# Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade
Systems

Ji Wang and Mamadou Diagne  J. Wang is with the Department of Automation,
Xiamen University, Xiamen, Fujian 361005, China (e-mail:jiwang9024@gmail.com).
M. Diagne is with the Department of Mechanical and Aerospace Engineering,
University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093
(mdiagne@ucsd.edu).

###### Abstract

This paper presents a delay-adaptive boundary control scheme for a $2\times 2$
coupled linear hyperbolic PDE-ODE cascade system with an unknown and
arbitrarily long input delay. To construct a nominal delay-compensated control
law, assuming a known input delay, a three-step backstepping design is used.
Based on the certainty equivalence principle, the nominal control action is
fed with the estimate of the unknown delay, which is generated from a batch
least-squares identifier that is updated by an event-triggering mechanism that
evaluates the growth of the norm of the system states. As a result of the
closed-loop system, the actuator and plant states can be regulated
exponentially while avoiding Zeno occurrences. A finite-time exact
identification of the unknown delay is also achieved except for the case that
all initial states of the plant are zero. As far as we know, this is the first
delay-adaptive control result for systems governed by heterodirectional
hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in
the control application of a deep-sea construction vessel with cable-payload
oscillations and subject to input delay.

###### Index Terms:

Hyperbolic PDEs, delay-adaptive control, event-triggered control, least-
squares identifier.

##  I Introduction

###  I-A Boundary control of coupled hyperbolic PDEs

Systems of transport partial differential equations (PDEs) appear in many
physical models, including road traffic [23, 60, 61], water management systems
[17, 18, 41, 42], flow of fluids in oil drilling systems [24, 25], and cable
vibration dynamics [49, 56]. As a result of the backstepping design [12, 48],
the sliding mode control approach [36] and the proportional-integral (PI)
controller design [47], the theoretical results on boundary control of coupled
first-order linear hyperbolic PDEs have emerged in the last decade. The
backstepping design was further extended to that of a $n+1$ system in [40],
and then to a more general coupled transport PDE system where the number of
PDEs in either direction is arbitrary [26]. Along the same lines, studies on
the design of an adaptive estimation framework have been proposed in [2, 3]
and extended to adaptive control in [4]. However, the problem of delay-
adaptive control for hyperbolic PDEs has gone unanswered in all these last
developments as traditional designs based on swapping identifiers, passive
identifiers, and Lyapunov functions remain difficult to exploit for such
systems.

###  I-B Delay-compensated control of finite- and infinite-dimensional plants

Time delays, which are well known to be detrimental to stability [22], often
exist in practical control systems. In order to compensate for arbitrarily
long delays, “avant-garde” backstepping-based delay compensation techniques
were first developed in [31, 33]. Bottom-line, the input delay is converted
into a transport PDE as an infinite-dimensional representation of the actuator
state. For ODE plants, a PDE/ODE cascade system ensues from this substitute
representation of the actuator state. The method has also been used to
compensate for the effect of sensor delays that oftentimes occur in ODE
plants. In comparison to many results [1, 11, 21], which only estimate plant
states, the approach proposed in [31, 33] enables estimation of both the plant
and the sensor states when designing a feedback loop. A number of results
considering delays that are described by complex transport actuation paths for
nonlinear ODE plants were developed in [15, 16] and the references therein.

While compensation for arbitrarily long delays is commonly available for
finite-dimensional systems, only very few examples for infinite-dimensional
systems were presented, where one pioneering result is [32] that is conceived
using backstepping. In recent years, researchers from the PDE control
community have shifted their attention to this topic, leading to many
interesting developments that can be found in [30, 37, 38, 44, 46]. By
treating the delay as a transport PDE, [34] presented the design of a boundary
controller for a pure wave PDE with compensation of an arbitrarily long input
delay while ensuring exponential stability for the closed-loop system. For
coupled heterodirectional hyperbolic PDEs, in [50], a delay-compensated
control scheme was designed for a sandwich hyperbolic PDE in the presence of a
sensor delay of arbitrary length. In the same spirit, [45] proposed a
distributed input delay compensation for traffic systems governed by coupled
hyperbolic PDEs (see [43] as well). In addition to the continuous-in-time
control law, on the basis of the event-triggered boundary control design of
PDEs [20, 19, 51], an event-triggered delay-compensated boundary control law
for coupled hyperbolic PDEs was presented in [53]. Although the above
substantial results emblematize a major step in the field, the prior knowledge
of the delay length is a mandatory weakening factor that mitigates their
viability for many practical applications.

Lyapunov designs have been employed to develop delay-adaptive controllers for
linear and nonlinear ODE plants [7, 9, 10, 62, 63, 35] via backstepping-based
certainty-equivalence compensators. The primary idea behind these
contributions is to estimate the unknown delay using input-output signals, and
then adjust a pre-designed nominal controller based on estimated parameters in
order to achieve convergence. In general, compared to other traditional
parameter identifier methods like swapping or passive identifiers, the
Lyapunov technique provides better transient performance properties. Recently,
the approach has been extended to linear reaction-diffusion PDEs with a
boundary or a distributed delayed input [57, 58], where asymptotic convergence
results are achieved. In the realm of advancing the design approach of [57,
58], a recent result has achieved the design of Lyapunov-based delay-adaptive
boundary control for a scalar Integral PDE [59]. As far as we are aware, the
three preceding contributions are the sole results on delay-adaptive control
for PDE plants. The method in the present contribution is different with both
the above delay-adaptive control results and traditional adaptive control
designs for hyperbolic PDEs [4]. More precisely, our design relies on a
triggered batch least-square identifier (BaLSI), a novel approach that was
initially introduced in [27, 28], which has at least two significant
advantages over the traditional adaptive control approaches: guaranteeing
exponential regulation of the states to zero, as well as finite-time
convergence of the estimates to the true values. This method has been applied
in adaptive control of a parabolic PDE [29], and first-order hyperbolic PDEs
in [52, 54, 55] with unknown plant parameters.

###  I-C Contributions

Comparison with some results on boundary control of PDEs with arbitrarily long
input delays is shown in Table I.

  * •

To the best of our knowledge, our result is the first delay-adaptive
compensator for coupled hyperbolic PDEs involving an unknown and arbitrarily
large input delay.

  * •

Different with the delay-robust stabilizing feedback control design for
coupled first-order hyperbolic PDEs that achieve robustness to small delays in
actuation [6], the present contribution ensures exact compensation of the
arbitrarily large unknown input delay.

  * •

The finite-time exact identification of the unknown delay is achieved. As a
result, the exponential regulation, instead of the asymptotic one in [57, 58],
is guaranteed in the closed-loop system. Basically, after the finite-time
identification of the unknown parameter, the delay-adaptive control signal is
identical to the nominal control action (with known input delay), which
ultimately improves substantially the resulting transient performance of the
whole closed-loop system’s dynamics.

TABLE I: Delay-compensated control of PDEs with arbitrarily long input delays | Known delay | Unknown delay
---|---|---
Parabolic PDEs |  [30][32][37][44] |  [57][58]
Hyperbolic PDEs |  [34][43][45][46] | This paper

###  I-D Organization

The problem formulation is shown in Section II. The nominal control design is
presented in Section III. The design of delay-adaptive control with piecewise-
constant parameter identification is proposed in Section IV. The main result
including the absence of a Zeno phenomenon, parameter convergence, and
exponential regulation of the states is proved in Section V. The effectiveness
of the proposed design is illustrated with a numerical simulation of a deep-
sea construction vessel (DCV) in Section VI. The conclusion and future work
are presented in Section VII.

###  I-E Notation

We adopt the following notation.

  * •

The symbol $\mathbb{Z}^{+}$ denotes the set of natural numbers including zero,
and the notation $\mathbb{N}$ for the set $\\{1,2,\cdots\\}$, i.e., the
natural numbers without $0$. We also use $\mathbb{R}_{+}:=[0,+\infty)$.

  * •

Let $U\subseteq\mathbb{R}^{m}$ be a set with non-empty interior and let
$\Omega\subseteq\mathbb{R}$ be a set. By $C^{0}(U;\Omega)$, we denote the
class of continuous mappings on $U$, which takes values in $\Omega$. By
$C^{k}(U;\Omega)$, where $k\geq 1$, we denote the class of continuous
functions on $U$, which have continuous derivatives of order $k$ on $U$ and
take values in $\Omega$.

  * •

We use the notation $L^{2}(0,1)$ for the standard space of the equivalence
class of square-integrable, measurable functions defined on $(0,1)$ and
$\|f\|=\left(\int_{0}^{1}f(x)^{2}dx\right)^{\frac{1}{2}}<+\infty$ for $f\in
L^{2}(0,1)$.

  * •

For an $I\subseteq\mathbb{R}_{+}$, the space $C^{0}(I;L^{2}(0,1))$ is the
space of continuous mappings $I\ni t\mapsto u[t]\in L^{2}(0,1)$.

  * •

Let $u:\mathbb{R}_{+}\times[0,1]\rightarrow\mathbb{R}$ be given. We use the
notation $u[t]$ to denote the profile of $u$ at certain $t\geq 0$, i.e.,
$(u[t])(x)=u(x,t)$, for all $x\in[0,1]$.

##  II Problem Formulation

Consider the potentially open-loop unstable plant governed by the following
$2\times 2$ linear hyperbolic PDE coupled with a linear ODE,

$\displaystyle\dot{X}(t)$ $\displaystyle=AX(t)+Bw(0,t),$ (1)
$\displaystyle{z_{t}}(x,t)$
$\displaystyle=-q_{1}{z_{x}}(x,t)+{d_{1}}z(x,t)+{d_{2}}w(x,t),$ (2)
$\displaystyle{w_{t}}(x,t)$
$\displaystyle=q_{2}{w_{x}}(x,t)+{d_{3}}z(x,t)+{d_{4}}w(x,t),$ (3)

with the boundary conditions:

$\displaystyle z(0,t)$ $\displaystyle=CX(t)-pw(0,t),$ (4) $\displaystyle
w(1,t)$ $\displaystyle=c_{0}U(t-D)+qz(1,t),$ (5)

where, $q,q_{1},q_{2},d_{1},d_{2},d_{3},d_{4},c_{0}$ and $p$ are arbitrary
parameters with $q_{1},q_{2}>0$ being transport speeds, and $p\neq 0$,
$c_{0}\neq 0$. Here, the matrix $A,B,C$ are known, $z(x,t)$ and $w(x,t)$ are
the PDE state variables, $X(t)\in\mathbb{R}^{m}$ is the linear ODE state, $U$
is the control variable and $D>0$ is the indiscriminately large and unknown
input delay. We assume that the initial conditions satisfy

$\displaystyle z^{0}(x),w^{0}(x)\in L^{2}(0,1),\ X^{0}\in\mathbb{R}^{m}$ (6)

and consider the following assumptions.

###### Assumption 1

The pair $A,B$ is controllable.

###### Assumption 2

Parameters $p,q$ satisfy

$\displaystyle{|pq|}{e^{{\max\left\\{\frac{2d_{4}}{q_{2}},\frac{2d_{1}}{q_{1}}\right\\}}}}\leq\frac{1}{\sqrt{2}}.$
(7)

###### Assumption 3

The bounds of the unknown input delay $D$ are known and arbitrary, i.e.,

$\displaystyle 0<\underline{D}\leq D\leq\overline{D}$ (8)

where positive constants $\underline{D}$, $\overline{D}$ are arbitrary.

Our goal is to design a delay-adaptive boundary control action, $U(t)$, that
exponentially regulates the system (1)–(5) despite the presence of an unknown
delay $D$ whose length is arbitrary. The plant (1)–(5) can be used to model
cable-payload oscillations in DCV, which are to be suppressed for the purpose
of accurate placement of the equipment to be installed on the sea floor. From
this application perspective, large-distance signal transmission in the water
through a set of acoustics devices and the actuation of the hydraulic actuator
for the ship-mounted crane are subject to delays, which are considered as an
unknown delay in the control input, the cable vibration dynamics are governed
by the $2\times 2$ hyperbolic PDE, and the vibration dynamics of the cage are
captured by the ODE system.

##  III Nominal delay-compensated control design

In order to design the nominal control law, we first construct an infinite-
dimensional representation of the actuator state by converting the delayed
input into transport PDE actuation dynamics. Define a new variable $v(x,t)$ as

$v(x,t)=\begin{cases}U\left(t-{D}x\right)&\mbox{if $t-{D}x\geq 0$}\\\
0&\mbox{if $t-{D}x<0$}\end{cases}$

then (5) is rewritten as

$\displaystyle w(1,t)$ $\displaystyle=c_{0}v(1,t)+qz(1,t),$ (9) $\displaystyle
v_{t}(x,t)$ $\displaystyle=-\frac{1}{D}v_{x}(x,t),$ (10) $\displaystyle
v(0,t)$ $\displaystyle=U(t),$ (11) $\displaystyle v(x,0)$ $\displaystyle=0$
(12)

for $x\in[0,1],t\in[0,\infty)$. Now, resulting from the new representation of
the actuator state, the function $U(t)$, which is defined as the boundary
condition (11) of the transport equation (10), is the delay-free control input
to be designed for the hyperbolic PDE-PDE-ODE cascade system consisting of
(1)–(4) combined with (9)–(12).

###  III-A First Step: Backstepping Transformation for the $2\times 2$ Coupled
Hyperbolic PDE-ODE

We introduce the following backstepping transformation [39] in order to remove
the in-domain coupling destabilizing terms from the $2\times 2$ hyperbolic PDE
system consisting of (2), (3) and make the ODE system matrix Hurwitz:

$\displaystyle\alpha(x,t)=$ $\displaystyle
z(x,t)-\int_{0}^{x}{\phi}(x,y)z(y,t)dy$
$\displaystyle-\int_{0}^{x}{\varphi}(x,y)w(y,t)dy-\gamma(x){X}(t),$ (13)
$\displaystyle\beta(x,t)=$ $\displaystyle
w(x,t)-\int_{0}^{x}{\Psi}(x,y)z(y,t)dy$
$\displaystyle-\int_{0}^{x}{\Phi}(x,y)w(y,t)dy-\lambda(x){X}(t)$ (14)

whose inverse is

$\displaystyle z(x,t)=$
$\displaystyle\alpha(x,t)-\int_{0}^{x}{\bar{\phi}}(x,y)\alpha(y,t)dy$
$\displaystyle-\int_{0}^{x}{\bar{\varphi}}(x,y)\beta(y,t)dy-\bar{\gamma}(x){X}(t),$
(15) $\displaystyle w(x,t)=$
$\displaystyle\beta(x,t)-\int_{0}^{x}{\bar{\Psi}}(x,y)\alpha(y,t)dy$
$\displaystyle-\int_{0}^{x}{\bar{\Phi}}(x,y)\beta(y,t)dy-\bar{\lambda}(x){X}(t)$
(16)

to convert (1)–(4), (9) into

$\displaystyle\dot{X}(t)=$ $\displaystyle A_{\rm m}X(t)+{B}\beta(0,t),$ (17)
$\displaystyle\alpha(0,t)=$ $\displaystyle-p\beta(0,t),$ (18)
$\displaystyle{\alpha_{t}}(x,t)=$
$\displaystyle-{q_{1}}{\alpha_{x}}(x,t)+{d_{1}}\alpha(x,t),$ (19)
$\displaystyle{\beta_{t}}(x,t)=$
$\displaystyle{q_{2}}{\beta_{x}}(x,t)+{d_{4}}\beta(x,t),$ (20)
$\displaystyle\beta(1,t)=$ $\displaystyle
c_{0}v(1,t)+q\alpha(1,t)+\left(\bar{\lambda}(1)-q\bar{\gamma}(1)\right){X}(t)$
$\displaystyle+\int_{0}^{1}\left({\bar{\Psi}}(1,y)-q{\bar{\phi}}(1,y)\right)\alpha(y,t)dy$
$\displaystyle+\int_{0}^{1}\left({\bar{\Phi}}(1,y)-q{\bar{\varphi}}(1,y)\right)\beta(y,t)dy.$
(21)

The gain vector $K$ is selected so that

$\displaystyle A_{\rm m}=A+BK^{T}$ (22)

is Hurwitz.

The conditions on the kernels ${\phi}(x,y)$, ${\varphi}(x,y)$, $\gamma(x)$,
${\Psi}(x,y)$, ${\Phi}(x,y)$, $\lambda(x)$ and ${\bar{\phi}}(x,y)$,
${\bar{\varphi}}(x,y)$, $\bar{\gamma}(x)$, ${\bar{\Psi}}(x,y)$,
${\bar{\Phi}}(x,y)$, $\bar{\lambda}(x)$ in the backstepping transformations
(13)–(16), which are obtained by matching the original system (1)–(5) and the
intermediate system (17)–(21), are shown in the part 1 of Appendix A, and the
well-posedness of the kernel conditions has been proved in Theorem 4.1 of
[39].

###  III-B Second Step: Transformation of the Actuator States

With the purpose of removing the integral terms and ODE state $X(t)$ from the
PDE boundary condition (21), we define the following change of coordinate

$\displaystyle u(x,t)=$ $\displaystyle
v(x,t)+\int_{0}^{1}K_{1}(x,y)\alpha(y,t)dy+\int_{0}^{1}K_{2}(x,y)\beta(y,t)dy$
$\displaystyle+\eta(x){X}(t)$ (23)

which enables one to map the actuator dynamics given by (10), (11), and (21)
into the following equations

$\displaystyle\beta(1,t)=$ $\displaystyle c_{0}u(1,t)+q\alpha(1,t),$ (24)
$\displaystyle{u_{t}}(x,t)=$
$\displaystyle-d{u_{x}}(x,t)+q_{2}K_{2}(x,1)c_{0}u(1,t),$ (25) $\displaystyle
u(0,t)=$ $\displaystyle U(t)+\int_{0}^{1}K_{1}(0,y)\alpha(y,t)dy$
$\displaystyle+\int_{0}^{1}K_{2}(0,y)\beta(y,t)dy+\eta(0){X}(t),$ (26)

where

$d=\frac{1}{D}.$

The detailed computation and conditions of the kernels
$K_{1}(x,y),K_{2}(x,y),\eta(x)$ are given in the part 2 of Appendix A.

###  III-C Third Step: Backstepping Transformation for the Resulting $u$-PDE

To remove the boundary nonlocal term $q_{2}K_{2}(x,1)c_{0}u(1,t)$ in the
transport PDE (25), we apply the following mapping

$\displaystyle u(x,t)=\hat{u}(x,t)+\int_{x}^{1}R(x,y)\hat{u}(y,t)dy$ (27)

which converts (24)–(26) into

$\displaystyle\beta(1,t)=$ $\displaystyle c_{0}\hat{u}(1,t)+q\alpha(1,t),$
(28) $\displaystyle{\hat{u}_{t}}(x,t)=$ $\displaystyle-{d}{\hat{u}_{x}}(x,t),$
(29) $\displaystyle\hat{u}(0,t)=$ $\displaystyle 0,$ (30)

with the nominal control input defined as

$\displaystyle U(t)=$
$\displaystyle-\int_{0}^{1}K_{1}(0,y)\alpha(y,t)dy-\int_{0}^{1}K_{2}(0,y)\beta(y,t)dy$
$\displaystyle+\int_{0}^{1}R(0,y)\hat{u}(y,t)dy-\eta(0){X}(t).$ (31)

The conditions of the kernel $R(x,y)$ are shown in the part 3 of Appendix A.
The inverse transformation of (27) can be found as

$\displaystyle\hat{u}(x,t)=u(x,t)+\int_{x}^{1}P(x,y)u(y,t)dy,$ (32)

where the conditions of $P(x,y)$ are given in the part 3 of Appendix A as
well.

Figure 1: The flow diagram of the nominal delay-compensated control design.

###  III-D Stability result of nominal delay-compensated control

The flow diagram of the nominal delay-compensated control is shown in Figure
1. In a nutshell, the prior transformations convert the original system that
consists of (1)–(4), (9)–(12) into the target system that consists of
(17)–(20), (28)–(30). The nominal control input (31) is rewritten with respect
to the original state variables as follows

$\displaystyle U(t)=$
$\displaystyle-\int_{0}^{1}M_{1}(y;D)z(y,t)dy-\int_{0}^{1}M_{2}(y;D)w(y,t)dy$
$\displaystyle+\int_{0}^{1}M_{3}(y;D)v(y,t)dy+M_{4}(D){X}(t),$ (33)

where the controller gains $M_{i}$, $\\{1,...,4\\}$ are given in Appendix B.
Writing $D$ after $``;"$ in $M_{i}(y;D)$ emphasizes the fact that these
functions are parameterized by the delay $D$.

The stability result of the nominal delay-compensated control is stated as
follows.

###### Theorem 1

For the known delay $D$, with arbitrary initial data $(z[0],w[0])^{T}\in
L^{2}(0,1)$, $X(0)\in\mathbb{R}^{m}$, considering the closed-loop system
consisting of the plant (1)–(4), (9)–(12) and the nominal controller (33), the
exponential stability of the closed-loop system is obtained in the sense that
there exist positive constants $\Upsilon,\lambda_{1}$ such that

$\displaystyle\Omega(t)\leq\Upsilon\Omega(0)e^{-\lambda_{1}t},~{}~{}t\geq 0,$
(34)

where $\Omega(t)$ is defined as

$\displaystyle\Omega(t)=\|z[t]\|^{2}+\|w[t]\|^{2}+\|v[t]\|^{2}+|{X}(t)|^{2}.$
(35)

###### Proof:

Define Lyapunov function $V(t)$ as

$\displaystyle V(t)=$
$\displaystyle\frac{r_{d}}{2}{X}^{T}(t)P_{1}X(t)+\frac{r_{a}}{2}{}\int_{0}^{1}{{e^{{\delta}x}}}\beta{(x,t)^{2}}dx$
$\displaystyle+\frac{1}{2}{}\int_{0}^{1}{{e^{-{\delta}x}}}\alpha{(x,t)^{2}}dx+\frac{r_{c}}{2}{}\int_{0}^{1}{{e^{-{}x}}}\hat{u}{(x,t)^{2}}dx,$
(36)

where a positive definite matrix $P_{1}={P_{1}}^{T}$ is the solution to the
Lyapunov equation $A_{\rm m}^{T}P_{1}+P_{1}A_{\rm m}=-Q_{1}$ for some
$Q_{1}={Q_{1}}^{T}>0$, and where $\delta,r_{a},r_{c},r_{d}$ satisfy

$\displaystyle{e^{-2{\max\left\\{\frac{2d_{4}}{q_{2}},\frac{2d_{1}}{q_{1}}\right\\}}}}>{e^{-2{\delta}}}>2{p^{2}q^{2}},$
(37)
$\displaystyle\frac{q_{1}}{2q_{2}q^{2}}{e^{-2{\delta}}}\geq{r_{a}}>\frac{p^{2}q_{1}}{q_{2}},$
(38) $\displaystyle{{r_{c}}}\geq 2\bar{D}q_{2}{r_{a}}{e^{{\delta}}}c_{0}^{2},$
(39) $\displaystyle 0<r_{d}\leq\frac{{\lambda_{\rm
min}(Q_{1})}}{2|P_{1}B|^{2}}\left(q_{2}{r_{a}}-{p^{2}q_{1}}\right).$ (40)

Please note that
${e^{-2{\max\left\\{\frac{2d_{4}}{q_{2}},\frac{2d_{1}}{q_{1}}\right\\}}}}>2{p^{2}q^{2}}$
holds in (37) under Assumption 2, and
$\frac{q_{1}}{2q_{2}q^{2}}{e^{-2{\delta}}}>\frac{p^{2}q_{1}}{q_{2}}$ holds in
(38) due to the right inequality in (37), which means the existence of
$\delta,r_{a}$ satisfying (37), (38). It is then straightforward to obtain
$r_{c},r_{d}$ by (39), (40), where the positiveness of the right-hand side of
(40) is ensured by the right inequality in (38). Therefore, there exists a
solution $\delta,r_{a},r_{c},r_{d}$ satisfying (37)–(40).

Following the calculation in Appendix C and using the norm estimate (C.10), we
have

$\displaystyle\xi_{1}\xi_{3}\Omega(t)\leq V(t)\leq\xi_{2}\xi_{4}\Omega(t)$
(41)

where $\xi_{1},\xi_{2}$ are given in (C.11), (C.12), and

$\displaystyle\xi_{3}$
$\displaystyle=\frac{1}{2}\min\left\\{{r_{d}}\lambda_{\rm
min}(P_{1}),{r_{a}},{{e^{-{\delta}}}},{r_{c}}{{e^{-{1}}}}\right\\},$ (42)
$\displaystyle\xi_{4}$
$\displaystyle=\frac{1}{2}\max\left\\{{r_{d}}\lambda_{\rm
max}(P_{1}),{r_{a}}{{e^{{\delta}}}},1,{r_{c}}\right\\},$ (43)

where ${\lambda_{\rm min}(P_{1})}$ is the smallest eigenvalue of $P_{1}$.

Taking the derivative of (36) along (17)–(20), (28)–(30), and applying Young’s
inequality and the Cauchy-Schwarz inequality, the following estimate holds for
all $t\geq 0$:

$\displaystyle\dot{V}(t)\leq$ $\displaystyle-\frac{r_{d}}{4}{\lambda_{\rm
min}(Q_{1})}|{X}(t)|^{2}$
$\displaystyle-\bigg{(}\frac{{1}}{2}q_{1}{e^{-{\delta}}}-q_{2}{r_{a}}{e^{{\delta}}}q^{2}\bigg{)}\alpha(1,t)^{2}$
$\displaystyle-\left(\frac{1}{2}q_{2}{r_{a}}-\frac{r_{d}|P_{1}B|^{2}}{{\lambda_{\rm
min}(Q_{1})}}-\frac{p^{2}q_{1}}{2}\right)\beta{(0,t)^{2}}$ $\displaystyle-
r_{a}\left(\frac{1}{2}{\delta}q_{2}-d_{4}\right)\int_{0}^{1}{{e^{{\delta}x}}}\beta{(x,t)^{2}}dx$
$\displaystyle-\left(\frac{1}{2}{\delta}q_{1}-d_{1}\right)\int_{0}^{1}{{e^{-{\delta}x}}}\alpha{(x,t)^{2}}dx$
$\displaystyle-\left(\frac{{r_{c}}}{2D}-q_{2}{r_{a}}{e^{{\delta}}}c_{0}^{2}\right)\hat{u}{(1,t)}^{2}$
$\displaystyle-\frac{{r_{c}}}{2D}\int_{0}^{1}{{e^{{}x}}}\hat{u}{(x,t)^{2}}dx.$
(44)

Recalling conditions (37)–(39) on ${\delta}$, $r_{a}$, $r_{c}$, and $r_{d}$,
there exists a sufficiently small positive constant $\lambda_{1}$, such that

$\displaystyle\dot{V}(t)\leq-\lambda_{1}V(t),$ (45)

where

$\displaystyle{\lambda_{1}=\min\left\\{\frac{{\lambda_{\rm
min}(Q_{1})}}{2\lambda_{\rm
max}(P_{1})},{\delta}q_{2}-2d_{4},{\delta}q_{1}-2d_{1},\frac{1}{D}\right\\}>0}.$
(46)

Recalling (41), we then obtain (34) where the positive constant $\Upsilon$ is
given as

$\displaystyle\Upsilon=\frac{\xi_{2}\xi_{4}}{\xi_{1}\xi_{3}}.$ (47)

The proof of the theorem is complete. ∎

Next, we will design a certainty equivalence delay-adaptive controller
considering the nominal control action (33) fed with an estimate $\hat{D}$,
which is given by an update law resulting from a triggered batch least-square
identifier of the unknown delay $D$.

##  IV  Delay-Adaptive controller

By replacing the unknown delay $D$ in the nominal continuous-in-time feedback
(33) with the estimate $\hat{D}(t_{i})$ that is generated with a triggered
batch least-squares identifier that is designed later, we derive the following
delay-compensated adaptive control law $U_{\rm d}$:

$\displaystyle U_{\rm d}(t)=$
$\displaystyle\int_{0}^{1}M_{1}(y;\hat{D}(t_{i}))z(y,t)dy+\int_{0}^{1}M_{2}(y;\hat{D}(t_{i}))w(y,t)dy$
$\displaystyle+\int_{0}^{1}M_{3}(y;\hat{D}(t_{i}))v(y,t)dy$
$\displaystyle+M_{4}(\hat{D}(t_{i}))X(t),\quad t\in[t_{i},t_{i+1}),$ (48)

where $\\{t_{i}\geq 0\\}_{i=0}^{\infty}$, $i\in\mathbb{Z}^{+}$ is the sequence
of time instants at which the delay estimate is updated. The triggering
mechanism that determines the sequence of time instants is shown next.

###  IV-A Triggering Mechanism

The sequence of time instants $\\{t_{i}\geq 0\\}_{i=0}^{\infty}$,
$i\in\mathbb{Z}^{+}$ ($t_{0}=0$), is defined as

$\displaystyle
t_{i+1}=\begin{cases}\min\bigg{\\{}\inf\big{\\{}t>t_{i}:\Omega(t)=(1+a)\hat{\Upsilon}({\hat{D}(t_{i})})\Omega(t_{i})\big{\\}},\\\
~{}~{}~{}~{}~{}~{}t_{i}+T\bigg{\\}},~{}~{}~{}~{}for~{}\mbox{
$\Omega(t_{i})\neq 0$}\\\
t_{i}+T,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}for~{}\mbox{$\Omega(t_{i})=0$}\end{cases}$
(49)

where the design parameters $a$, $T$ are positive and free, and the function
$\Omega(t)$ is given in (35). The function
${\hat{\Upsilon}({\hat{D}(t_{i})}})$ is the overshoot coefficient that is
associated with the system transient and is obtained by replacing the unknown
$D$ with $\hat{D}(t_{i})$ in $\Upsilon$ which is defined in (47) (please note
that $\xi_{1}$ and $\xi_{2}$ in (47) depend on the delay $D$ through the
delay-dependent kernel functions $K_{1},K_{2},\eta,R,P$ included in (C.11) and
(C.12). See Appendix C for further details).

###  IV-B Least-squares identifier for the unknown delay

Now, we design the identifier which stands as the update law of the estimated
delay $\hat{D}$. According to (10), for $\tau>0$ and $n=1,2,\cdots$, the
following equality holds:

$\displaystyle\quad D\frac{d}{d\tau}\int_{0}^{1}\sin({x\pi n})v(x,\tau)dx={\pi
n}\int_{0}^{1}\cos({x\pi n})v(x,\tau)dx.$ (50)

Integrating (50) from $0$ to $t$, yields

$\displaystyle D\int_{0}^{1}\sin({x\pi n})v(x,t)dx={\pi
n}\int_{0}^{t}\int_{0}^{1}\cos({x\pi n})v(x,\tau)dxd\tau$ (51)

where (12) has been recalled. Straightforwardly, (51) can be written as

$\displaystyle f_{n}(t)=Dg_{n}(t),$ (52)

where

$\displaystyle f_{n}(t)={\pi n}\int_{0}^{t}\int_{0}^{1}\cos({x\pi
n})v(x,\tau)dxd\tau,$ (53) $\displaystyle g_{n}(t)=\int_{0}^{1}\sin({x\pi
n})v(x,t)dx,$ (54)

for $n\in\mathbb{N}$. Define the function $h_{i,n}$ by the formula:

$\displaystyle h_{i,n}(\ell)=$
$\displaystyle\int_{\mu_{i+1}}^{t_{i+1}}(f_{n}(t)-\ell
g_{n}(t))^{2}dt,~{}i\in\mathbb{Z}^{+},n\in\mathbb{N},$ (55)

and time instant $\mu_{i+1}$ as

$\displaystyle\mu_{i+1}:=\min\\{{t_{g}}:g\in\\{0,\ldots,i\\},t_{g}\geq
t_{i+1}-\tilde{N}T\\},$ (56)

where the positive integer $\tilde{N}\geq 1$ is a free design parameter (in
practice, a lager $\tilde{N}$ means a bigger set of data used in the least-
squares identifier, which makes the identifier more robust with respect to
measurement errors), and where the positive constant $T$ is the maximum dwell
time according to (49). From (52), one can deduce that the function
$h_{i,n}(\ell)$ in (55) has a global minimum $h_{i,n}(D)=0$. Then, using
Fermat’s theorem (vanishing gradient at extrema), the following matrix
equation hold for every $i\in\mathbb{Z}^{+}$ and $n\in\mathbb{N}$:

$\displaystyle H_{n}(\mu_{i+1},t_{i+1})=G_{n}(\mu_{i+1},t_{i+1})D$ (57)

where

$\displaystyle H_{n}(\mu_{i+1},t_{i+1})$
$\displaystyle=\int_{\mu_{i+1}}^{t_{i+1}}g_{n}(t)f_{n}(t)dt,$ (58)
$\displaystyle G_{n}(\mu_{i+1},t_{i+1})$
$\displaystyle=\int_{\mu_{i+1}}^{t_{i+1}}g_{n}(t)^{2}dt.$ (59)

Indeed, (57) is obtained by differentiating the functions $h_{i,n}(\ell)$
defined by (55) with respect to $\ell$, and evaluating the derivative (zero)
at the global minimum $\ell=D$. Using (57)–(59), the following delay
identifier is constructed:

$\displaystyle\hat{D}(t_{i+1})={\rm
argmin}\bigg{\\{}|\ell-\hat{D}(t_{i})|^{2}:{\underline{D}\leq\ell\leq\overline{D}},$
$\displaystyle
H_{n}(\mu_{i+1},t_{i+1})=G_{n}(\mu_{i+1},t_{i+1})\ell,~{}~{}n=1,2,\cdots\bigg{\\}},~{}~{}i\in\mathbb{Z}^{+}.$
(60)

######  Remark 1 (Implementation of the identifier)

Implementation of the identifier begins with calculating
$H_{n}(\mu_{i+1},t_{i+1})$, $G_{n}(\mu_{i+1},t_{i+1})$ from $n=1$, $i=0$,
i.e., $H_{1}(\mu_{1},t_{1})$, $G_{1}(\mu_{1},t_{1})$, using (58), (59), (53),
(54). If $G_{1}(\mu_{1},t_{1})\neq 0$, it implies that $\ell$ belongs to a
singleton set, i.e., $\ell=\frac{H_{1}(\mu_{1},t_{1})}{G_{1}(\mu_{1},t_{1})}$.
It is followed that the output of the identifer (60) at $t_{1}$ is
$\hat{D}(t_{1})=\frac{H_{1}(\mu_{1},t_{1})}{G_{1}(\mu_{1},t_{1})}$. If
$G_{1}(\mu_{1},t_{1})=0$, we continue to calculate $H,G$ with $n=2$, $i=0$,
i.e., $H_{2}(\mu_{1},t_{1})$, $G_{2}(\mu_{1},t_{1})$, and then evaluate the
value of $G_{2}(\mu_{1},t_{1})$. Similarly, if $G_{2}(\mu_{1},t_{1})\neq 0$,
the output of the identifier at $t_{1}$ is
$\hat{D}(t_{1})=\frac{H_{2}(\mu_{1},t_{1})}{G_{2}(\mu_{1},t_{1})}$. If
$G_{2}(\mu_{1},t_{1})=0$, then move to calculate the case of $n=3$, $i=0$,
i.e., $H_{3}(\mu_{1},t_{1})$, $G_{3}(\mu_{1},t_{1})$. Repeating the above
steps, until we find a $G_{n}(\mu_{1},t_{1})\neq 0$ for a certain $n$, the
output of the identifier at $t_{1}$ is
$\hat{D}(t_{1})=H_{n}(\mu_{1},t_{1})/G_{n}(\mu_{1},t_{1})$. For saving the
computation time, we can set an upper limit $\bar{n}$ for $n$. That is, if
$G_{n}(\mu_{1},t_{1})=0$ for all $n=1,\cdots,\bar{n}$, we then stop the
seeking at the updating time $t_{1}$ and consider $\ell$ belongs to the
original set $\\{\ell\in\mathbb{R}:\underline{D}\leq\ell\leq\overline{D}\\}$
which leads to the output of the identifier is equal to the estimate at the
last time instant, i.e., $\hat{D}(t_{1})=\hat{D}(t_{0})$, according to (60).
The same computation process is followed for the subsequent updating time
instants $t_{2},t_{3},\ldots$. For many practical applications, such as
simulating a deep-sea construction vessel, locating non-zero values of
$G_{n}(\mu_{i+1},t_{i+1})$ is a straightforward task following the algorithm
described above.

Please note that even though the actuator states $v(x,t)$ are measurable in
this full-state feedback case, for the delay estimation, one cannot adopt the
“naive” method–that is, taking the time and spatial derivatives of the signal
$v(x,t)$ to calculate $d$ in (10) straightforwardly– because of the following
two reasons: 1) taking the time derivative of the measured signals always
leads to the undesired noise amplification in practice; 2) the possible zero
values of $v_{t}(x,t)$ accompanied with the unknown delay $D$ will engender
singularity.

######  Proposition 1 (Existence of solution in an interval)

For every $(z[t_{i}],w[t_{i}],v[t_{i}])^{T}\in L^{2}((0,1);\mathbb{R}^{3})$,
$X(t_{i})\in\mathbb{R}^{m}$, there exists a unique (weak) solution
$((z,w,v)^{T},X)\in C^{0}([t_{i},t_{i+1}];L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}([t_{i},t_{i+1}];\mathbb{R}^{m})$ to the system (1)–(4), (9)–(12), (48).

###### Proof:

Similar to the proof of Proposition 1 in [52], applying three transformations
to decouple PDEs and ODE, and recalling the result in the part 1 in Appendix
of [13], Proposition 1 can be obtained. We direct the reader to the proof of
Proposition 1 in [52] for details. ∎

##  V Main result

Before presenting the main theorem, we propose the following technical lemmas,
where when we say that $v(x,t)$ is equal to zero for
$x\in[0,1],t\in[\mu_{i+1},t_{i+1}]$, or not identically zero on the same
domain, we mean except possibly for finitely many discontinuities of the
functions $v(x,t)$. These discontinuities are isolated curves in the rectangle
$[0,1]\times[\mu_{i+1},t_{i+1}]$.

######  Lemma 1 ($G_{n}(\mu_{i+1},t_{i+1})=0$)

The sufficient and necessary condition of $G_{n}(\mu_{i+1},t_{i+1})=0$ for all
$n\in\mathbb{N}$ is $v[t]=0$ on $t\in[\mu_{i+1},t_{i+1}]$.

###### Proof:

Necessity: If $G_{n}(\mu_{i+1},t_{i+1})=0$ for all $n\in\mathbb{N}$, then the
definition (59) in conjunction with continuity of $g_{n}(t)$ for
$t\in[\mu_{i+1},t_{i+1}]$ (because of the definition (54) and the fact that
$v\in C^{0}([t_{i},t_{i+1}];L^{2}(0,1))$ in Proposition 1) implies

$\displaystyle g_{n}(t)=0,~{}~{}t\in[\mu_{i+1},t_{i+1}].$ (61)

According to the definition (54), the equation (61) implies

$\displaystyle\int_{0}^{1}\sin({x\pi
n})v(x,t)dx=0,~{}~{}t\in[\mu_{i+1},t_{i+1}]$ (62)

for all $n\in\mathbb{N}$. Since the set $\\{\sqrt{2}\sin(x\pi
n):n=1,2,\ldots\\}$ is an orthonormal basis of $L^{2}(0,1)$, we have $v[t]=0$
for $t\in[\mu_{i+1},t_{i+1}]$.

Sufficiency: If $v[t]=0$ on $t\in[\mu_{i+1},t_{i+1}]$, then
$G_{n}(\mu_{i+1},t_{i+1})=0$ for all $n\in\mathbb{N}$ is obtained directly by
recalling (59) and (54). ∎

######  Lemma 2 (The identifier properties at $t_{i+1}$)

For the adaptive estimates defined by (60), the following statements hold:

  * •

If $v[t]$ is not identically zero for $t\in[\mu_{i+1},t_{i+1}]$, then
$\hat{D}(t_{i+1})=D$.

  * •

If $v[t]$ is identically zero for $t\in[\mu_{i+1},t_{i+1}]$, then
$\hat{D}(t_{i+1})=\hat{D}(t_{i})$.

###### Proof:

Define a set

$\displaystyle S_{i}=\bigg{\\{}\underline{D}\leq\ell\leq\overline{D}:~{}$
$\displaystyle H_{n}(\mu_{i+1},t_{i+1})=G_{n}(\mu_{i+1},t_{i+1})\ell,$
$\displaystyle n=1,2,\cdots\bigg{\\}}.$ (63)

From (57), we know that $D\in S_{i}$. If $S_{i}$ is a singleton, it is nothing
else but the generated adaptive estimate $\hat{D}(t_{i+1})$ by (60), which is
equal to the true delay $D$.

  1. 1.

If $v[t]$ is not identically zero for $t\in[\mu_{i+1},t_{i+1}]$, recalling
Lemma 1, there exists $n\in{\mathbb{N}}$ such that
$G_{n}(\mu_{i+1},t_{i+1})\neq 0$. Now defining the index set $I$ as the set of
all $n\in{\mathbb{N}}$ with $G_{n}(\mu_{i+1},t_{i+1})\neq 0$, then (63)
implies that

$S_{i}=\left\\{\ell=\frac{H_{n}(\mu_{i+1},t_{i+1})}{G_{n}(\mu_{i+1},t_{i+1})},n\in
I\right\\}$

is a singleton, and therefore from (60) we get $\hat{D}(t_{i+1})=D$.

  2. 2.

If $v[t]$ is identically zero on $t\in[\mu_{i+1},t_{i+1}]$, according to (53),
(54), (58), (59), one obtains

$G_{n}(\mu_{i+1},t_{i+1})=H_{n}(\mu_{i+1},t_{i+1})=0,~{}~{}n\in\mathbb{N},$

and it follows that $S_{i}=\\{\underline{D}\leq\ell\leq\overline{D}\\}$. Then,
from (60) one arrive at $\hat{D}(t_{i+1})=\hat{D}(t_{i})$.

The proof is complete. ∎

######  Lemma 3 (The identifier properties for
$t\in[t_{i},\lim_{k\to\infty}(t_{k}))$)

If $\hat{D}(t_{i})=D$ for certain $i\in\mathbb{Z}^{+}$, then $\hat{D}(t)=D$
for all $t\in[t_{i},\lim_{k\to\infty}(t_{k}))$.

###### Proof:

According to Lemma 2, we have that $\hat{D}(t_{i+1})$ is equal to either $D$
or $\hat{D}(t_{i})$. Therefore, if $\hat{D}(t_{i})=D$, then
$\hat{D}(t_{i+1})=D$. Repeating this process, we then have $\hat{D}(t)=D$ for
all $t\in[t_{i},\lim_{k\to\infty}(t_{k}))$. The proof is complete. ∎

######  Lemma 4 (Existence of a minimum dwell-time)

There exists a positive constant $\tau_{d}$ such that
$t_{i+1}-t_{i}\geq\tau_{d}$ for all $i\in\mathbb{Z}^{+}$.

###### Proof:

The result is established by discussing the following two cases:

  * •

Case 1: The finite-time exact identification is not achieved. According to
Lemmas 2 and 3, we have $\hat{D}(t)=\hat{D}(0)$ in the control law. It implies
that there is no discontinuity in the control input, and thus $\Omega(t)$
defined in (35) is continuous and differentiable. Recalling the triggering
mechanism (49), the lower bound $\underline{\tau}_{i}$ of the dwell time is
given by

$\displaystyle\underline{\tau}_{i}=\begin{cases}\min\bigg{\\{}\frac{((1+a){\hat{\Upsilon}({\hat{D}(0)}})-1)\Omega(t_{i})}{\max_{t\in(t_{i},t_{i+1})}|\dot{\Omega}(t)|},T\bigg{\\}}>0,&\mbox{if
$\Omega(t_{i})\neq 0$}\\\ T,&\mbox{if $\Omega(t_{i})=0$}.\end{cases}$ (64)

Therefore, the lower bound of the minimal dwell time is written as

$\displaystyle\tau_{d}=\min\\{\underline{\tau}_{i}:i\in\mathbb{Z}^{+}\\}>0.$
(65)

  * •

Case 2: The finite-time exact identification is achieved.

Before the time instant when the estimate $\hat{D}(t)$ reaches the actual
value $D$, e.g., $t_{f}$, we know $\hat{D}(t)=\hat{D}(0)$ for $t<t_{f}$ by
applying Lemma 2 repeatedly. Through the same analysis in case 1, we have the
lower bound of the minimal dwell time before $t_{f}$ is $\tau_{d}$ given by
(64), (65).

After the time instant $t_{f}$, the exact identification is achieved and
$t_{i+1}-t_{i}=T\geq\tau_{d}$ for all $i\geq f$. We prove this as follows.
Once the exact delay identification is achieved, the delay-adaptive control
input is identical to the nominal delay-compensated control input in Section
III. When $\Omega(t_{i})\neq 0$, we have that
$\Omega(t)\leq\Upsilon\Omega(t_{i})$ for $t_{i}\leq t\leq t_{i+1}$, $i\geq f$
according to Theorem 1. It follows from
$\hat{\Upsilon}({\hat{D}(t_{i})})=\hat{\Upsilon}(D)=\Upsilon$ for all $i\geq
f$ that $\Omega(t)<(1+a)\hat{\Upsilon}({\hat{D}(t_{i})})\Omega(t_{i})$ for
$t_{i}\leq t\leq t_{i+1}$, $i\geq f$. Thus $t_{i+1}-t_{i}=T$ according to
(49). When $\Omega(t_{i})=0$, we straightforwardly have $t_{i+1}-t_{i}=T$
according to the second equation in (49) and therefore, $t_{i+1}-t_{i}=T$ for
all $i\geq f$.

We finally conclude that the lower bound of the minimal dwell time in case 2
is $\tau_{d}$ as well. ∎

######  Corollary 1 (Well-posedness of the closed-loop system)

No Zeno phenomenon occurs, i.e., $\lim_{i\to\infty}t_{i}=+\infty$, and the
closed-loop system is well-posed in the sense that for every
$(z[0],w[0])^{T}\in L^{2}((0,1);\mathbb{R}^{2})$, $X(0)\in\mathbb{R}^{m}$, and
$\hat{D}(0)\in[\underline{D},\overline{D}]$, there exists a unique (weak)
solution $((z,w,v)^{T},{X})\in
C^{0}(\mathbb{R}_{+};L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}(\mathbb{R}_{+};\mathbb{R}^{m})$, and
$\hat{D}(t)\in\\{\ell\in\mathbb{R}:\underline{D}\leq\ell\leq\overline{D}\\}$
for $t\in[0,\infty)$, to the system consisting of (1)–(4), (9)–(12), (48), and
(60).

###### Proof:

Recalling Lemma 4, we have that

$\displaystyle t_{i}\geq\tau_{d}i,~{}~{}~{}i\in\mathbb{Z}^{+},$

where $\tau_{d}>0$, that is,

$\displaystyle\lim_{i\to\infty}(t_{i})=+\infty,$ (66)

which implies a solution defined on $\mathbb{R}_{+}$ in the subsequent
analysis.

From the initial data $(z[0],w[0])^{T}\in L^{2}((0,1);\mathbb{R}^{2})$,
$X(0)\in\mathbb{R}^{m}$ and (12), recalling the result in Proposition 1 for
$i=0$, it follows that $((z,w,v)^{T},X)\in
C^{0}([t_{0},t_{1}];L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}([t_{0},t_{1}];\mathbb{R}^{m})$, which implies
$(z[t_{1}],w[t_{1}],v[t_{1}])^{T}\in L^{2}((0,1);\mathbb{R}^{3})$,
$X(t_{1})\in\mathbb{R}^{m}$. Recalling the result in Proposition 1 for $i=1$,
together with the solution obtained for $[t_{0},t_{1}]$, we have that
$((z,w,v)^{T},X)\in C^{0}([t_{0},t_{2}];L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}([t_{0},t_{2}];\mathbb{R}^{m})$. Repeating the above steps, we obtain
that $((z,w,v)^{T},X)\in C^{0}([t_{0},t_{i}];L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}([t_{0},t_{i}];\mathbb{R}^{m})$ for $i\in\mathbb{N}$. Applying (66), we
thus have $((z,w,v)^{T},X)\in
C^{0}(\mathbb{R}_{+};L^{2}(0,1);\mathbb{R}^{3})\times
C^{0}(\mathbb{R}_{+};\mathbb{R}^{m})$. It is straightforwardly obtained from
(60) that $\hat{D}(t)\in[\underline{D},\overline{D}]$ if
$\hat{D}(0)\in[\underline{D},\overline{D}]$.

Corollary 1 is thus obtained. ∎

######  Lemma 5 (Finite-time convergence of the update law)

If the control input $U_{\rm d}(t)$ is not identically zero on
$t\in[0,\infty)$, the estimate $\hat{D}$ converges to the true value in finite
time, i.e.,

$\displaystyle\hat{D}(t)=D,~{}\forall t\in[t_{f},\infty)$ (67) $\displaystyle
and~{}~{}$ $\displaystyle\hat{D}(t)=\hat{D}(0),~{}\forall
t\in[0,t_{f})~{}(for~{}f>0)$ (68)

where

$t_{f}=\min\\{t_{i}:i\in\mathbb{N},t_{i}>\min\left\\{t\geq 0:U_{\rm d}(t)\neq
0\right\\}\\}.$

If the control input $U_{\rm d}(t)$ is identically zero on $t\in[0,\infty)$,
$\hat{D}(t)\equiv\hat{D}(0)$ all the time.

###### Proof:

If the control input $U_{\rm d}(t)$ is not identically zero on
$t\in[0,\infty)$, there exists a time instant $t_{f}$ $(f>0)$ such that
$U_{\rm d}(t)$ is not identically zero in $t\in[\mu_{f},t_{f}]$. Recalling
(10), (11) where $U(t)$ has been replaced by $U_{\rm d}(t)$, we conclude that
the actuator state $v[t]$ is not identically zero on $t\in[\mu_{f},t_{f}]$.
Recalling Lemma 2, Lemma 3, and Corollary 1, we thus obtain
$\hat{D}(t)=D,~{}\forall t\in[t_{f},\infty)$ (67).

For $t\leq t_{f-1}$ $(f>1)$, $U_{\rm d}(t)=0$, i.e., $v[t]\equiv 0$, it
follows from Lemma 2 that $\hat{D}=\hat{D}(0)$ on $t\in[0,t_{f})$. If $f=1$,
we directly have $\hat{D}=\hat{D}(0)$ on $t\in[0,t_{f})$ because the
identifier has not been updated. We thus obtain (68).

If $U_{\rm d}(t)$ is identically zero on $t\in[0,\infty)$, we know that
$v[t]\equiv 0$ all the time. Recalling Lemma 2 and Corollary 1, we have
$\hat{D}(t)\equiv\hat{D}(0)$ on $t\in[0,\infty)$. ∎

By virtue of Lemma 5, a sufficient and necessary condition of exact
identification of the unknown delay is the fact that $U_{\rm d}(t;\hat{D}(0))$
is not identically zero on $t\in[0,\infty)$. Next, we show that we always can
find $\hat{D}(0)$ such that $U_{\rm d}(t)$ is not identically zero on
$t\in[0,\infty)$ if the exponential stability (34) is not ensured in the open
loop (if (34) is already achieved in the open loop, the control design would
be trivial). Define a set $\mathcal{D}$ of all
$\hat{D}(0)\in[\underline{D},\overline{D}]$ such that $U_{\rm
d}(t;\hat{D}(0))$ is not identically zero on $t\in[0,\infty)$. We show that
$\mathcal{D}$ is a nonempty set in the following lemma.

######  Lemma 6 (Nonempty set $\mathcal{D}$)

If the considered plant (1)–(5) in the open loop, i.e., with the identically
zero control input $U$, is not an exponentially stable system, the set
$\mathcal{D}$ of all $\hat{D}(0)\in[\underline{D},\overline{D}]$ such that
$U_{\rm d}(t;\hat{D}(0))$ is not identically zero on $t\in[0,\infty)$ is non-
empty.

###### Proof:

Suppose that $\mathcal{D}$ is an empty set, equivalently, $U_{\rm
d}(t;\hat{D}(0))$ is identically zero on $t\in[0,\infty)$ for all
$\hat{D}(0)\in[\underline{D},\overline{D}]$. It implies that $U_{\rm d}(t;D)$,
i.e., the nominal control input $U(t)$ in Theorem 1 that shows the exponential
stability of the nominal closed-loop system, is identically zero on
$t\in[0,\infty)$. Thus, the plant with the identically zero control input (the
open-loop system) has already been exponentially stable, which contradicts
with that the considered plant in open loop is not an exponentially stable
system. Therefore, Lemma 6 is obtained. ∎

######  Remark 2 (On the choice of $\hat{D}(0)$ in practice)

When $\Omega(0)\neq 0$, a simple way to set the initial condition $\hat{D}(0)$
is to find a $\hat{D}(0)$ such that $U_{\rm d}(0;\hat{D}(0))$ defined in (48)
is nonzero, which is available in most practical applications. As a result,
the exact parameter identification would be achieved at the first triggering
time. If $\Omega(0)=0$ that means that the initial values of all plant states
are zero, and consequently, the states of the plant (1)–(5) are identically
zero all the time under the zero control input. Therefore, the control input
vanishes with no need to construct an estimate of the delay, namely,
$\hat{D}$.

Now, we are in a position to state our main result in the following theorem,
i.e., exponential regulation of the plant and actuator states.

###### Remark 3

If the exponential stability has already been achieved in open loop, the
control design would be trivial, because the control input can just be set as
zero even though the unknown delay exists. We only discuss the case that the
open-loop system is not exponentially stable in the following theorem.

###### Theorem 2

For all initial data $(z[0],w[0])^{T}\in L^{2}(0,1)$, $X(0)\in\mathbb{R}^{m}$,
$\hat{D}(0)\in\mathcal{D}$, considering the closed-loop system consisting of
the plant (1)–(4), (9)–(12), the controller (48), the triggering mechanism
(49), and the least-squares identifier (60), the exponential regulation of the
closed-loop system is obtained in the sense that there exist positive
constants $M,\lambda_{1}$ such that

$\displaystyle\Omega(t)\leq M\Omega(0)e^{-\lambda_{1}t},~{}~{}t\geq 0,$ (69)

where $\Omega(t)$ is defined in (35).

###### Proof:

Replacing the nominal control law $U$ by the delay-adaptive control law
$U_{d}$ defined by (48) in (11), through the transformations in Section III,
the right boundary condition of the actuator PDE (30) in the target system
(17)–(20), (28)–(30) becomes

$\displaystyle\hat{u}(0,t)=\xi(t),$ (70)

where

$\displaystyle\xi(t)=$
$\displaystyle-\int_{0}^{1}(M_{1}(y;D)-M_{1}(y;\hat{D}))z(y,t)dy$
$\displaystyle-\int_{0}^{1}(M_{2}(y;D)-M_{2}(y;\hat{D}))w(y,t)dy$
$\displaystyle+\int_{0}^{1}(M_{3}(y;D)-M_{3}(y;\hat{D}))v(y,t)dy$
$\displaystyle+(M_{4}(D)-M_{4}(\hat{D})){X}(t).$ (71)

Taking the derivative of (36) along the target system states corresponding to
the even-based closed-loop system consisting of (17)–(20), (28), (29), and
(70), through a similar process in (44), recalling conditions (37)–(40) on
${\delta}$, $r_{a}$, $r_{c}$, and $r_{d}$ (we emphasize that conditions
(37)–(40) only depend on the known plant parameters and the known bounds of
the unknown parameters in Assumption 3), we obtain

$\displaystyle\dot{V}(t)\leq-\lambda_{1}V(t)+\frac{{r_{c}}}{2D}\xi(t)^{2},~{}~{}t\geq
0,$ (72)

where $\lambda_{1}$ is given in (46). From Lemma 5, 6 and Remark 3, the
finite-time convergence of the parameter estimate $\hat{D}$ to the actual
value $D$ is obtained directly. Now, using (67) and (71), one can establish
that

$\displaystyle\xi(t)\equiv 0,~{}~{}t\in[t_{f},\infty).$ (73)

We then have that

$\displaystyle\dot{V}(t)\leq-\lambda_{1}V(t),~{}~{}t\geq t_{f}.$ (74)

Multiplying both sides of (74) by $e^{\lambda_{1}t}$ and integrating the
resulting terms from $t_{f}$ to $t$ lead to the following inequality

$V(t)\leq V(t_{f})e^{-\lambda_{1}(t-t_{f})},~{}~{}t\geq t_{f},$

which, by virtue of (41), is equivalent to

$\displaystyle\Omega(t)\leq\Upsilon\Omega(t_{f})e^{-\lambda_{1}(t-t_{f})},~{}~{}t\geq
t_{f},$ (75)

where $\Omega$ is defined in (35) and the positive constant $\Upsilon$ is
given in (47)

Note that the norm estimate (75) is only true for $t\geq t_{f}$ and if
$t_{f}=0$, (69) is obtained directly.

Next, we extend our analysis for $t\in[0,t_{f}]$ with $t_{f}\neq 0$. With the
help of (41), (71), we obtain from (72) that

$\displaystyle\dot{V}(t)\leq$
$\displaystyle-\lambda_{1}V(t)+Q(\hat{D}(0))V(t),~{}~{}~{}t\in[0,t_{f}],$ (76)

where the positive constant $Q(\hat{D}(0))$ is

$\displaystyle Q(\hat{D}(0))=$
$\displaystyle\max\bigg{\\{}\max_{i=1,2,3;y\in[0,1]}\left\\{(M_{i}(y;D)-M_{i}(y;\hat{D}(0)))^{2}\right\\},$
$\displaystyle(M_{4}(D)-M_{4}(\hat{D}(0)))^{2}\bigg{\\}}\frac{2{r_{c}}}{D\xi_{1}\xi_{3}}$
(77)

which is derived by finding an upper bound for $\xi(t)^{2}$ given in (71) and
recalling (41).

Hence, the following holds

$\displaystyle\Omega(t)\leq\Upsilon\Omega(0)e^{\lambda_{2}(\hat{D}(0))t},~{}~{}t\in[0,t_{f}],$
(78)

where

$\lambda_{2}(\hat{D}(0))=|Q(\hat{D}(0))-\lambda_{1}|>0,$

and the positive constant $\Upsilon$ is given in (47). Therefore, it
straightforwardly follows that

$\displaystyle\Omega(t_{f})\leq\Upsilon
e^{\lambda_{2}(\hat{D}(0))t_{f}}\Omega(0).$ (79)

Considering (78), combining (75) and (79) yields

$\displaystyle\Omega(t)\leq\Upsilon^{2}e^{(\lambda_{2}(\hat{D}(0))+\lambda_{1})t_{f}}\Omega(0)e^{-\lambda_{1}t},~{}~{}~{}~{}t\geq
0,$ (80)

which is equivalent to (69) with

$M=\Upsilon^{2}e^{(\lambda_{2}(\hat{D}(0))+\lambda_{1})t_{f}}.$

The proof of the theorem is complete. ∎

##  VI Simulation

A deep-sea construction vessel (DCV) is used to place equipment to be
installed at the predetermined location on the seafloor, which is shown in
Figure 2. Different from [53] that deals with a known sensor delay that exists
in the large-distance transmission of the sensing signal from the seafloor to
the vessel on the ocean surface through a set of acoustics devices, we
consider all possible delays here (including the transmission of the sensing
signal, computation of the control law, and the delay in the hydraulic
actuator for the ship-mounted crane and so on) as an unknown delay in the
control input channel. By designing a control input at the top of the crane
cable, our goal is to reduce the oscillations of the crane cable with the
purpose of placing the payload attached at the bottom of the cable in the
target area, despite the presence of the unknown delay.

Figure 2: Deep-sea construction vessel.

###  VI-A Model

The following dynamic model of cable-payload lateral oscillations in DCV is
taken from [53],

$\displaystyle T_{0}u_{\bar{x}}(0,t)$ $\displaystyle=U(t-D),$ (81)
$\displaystyle\rho u_{tt}(\bar{x},t)$
$\displaystyle=T_{0}u_{\bar{x}\bar{x}}(\bar{x},t)-d_{c}{u_{t}}(\bar{x},t),$
(82) $\displaystyle u(L,t)$ $\displaystyle=b_{L}(t),$ (83) $\displaystyle
M_{L}\ddot{b}_{L}(t)$
$\displaystyle=-d_{L}\dot{b}_{L}(t)+T_{0}u_{\bar{x}}(L,t),$ (84)

$\forall(\bar{x},t)\in[0,L]\times[0,\infty)$. The state $u(\bar{x},t)$
describes the lateral oscillation displacement along the cable, and $b_{L}(t)$
denotes that of the payload. The control input $U$ is subject to the unknown
time delay $D$ mentioned above. The static tension $T_{0}$ is defined as

$T_{0}=M_{L}g-F_{\rm buoyant},$

where the buoyancy $F_{\rm buoyant}$ is

$F_{\rm buoyant}=\frac{1}{4}\pi D_{c}^{2}h_{c}\rho_{s}g.$

The physical parameters of the deep-sea construction vessel are shown in Table
II.

Like [53], after applying the Riemann transformations

$\displaystyle{z}(\bar{x},t)$
$\displaystyle={u_{t}}(\bar{x},t)-\sqrt{\frac{T_{0}}{\rho}}{u_{\bar{x}}}(\bar{x},t),$
(85) $\displaystyle{w}(\bar{x},t)$
$\displaystyle={u_{t}}(\bar{x},t)+\sqrt{\frac{T_{0}}{\rho}}{u_{\bar{x}}}(\bar{x},t),$
(86)

introducing a space normalization variable

$\displaystyle x=\frac{\bar{x}}{L}\in[0,1],$ (87)

and defining

$X(t)=\dot{b}_{L}(t),$

equations (81)–(84) are rewritten as the considered plant (1)–(5) with the
coefficients

$\displaystyle
c_{0}=2\sqrt{\frac{1}{T_{0}\rho}},~{}~{}{q_{1}}=q_{2}=\frac{1}{L}\sqrt{\frac{T_{0}}{\rho}},$
(88) $\displaystyle
d_{1}=d_{2}=d_{3}=d_{4}=\frac{{{-d_{c}}}}{2\rho},q=-1,~{}p=1,$ (89)
$\displaystyle
C=2,~{}{A}=\frac{-d_{L}}{M_{L}}+\frac{\sqrt{T_{0}\rho}}{M_{L}},~{}{B}=-\frac{\sqrt{{T_{0}}\rho}}{M_{L}},$
(90)

which is the simulation model in this section, where it can be checked that
the plan parameters in (88)–(90) satisfy Assumptions 1, 2 by recalling Table
II.

TABLE II: Physical parameters of the DCV. Parameters (units) | values
---|---
Cable length ${L}$ (m) | 1500
Cable linear density ${\rho}$ (kg/m) | 7.5
Payload mass ${M_{L}}$ (kg) | 3.5$\times$ $10^{5}$
Gravitational acceleration $g$ (m/s2) | 9.8
Cable material damping coefficient $d_{c}$ (N$\cdot$s/m) | 0.8
Height of payload modeled as a cylinder $h_{c}$ (m) | 7.5
Diameter of payload modeled as a cylinder $D_{c}$ (m) | 5
Damping coefficient at payload $d_{L}$ (N$\cdot$s/m) | 1.2$\times$$10^{5}$
Seawater density $\rho_{s}$ (kgm-3) | $1024$

The initial conditions are defined as

$z(x,0)=8\sin({5\pi x}(1-x)),~{}~{}w(x,0)=-8\cos(5{\pi x}),$

thereby,

$X(0)=1.13,$

recalling (4), which physically corresponds to the initial oscillation
velocities of the payload. The unknown delay $D$ is set as 1, and the known
bounds $\underline{D}$ and $\overline{D}$ are assumed as 0.01 and 2. We will
show the simulation results of the following four cases:

  * •

Open loop: the control input is zero;

  * •

Nonadaptive control: the nominal delay-compensated control with the unknown
delay $D$ replaced by its estimate $0.25$;

  * •

Delay-adaptive control with the initial delay estimate $\hat{D}(0)=0.25$,
where the design parameter $K$ in (22) is chosen as $K=-18$;

  * •

Delay-adaptive control with the initial delay estimate $\hat{D}(0)=1.5$, where
the design parameter $K$ in (22) is chosen as $K=-13$;

Other design parameters are

$\delta=-0.36,~{}r_{a}=1.02,~{}r_{c}=1,$
$r_{d}=0.02,~{}a=2,~{}T=3.12,~{}\tilde{N}=10,$

according to (37)–(40), where the last three parameters are free but positive.
The parameter $\bar{n}$ mentioned in Remark 1 is set as $\bar{n}=2$.

###### Remark 4

In addition to Remark 1 and Remark 2 about the implementation of the delay
identifier, some more things are worth noting in the simulation. 1)
Approximating the integration with respect to the space variable in the
identifier as the summation operator will cause a tiny error between the final
parameter estimate and the true value in the simulation result, which will be
seen in Fig. 4. The smaller space step adopted in the simulation will make the
error smaller. 2) The error of approximation in the simulation will also lead
to tiny differences between the outputs of the identifer at each updating time
even if the effective parameter deification has been achieved. Therefore, we
set a small margin to tolerate the approximation error, that is–if the
difference between the estimates from the identifier at two adjacent updating
times is smaller than $2\%$ of the true value, we consider that this
difference is caused by the approximation error in the simulation, and thus
keep the estimate value as same as the one at the former updating time.

###  VI-B Simulation result

The numerical computation is conducted using the finite difference method with
the step sizes of $t$ and $x$ as 0.001, and 0.02, respectively. The
approximate solutions of the kernel PDEs used in the control law, which is
defined by (48), (49), (60) where the integral operators are approximated by
sums, are also solved by the finite difference method based on the
discretization of the triangular domain into a uniformly spaced grid with the
interval of $0.02$.

The designed delay-adaptive control input and the estimate of the unknown
delay are shown in Figures 3 and 4, respectively, from which we know that the
identification of the unknown delay is achieved at the first triggering time,
no matter the initial delay estimate is less than ($\hat{D}(0)=0.25$) or
larger than ($\hat{D}(0)=1.5$) the true value $D=1$. As mentioned in Remark 4,
the tiny differences between the delay estimate and its true values come from
the error of approximation—that is, approximating the integration with respect
to the space variable from 0 to 1 in the identifier as the summation operator
for the 51 spatial discrete points with the fixed interval of $0.02$. The time
evolution of the ODE state $X(t)$ is shown in Figure 5, where the brown dashed
line, the red dashed line, the black solid line, and the blue dot-dash line
show the results of the four cases mentioned in Section VI-A, respectively.
Although both the nonadaptive delay-compensated controller and the delay-
adaptive controllers can attenuate the state of the ODE in comparison to the
open loop scenario, Figure 5 further reveals the “delay mismatch” in the non-
adaptive control law leads to slower convergence after the time point when the
exact delay estimate is obtained and the updated input signal reaches the ODE
through the transport PDEs. Even though the simulation model, like many
practical models that usually include damping, is not an open-loop unstable
plant, the proposed control design still shows improved convergence rates
under the proposed delay-adaptive controllers as compared to both the open-
loop case and nonadaptive delay-compensated controller. Similarly, it is shown
in Figures 6, 7, and 8 that the PDE plant states $z(x,t)$, $w(x,t)$ and the
actuator state $v(x,t)$ all converge to zero when the system is subject to the
proposed delay-adaptive control inputs with $\hat{D}(0)=0.25$ or
$\hat{D}(0)=1.5$.

Figure 3: The delay-adaptive control input $U_{\rm d}(t)$ with
$\hat{D}(0)=0.25$ or $\hat{D}(0)=1.5$ and the nonadaptive control input
$U_{0}(t)$. Figure 4: Estimate of the unknown delay $D$ under the initial
estimate $\hat{D}(0)=0.25$ or $\hat{D}(0)=1.5$. Figure 5: The evolution of
$X(t)$ under the delay-adaptive control $U_{\rm d}(t)$ with $\hat{D}(0)=0.25$
or $\hat{D}(0)=1.5$ and the nonadaptive control $U_{0}(t)$.

(a) $\hat{D}(0)=0.25$

(b) $\hat{D}(0)=1.5$

Figure 6: The evolution of the plant state $z(x,t)$ under the delay-adaptive
control $U_{\rm d}(t)$ with $\hat{D}(0)=0.25$ or $\hat{D}(0)=1.5$.

(a) $\hat{D}(0)=0.25$

(b) $\hat{D}(0)=1.5$

Figure 7: The evolution of the plant state $w(x,t)$ under the delay-adaptive
control $U_{\rm d}(t)$ with $\hat{D}(0)=0.25$ or $\hat{D}(0)=1.5$.

(a) $\hat{D}(0)=0.25$

(b) $\hat{D}(0)=1.5$

Figure 8: The evolution of the actuator state $v(x,t)$ under the delay-
adaptive control $U_{\rm d}(t)$ with $\hat{D}(0)=0.25$ or $\hat{D}(0)=1.5$.

It is easy to obtain the oscillation energy of the cable in DCV
$\frac{\rho}{2}\|u_{t}(\cdot,t)\|^{2}+\frac{T_{0}}{2}\|u_{x}(\cdot,t)\|^{2}=\frac{\rho}{8}\|w(\cdot,t)+z(\cdot,t)\|^{2}+\frac{\rho}{8}\|w(\cdot,t)-z(\cdot,t)\|^{2}$
by recalling (85)–(87). Therefore, it is known from the results $z(x,t)$ and
$w(x,t)$ in Figures 6 and 7 that the oscillation energy of the cable decreases
to zero fast under the proposed delay-adaptive controller. One can also
observe from Figure 5 that the regulation performance of the ODE, i.e., the
payload, is satisfied.

##  VII Conclusion and future work

In this paper, we have proposed a delay-adaptive control scheme for a $2\times
2$ hyperbolic PDE-ODE system, where the input delay is arbitrarily large and
unknown. The controller consists of a nominal delay-compensated control law, a
batch least-squares identifier for the unknown delay, and a triggering
mechanism to determine the update times of the identifier. We have proved that
the proposed control guarantees: 1) the avoidance of Zeno phenomenon; 2) the
identification of the unknown boundary input delay in finite time in most
situations; 3) the exponential regulation of both the plant and the actuator
states to zero. The effectiveness of the proposed design is verified by
numerical simulation in the control application of a deep-sea construction
vessel subject to input delay. This paper only deals with the state-feedback
adaptive-delay control design for coupled hyperbolic PDEs whose actuator
states and plant states are measurable. In our future work, the output-
feedback control design with unmeasurable actuator and plant states will be
considered.

##  VIII Appendix

###  VIII-A Gain kernels PDEs and their associated boundary conditions

(a) First-step transformation

The backstepping transformation (13) and (14) lead to the following PDE-ODE
system of kernel conditions for $\varphi,\phi,\Psi,\Phi,\gamma$ and $\lambda$.
These conditions are derived by mapping the original plant to the first
intermediate system.

$\displaystyle{q_{2}}{{\varphi}_{y}}(x,y)-{q_{1}}{{\varphi}_{x}}(x,y)-({d_{4}}-{d_{1}})\varphi(x,y)$
$\displaystyle-{d_{2}}\phi(x,y)=0,$ (A.1)
$\displaystyle{{q_{1}}{{\phi}_{x}}(x,y){+}{q_{1}}{{\phi}_{y}}(x,y)}{+}{d_{3}}\varphi(x,y)=0,$
(A.2)
$\displaystyle{q_{2}}{{\Psi}_{x}}(x,y)-{q_{1}}{{\Psi}_{y}}(x,y)+({d_{4}}-{d_{1}})\Psi(x,y)$
$\displaystyle-{d_{3}}\Phi(x,y)=0,$ (A.3)
$\displaystyle{{q_{2}}{\Phi_{x}}(x,y)+{q_{2}}{\Phi_{y}}(x,y)-{d_{2}}\Psi(x,y)}=0,$
(A.4)
$\displaystyle{q_{1}}{\gamma}^{\prime}(x)+{\gamma}(x)(A-{d_{1}}I_{n})+{q_{1}}{C}\phi(x,0)=0,$
(A.5)
$\displaystyle{q_{2}}\lambda^{\prime}(x)-{\lambda}(x)(A-{d_{4}}I_{n})-{q_{1}}{C}\Psi(x,0)=0,$
(A.6)

with the boundary conditions

$\displaystyle\varphi(x,x)=\frac{d_{2}}{{q_{1}}+{q_{2}}},$ (A.7)
$\displaystyle{q_{2}}\varphi(x,0)+{q_{1}}p\phi(x,0)={\gamma}(x)B,$ (A.8)
$\displaystyle\Psi(x,x)=\frac{-d_{3}}{{q_{1}}+{q_{2}}},$ (A.9)
$\displaystyle{q_{2}}\Phi(x,0)+{q_{1}}p\Psi(x,0)=\lambda(x)B,$ (A.10)
$\displaystyle{\lambda(0)=K^{T}},$ (A.11)
$\displaystyle{{\gamma}(0)=C-pK^{T}},$ (A.12)

where $I_{n}$ is an identity matrix with dimension $n$.

Similarly, the boundary conditions of the gain kernels associated with the
inverse backstepping transformation (15), (16), namely,
$\bar{\varphi},\bar{\phi},\bar{\gamma},\bar{\Psi},\bar{\Phi}$ and
$\bar{\lambda}$ are given by

$\displaystyle{q_{2}}{{\bar{\Psi}}_{x}}(x,y)-{q_{1}}{{\bar{\Psi}}_{y}}(x,y)+({d_{4}}-{d_{1}})\bar{\Psi}(x,y)$
$\displaystyle+{d_{3}}\bar{\phi}(x,y)=0,$ (A.13)
$\displaystyle{q_{1}}{{\bar{\phi}}_{x}}(x,y)+{q_{1}}{{\bar{\phi}}_{y}}(x,y)-{d_{2}}\bar{\Psi}(x,y)=0,$
(A.14)
$\displaystyle{q_{2}}{{\bar{\varphi}}_{y}}(x,y)-{q_{1}}{{\bar{\varphi}}_{x}}(x,y)-({d_{4}}-{d_{1}})\bar{\varphi}(x,y)$
$\displaystyle+{d_{2}}\bar{\Phi}(x,y)=0,$ (A.15)
$\displaystyle{q_{2}}{\bar{\Phi}_{y}}(x,y)+{q_{2}}{\bar{\Phi}_{x}}(x,y)+{d_{3}}\bar{\varphi}(x,y)=0,$
(A.16)
$\displaystyle{q_{1}}{\bar{\gamma}}^{\prime}(x)-{\bar{\gamma}}(x)(A+BK^{T}+{d_{1}}I_{n})-{d_{2}}\bar{\lambda}(x)=0,$
(A.17)
$\displaystyle{q_{2}}\bar{\lambda}^{\prime}(x)+\bar{\lambda}(x)(A+BK^{T}+d_{4}I_{n})+{d_{3}}{\bar{\gamma}}(x)=0,$
(A.18)

with the boundary conditions

$\displaystyle\bar{\Psi}(x,x)=-\frac{d_{3}}{{q_{1}}+{q_{2}}},$ (A.19)
$\displaystyle{q_{1}}p\bar{\phi}(x,0)+{q_{2}}\bar{\varphi}(x,0)={\bar{\gamma}}(x)B,$
(A.20) $\displaystyle\bar{\varphi}(x,x)=\frac{{d_{2}}}{{q_{1}}+{q_{2}}},$
(A.21)
$\displaystyle{q_{2}}\bar{\Phi}(x,0)+{q_{1}}p\bar{\Psi}(x,0)=\bar{\lambda}(x)B,$
(A.22) $\displaystyle{\bar{\lambda}(0)=-K^{T}},$ (A.23)
$\displaystyle{{\bar{\gamma}}(0)=pK^{T}-{C}.}$ (A.24)

The set of equations (A.1)–(A.12) and (A.13)–(A.24) are well-known for coupled
linear heterodirectional hyperbolic PDE-ODE systems, and their well-posedness
has been proved in Theorem 4.1 of [39].

(b) Second-step transformation

The gain kernels $K_{1},K_{2}$ and $\eta$ are defined below:

$\displaystyle dK_{1x}(x,y)+q_{1}K_{1y}(x,y)=-d_{1}K_{1}(x,y),$ (A.25)
$\displaystyle dK_{2x}(x,y)-q_{2}K_{2y}(x,y)=-d_{4}K_{2}(x,y),$ (A.26)
$\displaystyle{d}\eta^{\prime}(x)A_{\rm m}^{-1}+\eta(x)=0,$ (A.27)

with the boundary conditions

$\displaystyle
K_{1}(1,y)=\frac{1}{c_{0}}{\bar{\Psi}}(1,y)-\frac{1}{c_{0}}q{\bar{\phi}}(1,y),$
(A.28) $\displaystyle
K_{2}(1,y)=\frac{1}{c_{0}}{\bar{\Phi}}(1,y)-\frac{1}{c_{0}}q{\bar{\varphi}}(1,y),$
(A.29) $\displaystyle K_{1}(x,1)=\frac{qq_{2}}{q_{1}}K_{2}(x,1),$ (A.30)
$\displaystyle q_{1}pK_{1}(x,0)+q_{2}K_{2}(x,0)=\eta(x)B,$ (A.31)
$\displaystyle\eta(1)=-\frac{1}{c_{0}}q\bar{\gamma}(1)+\frac{1}{c_{0}}\bar{\lambda}(1).$
(A.32)

The proof of well-posedness of (A.25)–(A.32) is given in Lemma 2 in [5]. To
derive conditions (A.25)–(A.32), one needs to consider (21) and (23). Hence,
(24) holds straightforwardly under the conditions (A.28), (A.29), (A.32).
Taking the time and spatial derivatives of (23), inserting the results into
(25), recalling (17)–(20), (24), one obtains

$\displaystyle u_{t}(x,t)+du_{x}(x,t)-q_{2}K_{2}(x,1)c_{0}u(1,t)$
$\displaystyle=$ $\displaystyle
v_{t}(x,t)+\int_{0}^{1}K_{1}(x,y)\alpha_{t}(y,t)dy+\int_{0}^{1}K_{2}(x,y)\beta_{t}(y,t)dy$
$\displaystyle+dv_{x}(x,t)+d\int_{0}^{1}K_{1x}(x,y)\alpha(y,t)dy$
$\displaystyle+d\int_{0}^{1}K_{2x}(x,y)\beta(y,t)dy$
$\displaystyle+\eta(x)\dot{X}(t)+d\eta^{\prime}(x){X}(t)-q_{2}K_{2}(x,1)c_{0}u(1,t)$
$\displaystyle=$ $\displaystyle
v_{t}(x,t)-q_{1}\int_{0}^{1}K_{1}(x,y)\alpha_{x}(y,t)dy$
$\displaystyle+d_{1}\int_{0}^{1}K_{1}(x,y)\alpha(y,t)dy$
$\displaystyle+q_{2}\int_{0}^{1}K_{2}(x,y)\beta_{x}(y,t)dy+d_{4}\int_{0}^{1}K_{2}(x,y)\beta(y,t)dy$
$\displaystyle+dv_{x}(x,t)+d\int_{0}^{1}K_{1x}(x,y)\alpha(y,t)dy$
$\displaystyle+d\int_{0}^{1}K_{2x}(x,y)\beta(y,t)dy+\eta(x)(A_{\rm
m}{X}(t)+B\beta(0,t))$
$\displaystyle+d\eta^{\prime}(x){X}(t)-q_{2}K_{2}(x,1)c_{0}u(1,t)$
$\displaystyle=$
$\displaystyle\left(q_{2}K_{2}(x,1)q-q_{1}K_{1}(x,1)\right)\alpha(1,t)$
$\displaystyle+\int_{0}^{x}\left(q_{1}K_{1y}(x,y)+d_{1}K_{1}(x,y)+dK_{1x}(x,y)\right)\alpha(y,t)dy$
$\displaystyle-\left(q_{2}K_{2}(x,0)+q_{1}K_{1}(x,0)p-\eta(x)B\right)\beta(0,t)$
$\displaystyle+\int_{0}^{x}\left(d_{4}K_{2}(x,y)-q_{2}K_{2y}(x,y)+dK_{2x}(x,y)\right)\beta(y,t)dy$
$\displaystyle+\left(\eta(x)A_{\rm m}+d\eta^{\prime}(x)\right){X}(t)=0.$
(A.33)

The necessary and sufficient conditions for (A.33) to hold are given as
(A.25)–(A.27), (A.30), (A.31).

(c) Third-step transformation

The derivation of the gain kernels PDE $R$ and $R^{I}$ is performed as
follows. Substituting the time and spatial derivatives of (27) into (25) and
recalling (28)–(30), we have

$\displaystyle u_{t}(x,t)+du_{x}(x,t)-q_{2}K_{2}(x,1)c_{0}u(1,t)$
$\displaystyle=$
$\displaystyle\hat{u}_{t}(x,t)+\int_{x}^{1}R(x,y)\hat{u}_{t}(y,t)dy+d\hat{u}_{x}(x,t)$
$\displaystyle+d\int_{x}^{1}R_{x}(x,y)\hat{u}(y,t)dy-dR(x,x)\hat{u}(x,t)$
$\displaystyle-q_{2}K_{2}(x,1)c_{0}\hat{u}(1,t)$ $\displaystyle=$
$\displaystyle-d\int_{x}^{1}R(x,y)\hat{u}_{x}(y,t)dy+d\int_{x}^{1}R_{x}(x,y)\hat{u}(y,t)dy$
$\displaystyle-dR(x,x)\hat{u}(x,t)-q_{2}K_{2}(x,1)c_{0}\hat{u}(1,t)$
$\displaystyle=$ $\displaystyle-dR(x,1)\hat{u}(1,t)+dR(x,x)\hat{u}(x,t)$
$\displaystyle+d\int_{x}^{1}R_{y}(x,y)\hat{u}(y,t)dy+d\int_{x}^{1}R_{x}(x,y)\hat{u}(y,t)dy$
$\displaystyle-dR(x,x)\hat{u}(x,t)-q_{2}K_{2}(x,1)c_{0}\hat{u}(1,t)$
$\displaystyle=$ $\displaystyle-(dR(x,1)+q_{2}K_{2}(x,1)c_{0})\hat{u}(1,t)$
$\displaystyle+d\int_{x}^{1}(R_{x}(x,y)+R_{y}(x,y))\hat{u}(y,t)dy$
$\displaystyle=$ $\displaystyle 0.$ (A.34)

For (A.34) to hold, the following equality must be satisfied:

$\displaystyle R_{x}(x,y)+R_{y}(x,y)=0,$ (A.35) $\displaystyle
dR(x,1)=-q_{2}c_{0}K_{2}(x,1),$ (A.36)

which obviously admits a unique solution

$R(x,y)=-\frac{q_{2}c_{0}}{d}K_{2}(x-y+1,1).$

Similarly, substituting the time and spatial derivatives of (32) into (29) and
recalling (25), we have

$\displaystyle\hat{u}_{t}(x,t)+d\hat{u}_{x}(x,t)$ $\displaystyle=$
$\displaystyle u_{t}(x,t)+\int_{x}^{1}P(x,y)u_{t}(y,t)dy+du_{x}(x,t)$
$\displaystyle+d\int_{x}^{1}P_{x}(x,y)u(y,t)dy-dP(x,x)u(x,t)$ $\displaystyle=$
$\displaystyle q_{2}K_{2}(x,1)c_{0}u(1,t)-d\int_{x}^{1}P(x,y)u_{x}(y,t)dy$
$\displaystyle+\int_{x}^{1}P(x,y)q_{2}K_{2}(y,1)c_{0}dyu(1,t)$
$\displaystyle+d\int_{x}^{1}P_{x}(x,y)u(y,t)dy-dP(x,x)u(x,t)$ $\displaystyle=$
$\displaystyle q_{2}K_{2}(x,1)c_{0}u(1,t)-dP(x,1)u(1,t)+dP(x,x)u(x,t)$
$\displaystyle+d\int_{x}^{1}P_{y}(x,y)u(y,t)dy+\int_{x}^{1}P(x,y)q_{2}K_{2}(y,1)c_{0}dyu(1,t)$
$\displaystyle+d\int_{x}^{1}P_{x}(x,y)u(y,t)dy-dP(x,x)u(x,t)$ $\displaystyle=$
$\displaystyle\big{(}q_{2}K_{2}(x,1)c_{0}-dP(x,1)$
$\displaystyle+\int_{x}^{1}P(x,y)q_{2}K_{2}(y,1)c_{0}dy\big{)}u(1,t)$
$\displaystyle+d\int_{x}^{1}(P_{y}(x,y)+P_{x}(x,y))u(y,t)dy=0.$

The equation above suggests that the kernel function $P$ in the inverse
transformation (32) satisfies the following PDE with the corresponding
boundary value:

$\displaystyle P_{y}(x,y)-P_{x}(x,y)=0,$ (A.37) $\displaystyle
P(x,1)=\frac{q_{2}}{d}K_{2}(x,1)c_{0}+\frac{1}{d}\int_{x}^{1}P(x,y)q_{2}K_{2}(y,1)c_{0}dy,$
(A.38)

whose well-posedness can be obtained by the method of characteristics.

###  VIII-B Expressions of the controller gain functions $M_{1}$, $M_{2}$,
$M_{3}$, $M_{4}$

The functions $M_{1}$, $M_{2}$, $M_{3}$, and $M_{4}$ are given as follows,

$\displaystyle M_{1}(y)=$ $\displaystyle\int_{0}^{1}R(0,s)K_{1}(s,y)ds-
K_{1}(0,y)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},s)K_{1}(s,y)dsd{s_{1}}$
$\displaystyle-\int_{y}^{1}\bigg{[}\int_{0}^{1}R(0,y)K_{1}(y,s)dy-K_{1}(0,s)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},y)K_{1}(y,s)dyd{s_{1}}\bigg{]}{\phi}(s,y)ds$
$\displaystyle-\int_{y}^{1}\bigg{[}\int_{0}^{1}R(0,y)K_{2}(y,s)dy-K_{2}(0,s)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},y)K_{2}(y,s)dyd{s_{1}}\bigg{]}{\Psi}(s,y)ds,$
$\displaystyle M_{2}(y)=$
$\displaystyle-\int_{y}^{1}\bigg{[}\int_{0}^{1}R(0,y)K_{1}(y,s)dy-K_{1}(0,s)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},y)K_{1}(y,s)dyd{s_{1}}\bigg{]}{\varphi}(s,y)ds$
$\displaystyle+\int_{0}^{1}R(0,s)K_{2}(s,y)ds-K_{2}(0,y)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},s)K_{2}(s,y)dsd{s_{1}}$
$\displaystyle-\int_{y}^{1}\bigg{[}\int_{0}^{1}R(0,y)K_{2}(y,s)dy-K_{2}(0,s)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},y)K_{2}(y,s)dyd{s_{1}}\bigg{]}{\Phi}(s,y)ds,$
$\displaystyle M_{3}(y)=$ $\displaystyle R(0,y)+\int_{0}^{y}R(0,s)P(s,y)ds,$
$\displaystyle M_{4}=$ $\displaystyle\int_{0}^{1}R(0,y)\eta(y)dy-\eta(0)$
$\displaystyle+\int_{0}^{1}R(0,y)\int_{y}^{1}P(y,s)\eta(s)dsdy$
$\displaystyle-\int_{0}^{1}\bigg{[}\int_{0}^{1}R(0,s)K_{2}(s,y)ds-K_{2}(0,y)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},s)K_{2}(s,y)dsd{s_{1}}\bigg{]}\lambda(y)dy$
$\displaystyle-\int_{0}^{1}\bigg{[}\int_{0}^{1}R(0,s)K_{1}(s,y)ds-K_{1}(0,y)$
$\displaystyle+\int_{0}^{1}R(0,s_{1})\int_{s_{1}}^{1}P(s_{1},s)K_{1}(s,y)dsd{s_{1}}\bigg{]}\gamma(y)dy,$

where $K_{1},K_{2},\eta,R,P$ are parametrized by the unknown delay
$D=\frac{1}{d}$ according to the conditions defined in Appendix A.

###  VIII-C Norms equivalence between the original and the target systems’
states

From (13)–(16), (27), (32), we get

$\displaystyle\|\alpha(\cdot,t)\|^{2}\leq$
$\displaystyle\eta_{1}\bigg{(}\|z(\cdot,t)\|^{2}+\|w(\cdot,t)\|^{2}+|X(t)|^{2}\bigg{)},$
(C.1) $\displaystyle\|\beta(\cdot,t)\|^{2}\leq$
$\displaystyle\eta_{2}\bigg{(}\|z(\cdot,t)\|^{2}+\|w(\cdot,t)\|^{2}+|X(t)|^{2}\bigg{)},$
(C.2) $\displaystyle\|z(\cdot,t)\|^{2}\leq$
$\displaystyle\eta_{3}\bigg{(}\|\alpha(\cdot,t)\|^{2}+\|\beta(\cdot,t)\|^{2}+|X(t)|^{2}\bigg{)},$
(C.3) $\displaystyle\|w(\cdot,t)\|^{2}\leq$
$\displaystyle\eta_{4}\bigg{(}\|\alpha(\cdot,t)\|^{2}+\|\beta(\cdot,t)\|^{2}+|X(t)|^{2}\bigg{)},$
(C.4) $\displaystyle\|u(x,t)\|^{2}\leq$
$\displaystyle\eta_{5}\|\hat{u}(x,t)\|^{2},$ (C.5)
$\displaystyle\|\hat{u}(x,t)\|^{2}\leq$ $\displaystyle\eta_{6}\|u(x,t)\|^{2},$
(C.6)

where

$\displaystyle\eta_{1}=$ $\displaystyle
4\bigg{(}1+\int_{0}^{1}\int_{0}^{x}{\phi}(x,y)^{2}dydx+\int_{0}^{1}\int_{0}^{x}{\varphi}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\gamma(x)^{2}dx\bigg{)},$ $\displaystyle\eta_{2}=$
$\displaystyle
4\bigg{(}1+\int_{0}^{1}\int_{0}^{x}{\Psi}(x,y)^{2}dydx+\int_{0}^{1}\int_{0}^{x}{\Phi}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\lambda(x)^{2}dx\bigg{)},$ $\displaystyle\eta_{3}=$
$\displaystyle
4\bigg{(}1+\int_{0}^{1}\int_{0}^{x}{\bar{\phi}}(x,y)^{2}dydx+\int_{0}^{1}\int_{0}^{x}{\bar{\varphi}}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\bar{\gamma}(x)^{2}dx\bigg{)},$
$\displaystyle\eta_{4}=$ $\displaystyle
4\bigg{(}1+\int_{0}^{1}\int_{0}^{x}{\bar{\Psi}}(x,y)^{2}dydx+\int_{0}^{1}\int_{0}^{x}{\bar{\Phi}}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\bar{\lambda}(x)^{2}dx\bigg{)},$
$\displaystyle\eta_{5}=$ $\displaystyle
2\bigg{(}1+\int_{0}^{1}\int_{x}^{1}R(x,y)^{2}dydx\bigg{)},$
$\displaystyle\eta_{6}=$ $\displaystyle
2\bigg{(}1+\int_{0}^{1}\int_{x}^{1}P(x,y)^{2}dydx\bigg{)}.$

Recalling (23), together with (C.1), (C.2), (C.5), (C.6), yields

$\displaystyle||v(\cdot,t)||^{2}\leq$ $\displaystyle
4\bigg{(}\eta_{5}+\int_{0}^{1}\int_{0}^{1}K_{1}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}K_{2}(x,y)^{2}dydx+\int_{0}^{1}\eta(x)^{2}dx\bigg{)}\big{(}\|\hat{u}(\cdot,t)\|^{2}$
$\displaystyle+||\alpha(\cdot,t)||^{2}+||\beta(\cdot,t)||^{2}+|{X}(t)|^{2}\big{)},$
(C.7) $\displaystyle||\hat{u}(\cdot,t)||^{2}\leq$ $\displaystyle
4\eta_{6}(\eta_{1}+\eta_{2}+1)\bigg{(}1+\int_{0}^{1}\int_{0}^{1}K_{1}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}K_{2}(x,y)^{2}dydx+\int_{0}^{1}\eta(x)^{2}dx\bigg{)}\big{(}||v(\cdot,t)||^{2}$
$\displaystyle+\|z(\cdot,t)\|^{2}+\|w(\cdot,t)\|^{2}+|X(t)|^{2}\big{)}.$ (C.8)

Defining

$\displaystyle\bar{\Omega}(t)=\|\alpha[t]\|^{2}+\|\beta[t]\|^{2}+\|\hat{u}[t]\|^{2}+{X}(t)^{2}$
(C.9)

one obtains

$\displaystyle\xi_{1}\Omega(t)\leq\bar{\Omega}(t)\leq\xi_{2}\Omega(t)$ (C.10)

where

$\displaystyle\xi_{1}=$ $\displaystyle
1/\bigg{(}1+\eta_{3}+\eta_{4}+4\eta_{5}+4\int_{0}^{1}\int_{0}^{1}K_{1}(x,y)^{2}dydx$
$\displaystyle+4\int_{0}^{1}\int_{0}^{1}K_{2}(x,y)^{2}dydx+4\int_{0}^{1}\eta(x)^{2}dx\bigg{)},$
(C.11) $\displaystyle\xi_{2}=$ $\displaystyle 1+\eta_{1}+\eta_{2}$
$\displaystyle+4\eta_{6}(\eta_{1}+\eta_{2}+1)\bigg{(}1+\int_{0}^{1}\int_{0}^{1}K_{1}(x,y)^{2}dydx$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}K_{2}(x,y)^{2}dydx+\int_{0}^{1}\eta(x)^{2}dx\bigg{)}.$
(C.12)

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