# Output Feedback Periodic Event-triggered Control of Coupled $2\times 2$
Linear Hyperbolic PDEs

Eranda Somathilake1, Bhathiya Rathnayake2, and Mamadou Diagne1 1E. Somathilake
and Mamadou Diagne are with the Department of Mechanical and Aerospace
Engineering, University of California San Diego, CA 92093, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>Rathnayake is with the Department
of Electrical and Computer Engineering,University of California San Diego, CA
92093, USA<EMAIL_ADDRESS>
###### Abstract

This article introduces an observer-based periodic event-triggered control
(PETC) strategy for boundary control of a system characterized by $2\times 2$
linear hyperbolic partial differential equations (PDEs). An anti-collocated
actuation and sensing configuration is considered, and an exponentially
convergent observer for state estimation from boundary data is designed.
Initially, a continuous-time dynamic event-triggering mechanism requiring
constant monitoring of the triggering function is developed. This mechanism is
subsequently adapted into a periodic event-triggering scheme, which
necessitates only periodic monitoring to identify when the control input needs
updating. The underlying control approach is the PDE backstepping boundary
control, implemented in a zero-order hold manner between events. This result
marks a substantial improvement over conventional observer-based continuous-
time event-triggered control for linear coupled hyperbolic PDEs by removing
the requirement for constant monitoring of the triggering function. With the
triggering function evaluated periodically, the closed-loop system is
inherently free from Zeno behavior. It is demonstrated that under the proposed
PETC, the closed-loop system globally exponentially converges to zero in the
spatial $L^{2}$ norm. A simulation study illustrating the theoretical results
is presented.

##  I Introduction

Hyperbolic partial differential equations (PDEs) provide a mathematical
description of transport processes. Via first-principal modeling of physical
systems, these PDEs are useful for estimation, prediction, and control of
various systems such as open channel fluid flow [8, 10, 14, 7], blood flow
dynamics [4], traffic systems [20], and heat exchanger systems [19], among
others. From a boundary control perspective, several noteworthy contributions
include [5, 2, 3, 17, 6]. In [5], an important advancement in local
exponential stabilization was proposed for controlling irrigation canals
modeled by Saint-Venant equations. This work was further developed in [2, 3]
as well as [17] for a $2\times 2$ linear hyperbolic systems via boundary
control. Contribution [17] employs PDE backstepping control design assuming
collocated actuation and sensing. The resulting output feedback boundary
control law achieves exponential stability of the closed-loop system. Later,
this design was extended to a system of $n+1$ PDEs in [6] but with an anti-
collocated observer with a control input that exponentially stabilizes the
system. In this work, we rely on the PDE backstepping boundary control
proposed in [17] with slight modifications to accommodate the change in the
observer from collocated to anti-collocated.

The control designs mentioned above require the boundary controller to be
implemented in continuous-time. However, in many situations, this continuous-
time control (CTC) approach is not practical, especially when the controller
operates on a digital platform that can only process discrete signals. A
viable alternative is sampled-data control, where the control input is updated
only at specific time instants governed by an appropriate sampling schedule
determined offline. For greater flexibility in selecting the sampling
schedule, a more effective approach is the continuous-time event-triggered
control (CETC), where the control input is updated only when a specific
condition, termed an _event,_ is satisfied. CETC results in less frequent
updates of the control input as less conservative dwell-times may be possible
depending on the current state as opposed to conventional sampled-data control
where the update frequency is bounded by the worst-case scenario. Recent
developments in CETC design for hyperbolic PDEs using the backstepping
approach are notably, [12, 11], which use the backstepping approach to achieve
convergence of a system of $2\times 2$ linear hyperbolic PDEs in [12] under
full state feedback and in [11] under observer-based feedback control. In
addition, as the only result on event-triggered control for non-linear PDEs,
[9] proposes a static triggering condition to control a non-linear hyperbolic
PDE.

Although the CETC approach provides a method to implement the control input at
discrete time points, it still requires the triggering condition to be
continuously evaluated in time in order to detect events, which is not
realistic on a digital platform. One solution is to evaluate the triggering
function periodically, leading to what is known as periodic event-triggered
control (PETC) [13]. Developments in PETC for PDEs include [18, 15, 16]. PETC
for PDEs is achieved via state feedback in [18] while observer based PETC with
a boundary input is achieved in [15, 16] where the closed loop system is
globally exponentially convergent in the spatial $L^{2}$ norm.

In this paper, we introduce an observer-based periodic event-triggered
boundary controller for a system of $2\times 2$ linear hyperbolic PDEs, a
first of its kind. Moreover, unlike the approaches in [17, 11], which focus on
collocated sensing-actuation configurations, our method utilizes an anti-
collocated sensing and actuation setup. Initially, by emulating a continuous-
time observer-based backstepping boundary controller, we construct a sampled-
data feedback law. We then derive a continuous-time dynamic event-triggering
function that requires continuous evaluations to determine control update
times. Finally, we adapt the continuous-time event-triggering function into a
periodic event-triggering mechanism that necessitates only periodic
evaluations. It is worth noting that [15] only considers a scalar reaction-
diffusion systems whereas our design is for $2\times 2$ linear hyperbolic PDE
systems. We prove the global exponential convergence of the system’s states to
the origin in the spatial $L^{2}$ norm using Lyapunov arguments. The closed-
loop system is inherently Zeno-free, as the triggering function is evaluated
only periodically.

The article is organized as follows. Section II introduces the class of
problems under consideration along with the CTC design. Sections III and IV
discuss the CETC and PETC designs respectively. Section V provides numerical
simulations to illustrate the theoretical results, followed by concluding
remarks in Section VI.

_Notation:_ Let $\mathds{R}$ be the set of real numbers and $\mathds{R}^{+}$
be the set of nonnegative real numbers. Let $\mathds{N}$ be the set of natural
numbers including $0$. By $L^{2}(0,1)$ we denote the equivalence class of
Lebesgue measurable functions $f:[0,1]\rightarrow\mathds{R}$ such that
$\|f\|_{L^{2}}=\left(\int_{0}^{1}|f(x)|^{2}\right)^{1/2}$. Define
$\mathcal{C}^{0}(I;L^{2}(0,1))$ as the space of continuous functions
$u(t,\cdot)$ for an interval $I\subseteq\mathds{R}^{+}$ such that $I\ni
t\rightarrow u(t,\cdot)\in L^{2}(0,1)$. For convenience, we set
$\rVert\chi_{1}(\cdot),\chi_{2}(\cdot)\rVert=\rVert\chi_{1}(\cdot),\chi_{2}(\cdot)\rVert_{L^{2}((0,1);\mathds{R}^{2})}$.

##  II Preliminaries and Problem statement

We consider the following $2\times 2$ linear hyperbolic PDE system in the
canonical form where the independent variables $t\geq 0$ and $x\in[0,1]$ are
the time and spatial domains, respectively, and the PDE states are denoted by
$(u(t,x),v(t,x))^{T}$ satisfy

$\displaystyle\partial_{t}u(t,x)=$
$\displaystyle-\lambda_{1}\partial_{x}u(t,x)+c_{1}(x)v(t,x),$ (1)
$\displaystyle\partial_{t}v(t,x)=$
$\displaystyle\lambda_{2}\partial_{x}v(t,x)+c_{2}(x)u(t,x),$ (2)

with boundary conditions

$\displaystyle u(t,0)$ $\displaystyle=qv(t,0),$ (3) $\displaystyle v(t,1)$
$\displaystyle=\rho v(t,1)+U(t).$ (4)

Here, $U(t)$ is the continuous time boundary control input, and
$\lambda_{1},\lambda_{2}>0$,
$c_{1}(x),c_{2}(x)\in\mathcal{C}^{0}((0,1),\mathds{R})$. Further, $q\neq 0$ is
the distal reflection term, and $\rho\neq 0$ is the proximal reflection term.
The initial conditions are chosen such that $(v^{0}(x),u^{0}(x))^{T}\in
L^{2}((0,1);\mathds{R}^{2})$. We make the following assumption on the
reflection terms.

###### Assumption 1

[11] The reflection terms are small enough such that the following inequality
holds:

$\displaystyle|\rho q|\leq 0.5.$ (5)

Our objective is to devise a PETC strategy where the triggering function is
evaluated only periodically to detect events, and the control input is updated
solely at the times of these events, which is held constant in between. To
achieve this, we first derive a stabilizing continuous boundary feedback
control that will be applied in a zero-order hold fashion between events
generated by a suitable continuous-time triggering mechanism, leading to a
CETC strategy. Then, we convert the continuous-time triggering function into a
periodic triggering function that requires only periodic evaluations, leading
to the proposed PETC design.

###  II-A Continuous-time output feedback control and its emulation for ETC

In this subsection, we develop a continuous-time backstepping output feedback
control $U(t)$ capable of exponentially stabilizing the closed-loop system
consisting of the plant (1)-(4) with initial conditions
$(v^{0}(x),u^{0}(x))^{T}\in L^{2}((0,1);\mathds{R}^{2})$ and an observer,
using $v(t,0)$ as the available boundary measurement. Since the actuation and
measurement are located at opposite boundaries, this setup is referred to as
anti-collocated sensing and actuation, which differs from the configuration in
[11], where both the actuation and measurement are located at the same
boundary.

We design an observer consisting of the copy of the plant plus some output
injection terms with the observer states denoted by
$(\hat{u}(t,x),\hat{v}(t,x))^{T}$. Defining the errors as
$(\tilde{u}(t,x),\tilde{v}(t,x))^{T}=(u(t,x),v(t,x))^{T}-(\hat{u}(t,x),\hat{v}(t,x))^{T}$,
the following observer is proposed:

$\displaystyle\partial_{t}\hat{u}(t,x)=$
$\displaystyle-\lambda_{1}\partial_{x}\hat{u}(t,x)+c_{1}(x)\hat{v}(t,x)+p_{1}(x)\tilde{v}(t,0),$
(6) $\displaystyle\partial_{t}\hat{v}(t,x)=$
$\displaystyle\lambda_{2}\partial_{x}\hat{v}(t,x)+c_{2}(x)\hat{u}(t,x)+p_{2}(x)\tilde{v}(t,0),$
(7)

with boundary conditions

$\displaystyle\hat{u}(t,0)$ $\displaystyle=qv(t,0),$ (8)
$\displaystyle\hat{v}(t,1)$ $\displaystyle=\rho\hat{u}(t,1)+U(t),$ (9)

and initial conditions defined such that
$(\hat{v}^{0}(x),\hat{u}^{0}(x))^{T}\in L^{2}((0,1);\mathds{R}^{2})$. The
functions $p_{1}(x)$ and $p_{2}(x)$ are the observer output injection gains,
which are to be determined through backstepping design to ensure the
convergence of the estimated states to the plant states. It can be easily
verified that the dynamics of the observer errors
$(\tilde{u}(t,x),\tilde{v}(t,x))^{T}$ satisfy

$\displaystyle\partial_{t}\tilde{u}(t,x)=$
$\displaystyle-\lambda_{1}\partial_{x}\tilde{u}(t,x)+c_{1}(x)\tilde{u}(t,x)-p_{1}(x)\tilde{v}(t,0),$
(10) $\displaystyle\partial_{t}\tilde{v}(t,x)=$
$\displaystyle\lambda_{2}\partial_{x}\tilde{v}(t,x)+c_{2}(x)\tilde{u}(t,x)-p_{2}(x)\tilde{v}(t,0),$
(11)

with boundary conditions

$\displaystyle\tilde{u}(t,0)$ $\displaystyle=0,$ (12)
$\displaystyle\tilde{v}(t,1)$ $\displaystyle=\rho\tilde{u}(t,1).$ (13)

Consider the following observer error backstepping transformations:

$\displaystyle\tilde{u}(t,x)$
$\displaystyle=\tilde{\alpha}(t,x)-\int_{0}^{x}P^{\alpha\alpha}(x,\xi)\tilde{\alpha}(t,\xi)d\xi$
$\displaystyle-\int_{0}^{x}P^{\alpha\beta}(x,\xi)\tilde{\beta}(t,\xi)d\xi,$
(14) $\displaystyle\tilde{v}(t,x)$
$\displaystyle=\tilde{\beta}(t,x)-\int_{0}^{x}P^{\beta\alpha}(x,\xi)\tilde{\alpha}(t,\xi)d\xi$
$\displaystyle-\int_{0}^{x}P^{\beta\beta}(x,\xi)\tilde{\beta}(t,\xi)d\xi,$
(15)

defined over the triangular domain $\mathcal{T}=0\leq\xi\leq x\leq 1$, where
the kernels $P^{\alpha\alpha},P^{\alpha\beta},P^{\beta\alpha},$ and
$P^{\beta\beta}$ satisfy the following PDEs:

$\displaystyle\lambda_{1}\partial_{x}P^{\alpha\alpha}(x,\xi)+\lambda_{1}\partial_{\xi}P^{\alpha\alpha}(x,\xi)=$
$\displaystyle c_{1}(x)P^{\beta\alpha}(x,\xi),$ (16)
$\displaystyle\lambda_{1}\partial_{x}P^{\alpha\beta}(x,\xi)-\lambda_{2}\partial_{\xi}P^{\alpha\beta}(x,\xi)=$
$\displaystyle c_{1}(x)P^{\beta\beta}(x,\xi),$ (17)
$\displaystyle\lambda_{2}\partial_{x}P^{\beta\alpha}(x,\xi)-\lambda_{1}\partial_{\xi}P^{\beta\alpha}(x,\xi)=$
$\displaystyle-c_{2}(x)P^{\alpha\alpha}(x,\xi),$ (18)
$\displaystyle\lambda_{2}\partial_{x}P^{\beta\alpha}(x,\xi)+\lambda_{2}\partial_{\xi}P^{\beta\alpha}(x,\xi)=$
$\displaystyle-c_{2}(x)P^{\alpha\beta}(x,\xi),$ (19)

with boundary conditions

$\displaystyle P^{\alpha\alpha}(1,\xi)=\frac{1}{\rho}P^{\alpha\beta}(1,\xi),$
$\displaystyle
P^{\alpha\beta}(x,x)=-\frac{c_{1}(x)}{\lambda_{1}+\lambda_{2}},$ (20)
$\displaystyle P^{\beta\beta}(1,\xi)=\rho P^{\alpha\beta}(1,\xi),$
$\displaystyle P^{\beta\alpha}(x,x)=\frac{c_{2}(x)}{\lambda_{1}+\lambda_{2}}.$
(21)

The output gain terms are chosen as

$\displaystyle p_{1}(x)=-\lambda_{2}P^{\alpha\beta}(x,0),\quad
p_{2}(x)=-\lambda_{2}P^{\beta\beta}(x,0).$ (22)

The backstepping transformation (14)-(15) transforms the observer error system
(10)-(13) into the following target system:

$\displaystyle\partial_{t}\tilde{\alpha}(t,x)=$
$\displaystyle-\lambda_{1}\partial_{x}\tilde{\alpha}(t,x),$ (23)
$\displaystyle\partial_{t}\tilde{\beta}(t,x)=$
$\displaystyle\lambda_{2}\partial_{x}\tilde{\beta}(t,x),$ (24)

with boundary conditions

$\displaystyle\tilde{\alpha}(t,0)$ $\displaystyle=0,$ (25)
$\displaystyle\tilde{\beta}(t,1)$ $\displaystyle=\rho\tilde{\alpha}(t,1).$
(26)

The inverse transformations of (14)-(15) are given by

$\displaystyle\tilde{\alpha}(t,x)=$
$\displaystyle\tilde{u}(t,x)+\int_{0}^{x}R^{uu}(x,\xi)\tilde{u}(t,\xi)d\xi$
$\displaystyle+\int_{0}^{x}R^{uv}(x,\xi)\tilde{v}(t,\xi)d\xi,$ (27)
$\displaystyle\tilde{\beta}(t,x)=$
$\displaystyle\tilde{v}(t,x)+\int_{0}^{x}R^{vu}(x,\xi)\tilde{u}(t,\xi)d\xi$
$\displaystyle+\int_{0}^{x}R^{vv}(x,\xi)\tilde{v}(t,\xi)d\xi,$ (28)

defined over the triangular domain $0\leq\xi\leq x\leq 1$, where the kernels
$R^{uu},R^{uv},R^{vu},$ and $R^{vv}$ satisfy the following PDEs:

$\displaystyle\lambda_{1}\partial_{x}R^{uu}(x,\xi)+\lambda_{1}\partial_{\xi}R^{uu}(x,\xi)=$
$\displaystyle-c_{2}(\xi)R^{uv}(x,\xi),$ (29)
$\displaystyle\lambda_{1}\partial_{x}R^{uv}(x,\xi)-\lambda_{2}\partial_{\xi}R^{uv}(x,\xi)=$
$\displaystyle-c_{1}(\xi)R^{uu}(x,\xi),$ (30)
$\displaystyle\lambda_{2}\partial_{x}R^{vu}(x,\xi)-\lambda_{1}\partial_{\xi}R^{vu}(x,\xi)=$
$\displaystyle c_{2}(\xi)R^{vv}(x,\xi),$ (31)
$\displaystyle\lambda_{2}\partial_{x}R^{vu}(x,\xi)+\lambda_{2}\partial_{\xi}R^{vv}(x,\xi)=$
$\displaystyle c_{1}(\xi)R^{vu}(x,\xi),$ (32)

with boundary conditions

$\displaystyle R^{uu}(1,\xi)=\frac{1}{\rho}R^{vu}(1,\xi),$ $\displaystyle
R^{uv}(x,x)=-\frac{c_{1}(x)}{\lambda_{1}+\lambda_{2}},$ (33) $\displaystyle
R^{vv}(1,\xi)=\rho R^{uv}(1,\xi),$ $\displaystyle
R^{vu}(x,x)=\frac{c_{2}(x)}{\lambda_{1}+\lambda_{2}}.$ (34)

In order to derive a stabilizing control law via PDE backstepping, considering
the observer system (6)-(9), the following backstepping transformations are
introduced:

$\displaystyle\hat{\alpha}(t,x)=$
$\displaystyle\hat{u}(t,x)-\int_{0}^{x}K^{uu}(x,\xi)\hat{u}(t,\xi)d\xi$
$\displaystyle-\int_{0}^{x}K^{uv}(x,\xi)\hat{v}(t,\xi)d\xi,$ (35)
$\displaystyle\hat{\beta}(t,x)=$
$\displaystyle\hat{v}(t,x)-\int_{0}^{x}K^{vu}(x,\xi)\hat{u}(t,\xi)d\xi$
$\displaystyle-\int_{0}^{x}K^{vv}(x,\xi)\hat{v}(t,\xi)d\xi,$ (36)

which are defined over the triangular domain $0\leq\xi\leq x\leq 1$, with the
kernels $K^{uu},K^{uv},K^{vu},$ and $K^{vv}$ satisfying the following PDEs:

$\displaystyle\lambda_{1}\partial_{x}K^{uu}(x,\xi)+\lambda_{1}\partial_{\xi}K^{uu}(x,\xi)=$
$\displaystyle-c_{2}(\xi)K^{uv}(x,\xi),$ (37)
$\displaystyle\lambda_{1}\partial_{x}K^{uv}(x,\xi)-\lambda_{2}\partial_{\xi}K^{uv}(x,\xi)=$
$\displaystyle-c_{1}(\xi)K^{uu}(x,\xi),$ (38)
$\displaystyle\lambda_{2}\partial_{x}K^{vu}(x,\xi)-\lambda_{1}\partial_{\xi}K^{vu}(x,\xi)=$
$\displaystyle c_{2}(\xi)K^{vv}(x,\xi),$ (39)
$\displaystyle\lambda_{2}\partial_{x}K^{vu}(x,\xi)+\lambda_{2}\partial_{\xi}K^{vu}(x,\xi)=$
$\displaystyle c_{1}(\xi)K^{vu}(x,\xi),$ (40)

and boundary conditions

$\displaystyle K^{uu}(x,0)=\frac{\lambda_{2}}{q\lambda_{1}}K^{uv}(x,0),\quad$
$\displaystyle K^{uv}(x,x)=\frac{c_{1}(x)}{\lambda_{1}+\lambda_{2}},$ (41)
$\displaystyle K^{vv}(x,0)=\frac{q\lambda_{1}}{\lambda_{2}}K^{vu}(x,0),\quad$
$\displaystyle K^{vu}(x,x)=-\frac{c_{2}(x)}{\lambda_{1}+\lambda_{2}}.$ (42)

The inverse transformations of (35),(36) are given by

$\displaystyle\hat{u}(t,x)=$
$\displaystyle\hat{\alpha}(t,x)+\int_{0}^{x}L^{\alpha\alpha}(x,\xi)\hat{\alpha}(t,\xi)d\xi$
$\displaystyle+\int_{0}^{x}L^{\alpha\beta}(x,\xi)\hat{\beta}(t,\xi)d\xi,$ (43)
$\displaystyle\hat{v}(t,x)=$
$\displaystyle\hat{\beta}(t,x)+\int_{0}^{x}L^{\beta\alpha}(x,\xi)\hat{\alpha}(t,\xi)d\xi$
$\displaystyle+\int_{0}^{x}L^{\beta\beta}(x,\xi)\hat{\beta}(t,\xi)d\xi,$ (44)

defined over the triangular domain $0\leq\xi\leq x\leq 1$, where the kernels
$L^{\alpha\alpha},L^{\alpha\beta},L^{\beta\alpha},L^{\beta\beta}$ satisfy the
following PDEs:

$\displaystyle\lambda_{1}\partial_{x}L^{\alpha\alpha}(x,\xi)+\lambda_{1}\partial_{\xi}L^{\alpha\alpha}(x,\xi)=$
$\displaystyle c_{1}(x)L^{\beta\alpha}(x,\xi),$ (45)
$\displaystyle\lambda_{1}\partial_{x}L^{\alpha\beta}(x,\xi)-\lambda_{2}\partial_{\xi}L^{\alpha\beta}(x,\xi)=$
$\displaystyle c_{1}(x)L^{\beta\beta}(x,\xi),$ (46)
$\displaystyle\lambda_{2}\partial_{x}L^{\beta\alpha}(x,\xi)-\lambda_{1}\partial_{\xi}L^{\beta\alpha}(x,\xi)=$
$\displaystyle-c_{2}(x)L^{\alpha\alpha}(x,\xi),$ (47)
$\displaystyle\lambda_{2}\partial_{x}L^{\beta\beta}(x,\xi)+\lambda_{2}\partial_{\xi}L^{\beta\beta}(x,\xi)=$
$\displaystyle-c_{2}(x)L^{\alpha\beta}(x,\xi),$ (48)

with boundary conditions

$\displaystyle
L^{\alpha\alpha}(x,0)=\frac{\lambda_{2}}{q\lambda_{1}}L^{\alpha\beta}(x,0),$
$\displaystyle L^{\alpha\beta}(x,x)=\frac{c_{1}(x)}{\lambda_{1}+\lambda_{2}},$
(49) $\displaystyle
L^{\beta\beta}(x,0)=\frac{q\lambda_{1}}{\lambda_{2}}L^{\beta\alpha}(x,0),$
$\displaystyle
L^{\beta\alpha}(x,x)=-\frac{c_{2}(x)}{\lambda_{1}+\lambda_{2}}.$ (50)

It can easily be verified that if the control law $U(t)$ is chosen as

$U(t)=\int_{0}^{1}N^{\alpha}(\xi)\hat{\alpha}(t,\xi)+\int_{0}^{1}N^{\beta}(\xi)\hat{\beta}(t,\xi)d\xi,$
(51)

where

$\displaystyle N^{\alpha}(\xi)=$ $\displaystyle L^{\beta\alpha}(1,\xi)-\rho
L^{\alpha\alpha}(1,\xi),$ (52) $\displaystyle N^{\beta}(\xi)=$ $\displaystyle
L^{\beta\beta}(1,\xi)-\rho L^{\alpha\beta}(1,\xi),$ (53)

the observer (6)-(9) gets transformed into the following target system via the
transformations (35)-(36):

$\displaystyle\partial_{t}\hat{\alpha}(t,x)=$
$\displaystyle-\lambda_{1}\partial_{x}\hat{\alpha}(t,x)+\bar{p}_{1}(x)\tilde{\beta}(t,0),$
(54) $\displaystyle\partial_{t}\hat{\beta}(t,x)=$
$\displaystyle\lambda_{2}\partial_{x}\hat{\beta}(t,x)+\bar{p}_{2}(x)\tilde{\beta}(t,0),$
(55)

with boundary conditions

$\displaystyle\hat{\alpha}(t,0)=$ $\displaystyle
q\hat{\beta}(t,0)+q\tilde{\beta}(t,0),$ (56) $\displaystyle\hat{\beta}(t,1)=$
$\displaystyle\rho\hat{\alpha}(t,1),$ (57)

where

$\displaystyle\bar{p}_{1}(x)=$ $\displaystyle
p_{1}(x)-\int_{0}^{x}K^{uu}(x,\xi)p_{1}(\xi)d\xi$
$\displaystyle-\int_{0}^{x}K^{uv}(x,\xi)p_{2}(\xi)d\xi,$ (58)
$\displaystyle\bar{p}_{2}(x)=$ $\displaystyle
p_{2}(x)-\int_{0}^{x}K^{vu}(x,\xi)p_{1}(\xi)d\xi$
$\displaystyle-\int_{0}^{x}K^{vv}(x,\xi)p_{2}(\xi)d\xi.$ (59)

The well-posedness of the kernel PDEs (16)-(21), (29)-(34), (37)-(42), and
(45)-(50) is established in [17]. The closed loop system (1)-(4),(6)-(9) with
the control input (51), is exponentially stable in the $L^{2}$-norm sense. We
omit the proof of stability due to page limitations. Since our goal is to
derive event-triggered control schemes (CETC and PETC), we sample the
continuous feedback (51) and hold it in between events, leading to the
following sampled-data control:

$\displaystyle U_{k}^{\omega}(t):=$ $\displaystyle U(t_{k}^{\omega}),$ (60)

$\forall~{}t\in[t_{k}^{\omega},t_{k+1}^{\omega}),k\in\mathbb{N}$, where
$\omega=\\{``c",``p"\\}$. The increasing sequence of times
$I^{c}=\\{t_{k}^{c}\\}_{k\in\mathbb{N}}$ and
$I^{p}=\\{t_{k}^{p}\\}_{k\in\mathbb{N}}$ with $t_{0}^{\omega}=0$ are CETC and
PETC event times respectively, and $U_{k}^{c}$ and $U_{k}^{p}$ are CETC and
PETC control inputs, respectively. In subsequent sections, we present
triggering mechanisms to determine the event sequences $I^{c}$ and $I^{p}$.
The deviation between the sampled-data control (60) and the continuous control
(51), is referred to as the input holding error, $d(t)$ defined as

$\displaystyle d(t):=$
$\displaystyle\int_{0}^{1}\left(N^{\alpha}(\xi)\hat{\alpha}(t_{k}^{\omega},\xi)+N^{\beta}(\xi)\hat{\beta}(t_{k}^{\omega},\xi)\right)d\xi$
$\displaystyle-\int_{0}^{1}\left(N^{\alpha}(\xi)\hat{\alpha}(t,\xi)+N^{\beta}(\xi)\hat{\beta}(t,\xi)\right)d\xi,$
(61)

for $t\in[t_{k}^{\omega},t_{k+1}^{\omega}),k\in\mathbb{N}$. Then, the boundary
conditions (4),(9), and (57) become

$\displaystyle v(t,1)$ $\displaystyle=\rho v(t,1)+U_{k}^{\omega}(t),$ (62)
$\displaystyle\hat{v}(t,1)$
$\displaystyle=\rho\hat{v}(t,1)+U_{k}^{\omega}(t),$ (63)
$\displaystyle\hat{\beta}(t,1)$ $\displaystyle=\rho\hat{\alpha}(t,1)+d(t),$
(64)

respectively. The well-posedness of the systems (1)-(3),(62) and (6)-(8),(63)
is established in the following proposition.

###### Proposition 1

For a given $(u^{0}(x),v^{0}(x))^{T}\in L^{2}((0,1);\mathds{R}^{2})$ and
$(\hat{u}^{0}(x),\hat{v}^{0}(x))^{T}\in L^{2}((0,1);\mathds{R}^{2})$, there
exists unique solutions such that
$(u(t,x),v(t,x))^{T}\in\mathcal{C}^{0}([t_{k}^{\omega},t_{k+1}^{\omega}];L^{2}((0,1);\mathds{R}^{2}))$
and
$(\hat{u}(t,x),\hat{v}(t,x))^{T}\in\mathcal{C}^{0}([t_{k}^{\omega},t_{k+1}^{\omega}];L^{2}((0,1);\mathds{R}^{2}))$
to the systems (1)-(3),(62) and (6)-(8),(63) respectively between two time
instants $t_{k}^{\omega}$ and $t_{k+1}^{\omega}$.

The proof of Proposition 1 is similar to [11, Proposition 1].

##  III Continuous-time event-triggered control

The triggering mechanism of CETC determines the increasing sequence of times
$I^{c}$ at which the control input $U_{k}^{c}(t)$ must be updated by
continuously evaluating an appropriately chosen triggering function
$\Gamma^{c}(t)$ to verify if a specific condition is met. Below, we present
the proposed CETC triggering mechanism.

###### Definition 1

Let $\sigma\in(0,1)$, $\theta,\eta,\mu>0$,
$\kappa_{0},\kappa_{1},\kappa_{2},\kappa_{3}\geq 0$, and $A,B,C>0$ be some
parameters to be determined. Then, the increasing sequence of times $I^{c}$ at
which events are generated is defined according to the following rule:

$\displaystyle t_{k+1}^{c}:=$
$\displaystyle\inf\\{t\in\mathds{R}|t>t_{k}^{c},\
\Gamma^{c}(t)>0,k\in\mathds{N}\\},$ (65) $\displaystyle\Gamma^{c}(t):=$
$\displaystyle\theta CBe^{\frac{\mu}{\lambda_{2}}}d^{2}(t)+m(t).$ (66)

Here, $d(t)$ is given by (61), and the dynamic variable $m(t)$ evolves
according to the following ODE:

$\displaystyle\dot{m}(t)=$ $\displaystyle-\eta
m(t)+2CBe^{\frac{\mu}{\lambda_{2}}}d^{2}(t)-\kappa_{0}\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}$
$\displaystyle-\kappa_{1}\hat{\alpha}^{2}(t,1)-\kappa_{2}\hat{\beta}^{2}(t,0)-\kappa_{3}\tilde{\beta}^{2}(t,0),$
(67)

for $t\in(t_{k}^{c},t_{k+1}^{c}),k\in\mathds{N}$, with $m(0)=m^{0}<0$,
$m(t_{k}^{c^{-}})=m(t_{k}^{c})=m(t_{k}^{c^{+}}),k\in\mathds{N}$.

Before proceeding to the CETC results, which include Zeno-free behavior and
global exponential convergence, we present several important findings in
Lemmas 1 and 2, that are instrumental in obtaining these results.

###### Lemma 1

For $d(t)$ given by (61), the following inequality holds
$\forall~{}t\in(t_{k}^{c},t_{k+1}^{c}),~{}k\in\mathds{N}$:

$\displaystyle(\dot{d}(t))^{2}\leq$
$\displaystyle\epsilon_{0}\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}+\epsilon_{1}\hat{\alpha}^{2}(t,1)$
$\displaystyle+$
$\displaystyle\epsilon_{2}\hat{\beta}^{2}(t,0)+\epsilon_{3}\tilde{\beta}^{2}(t,0)+\epsilon_{4}d^{2}(t),$
(68)

where $\epsilon_{0},\epsilon_{1},\epsilon_{2},\epsilon_{3},$ and
$\epsilon_{4}\geq 0$ for $t\in(t_{k}^{c},t_{k+1}^{c}),k\in\mathds{N}$.

###### Proof:

Taking the time derivative of (61) for
$t\in(t_{k}^{c},t_{k+1}^{c}),k\in\mathds{N}$ and using the equations (54)-(57)
we obtain

$\displaystyle\dot{d}(t)=(\lambda_{1}N^{\alpha}(1)-\rho\lambda_{2}N^{\beta}(1))\hat{\alpha}(t,1)$
$\displaystyle+(\lambda_{2}N^{\beta}(0)-q\lambda_{1}N^{\alpha}(0))\hat{\beta}(t,0)$
$\displaystyle-\tilde{\beta}(t,0)\left(\int_{0}^{1}N^{\alpha}(\xi)\bar{p}_{1}(\xi)d\xi+\int_{0}^{1}N^{\beta}(\xi)\bar{p}_{2}(\xi)d\xi\right)$
$\displaystyle-\lambda_{1}\int_{0}^{1}\partial_{\xi}N^{\alpha}(\xi)\hat{\alpha}(t,\xi)d+\lambda_{2}\int_{0}^{1}\partial_{\xi}N^{\beta}(\xi)\hat{\beta}(t,\xi)d\xi$
$\displaystyle-q\lambda_{1}N^{\alpha}(0)\tilde{\beta}(t,0)-\lambda_{2}N^{\beta}(1)d(t).$
(69)

Further, taking the square of $\dot{d}(t)$ and using the Young’s inequality
and Cauchy Schwartz inequality, we can derive (68), where

$\displaystyle\epsilon_{0}=$ $\displaystyle
6\max\left\\{\lambda_{1}^{2}\int_{0}^{1}(\partial_{\xi}N^{\alpha}(\xi))^{2}d\xi,\lambda_{2}^{2}\int_{0}^{1}(\partial_{\xi}N^{\beta}(\xi))^{2}d\xi\right\\},$
(70) $\displaystyle\epsilon_{1}=$ $\displaystyle
6(\lambda_{1}N^{\alpha}(1)-\rho\lambda_{2}N^{\beta}(1))^{2},$ (71)
$\displaystyle\epsilon_{2}=$ $\displaystyle
6(\lambda_{2}N^{\beta}(0)-q\lambda_{1}N^{\alpha}(0))^{2},$ (72)
$\displaystyle\epsilon_{3}=$ $\displaystyle
6\Bigg{(}\int_{0}^{1}\left(N^{\alpha}(\xi)\bar{p}_{1}(\xi)d\xi+N^{\beta}(\xi)\bar{p}_{2}(\xi)\right)d\xi^{2}$
$\displaystyle\mkern 200.0mu+q\lambda_{1}N^{\alpha}(0)\Bigg{)}^{2},$ (73)
$\displaystyle\epsilon_{4}=$ $\displaystyle 6(\lambda_{2}N^{\beta}(1))^{2},$
(74)

which concludes the proof. ∎

The event-triggering condition guarantees that $\Gamma^{c}(t)<0$
$\forall~{}t\in[0,\sup\\{I^{c}\\})$, hence the following lemma holds.

###### Lemma 2

Under the CETC strategy (65)-(67), it holds that, $m(t)<0$
$\forall~{}t\in[0,\sup\\{I^{c}\\})$.

The proof is very similar to that in [11, Lemma 1], and is therefore omitted.

The existence of a minimum dwell-time between events is shown in the following
lemma.

###### Lemma 3

Consider the CETC strategy (65)-(67), with the parameters $\theta,C,B,\mu>0$
and $\kappa_{0},\kappa_{1},\kappa_{2},\kappa_{3}\geq 0$ and $\sigma\in(0,1)$
selected such that

$\displaystyle\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{i}\leq(1-\sigma)\kappa_{i},\text{
}i=0,1,2,3.$ (75)

Then, there exists a minimum dwell-time $\tau>0$ such that
$t_{k+1}^{c}-t_{k}^{c}\geq\tau,k\in\mathds{N}$, which ensures that the CETC
closed-loop system is Zeno-free.

###### Proof:

Let events be triggered at times $t=t_{k}^{c},~{}k\in\mathds{N}$. Considering
the control input is updated as defined in (65)-(67), let $\psi(t)$ with
$\psi(t_{k+1}^{c^{-}})=1$ be

$\displaystyle\psi(t):=\frac{\theta
CBe^{\frac{\mu}{\lambda_{2}}}d^{2}(t)+(1-\sigma)m(t)}{-\sigma m(t)}.$ (76)

Then, $d(t_{k}^{c})=0$ and since $\sigma\in(0,1)$ and $m(t_{k}^{c})<0$, we
have $\psi(t_{k}^{c})<0$. Also note that $d(t)$ and $m(t)$ are continuous on
$t\in[t_{k}^{c},t_{k+1}^{c})$ making $\psi(t)$ a continuous function on
$t\in[t_{k}^{c},t_{k+1}^{c})$. Therefore, from the intermediate value theorem,
$\exists~{}t_{k}^{\prime}\in(t_{k}^{c},t_{k+1}^{c})$ such that
$\psi(t)\in[0,1]$ for $t\in[t_{k}^{\prime},t_{k+1}^{c^{-}}]$. Taking the time
derivative of $\psi(t)$ on $t\in[t_{k}^{\prime},t_{k+1}^{c})$, we obtain the
following expression:

$\displaystyle\dot{\psi}(t)=\frac{2\theta
CBe^{\frac{\mu}{\lambda_{2}}}d\dot{d}+(1-\sigma)\dot{m}}{-\sigma
m}-\frac{\dot{m}}{m}\psi.$ (77)

Using Young’s inequality and from (67),(68) along with the conditions given in
(75) and using the fact that $m(t)<0$, we arrive at the following inequality:

$\displaystyle\dot{\psi}(t)\leq a_{0}+a_{1}\psi(t)+a_{2}\psi^{2}(t),$ (78)

where $a_{0},a_{1},a_{2}>0$ are defined as

$\displaystyle a_{0}=$
$\displaystyle\left(1+\epsilon_{4}+\frac{2(1-\sigma)}{\theta}\right)\frac{(1-\sigma)}{\sigma}+\frac{1-\sigma}{\sigma}\eta,$
(79) $\displaystyle a_{1}=$ $\displaystyle
1+\epsilon_{4}+\frac{4(1-\sigma)}{\theta}+\eta,$ (80) $\displaystyle a_{2}=$
$\displaystyle\frac{2\sigma}{\theta}.$ (81)

Therefore, using the comparison principle, the time required for $\psi(t)$ to
change from $\psi(t_{k}^{\prime})=0$ to $\psi(t_{k+1}^{c^{-}})=1$ is at least

$\displaystyle\tau=\int_{0}^{1}\frac{1}{a_{0}+a_{1}s+a_{2}s^{2}}ds.$ (82)

Solving the integral (82) using partial fraction expansion, we obtain the
following explicit formula for $\tau$:

$\displaystyle\tau=\frac{1}{a}\ln\left(1+\frac{a\theta\sigma}{(a\theta+2)(1-\sigma)}\right),$
(83)

where $a=1+\epsilon_{4}+\eta>0$. Since $t_{k+1}^{c}-t_{k}^{\prime}\geq\tau$
and $t_{k+1}^{c}-t_{k}^{c}\geq t_{k+1}^{c}-t_{k}^{\prime}$, we conclude that
$t_{k+1}^{c}-t_{k}^{c}\geq\tau>0$. Hence, $\tau$ is the lower bound for the
time between two triggering events, also known as the minimum dwell-time. In
conclusion, there exists a minimum dwell-time $\tau>0$ for the CETC triggering
mechanism (65)-(67), and thus the Zeno phenomenon is excluded. ∎

We establish the convergence of the closed-loop system under CETC in the
following theorem.

###### Theorem 1

Let $\sigma\in(0,1)$ and $\eta>\mu(1-\sigma)$ be design parameters, and under
Assumption 1 choose the parameters $\mu,A,B,C,r_{v},\delta$ such that

$\displaystyle\mu\in$
$\displaystyle\left(0,\frac{2\lambda_{1}\lambda_{2}}{\lambda_{1}+\lambda_{2}}\ln\left(\frac{1}{2|q\rho|}\right)\right),$
$\displaystyle A=$
$\displaystyle\rho^{2}e^{\frac{\mu}{\lambda_{1}}+\frac{\mu}{\lambda_{2}}}+\frac{e^{\frac{\mu}{\lambda_{1}}}}{2},\quad
B=q^{2}e^{\frac{\mu}{\lambda_{1}}}+\frac{1}{2},$ $\displaystyle 1/C=$
$\displaystyle\frac{r}{\sigma\mu}\int_{0}^{1}\left(\frac{A}{\lambda_{1}}e^{-\frac{\mu}{\lambda_{1}}x}\bar{p}_{1}^{2}(x)+\frac{B}{\lambda_{2}}e^{\frac{\mu}{\lambda_{2}}x}\bar{p}_{2}^{2}(x)\right)dx,$
$\displaystyle r_{v}=$
$\displaystyle\max\left\\{\frac{A}{\lambda_{1}},\frac{Be^{\frac{\mu}{\lambda_{2}}}}{\lambda_{2}}\right\\},$
$\displaystyle\delta=$
$\displaystyle\frac{r}{2\sigma\mu}\left(1-4\rho^{2}q^{2}e^{\mu\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\right)}\right),$

where $r$ is selected large enough such that $C<1$ and
$1-\frac{1}{\sigma\mu\delta}>0$. Functions $\bar{p}_{1}(x)$ and
$\bar{p}_{2}(x)$ are given in (58) and (59) respectively. For $\kappa_{0}$
$\kappa_{1}$, $\kappa_{2}$, and $\kappa_{3}$ that satisfy the inequalities

$\displaystyle 0\leq\kappa_{0}\leq\frac{\sigma\mu
C}{r_{v}}\left(1-\frac{1}{\delta\sigma\mu}\right),$ (84) $\displaystyle
0\leq\kappa_{1}\leq\frac{C}{2}\left(1-4\rho^{2}q^{2}e^{\mu\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\right)}\right),$
(85) $\displaystyle
0\leq\kappa_{2}\leq\frac{C}{2}\left(1-4\rho^{2}q^{2}e^{\mu\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\right)}\right),$
(86) $\displaystyle
0\leq\kappa_{3}\leq(1-C)\left(q^{2}e^{\frac{\mu}{\lambda_{1}}}+2\rho^{2}q^{2}e^{\mu\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\right)}\right),$
(87)

and $\theta$ such that (75) hold, the observer based event-triggered closed
loop system (1)-(3),(62), (6)-(8),(63) with $U_{k}^{c}(t)$ given in (60) has a
unique solution $\forall~{}t>0$ and globally exponentially converges to $0$ in
the spatial $L^{2}$ norm under the triggering condition given (65)-(67).

###### Proof:

The existence of a unique solution to the system (1)-(3),(62), (6)-(8),(63)
$\forall~{}t>0$ is shown by applying Proposition 1 successively for all
intervals as Zeno behavior is excluded. Consider the following Lyapunov
function to the target systems (23)-(26) and (54)-(56),(64):

$\displaystyle W(t)=V_{1}(t)+CV_{2}(t)-m(t),$ (88)

where

$\displaystyle V_{1}(t)=$
$\displaystyle\int_{0}^{1}\left(\frac{A}{\lambda_{1}}\tilde{\alpha}^{2}(t,x)e^{-\frac{\mu}{\lambda_{1}}x}+\frac{B}{\lambda_{2}}\tilde{\beta}^{2}(t,x)e^{\frac{\mu}{\lambda_{2}}x}\right)dx,$
(89) $\displaystyle V_{2}(t)=$
$\displaystyle\int_{0}^{1}\left(\frac{A}{\lambda_{1}}\hat{\alpha}^{2}(t,x)e^{-\frac{\mu}{\lambda_{1}}x}+\frac{B}{\lambda_{2}}\hat{\beta}^{2}(t,x)e^{\frac{\mu}{\lambda_{2}}x}\right)dx,$
(90)

Differentiating (88) with respect to time and using (23)-(26), (54)- (56),(64)
and applying Young’s inequality along with the given values for $\mu$, $A$,
$B$, $C$, $r$, $r_{v}$, $\delta$, $\kappa_{0}$, $\kappa_{1}$, $\kappa_{2}$,
and $\kappa_{3}$, we obtain the following inequality:

$\displaystyle\dot{W}\leq-\mu(1-\sigma)W(t).$ (91)

Using the comparison principle and the continuity of $W(t)$, the following
estimate can be obtained for $W(t)$:

$\displaystyle W(t)\leq W(0)e^{-\mu(1-\sigma)t},$ (92)

hence it can be seen that the target systems globally exponentially converge
to $0$. The backstepping transformations (14)-(15) and (43)-(44) show that the
observer based event-triggered closed loop system (6)-(9), (10)-(13) with
$U_{k}^{c}(t)$ given in (60) globally exponentially converges to $0$. ∎

##  IV Periodic event-triggered control

Unlike the CETC approach, in PETC, the triggering function $\Gamma^{p}(t)$ is
only evaluated periodically. The increasing sequence of times $I^{p}$ at which
the control input $U_{k}^{p}(t)$ is updated is determined according the to
triggering mechanism proposed below.

###### Definition 2

Let the event-triggering function $\Gamma^{p}(t)$ be evaluated periodically,
with a maximum allowable sampling period of $h>0$. The increasing sequence of
times $I^{p}$ at which events are generated is defined according to the
following rule:

$\displaystyle t_{k+1}^{p}:=$
$\displaystyle\inf\\{t\in\mathds{R}|t>t_{k}^{p},\Gamma^{p}(t)>0,k\in\mathds{N},$
$\displaystyle\mkern 40.0mut=nh,~{}n\in\mathds{N},h>0\\},$ (93)
$\displaystyle\Gamma^{p}(t):=$ $\displaystyle(e^{ah}(2+\theta
a)-2)CBe^{\frac{\mu}{\lambda_{2}}}d^{2}(t)+m(t).$ (94)

The constants $a,\mu,\theta,B,C>0$ are defined in the CETC counterpart in
Section III. The function $d(t)$ is given in (61) and $m(t)$ is defined in
(67) with $m(0)=m^{0}<0$,
$m(t_{k}^{p^{-}})=m(t_{k}^{p})=m(t_{k}^{p^{+}}),k\in{\mathds{N}}$.

Assume that the PETC conditions (93)-(94) are defined such that
$\Gamma^{c}(t)$ given by (66) in CETC counterpart satisfies $\Gamma^{c}(t)<0$
$\forall~{}t\in[t_{k}^{p},t_{k+1}^{p}),k\in\mathds{N}$, leading to $m(t)<0$,
$\forall~{}t>0$. Let an event be triggered at $t=t_{k}^{p}$. Then, due to the
existence of the minimum dwell time $\tau$, we know that $\Gamma^{c}(t)<0$ for
$t_{k}^{p}\leq t<t_{k}^{p}+\tau$. Therefore, select the sampling period as,
$0<h<\tau$ such that $\Gamma^{c}(t)$ remains negative at least until the next
evaluation of $\Gamma^{p}(t)$.

Let us now delve into design of the triggering function $\Gamma^{p}(t)$. It is
worth noting that from Lemma 1, $(\dot{d}(t))^{2}$ can be bounded
$\forall~{}t\in[nh,(n+1)h),~{}n\in\mathds{N}$ as given by (68), (70)-(74).

###### Proposition 2

Let the PETC approach (93)-(94) have a sampling period of $h<\tau$. Let
$\kappa_{0}$, $\kappa_{1}$, $\kappa_{2}$, and $\kappa_{3}$ satisfy (75). Then
the following inequality holds for $\Gamma^{c}(t)$ given in (66):

$\displaystyle\Gamma^{c}(t)\leq$
$\displaystyle\bigg{[}\left(e^{a(t-nh)}(2+\theta
a)-2\right)CBe^{\frac{\mu}{\lambda_{2}}}d^{2}(nh)$ $\displaystyle\mkern
30.0mu+am(nh)\bigg{]}\frac{e^{-\eta(t-nh)}}{a},$ (95)

$\forall~{}t\in[nh,(n+1)h),~{}n\in\mathds{N}$.

###### Proof:

Note that $d(t)$, $\Gamma^{c}$, and $m(t)$ given by (61),(66), and (67)
respectively are continuous $\forall~{}t\in[nh,(n+1)h),~{}n\in\mathds{N}$ for
the PETC design given in (93)-(94) where $h<\tau$ is the sampling period. The
time derivative of (66) for $\forall~{}t\in[nh,(n+1)h),~{}n\in\mathds{N}$ is

$\displaystyle\dot{\Gamma}^{c}(t)=\theta
CBe^{\frac{\mu}{\lambda_{2}}}2d(t)\dot{d}(t)+\dot{m}(t).$ (96)

Using Young’s inequality and (66), (67), (68) and noticing that there always
exists a continuous function $\iota(t)>0$ such that

$\displaystyle\dot{\Gamma}^{c}(t)=\left(1+\epsilon_{4}+\frac{2}{\theta}\right)\Gamma^{c}(t)-\left(a+\frac{2}{\theta}\right)m(t)$
$\displaystyle+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{0}-\kappa_{0}\right)\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}$
$\displaystyle+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{1}-\kappa_{1}\right)\hat{\alpha}^{2}(t,1)+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{2}-\kappa_{3}\right)\hat{\beta}^{2}(t,0)$
$\displaystyle+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{3}-\kappa_{3}\right)\tilde{\beta}^{2}(t,0)-\iota(t),$
(97)

we obtain and expression for $\dot{\Gamma}^{c}(t)$. Using (66) and (67), the
following ODE holds:

$\displaystyle\dot{m}(t)=$
$\displaystyle-\kappa_{0}\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}-\kappa_{1}\hat{\alpha}^{2}(t,1)+\frac{2}{\theta}\Gamma^{c}(t)$
$\displaystyle-\left(\eta+\frac{2}{\theta}\right)m(t)-\kappa_{2}\hat{\beta}^{2}(t,0)-\kappa_{3}\tilde{\beta}^{2}(t,0).$
(98)

Defining

$\displaystyle\mathbf{z}(t)=\begin{pmatrix}\Gamma^{c}(t)\\\
m(t)\end{pmatrix},\quad\mathbf{A}=\begin{pmatrix}\left(1+\epsilon_{4}+\frac{2}{\theta}\right)&\left(-a-\frac{2}{\theta}\right)\\\
\frac{2}{\theta}&\left(-\eta-\frac{2}{\theta}\right)\end{pmatrix},$ (99)

and $\mathbf{B}(t)$ be as given in (100),

$\displaystyle\mathbf{B}(t)=\begin{pmatrix}\begin{aligned} &\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{0}-\kappa_{0}\right)\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{1}-\kappa_{1}\right)\hat{\alpha}^{2}(t,1)+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{2}-\kappa_{3}\right)\hat{\beta}^{2}(t,0)\\\
&\mkern 450.0mu+\left(\theta
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{3}-\kappa_{3}\right)\tilde{\beta}^{2}(t,0)-\iota(t),\\\
&\kappa_{0}\|\hat{\alpha}(t,\cdot),\hat{\beta}(t,\cdot)\|^{2}-\kappa_{1}\hat{\alpha}^{2}(t,1)-\kappa_{2}\hat{\beta}^{2}(t,0)-\kappa_{3}\tilde{\beta}^{2}(t,0)\end{aligned}\end{pmatrix}$
(100)

we obtain the following ODE system:

$\displaystyle\dot{\mathbf{z}}(t)=\mathbf{A}\mathbf{z}(t)+\mathbf{B}(t).$
(101)

The solution to (101) $\forall t\in[nh,(n+1)h)~{},n\in\mathds{N}$ is

$\displaystyle\mathbf{z}(t)=e^{\mathbf{A}(t-nh)}\mathbf{z}(nh)+\int_{nh}^{t}e^{\mathbf{A}(t-\xi)}\mathbf{B}(\xi)d\xi.$
(102)

Let $\mathbf{C}=\begin{pmatrix}1&0\end{pmatrix}$, since
$\Gamma^{c}(t)=\mathbf{C}\mathbf{z}(t)$, $\Gamma^{c}(t)$ can be calculated to
be

$\displaystyle\Gamma^{c}(t)=\mathbf{C}e^{\mathbf{A}(t-nh)}\mathbf{z}(nh)+\int_{nh}^{t}\mathbf{C}e^{\mathbf{A}(t-\xi)}\mathbf{B}(\xi)d\xi.$
(103)

Finding the eigenvalues and eigenvectors of $\mathbf{A}$, $e^{\mathbf{A}(t)}$
can be expressed as

$\displaystyle e^{\mathbf{A}t}=$ $\displaystyle\frac{e^{-\eta
t}}{a}\begin{pmatrix}\frac{2}{\theta}(e^{at}-1)+ae^{at}&\left(a+\frac{2}{\theta}\right)(1-e^{at})\\\
\frac{2}{\theta}(e^{at}-1)&\frac{2}{\theta}(1-e^{at})+a\end{pmatrix}.$

Also, $\mathbf{C}e^{\mathbf{A}(t-\xi)}\mathbf{B}(\xi)$ can be written as

$\displaystyle\mathbf{C}e^{\mathbf{A}(t-\xi)}\mathbf{B}(\xi)=$
$\displaystyle\frac{e^{-\eta(t-\xi)}}{a}\Bigg{[}(g(t-\xi)\epsilon_{0}-\kappa_{0}a)\|\hat{\alpha}(t-\xi,\cdot),\hat{\beta}(\xi,\cdot)\|^{2}$
$\displaystyle+(g(t-\xi)\epsilon_{1}-\kappa_{1}a)\hat{\alpha}^{2}(\xi,1)+(g(t-\xi)\epsilon_{2}-\kappa_{2}a)\hat{\beta}^{2}(\xi,0)$
$\displaystyle+(g(t-\xi)\epsilon_{3}-\kappa_{3}a)\tilde{\beta}^{2}(\xi,0)-\frac{g(t-\xi)}{\theta
CBe^{\frac{\mu}{\lambda_{2}}}}e^{-\eta(t-\xi)}\iota(\xi)\Bigg{]},$ (104)

where the increasing function $g(t)$ is

$\displaystyle 0\leq
g(t)=\left(e^{at}(2+a\theta\right)-2)CBe^{\frac{\mu}{\lambda_{2}}}.$ (105)

For $nh\leq\xi\leq t<(n+1)h$, $g(t-\xi)\epsilon_{i}-\kappa_{i}a$ for
$i=0,1,2,3$ can be approximated using (75) and (83) as

$\displaystyle g(t-\xi)\epsilon_{i}-\kappa_{i}a\leq
CBe^{\frac{\mu}{\lambda_{2}}}\epsilon_{i}(2+\theta a)(e^{ah}-e^{a\tau}).$
(106)

Since $h<\tau$, $(e^{ah}-e^{a\tau})<0$ hence it can be seen that
$g(t-\xi)\epsilon_{i}-\kappa_{i}a<0$ for $i=0,1,2,3$. Therefore,
$\mathbf{C}e^{\mathbf{A}(t-\xi)}\mathbf{B}(\xi)\leq 0$ leading to the
following inequality:

$\displaystyle\Gamma^{c}(t)\leq\mathbf{C}e^{\mathbf{A}(t-nh)}\mathbf{z}(nh),$
(107) $\displaystyle\mkern
10.0mu\leq\frac{e^{-\eta(t-nh)}}{a}\Bigg{[}\left(\frac{2}{\theta}(e^{a(t-nh)}-1)+ae^{a(t-nh)}\right)\Gamma^{c}(nh)$
$\displaystyle\mkern
100.0mu+\left(a+\frac{2}{\theta}\right)(1-e^{a(t-nh)})m(nh)\Bigg{]}.$ (108)

Substituting for $\Gamma^{c}(nh)$ using (66) we obtain (95) to conclude the
proof. ∎

###### Lemma 4

Let the PETC event triggering mechanism be as defined in (93)-(94) with a
sample period of $h<\tau$. Events at an increasing sequence of times $I^{p}$
can be generated at which the system is actuated by the control input given in
(60). Then, it holds that $\Gamma^{c}(t)<0$ and $m(t)<0$, $\forall~{}t>0$.

###### Proof:

Assume that two successive events are triggered at $t=t_{k}^{p}$ and
$t=t_{k+1}^{p}$, $k\in\mathds{N}$ under the PETC design (93)-(94) with a
sampling period $h$ with $m(t_{k}^{p})<0$. At $t=t_{k}^{p}$,
$\Gamma^{c}(t_{k}^{p})=\Gamma^{p}(t_{k}^{p})=m(t_{k}^{p})<0$. Substituting
$t=(n+1)h,~{}n\in\mathds{N}$ to (95), we see that $\Gamma^{c}((n+1)h)<0$ given
that $\Gamma^{p}(nh)<0$. Thus, $\Gamma^{c}(t)$ remains negative, at least
until $t=t_{k+1}^{p^{-}}$ where $\Gamma^{p}(t_{k+1}^{p^{-}})\geq 0$. Since
$\Gamma^{c}(t)<0$ for $t\in[t_{k}^{p},t_{k+1}^{p})$, we know that $m(t)<0$ for
$t\in[t_{k}^{p},t_{k+1}^{p})$. By definition, since
$m(t_{k+1}^{p^{-}})=m(t_{k+1}^{p})=m(t_{k+1}^{p^{+}})$, $m(t_{k+1}^{p})<0$.
Finally, after, the control input has been updated at $t=t_{k+1}^{p}$, we have
$\Gamma^{c}(t_{k+1}^{p})=\Gamma^{p}(t_{k+1}^{p})=m(t_{k+1}^{p})<0$. Hence,
using this reasoning for all intervals in $I^{p}$ and noting that $m^{0}<0$,
we see that $\Gamma^{c}(t)<0$ and $m(t)<0$, $\forall~{}t>0$. ∎

We establish the convergence of the closed-loop system under PETC in the
following theorem.

###### Theorem 2

Let the parameters $\theta$, $\sigma$, $\mu$, $\eta$, $A$, $B$, $C$, $r_{v}$,
$\delta$, $\kappa_{0}$, $\kappa_{1}$, $\kappa_{2}$, and $\kappa_{3}$ be as
defined in Theorem 1. The observer based event-triggered closed-loop system
(1)-(3),(62), (6)-(8),(63) with $U_{k}^{p}(t)$ given in (60) has a unique
solution $\forall~{}t>0$ and globally exponentially converges to $0$ in the
$L^{2}$-norm sense under the triggering condition (93)-(94) with a sampling
period $h<\tau$.

###### Proof:

The existence of a unique solution to the system (1)-(3),(62), (6)-(8),(63)
$\forall~{}t>0$ is shown by applying Proposition 1 successively for all
intervals as Zeno behavior is inherently excluded due to the periodic
evaluation of $\Gamma^{p}(t)$. Following Lemma 4, we know that $\Gamma^{c}<0$
and $m(t)<0$, $\forall~{}t>0$ . Hence, we see that the PETC conditions
(93)-(94) guarantee that the CETC conditions (65)-(67) are satisfied
$\forall~{}t>0$. Therefore, using Theorem 1, we can conclude that the systems
(1)-(3),(62), (6)-(8),(63) with $U_{k}^{p}(t)$ given in (60) globally
exponentially converges to $0$ in the $L^{2}$-norm sense under the triggering
condition (93)-(94) . ∎

##  V Numerical Simulations

Consider the system (1)-(4) with parameters identical to the system simulated
in [11], where $\lambda_{1}=1$, $\lambda_{2}=2$, $c_{1}(x)=c_{2}(x)=2$,
$q=0.5$, and $\rho=0.9$. The initial conditions $v(0,x)=10(1-x)$,
$u(0,x)=qv(0,x)$, $\hat{u}(0,x)=1.5u(0,x)$, and $\hat{v}(0,x)=1.5v(0,x)$
$\forall~{}x\in[0,1]$. The kernels given in (16)-(21), (29)-(34), (37)-(42),
and (45)-(50) are solved using the method proposed in [1, Appendix]. Let
$\mu=0.11238$, $\sigma=0.5$, $r=200$ and $\eta=50$, leading to $A=1.5182$,
$B=0.77974$, $\delta=73.4416$ and $C=4.1088\times 10^{-5}$. This then yields
$r_{v}=1.5182$. Using the equations in (70)-(74) we obtain
$\epsilon_{0}=6.4506$, $\epsilon_{1}=3.0531$, $\epsilon_{2}=0$,
$\epsilon_{3}=416.3318$, and $\epsilon_{4}=49.3067$. Next we define
$\kappa_{0}=1.1523\times 10^{-6}$, $\kappa_{1}=6.966\times 10^{-7}$,
$\kappa_{2}=4.2391\times 10^{-7}$, and $\kappa_{3}=7.4369\times 10^{-5}$ such
that the inequalities (84)-(87) are satisfied. Finally, we define
$\theta=2.6354\times 10^{-3}$ small enough such that (75) is satisfied. The
minimum dwell-time and the sampling period are $\tau=0.006317s$ and $h=0.005s$
respectively. Variation of the control input and the system norm for the
states $u(t,x),v(t,x)$ with time for CTC, CETC, and PETC is given in Fig. 1a
and Fig. 1b respectively.

(a) Time evolution $U(t)$ and $U_{k}^{\omega}(t)$ (b) Time evolution of the
norm

Figure 1: Variation of the control input and state norms with time for CT,
CETC and PETC

##  VI Conclusion

This work has proposed an anti-collocated observer-based periodic event-
triggered boundary control for a class of $2\times 2$ hyperbolic PDEs with
reflection terms at boundaries. We have developed a method to extend the
existing CETC design to a PETC design with an explicit upper bound for the
sampling period. Unlike CETC, which requires continuous evaluation of the
triggering function to detect events, the proposed PETC strategy necessitates
only periodic evaluation of the triggering function. The existence of a unique
solution and global $L^{2}$-exponential convergence to zero have been proven
under the proposed control strategy.

For future work, we may consider development of a self-triggered control
strategy for $2\times 2$ linear hyperbolic PDEs.

##  VII Acknowledgments

This work was funded by the NSF CAREER Award CMMI-2302030.

## References

  * [1] H. Anfinsen and O. M. Aamo, _Adaptive control of hyperbolic PDEs_. Springer, 2019.
  * [2] G. Bastin and J.-M. Coron, “Further results on boundary feedback stabilisation of 2$\times$ 2 hyperbolic systems over a bounded interval,” _IFAC Proceedings Volumes_ , vol. 43, no. 14, pp. 1081–1085, 2010.
  * [3] ——, “On boundary feedback stabilization of non-uniform linear $2\times\texttimes 2$ hyperbolic systems over a bounded interval,” _Systems & Control Letters_, vol. 60, no. 11, pp. 900–906, 2011.
  * [4] N. Bekiaris-Liberis, “On 1-D PDE-based cardiovascular flow bottleneck modeling and analysis: A vehicular traffic flow-inspired approach,” _IEEE Transactions on Automatic Control_ , 2022.
  * [5] J.-M. Coron, B. d’Andréa Novel, and G. Bastin, “A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations,” in _1999 European control conference (ECC)_. IEEE, 1999, pp. 3178–3183.
  * [6] F. Di Meglio, R. Vazquez, and M. Krstic, “Stabilization of a system of $n+1$ coupled first-order hyperbolic linear PDEs with a single boundary input,” _IEEE Transactions on Automatic Control_ , vol. 58, no. 12, pp. 3097–3111, 2013.
  * [7] A. Diagne, G. Bastin, and J.-M. Coron, “Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws,” _Automatica_ , vol. 48, no. 1, pp. 109–114, 2012.
  * [8] A. Diagne, M. Diagne, S. Tang, and M. Krstic, “Backstepping stabilization of the linearized Saint-Venant-Exner model,” _Automatica_ , vol. 76, pp. 345–354, 2017.
  * [9] M. Diagne and I. Karafyllis, “Event-triggered boundary control of a continuum model of highly re-entrant manufacturing systems,” _Automatica_ , vol. 134, p. 109902, 2021.
  * [10] M. Diagne, S.-X. Tang, A. Diagne, and M. Krstic, “Control of shallow waves of two unmixed fluids by backstepping,” _Annual Reviews in Control_ , vol. 44, pp. 211–225, 2017.
  * [11] N. Espitia, “Observer-based event-triggered boundary control of a linear 2 × 2 hyperbolic systems,” _Systems & Control Letters_, vol. 138, p. 104668, Apr. 2020.
  * [12] N. Espitia, A. Girard, N. Marchand, and C. Prieur, “Event-based boundary control of a linear $2\times 2$ hyperbolic system via backstepping approach,” _IEEE Transactions on Automatic Control_ , vol. 63, no. 8, pp. 2686–2693, 2018.
  * [13] W. P. M. H. Heemels, M. C. F. Donkers, and A. R. Teel, “Periodic event-triggered control for linear systems,” _IEEE Transactions on Automatic Control_ , vol. 58, no. 4, pp. 847–861, Apr. 2013.
  * [14] V. D. S. Martins, M. Rodrigues, and M. Diagne, “A multi-model approach to Saint-Venant equations: A stability study by lmis,” _International Journal of Applied Mathematics and Computer Science_ , vol. 22, no. 3, pp. 539–550, 2012.
  * [15] B. Rathnayake and M. Diagne, “Observer-based periodic event-triggered and self-triggered boundary control of a class of parabolic PDEs,” _arXiv preprint arXiv:2312.01313_ , 2023.
  * [16] ——, “Observer-based periodic event-triggered boundary control of the one-phase Stefan problem,” _IFAC-PapersOnLine_ , vol. 56, no. 2, pp. 11 415–11 422, 2023, 22nd IFAC World Congress.
  * [17] R. Vazquez, M. Krstic, and J.-M. Coron, “Backstepping boundary stabilization and state estimation of a $2\times 2$ linear hyperbolic system,” in _2011 50th IEEE Conference on Decision and Control and European Control Conference_ , 2011, pp. 4937–4942.
  * [18] M. Wakaiki and H. Sano, “Event-triggered control of infinite-dimensional systems,” _SIAM Journal on Control and Optimization_ , vol. 58, no. 2, pp. 605–635, 2020.
  * [19] C.-Z. Xu and G. Sallet, “Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,” _ESAIM: Control, Optimisation and Calculus of Variations_ , vol. 7, pp. 421–442, 2002.
  * [20] H. Yu and M. Krstic, “Traffic congestion control for Aw–Rascle–Zhang model,” _Automatica_ , vol. 100, pp. 38–51, 2019.


The Weak Supervision Landscape

Partially supported by project TEC2017-83838-R, funded by FEDER /Ministerio de Ciencia, Innovación y Universidades – AEI;
by the SPHERE Next Steps Project funded by EPSRC [grant EP/R005273/1]; RP and RSR are funded by
the UKRI Turing AI Fellowship EP/V024817/1.

Rafael Poyiadzi1, Daniel Bacaicoa-Barber2, Jesus Cid-Sueiro2,

Miquel Perello-Nieto1, Peter Flach1, Raul Santos-Rodriguez1
1Intelligent Systems Lab, University of Bristol, Bristol, UK

2Signal Theory and Communications Dept., Universidad Carlos III of Madrid, Spain

Contact Email<EMAIL_ADDRESS>
Many ways of annotating a dataset for machine learning classification tasks that go beyond the usual class labels exist in practice. These are of interest as they can simplify or facilitate the collection of annotations, while not greatly affecting the resulting machine learning model. Many of these fall under the umbrella term of weak labels or annotations. However, it is not always clear how different alternatives are related. In this paper we propose a framework for categorising weak supervision settings with the aim of: (1) helping the dataset owner or annotator navigate through the available options within weak supervision when prescribing an annotation process, and (2) describing existing annotations for a dataset to machine learning practitioners so that we allow them to understand the implications for the learning process. To this end, we identify the key elements that characterise weak supervision and devise a series of dimensions that categorise most of the existing approaches. We show how common settings in the literature fit within the framework and discuss its possible uses in practice.

weak supervision, weak labels, annotation process

§ INTRODUCTION

A machine learning classification task requires having a dataset of pairs of instances and labels. Obtaining labels of good quality can be expensive, time-consuming, or difficult in general. These constraints have led to the development and study of several flexible settings where annotations are assumed to not be perfect, but still suitable for the learning process. These are usually referred to as weak supervision (WS). For example, specialised products like Amazon's Mechanical Turk  provide access to pools of (non-expert) annotators whose labels are sometimes ambiguous and noisy. Interestingly, weak labels can take several forms and not only be the result of the annotation process (e.g., the data not annotated by an expert, or automatically extracted from the web) but they can come from the choices made by the dataset owner when deciding on the annotation process (e.g., allow annotators to provide more than one candidate classes when uncertain).

In machine learning, weakly supervised classification refers the task of obtaining a classifier from a given weak dataset, such that it has a low generalisation error with respect to the true data distribution. Even though weakly supervised learning encompasses settings which are widely applicable and studied, it is yet to become a standard machine learning setting such as traditional supervised classification or clustering <cit.>. They have been previously referred to as indirect supervision <cit.>, distant supervision <cit.>, inaccurate, incomplete, or inexact supervision <cit.>, learning from measurements <cit.>, learning imprecise and fuzzy observations <cit.> and many more. In this work we refer to all the approaches where the observed label is not perfect as . Where applicable we will also refer to the observed label as , as opposed to the unobserved . We will also be referring to the process by which a clean label is transformed to a weak label as the .

In this paper we introduce a framework for categorising weak supervision settings by identifying the key set of dimensions that should be used to describe the different instantiations that exist in the literature and in practice. This framework can then be used by both dataset owners / annotators and machine learning practitioners / researchers. For the former, it will be a tool to navigate the landscape of options when designing the data collection or annotation process or to describe an existing dataset. In both cases it will aid communication and can help future users understand implications of the type of annotations present in the dataset. For the later, it can be used to find the right place for a new or existing technique to learn from weak labels and identify (open) research problems in the field and have a clearer understanding of the generalisation of their contributions.

The remainder of the paper is structured as follows: in Section <ref> we introduce the framework and in Section <ref> we present several well studied WSL settings and discuss how they fit within our framework. Lastly in Section <ref> we discuss the main implications and limitations of the work and directions for future research.

Framework for categorising weak supervision settings constituted by categories and dimensions. The third column lists options for each. Individual settings can be identified by selecting one option per dimension. We note that certain combinations of options are not compatible. The final column summarizes the meaning of each of the dimensions for use in the prescription or description of the corresponding dimension.

Category                                Dimension                                                                  Options                  1c|[HTML]EFEFEFQuestion
c]@c@Number of
classes               Binary / Multi-class     How many classes does the task involve?
-3*True Label Space      Multi-label                                                                Yes / No                 Can instances belong to more than one class?
Unsupervised                                                               Yes / No                 c]@l@Are annotators allowed to not annotate certain samples?
Soft labels                                                               Yes / No                 c]@l@Are annotators allowed to use soft or probabilistic labels?
c]@c@Number of
Annotators              $1~/~\textgreater1$                  How many annotators will annotate the data?
-5*Weak Label space      c]@c@Number of
candidate classes     $1~/~\textgreater1$                     Are annotators allowed to provide annotations covering more than one class?
Aggregation                                                        Yes / No                 Are samples annotated individually or as a group?
Class dependent                                                            Yes / No                 Are classes equally prone to annotation errors?
-3*Weakening Process     Instance dependent                                                                 Yes / No                 Are samples equally prone to annotation errors?

§ DIMENSIONS OF WEAK SUPERVISION

In this section we present the dimensions of weak supervision, which are the building blocks of our framework. We separate the dimensions into three groups based on whether they refer to the true label space, the weak label space or the weakening process. Table <ref> condenses all this information together with questions to illustrate the meaning of each dimension and the options that they offer. This table can be used as a standalone tool to plan or understand weak annotations.

§.§ True Label Space

Here we present dimensions that derive directly form the nature and description of the task. Classification tasks can take different forms, from binary classification, where we might be interested in identifying an image as a dog or a cat, to multi-class, where we might aim to recognize images of dogs according to their breed. This is summarized in the first dimension – number of classes. Additionally, we accept that it is sometimes the case that more than one categories can be assigned to a single instance as in multi-label settings, which forms the second dimension. The types of true/clean label that we consider are then as follows.

\begin{align}
% &\textrm{Binary}\footnotemark:\\
% &\mathcal{Y}_2~=~ \big\{y~\mid~y\in\{0,~1\}\big\}~=~ \big\{\boldsymbol{y}~\mid~\boldsymbol{y}\in\{0,~1\}^2,~\boldsymbol{1}^\top \boldsymbol{y}=1\big\}\\
&\textrm{Number of classes:}\nonumber\\
&\hspace{15pt}\mathcal{Y}_k~=~ \big\{\boldsymbol{y}~\mid~\boldsymbol{y}\in\{0,~1\}^k,~\boldsymbol{1}^\top \boldsymbol{y}=1\big\}\label{eq:multiclass_label_space}\\
% &\textrm{Multi-class with unlabelled}:\\
% &\mathcal{Y}_{k+}~=~ \big\{\boldsymbol{y}~\mid~\boldsymbol{y}\in\{0,~1\}^k,~\boldsymbol{1}^\top \boldsymbol{y}\in\{0,1\}\big\}\\
&\hspace{15pt}\mathcal{Y}_{m, k} ~=~ \big\{\boldsymbol{y}~|~\boldsymbol{y}\in\{0,~1\}^k,~1\leq\boldsymbol{1}^{\top}\boldsymbol{y}\leq m\leq k\big\}\label{eq:multilabel_label_space}
\end{align}

Although this can be extended to structured data, we do not explore it here for simplicity. We also assume that the dataset only contains what is defined by the task, e.g., in the case of the task being classifying dogs vs cats, there would not be images of other animals in the dataset. Even though we do not discuss the learning stage in this paper, the framework is constructed  keeping in mind that part of the task would be to obtain a classifier: $f:~\mathcal{X}~\to~\mathcal{Y}_{\textrm{Clean}}$ and in minimizing ${\mathbb{E}_{X,Y}\ell(Y, f(X))}$, where $\ell(.,.)$ is a loss function and the expectation is over the clean label data distribution.

§.§ Weak Label Space

In this section we focus on dimensions concerned with the weak label space, $\mathcal{Y}_{Weak}$ (See Eq. <ref>) and the possible forms that it might take. These characterise the degrees of freedom of the annotator.

Access to unlabelled data. In certain situations, besides the potentially weakly labelled dataset, we also have access to a separate unlabelled dataset. This could be either because annotators are allowed to return an empty annotation, or because this set of data was just not chosen for annotation. We refer to this dimension as unsupervised.

Access to multiple annotators.
We usually assume a dataset is annotated by one annotator, but it might be the case that a number of annotators provide annotations for the same dataset. Annotations on the same instance will not necessarily agree, potentially creating ambiguity.

Restriction on number of assigned candidate classes. In binary and multi-class classification, the classes are mutually exclusive and exhaustive, which means that every instance is associated with one true class only. In the weak label space, we could allow an annotator to provide a set of candidate classes, instead of just one (number of candidate classes).

An example of a weak label and how this set of dimensions can affect what we get to observe is shown in Figure <ref>. It shows how these three dimensions can increase the complexity of a weak label, as we can have several annotators, allow them to provide no annotation, and also allow them to assign more than one class to each sample.

Soft labels. Also known as probabilistic labels, the soft labels relaxation allows for more flexibility in the annotation process by letting an annotator express a degree of belief. For example, instead of resorting to `dog' for an image, they could say "70% confident this image contains a dog" <cit.>.

An example of a non-aggregate weak label. For this case we consider six base classes and access to four annotators. This example showcases three dimensions: (1) having access to more than one annotators, (2) annotators can provide an empty annotation set (see annotator 3), and (3) annotators can provide an annotation set with more than one candidate class (see annotators 1 and 2).

§.§ Weakening Process

The final set of dimensions captures the different aspects of the weakening process, i.e., the (usually unknown) transformation the maps true/clean to weak labels. Interestingly, the weakening process depends on the annotator and how they make suboptimal annotation decisions, but it can also depend on choices made by the dataset owner or on the task itself.

A key dimension is whether the labels provided correspond to a single instance, or whether they depend on, and correspond to, a set of instances through an aggregation mechanism. For the aggregate cases, the labels provided correspond to a set of instances which we will refer to as a `bag'. Instead of being provided with a set of instance-label pairs $\{(x_i,~y_i)\}_{i=1}^n$, we are provided with a set of $\{(x_i,~b_i)\}_{i=1}^n$ where $b_i$'s are bag indicators and  $\{(b_j,~t_j)\}_{j=1}^m$, where $t_j$'s are the corresponding labels. The labels in this case are of the form:

\begin{equation}
\label{eq:aggregate_function}
t_j~=~g\Big(\big\{y_i~\mid~i\in b_j\big\}\Big)
\end{equation}

where we use $i\in b_j$ to imply that the $i^{th}$ instance belongs to the $j^{th}$ bag, and $g$ is the label aggregation function. It should be noted that $t_j$ is the label for all instances $x_i,~i\in b_j$.
The distinction between the two is shown pictorially in Fig. <ref>.

Depiction of the difference between non-aggregate and aggregate settings. In the case of non-aggregate settings, there is a one-to-one relationship between instances and labels, while in the case of aggregate noise settings we observe one label per group. We note that a label could be one class, multiple or none, as we describe below.

For non-aggregate settings, we will refer to $\mathbb{P}(\tilde{Y}\mid X,Y)$, where $\tilde{Y}$ denotes the random variable for the weak label, as the weakening function, which upon conditioning has the form,

\begin{equation}
\label{eq:noise_process}
\tau(y;x)=\mathbb{P}\big(\tilde{Y}\mid X=x,Y=y\big)
\end{equation}

\begin{equation}
\label{eq:noise_process_2}
\tau(y;x): \mathcal{Y}_{Clean}\to\triangle_{\mathcal{Y}_{Weak}}
\end{equation}

where $\triangle_{\mathcal{Y}_{Weak}}$ is the probability simplex over $\mathcal{Y}_{Weak}$. Eqs. <ref> & <ref> involve three objects:

* the clean label space $\mathcal{Y}_{Clean}$ (covered in Sec. <ref>),

* the weak label space $\mathcal{Y}_{Weak}$ (covered in Sec. <ref>) and

* the weakening function $\tau(y;x)$.

The weakening function acts between discrete sets and can therefore be described as a Categorical () distribution. This distribution is parametrized by a column of a mixing matrix or transition matrix, $\boldsymbol{T}$, which is a non-negative column-wise stochastic matrix. The weakening process is modelled as,

\begin{equation}
\label{eq:noise_categorigal}
\tilde{Y}~\mid~Y=\boldsymbol{y},~X=\boldsymbol{x}~\sim~\texttt{Cat}\big(\boldsymbol{T}\boldsymbol{y}\big)
\end{equation}

With regards to dimensions arising of the weakening function we will consider two, the dependence on the instance and on the true class.

Dependence on instance. We consider two cases, instance dependence or instance independence <cit.>.

\begin{align*}
&\textrm{Instance-independent noise (IIN)}:\\
&\hspace{10pt} \mathbb{P}(\tilde{Y}=\tilde{y}~\mid~X=x,~Y=y)~=~\mathbb{P}(\tilde{Y}=\tilde{y}~\mid~Y=y),\\
&\textrm{Instance-dependent noise (IDN)}:\\
&\hspace{10pt} \mathbb{P}(\tilde{Y}=\tilde{y}~\mid~X=x,~Y=y)~\ne~\mathbb{P}(\tilde{Y}=\tilde{y}~\mid~Y=y).
% \textrm{posterior-instance-dependent (PIDN)}:&\hspace{10pt} \mathbb{P}(\tilde{Y}=\tilde{y}~\mid~X=x,~Y=y) = \mathbb{P}(\tilde{Y}=\tilde{y}~\mid~\eta(x)=\eta,~Y=y)
\end{align*}

Dependence on true class. With regards to the class dependence we consider symmetric (uniform) or asymmetric (class-conditional) with respect to the original classes <cit.>. In the case of multi-class classification, symmetric noise would imply:

\begin{align}
\label{eq:multiclass_cnn}
&\mathbb{P}(\tilde{Y}=e_u\mid X=x,Y=e_i)=\mathbb{P}(\tilde{Y}=e_v\mid X=x,Y=e_i)\\
&~~\forall u,v,i \in [k],~u\neq v\neq i. \nonumber
\end{align}

We have asymmetric label noise when this does not hold.

§ THE DIMENSIONS IN PRACTICE

We now present several well-studied settings within the field of weak supervision and discuss how they fit in the proposed framework. We first start by exploring examples that belong to the the non-aggregate category and then move to aggregate, as this separation simplifies the formulation of the settings.

Non-aggregate WSL settings according to the label-space of true labels and weak labels, and the mixing matrix used to model the weakening process.

[HTML]EFEFEFName     [HTML]EFEFEFc]@c@True
Label Space     [HTML]EFEFEFc]@c@Weak
Label Space     [HTML]EFEFEFMixing matrix
Noisy Labels     $\mathcal{Y}_k$     $\mathcal{Y}_k$     $\boldsymbol{T}\in\mathbb{R}^{k\times k}$
Partial Labels     $\mathcal{Y}_k$     $\mathcal{Y}_{m, k}$     $\boldsymbol{T}\in\mathbb{R}^{(2^k-2)\times k}$
Superset Learning     $\mathcal{Y}_k$     $\mathcal{Y}_{m, k}(z)$     $\boldsymbol{T}\in\mathbb{R}^{(2^k-2)\times k} $

Learning     $\mathcal{Y}_k$     $\mathcal{Y}_{k+1}$     $\boldsymbol{T}\in\mathbb{R}^{(k+1)\times k}$
Positive-Unlabelled     $\mathcal{Y}_2$     $\mathcal{Y}_3$     $\boldsymbol{T}\in\mathbb{R}^{3\times 2}$

Annotators     $\mathcal{Y}_k$     $\mathcal{Y}_k^n$     $\Big\{\boldsymbol{T}\in\mathbb{R}^{k\times k}\Big\}_{i=1}^n$

§.§ Non-aggregate WS Settings

For non-aggregate WS settings, the selected examples are summarized in Table <ref> to illustrate their constraints in the true label space, weak label space and weakening process.

Noisy labels (Flipping noise) In this setting, instances switch labels with a certain probability. This type of noise could be introduced when the data is labelled by a non-expert annotator, and nicely extends to having multiple such annotators. In this setting the clean and noisy label spaces are the same:

\begin{align*}
\mathcal{Y}_k&~\to~\mathcal{Y}_k\\
\boldsymbol{T}&~\in~\mathbb{R}^{k\times k}
\end{align*}

An example of a mixing matrix for binary classification:

\begin{equation*}
\centering
\bordermatrix{& \textcolor{blue}{\textrm{+}} & \textcolor{blue}{\textrm{-}}\cr
\textcolor{blue}{\textrm{+}} & 1-\gamma_0 & \gamma_1 \cr
\textcolor{blue}{\textrm{-}} & \gamma_0 & 1-\gamma_1}
\end{equation*}

where if $\gamma_0=\gamma_1$ it implies symmetric label noise, and asymmetric label noise otherwise. In the case of instance-independent noise, $\gamma_0$ and $\gamma_1$ are constants, i.e., all instances have their label flipped with same probability. On the other hand, with instance-dependent noise, $\gamma_0$ and $\gamma_1$ could be functions of $x$, where we could have rates differ $\gamma_0(x_i)\neq\gamma_1(x_j)$ for $x_i\ne x_j$. A comprehensive review of classification in the presence of label noise is given by <cit.>.

Superset Learning In this setting a weak label could be any combination of class labels, subject to the constraint that this combination contains the true label. For example, in the case of multi-class classification with three classes, for a true label $[001]$ (class 3 in one-hot encoding), we could observe $[011]$ or $[101]$, but not $[110]$ (represented as the binary OR operator on the true one-hot encoding). Superset learning has been studied in the literature under different names such as `learning with partial-labels' (see below), `learning with ambiguous labels' and `learning from complementary labels' <cit.>. Some of the earlier works on the topic include <cit.> and <cit.> where the setting is referred to as `partial-labels' and `multiple labels' respectively. An example of a mixing matrix:

\begin{equation*}
\bordermatrix{
& \textcolor{blue}{\textrm{001}} & \textcolor{blue}{\textrm{010}} & \textcolor{blue}{\textrm{100}}\cr
\textcolor{blue}{\textrm{001}} & \alpha_0 & 0 & 0\cr                % 001
\textcolor{blue}{\textrm{010}} & 0 & \beta_0 & 0\cr               % 010
\textcolor{blue}{\textrm{100}} & 0 & 0 & \gamma_0\cr                % 100
\textcolor{blue}{\textrm{110}} & 0 & \beta_1 & \gamma_1\cr         % 011
\textcolor{blue}{\textrm{101}} & \alpha_1 & 0 & 1-\gamma_0-\gamma_1\cr        % 101
\textcolor{blue}{\textrm{011}} & 1-\alpha_0-\alpha_1 & 1-\beta_0-\beta_1 & 0\cr         % 110
%   \textcolor{blue}{\textrm{111}} & 1 - \sum_i \alpha_i & 1 - \sum_i \beta_i  & 1 - \sum_i \gamma_i\cr    % 111
\end{equation*}

\begin{align*}
\mathcal{Y}_k&~\to~\mathcal{Y}_{m,k}(z)\\
\boldsymbol{T}&~\in~\mathbb{R}^{(2^k-2)\times k}
\end{align*}

\begin{equation}
\label{eq:superset_label_space}
\mathcal{Y}_{m, k}(z)= \big\{\boldsymbol{y}|\boldsymbol{y}\in\{0,1\}^k,~1\leq\boldsymbol{1}^{\top}\boldsymbol{y}\leq m\leq k,~z\in \boldsymbol{y}\big\}
\end{equation}

We abuse notation and use $\boldsymbol{y}$ both as a vector and as a set and with $z\in\boldsymbol{y}$ we imply that $\boldsymbol{y}$ covers $z$, i.e. has a non-zero entry wherever $z$ has a non-zero entry.

Partial Labels (PLL) PLL is sometimes used to refer to superset learning. This setting is similar to that of superset learning but where there is no restriction that the observed weak label includes the true label<cit.>.

\begin{align*}
\mathcal{Y}_k&~\to~\mathcal{Y}_{m,k}\\
\boldsymbol{T}&~\in~\mathbb{R}^{(2^k-2)\times k}
\end{align*}

\begin{equation}
\label{eq:partial_label_space}
\mathcal{Y}_{m, k} ~=~ \big\{\boldsymbol{y}~|~\boldsymbol{y}\in\{0,1\}^k,~1\leq\boldsymbol{1}^{\top}\boldsymbol{y}\leq m\leq k\big\}
\end{equation}

In both settings, the set of potential observations extends from $k$ to $2^k-2$. While standard presentations of the settings consider $2^k-1$, we choose to exclude the all inclusive potential observation and instead consider it as an extra dimension.

Semi-supervised Learning (SSL) In semi-supervised learning <cit.>, on top of the usual supervised dataset, we are also provided with an unlabelled dataset.

\begin{align*}
\mathcal{Y}_k&~\to~\mathcal{Y}_{k+}\\
\boldsymbol{T}&~\in~\mathbb{R}^{(k+1)\times k}
\end{align*}

\begin{equation}
\label{eq:ssl_labelspace}
\mathcal{Y}_{k+}~=~ \big\{\boldsymbol{y}~\mid~\boldsymbol{y}\in\{0,1\}^k,~\boldsymbol{1}^\top \boldsymbol{y}\in\{0,1\}\big\}
\end{equation}

and is an extension of the multi-class label space (Eq. <ref>) that allows for no classes to be provided. An example of a mixing matrix for binary classification:

\begin{equation}
\label{eq:ssl_mixing_matrix}
\bordermatrix{& \textcolor{blue}{\textrm{+}} & \textcolor{blue}{\textrm{-}}\cr
\textcolor{blue}{\textrm{+}} & 1-\gamma_0 & 0 \cr
\textcolor{blue}{\textrm{-}} & 0 & 1-\gamma_1 \cr
\textcolor{blue}{\textrm{na}} & \gamma_0 & \gamma_1
\end{equation}

Positive-Unlabelled (PU) Learning
Learning with positive and unlabelled instances <cit.> is the setting of binary classification where the training dataset only consists of positive and unlabelled instances. Situations where PU learning arises include medical records where only known previous diseases are listed and personalised advertising where visited pages and clicks are the positive cases <cit.>.

\begin{align*}
\mathcal{Y}_2&~\to~\mathcal{Y}_{2+}\\
\boldsymbol{T}&~\in~\mathbb{R}^{3\times 2}
\end{align*}

An example of a mixing matrix for PU learning:

\begin{equation*}
\bordermatrix{& \textcolor{blue}{\textrm{+}} & \textcolor{blue}{\textrm{-}}\cr
\textcolor{blue}{\textrm{+}} & 1-\gamma_0 & 0 \cr
\textcolor{blue}{\textrm{-}} & 0 & 0 \cr
\textcolor{blue}{\textrm{na}} & \gamma_0 & 1}
\end{equation*}

PU learning can be seen as a special case of semi-supervised learning where $\gamma_1 = 1$ (Eq. <ref>).
It has been extended to the multi-class case, under the name Multi-Positive and Unlabeled learning, where labeled data from multiple positive classes are provided for training along with unlabeled data from a mixture of the positive classes and a negative class <cit.>.

Multiple annotators Assuming we have $m$ annotators, we also have potentially $m$ distinct mixing matrices <cit.>. Therefore, $\tilde{Y}^n~=~\big\{\tilde{Y} \sim \texttt{Cat}\big(\boldsymbol{T}_j\boldsymbol{y}\big)\big\}_{j=1}^n$, where $\boldsymbol{T}_j$ denotes the $j^{th}$ annotator's mixing matrix.

\begin{align*}
\mathcal{Y}_k&~\to~\mathcal{Y}^{n}_{k}\\
\Big\{\boldsymbol{T}&~\in~\mathbb{R}^{k\times k}\Big\}_{j=1}^n
\end{align*}

\begin{equation}
\label{eq:multiannotator_labelspace}
\mathcal{Y}_{k}^n ~=~ \big\{\boldsymbol{y}~|~\boldsymbol{y}\in\{0,~1\}^{k\times n},~\boldsymbol{y}^{\top}\boldsymbol{1}=\boldsymbol{1}\big\}
\end{equation}

where the equality should be understood as elementwise.

§.§ Aggregate WSL Settings

Aggregate settings have comparatively received much less attention in the literature. Here, we present multiple instance learning and learning from label proportions as the main representatives.

Multiple Instance Learning Multiple instance learning (MIL) is usually considered in the binary classification case. The label provided for a bag of samples is an indicator of the presence of the positive class. In other words, is there at least one positive instance in the set? It was first introduced in <cit.> with the motivation of drug activity prediction.

\begin{equation*}
g(y_1, \dots, y_n)=max\big(y_{[1..n]}\big)
\end{equation*}

In <cit.> the authors extend MIL, from requiring at least one positive instance in a bag, to requiring $r$.

\begin{equation*}
g(y_1, \dots, y_m)~=~\boldsymbol{1}\Big\{\sum_{i=1}^{m} y_{i} \geq r\Big\}
\end{equation*}

Learning from Label Proportions (LLP)
In LLP, the aggregation function is the count function (or proportion) for each of the classes present in a bag of samples. LLP was introduced in <cit.> with the motivation of learning with mass spectrometry data.

\begin{equation*}
g(y_1, \dots, y_m)=\Bigg[\sum_{i=1}^{m} y_{i,0},~\cdots,~\sum_{i=1}^{m} y_{i,c}\Bigg]
\end{equation*}

Interestingly, PU Learning was presented as being a non-aggregate setting, but under certain conditions it can also be seen as a case of learning from label proportions. In PU Learning, we are provided with a set of positive data and a set of unlabelled data. If we know the portion of positives and negatives in the unlabelled set, we can view this as an instance of LLP with two bags.

§.§ Towards a Unified Formulation

For non-aggregate settings, we can now see how to find a common formulation for weakly supervised settings. Starting with the label space corresponding to multiple annotators in Eq. <ref>, we can extend it to allow annotators to provide an empty set for a sample (e.g., when they find an instance difficult to annotate):

\begin{equation}
\label{eq:multiannotator_w_unlabelled_labelspace}
\mathcal{Y}_{k+}^n ~=~ \big\{\boldsymbol{y}~|~\boldsymbol{y}\in\{0,~1\}^{k\times n},~\boldsymbol{y}^{\top}\boldsymbol{1}\in\{0,1\}\big\}
\end{equation}

and then it could be extended to allow annotators to provide an annotation set with more than one candidate,

\begin{equation}
\label{eq:multiannotator_multlabel_labelspace}
\mathcal{Y}_{m,k+}^n ~=~ \big\{\boldsymbol{y}~|~\boldsymbol{y}\in\{0,~1\}^{k\times n},~\boldsymbol{0}\leq\boldsymbol{y}^{\top}\boldsymbol{1}\leq \boldsymbol{m}\leq \boldsymbol{k}\big\}
\end{equation}

With regards to describing the annotation process for a dataset, we could model the weakening process through its dependence on the instance and the true class.

Aggregate settings can also be extended to the case of having access to unsupervised data or multiple annotators. Instance-dependence has a different meaning in this case though. While previously it had to do with whether the weakening function was uniform across all instances, in this case it has to do with bag creation and intra-bag similarities <cit.>

§ CONCLUSION & FUTURE WORK

We have presented a framework with dimensions that can help in navigating the weak supervision field, but that can also help in understanding exploiting the flexibility of the annotation process.

However, this is nothing but a first step towards categorizing weak supervision. In future work we wish to extend this work in three directions. First, we aim to complement our framework with corresponding algorithms where a practitioner can turn to after identifying the characteristics of their problem. This would also allow for understanding the implications that certain choices on the annotation process have on available algorithms, their theoretical guarantees and practical considerations. Second, we want to expand this work into a transparent process for documenting the annotation of a dataset. This very much aligns with the proposal of accompanying a dataset with a datasheet that documents its motivation, composition, collection process and recommended uses in Datasheets for Datasets <cit.>. Third, we want to strengthen the framework itself. Non-aggregate weak supervision settings have been more widely studied and hence the difference in weight they have received in this paper. As seen, they can be unified through Eq. <ref> and the mixing matrix which can be used to reverse the noise process and make learning unbiased <cit.>. An important aspect in these settings is whether the mixing matrix is known <cit.> or whether it has to be estimated <cit.>. Also, the unified formulation is not only a matter of aesthetics, but more importantly allows for the transferability of methods and theory. In the case of aggregate WSL settings, while we have Eq. <ref> showing how aggregation is abstracted away, it does not improve our theoretical understanding, or allow for algorithms to be applied across settings yet.


Department of Physics, Indiana University - Bloomington, Indiana, USA

# Amplitude analysis of vector-pseudoscalar final states at GlueX

A.M. Schertz for the GlueX Collaboration<EMAIL_ADDRESS>
###### Abstract

Excited vector mesons, some of which are predicted to include gluonic
excitation in their wavefunctions, and certain classes of axial vector mesons,
some of which are implicated in the decay of spin-exotic mesons, decay
dominantly to vector-pseudoscalar final states. In particular, the axial
vector $b_{1}(1235)$ decays dominantly to $\omega\pi$, and is produced
copiously at the GlueX experiment. It provides a testing ground to explore
strategies for amplitude analysis of further vector-pseudoscalar channels
where the intermediate resonances may be harder to isolate. The formalism for
amplitude analysis of $b_{1}^{-}$ production via charge exchange with a
recoiling $\Delta^{++}$ is presented here, and compared with the proton recoil
case. A strategy for a systematic analysis of reactions involving
$\Delta^{++}$ recoil is outlined.

##  1 Introduction

The GlueX experiment [1, 2] is a photoproduction experiment at Jefferson Lab
in Virginia, USA with the goal of exploring the light-quark meson spectrum and
searching for and understanding the nature of spin-exotic mesons, which have
quantum numbers that are not allowed in the quark model. Theoretical
predictions [3] and experimental observations [4, 5, 6] indicate that spin-
exotic mesons exist, but they have yet to be observed in photoproduction.
Through amplitude analyses, the quantum numbers and relative strengths of each
amplitude contributing to a given final state can be extracted, allowing for
identification of spin-exotic states. These proceedings focus on amplitude
analysis of a resonance decaying to a vector and a pseudoscalar meson, in
particular, $\omega\pi^{0}$ with a recoil proton, and $\omega\pi^{-}$ with a
recoil $\Delta^{++}$ baryon. The $\omega\pi$ mass spectra for both processes
are shown in Fig. 1.

Figure 1: Left: Mass spectrum of $\omega\pi^{0}$ in the reaction $\gamma
p\to\omega\pi^{0}p$. Right: Mass spectrum of $\omega\pi^{-}$ in the reaction
$\gamma p\to\omega\pi^{+}\pi^{-}p$. A peak at the $b_{1}$ mass of 1.235 GeV is
clear in both plots.

The most dominant contribution to the $\omega\pi$ channels is the
$b_{1}(1235)$, whose properties are documented in Ref. [7]. Its dominance of
these channels allows for analysis of its decay properties, such as the ratio
of $D$\- to $S$-wave, which was measured to be $D/S=0.269\pm
0.009_{\tx{stat}}\pm 0.01_{\tx{sys}}$ [8]. Decay properties are not sensitive
to production mechanisms, so measurement of them provides a testing ground for
the amplitude analysis model used here. Vector-pseudoscalar channels also give
access to excited vectors, with quantum numbers $J^{PC}=1^{--}$ [9, 10], some
of which are predicted to contain gluonic excitation in their wavefunctions
[3], and to excited axial vectors, with quantum numbers $J^{PC}=1^{+-}$.

Figure 2: Photoproduction of a resonance $X$ in neutral (left) and charged
(right) $t$-channel exchange. Quantities in parentheses are the helicities of
the corresponding particles.

These reactions, $\gamma p\to\omega\pi^{0}p$ and $\gamma
p\to\omega\pi^{-}\Delta^{++}$, are analyzed in the low momentum transfer
region where they are assumed to proceed via exchange of a virtual particle in
the t-channel, illustrated in Fig. 2. The virtual exchange particle can be
electrically neutral, producing a neutral resonance recoiling off a proton, or
it can carry an electric charge, producing a charged resonance recoiling off
an unstable baryon, in this case a $\Delta^{++}$. Charged exchange breaks the
requirement that $C$-parity be conserved, therefore certain charged mesons
such as the $a^{-}_{2}(1320)$ have larger cross sections than their
electrically neutral isospin partners [11]. In the case of $\omega\pi$, the
neutral $\omega\pi^{0}$ channel shows $b_{1}$ production with little else,
while the $\omega\pi^{-}$ channel still has a dominant $b_{1}^{-}$
contribution but also shows a slight shoulder around an $\omega\pi^{-}$ mass
of 1.6 GeV. This enhancement has been previously attributed to the
$\rho_{3}(1690)$ [8], but confirmation of that attribution will require a
rigorous amplitude analysis to properly determine its quantum numbers. The
formalism for such an analysis is presented here.

##  2 Formalism

To determine the various amplitudes in a reaction such as $\gamma
p\to\omega\pi^{-}\Delta^{++}$, an extended unbinned maximum likelihood fit is
performed to the measured intensity distribution as a function of the multi-
dimensional angular variables measured in experiment.

###  2.1 Proton recoil

Appendix D of Ref. [12] derives the intensity distribution for $\eta\pi^{0}$
resonances produced in the reaction $\gamma p\to\eta\pi^{0}p$. By substituting
an appropriate decay amplitude111here $F^{j}_{\lambda}$ is the lineshape of
the resonance decaying to $\omega\pi$, and $G$ is the Dalitz function
describing the $\omega$ decay,

$Z^{j}_{m}(\Phi,\Omega)=e^{-i\Phi}\sum_{\lambda}F^{j}_{\lambda}D^{J_{j}*}_{m,\lambda}(\Omega_{\omega})Y^{1}_{\lambda}(\Omega_{H})G,$
(1)

for the phase-rotated spherical harmonics used to describe the $\eta\pi^{0}$
system, that intensity distribution can be rewritten in the form

$\displaystyle I\propto\sum_{k}\bigg{\\{}(1-P_{\gamma})$
$\displaystyle\left[\left|\sum_{j,m}[J^{P}]^{(-)}_{m,k}\tx{Im}Z^{j}_{m}(\Phi,\Omega)\right|^{2}+\left|\sum_{j,m}[J^{P}]^{(+)}_{m,k}\tx{Re}Z^{j}_{m}(\Phi,\Omega)\right|^{2}\right]$
$\displaystyle+(1+P_{\gamma})$
$\displaystyle\left[\left|\sum_{j,m}[J^{P}]^{(+)}_{m,k}\tx{Im}Z^{j}_{m}(\Phi,\Omega)\right|^{2}+\left|\sum_{j,m}[J^{P}]^{(-)}_{m,k}\tx{Re}Z^{j}_{m}(\Phi,\Omega)\right|^{2}\right]\bigg{\\}},$
(2)

which is suitable for vector-pseudoscalar analysis. The quantum numbers
$J^{P}$ of the resonance are indexed by $j$. The intensity is expressed in a
basis of definite reflectivity, $\varepsilon=\pm 1$. In the high-energy limit
for $t$-channel processes, $\varepsilon$ is equal to the naturality,
$\tau=P(-1)^{J}$, of the virtual exchange particle. The production amplitudes
$[J^{P}]^{(\varepsilon)}_{m,k}$ are estimated by fitting $I(\Phi,\Omega)$ to
the data. Notably, the recoil proton only enters into this formulation via the
helicity flip between the target and recoil proton, $k$. This will be
discussed further in Sec. 3.2.

###  2.2 Unstable $\Delta^{++}$ recoil

The intensity function for a vector-pseudoscalar system recoiling off a
$\Delta^{++}$ is

$\displaystyle I\propto\sum_{\lambda_{2}}\Bigg{\\{}(1-P_{\gamma})$
$\displaystyle\left|\sum_{j,m,\lambda_{\Delta}}D^{3/2*}_{\lambda_{\Delta},\lambda_{2}}(\Omega_{p})\left([J^{P}]^{(+)}_{m,\lambda_{\Delta}}\tx{Re}Z^{j}_{m}(\Phi,\Omega)+i[J^{P}]^{(-)}_{m,\lambda_{\Delta}}\tx{Im}Z^{j}_{m}(\Phi,\Omega)\right)\right|^{2}$
$\displaystyle+(1-P_{\gamma})$
$\displaystyle\left|\sum_{j,m,\lambda_{\Delta}}D^{3/2}_{\lambda_{\Delta},\lambda_{2}}(\Omega_{p})\left([J^{P}]^{(+)}_{m,\lambda_{\Delta}}\tx{Re}Z^{j}_{m}(\Phi,\Omega)-i[J^{P}]^{(-)}_{m,\lambda_{\Delta}}\tx{Im}Z^{j}_{m}(\Phi,\Omega)\right)\right|^{2}$
$\displaystyle+(1+P_{\gamma})$
$\displaystyle\left|\sum_{j,m,\lambda_{\Delta}}D^{3/2*}_{\lambda_{\Delta},\lambda_{2}}(\Omega_{p})\left([J^{P}]^{(+)}_{m,\lambda_{\Delta}}\tx{Im}Z^{j}_{m}(\Phi,\Omega)-i[J^{P}]^{(-)}_{m,\lambda_{\Delta}}\tx{Re}Z^{j}_{m}(\Phi,\Omega)\right)\right|^{2}$
$\displaystyle+(1+P_{\gamma})$
$\displaystyle\left|\sum_{j,m,\lambda_{\Delta}}D^{3/2}_{\lambda_{\Delta},\lambda_{2}}(\Omega_{p})\left([J^{P}]^{(+)}_{m,\lambda_{\Delta}}\tx{Im}Z^{j}_{m}(\Phi,\Omega)+i[J^{P}]^{(-)}_{m,\lambda_{\Delta}}\tx{Re}Z^{j}_{m}(\Phi,\Omega)\right)\right|^{2}\Bigg{\\}}$
(3)

where the additional Wigner D-function describes the $\Delta^{++}\to p\pi^{+}$
decay, and amplitudes with opposite reflectivities appear in the same coherent
sum, unlike in the case with a stable recoil particle.

##  3 Implications

###  3.1 Reflectivity interference

When the recoil particle is stable, amplitudes with opposite reflectivities
appear in separate coherent sums in the intensity function and cannot
interfere with each other. In other words there is no interference between
amplitudes produced by natural and unnatural exchange, and the notion of
independent cross sections for natural and unnatural exchange exists, as are
measured for the $a_{2}(1320)$ in the reaction $\gamma p\to\eta\pi^{0}p$ in
Ref. [11]. When the recoil particle is unstable, inclusion of its decay in the
intensity function leads to positive and negative reflectivity amplitudes
appearing in the same coherent sum. Since opposite reflectivity amplitudes add
coherently, it is possible that interference between them may contribute to
the intensity, and it does not make sense to extract separate cross sections
for natural and unnatural parity exchange in the case of a polarized beam.

###  3.2 Separation of baryon spin flip

In $t$-channel exchanges such as these, a quantity that is considered is the
helicity flip, $k$, between the target proton and the recoil baryon. In the
case of a recoil proton, $k$ can take on two possible values, related to
proton helicity flip and non-flip, but in measurements taken at GlueX, it is
ambiguous which value of $k$ corresponds to flip or non-flip, since it does
not appear in the decay amplitude. This quantity is summed over incoherently
in the intensity function. If the recoil baryon is an unstable $\Delta^{++}$,
there are four possible values of $k$, one for each value of $\Delta^{++}$
helicity, $\lambda_{\Delta}$, which is summed coherently in the intensity
function. This coherent sum and the fact that the decay amplitudes used in the
intensity function have an explicit dependence on $\lambda_{\Delta}$, mean
that if the recoil particle is unstable, amplitudes with different values of
$k$ are distinguishable and it is in principle possible to determine which, if
any, is dominant.

###  3.3 Analysis strategy

In parallel with the analysis of $\gamma p\to\omega\pi^{-}\Delta^{++}$, it is
informative to analyze other channels with a recoiling $\Delta^{++}$. Equation
3 can still be used with an appropriate selection of the resonance decay
amplitude $Z(\Phi,\Omega)$. The simplest of these channels is $\gamma
p\to\pi^{-}\Delta^{++}$, where preliminary amplitude fits show good agreement
with the spin-density matrix elements (SDME) of $\pi^{-}\Delta^{++}$.
Extraction of the SDMEs of the target proton to recoil $\Delta^{++}$
transition provide an independent determination of which amounts of natural or
unnatural exchanges are present, as done for the $\Lambda(1520)$ in Ref. [13].
This is an important cross-check of this intensity formulation and may provide
information as to which baryon spin flip and exchange naturality combinations,
for example, are dominant or negligible.

Further instructive cases include $\gamma p\to\eta^{\prime}\pi^{-}\Delta^{++}$
and $\gamma p\to\pi^{0}\pi^{-}\Delta^{++}$. Resonances decaying to
$\eta^{\prime}\pi^{-}$ may include the spin-exotic $\pi_{1}^{-}(1600)$ [5,
14], which would decay in a $P$-wave. Any odd-$L$ contributions to the
$\eta^{\prime}\pi$ channel must have spin-exotic quantum numbers [4], so
observation of a resonant $P$-wave would be clear indication of a spin-exotic
state. The dominant contribution to the $\pi^{0}\pi^{-}$ channel is the
$\rho^{-}$, which decays in a $P$-wave. Thus, analysis of $\gamma
p\to\pi^{0}\pi^{-}\Delta^{++}$ will provide a picture of what a resonance
decaying to two pseudoscalars in a $P$-wave looks like, and will inform
analysis of $\gamma p\to\eta^{\prime}\pi^{-}\Delta^{++}$.

##  4 Analysis Status and Outlook

The channel $\gamma p\to\omega\pi^{0}p$ is dominated by the axial vector
resonance $b_{1}$, which allows for study of the properties of that meson. The
intensity function described in Eqn. 3 is being tested in multiple channels
involving $\Delta^{++}$ recoil at GlueX, in particular, $\gamma
p\to\omega\pi^{-}\Delta^{++}$. Monte Carlo samples are generated using the
full unstable recoil model and fit using the full model as well as the stable
proton recoil model described in Eqn. 2 and an SDME formulation similar to the
one used for $\pi^{-}\Delta^{++}$. Since Eqn. 3 is an extension of Eqn. 2, it
should be returned if the $\Delta^{++}$ decay angles in Eqn. 3 are integrated
over. Likewise, if the resonance angles in Eqn. 3 are integrated over, the
SDME formulation for the baryon vertex is returned, producing a relationship
between the amplitude and SDME formulations. Thus, it is expected that fits
using the SDME formulation or Eqn. 2 will be able to extract at least some
correct information from the data, and fits to Monte Carlo samples will
indicate if they can. Quantities under investigation include the reflectivity
of the reaction, the helicity of the intermediate resonance, and the helicity
flip between the target proton and recoil baryon. All of these quantities will
be critical in the eventual amplitude analysis.

###### Acknowledgements.

Thanks to F. Afzal, V. Mathieu, M.R. Shepherd, and J.R. Stevens for the many
productive discussions involved in this work. This work was funded in part by
the US Department of Energy Office of Nuclear Physics under grant DE-
FG02-05ER41374.

## References

  * [1] Austregesilo A. these proceedings.
  * [2] Adhikari S. et al. (GlueX) Nucl. Instrum. & Meth. A9872021164807.
  * [3] Dudek J. J. et al. (HadSpec) Phys. Rev. D8820139, 094505.
  * [4] Alexeev M. G. et al. (COMPASS) Phys. Rev. D10520221, 012005.
  * [5] Rodas A. et al. (JPAC) Phys. Rev. Lett.12220194, 042002.
  * [6] Ablikim M. et al. (BESIII) Phys. Rev. Lett.129202219, 192002.
  * [7] Workman R. L. et al. (PDG) Prog. Theor. Exp. Phys.20222022083C01.
  * [8] Nozar M. et al. (E852) Phys. Lett. B541200235-44.
  * [9] Ablikim M. et al. (BESIII) Phys. Lett. B8132021136059.
  * [10] Achasov M. N. et al. (SND) Phys. Rev. D942016112001.
  * [11] Albrecht M. these proceedings.
  * [12] Mathieu V. et al. (JPAC) Phys. Rev. D10020195, 054017.
  * [13] Adhikari S. et al. (GlueX) Phys. Rev. C1052022035201.
  * [14] Woss A. J. et al. (HadSpec) Phys. Rev. D10320215, 054502.