adam | $e^{-5}$ | 4 | 32
Table 2: Tuned hyperparameters used for training on arXiv datasets. Dataset | Optimizer | Learning Rate | #Epochs | Minibatch
---|---|---|---|---
LSW | Adam | $e^{-5}$ | 10 | 32
Baseline #1 | Adam | $e^{-5}$ | 10 | 32
Baseline #2 | Adam | $e^{-5}$ | 10 | 32
Baseline #3 | Adam | $e^{-5}$ | 10 | 32
Table 3: Datasets information.

Dataset | #Documents | #Classes | Sections
---|---|---|---
Elsevier | 120,000 | 52 | [Abstract, Title, Keywords]
arXiv | 306,114 | 18 | [Abstract, Title]

##  3 Experimental Setups and Results

In this section, we discuss firstly baselines and how we fine-tuned the
network then, and secondly the datasets used for our experiments. Finally, we
show the results and a few section weights plots which illustrate respectively
that our proposed LSW network can achieve both state-of-the-art results and
bring explainability by measuring the contribution of each section.

###  3.1 Baselines

Baseline #1 : This architecture is quite similar to our proposed LSW network
except for the section weights, which have been removed. Please note that in
this case, BERT trainable parameters are not frozen, and they are updated due
to classification errors (similar to our proposed LSW).

Table 4: Multi-label classification results on arXiv dataset. Bold numbers
represent the best performances.

Method | Macro F-1 | Macro Precision | Macro Recall | Micro F-1 | Micro Precision | Micro Recall
---|---|---|---|---|---|---
LSW |  94.8% | 94.9% |  95.0% |  96.5% |  96.4% |  96.6%
Baseline #1 | 94.4% |  95.3% | 93.7% | 96.3% | 96.2% | 96.4%
Baseline #2 | 93.9% | 94.4% | 91.2% | 94.5% | 92.2% | 92.3%
Baseline #3 | 89.1% | 89.3% | 89.2% | 90.1% | 90.0% | 90.3%

Table 5: Multi-label classification results on Elsevier dataset. Bold numbers
represent the best performances.

Method | Macro F-1 | Macro Precision | Macro Recall | Micro F-1 | Micro Precision | Micro Recall
---|---|---|---|---|---|---
LSW |  58.0% | 69.2% |  52.8% |  82.2% |  84.2% |  80.3%
Baseline #1 | 56.7% |  70.3% | 52.7% | 81.8% | 83.8% | 79.9%
Baseline #2 | 57.4% | 67.1% |  52.8% | 80.0% | 81.0% | 79.1%
Baseline #3 | 54.8% | 66.0% | 51.5% | 77.7% | 78.3% | 76.6%

Baseline #2 : This baseline concatenates all CLS BERT representations and then
applies a classification layer on top of these representations. Please note
that BERT trainable parameters are not frozen, and they are updated at
training time (similar to our proposed LSW).

Baseline #3 : The architecture of this baseline is identical to baseline #1,
with the difference that the BERT model parameters have now been frozen (i.e.,
simulates zero-shot learning).

###  3.2 Dataset

We have used both private (Elsevier) and public (arXiv) randomly sampled
scientific article datasets to evaluate the performance of our proposed
LSW111https://tinyurl.com/y4su8pz2 approach. Tab.3 includes information
regarding these two datasets. Note that each document contains an average of
200 words, giving rise to datasets comprising more than 20 million tokens
each.

To fine-tune our proposed LSW network and the baselines mentioned above, we
have used 10% of our dataset as validation data. The hyperparameters that we
have tuned are: 1) the choice of optimizer, 2) the learning rate, 3) batch-
size and 4) the number of epochs. We report the best performing configurations
in Tables 1–2.

(a)

(b)

(c)

Figure 2: Distribution of section weights indicating the importance of
different sections. (a) shows section weights pertaining to a random document
from arXiv dataset while (b) is related to section weights averaged on all
arXiv test data. (c) shows section weights averaged on all Elsevier test data.

###  3.3 Results

We have evaluated the performance of our proposed LSW network and baselines
mentioned in Sec. 3.1 using six different metrics, including Micro/Macro
F1-score, Micro/Macro Precision and Micro/Macro Recall. The results shown in
Tab. 4 and Tab. 5 indicate that our proposed algorithm can outperform
baselines on the majority of the metrics. Indeed, we can notice that our
proposed LSW approach is able to consistently outperform all baseline on all
metrics except Macro Precision. The ability to surpass baselines on both arXiv
(public) dataset and our private (Elsevier) dataset proves the stability of
our proposed LSW approach meaning that LSW is able to handle real world case
problems. Moreover, Fig. 2 indicates that our proposed LSW network can assess
the contribution of each section per document independently. As it can be
noticed, per document, different sections contribute to the multi-label
classification task differently, with the average section weights across all
documents showing that, in general, the abstract plays the most important role
in the tagging of the documents.

##  4 Conclusions

In this paper, we presented our proposed LSW network, which can assess the
contribution of each section of an article in the multi-labelling
classification downstream task. Indeed, this network can classify the
underlying document and add explainability regarding the importance of each
section. The section weights are updated and computed using gradient descent
and backpropagation, which helps to obtain better results and utilise sections
such that the classification error is minimised. The results of utilising this
approach on both our Elsevier and arXiv datasets indicate that our proposed
LSW network can achieve state-of-the-art results compared to different
baselines. However, for future work, we will apply the proposed architecture
to different tasks (e.g., clustering).

##  5 Acknowledgements

We extend our gratitude to the Elsevier Life Sciences Department, whose
sponsorship and support made this research possible. We also extend our
gratitude to all the many colleagues who contributed with their feedback to
earlier drafts of this paper.

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bounded box $B^{\prime}\subset B$ such that $\langle\varphi_{n},G(B\setminus
B^{\prime})\varphi_{n}\rangle\leq\frac{\epsilon}{2}$ for all
$n=1,\dots,n^{\prime}$. It follows $m(B\setminus B^{\prime})<\epsilon$. Hence
$m$ is $\sigma$-additive by (24.6).
Thus, by (24.6), there is the unique POM $T$ as asserted, which is an PM if
$G$ is projection-valued.
As to the last part of the assertion one has $m(M_{i})\to
0\;\Leftrightarrow\;m(S(M_{i}))\to 0$ for all $(M_{i})$ in
${\mathcal{Q}^{d}}$. Let $\Delta\subset\mathbb{R}^{d}$ be Borel,
$(M_{ij})\in\mathcal{C}(\Delta)$, and $(L_{n}):=(M_{i_{n}j_{n}})$ such that
$\hat{m}(\Delta\vartriangle L_{n})\to 0$ (cf. the proof of (24.6), in
particular (1) and (2)). Therefore $\hat{m}\big{(}S(\Delta)\vartriangle
S(L_{n})\big{)}\to 0$, whence easily
$\big{(}S(M_{ij})\big{)}\in\mathcal{C}\big{(}S(\Delta)\big{)}$. Therefore, by
(3) of the proof of (24.6),
$T\big{(}S(\Delta)\big{)}=\lim_{i}\lim_{j}T\big{(}S(M_{ij})\big{)}=\lim_{i}\lim_{j}U(S)T(M_{ij})U(S)^{-1}=U(S)T(\Delta)U(S)^{-1}$.∎

The following example shows that $G$ in (24.6) need not be $\sigma$-additive
without the assumption that $G$ is determinate by its values at bounded boxes.
More precisely, there is a translation-covariant non-$\sigma$-additive
projection-valued content $G$ on $(\mathbb{R}^{d},\mathcal{Q}^{d})$ in any
Hilbert space $\neq\\{0\\}$. Hence, in particular, [1, Theorem A5] is not
generally valid.
Let $R_{n}:=\\{x:x_{i}\geq n,i=1,\dots,d\\},n\in\mathbb{N}$. For
$M\in\mathcal{Q}^{d}$ set $G(M):=I$ if $R_{n}\subset M$ for some $n$ and
$G(M):=0$ otherwise. Now let $M=B_{1}\cup\dots\cup B_{k}$ with disjoint boxes
$B_{l}$. The claim is that $R_{n_{0}}\subset M$ for some $n_{0}$ if and only
if there is just one box $B_{l}$ containing some $R_{m}$. A moment of
reflection shows that this implies the additivity of $G$. — As to the claim
note first that at most one box may contain some $R_{m}$ since the boxes are
disjoint. Now let $S:=\\{(m,\dots,m):m\in\mathbb{N},m\geq n_{0}\\}$. As
$S\subset M$ there is some $B_{l}=\langle a,b[$ such that $B_{l}\cap S$ is
infinite. This implies for every $i$ that $m<b_{i}$ for infinitely many
$m\in\mathbb{N}$. Hence $b_{i}=\infty$ for $i=1,\dots,d$. Thus $R_{m}\subset
B_{l}$ for $m\geq\max_{i}a_{i}$ proving the claim. — Now let $U(b)$ for
$b\in\mathbb{R}^{d}$ be any unitary operator. Since obviously
$G(b+M)=0\Leftrightarrow G(M)=0$, translational covariance holds. — Finally,
$G$ is not $\sigma$-additive, as e.g. the boxes
$B_{n}:=\langle(n-1,\dots,n-1),(n,\dots,n)[$, $n\in\mathbb{N}$ are disjoint,
their union is $B:=\langle(0,\dots,0),(\infty,\dots,\infty)[$, but
$G(B_{n})=0$, $G(B)=I$.

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