# Periodic Event-Triggered Boundary Control of Neuron Growth with Actuation at
Soma

Cenk Demir1, Mamadou Diagne1 and Miroslav Krstic1 1Department of Mechanical
and Aerospace Engineering, UC San Diego, 9500 Gilman Drive, La Jolla, CA,
92093-0411, cdemir@ucsd,edu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Exploring novel strategies for the regulation of axon growth, we introduce a
periodic event-triggered control (PETC) to enhance the practical
implementation of the associated PDE backstepping control law. Neurological
injuries may impair neuronal function, but therapies like Chondroitinase ABC
(ChABC) have shown promise in improving axon elongation by influencing the
extracellular matrix. This matrix, composed of extracellular macromolecules
and minerals, regulates tubulin protein concentration, potentially aiding in
neuronal recovery. The concentration and spatial distribution of tubulin
influence axon elongation dynamics. Recent research explores feedback control
strategies for this model, leading to the development of an event-triggering
control (CETC) approach. In this approach, the control law updates when the
monitored triggering condition is met, reducing actuation resource
consumption. Through the meticulous redesign of the triggering mechanism, we
introduce a periodic event-triggering control (PETC), updating control inputs
at specific intervals, but evaluating the event-trigger only periodically, an
ideal tool for standard time-sliced actuators like ChABC. PETC is a step
forward to the design of practically feasible feedback laws for the neuron
growth process. The PETC strategy establishes an upper bound on event triggers
between periodic examinations, ensuring convergence and preventing Zeno
behavior. Through Lyapunov analysis , we demonstrate the local exponential
convergence of the system with the periodic event-triggering mechanism in the
$L^{2}$-norm sense. Numerical examples are presented to confirm the
theoretical findings.

##  I Introduction

Neurons, as fundamental components of neural networks, are intricately
involved in sensory processing [37]. They play a critical role in acquiring
and interpreting sensory information by transmitting electrical signals
through their specialized structures. This communication occurs between
neurons via their axons, which function as cellular wires. Axons consist of
groups of proteins called “tubulin.” The dynamics of these proteins facilitate
the elongation of axons, allowing them to reach the target neuron, establish
synaptic connections, and complete the transmission process. However,
neurological diseases or injuries can disrupt or completely halt this
transmission such as Alzheimer’s disease [27] and spinal cord injuries [26].
In such complications, neurons may degenerate, leading to axonal shrinkage or
an inability to reach target neurons for signal transmission. Until recently,
the prevailing belief was that injured neurons could not regenerate to
complete the transmission process [19]. However, recent research has clarified
that regeneration is possible under certain conditions. A recently developed
therapy method, Chondroitinase ABC, shows promising potential for regenerating
neurons [3, 21]. With this therapy, it is possible to stimulate neuron
elongation.

Advancements in ChABC and other therapy methods could help neuron
regeneration, but the key factor of this regeneration will be the degree of
neuronal elongation which is a process regulated by tubulin dynamics. This
dynamical process is modeled with different mathematical approaches and
assumptions [40, 15]. However, a more comprehensive model for tubulin-based
axon elongation is proposed in [30]. This model comprises a diffusion-
reaction-advection partial differential equation (PDE) governing tubulin
evolution along the axon, alongside an ordinary differential equation
describing tubulin evolution at the axon’s far end, namely the growth cone,
and the axon’s length over time. This specific model takes the form of a
Stefan-type PDE, a well-studied model in the literature [16]. Furthermore,
mathematical stability analysis and the effect of model parameters of the axon
elongation are detailed in [29].

In recent times, PDE systems have garnered significant attention across
various disciplines. Control system engineering, in particular, has emerged as
a prominent field offering control strategies for these systems. A pioneering
research direction in this domain is boundary control of PDE systems which is
based on the work [25]. Following these important initial studies, researchers
have broadened their focus to include backstepping-based boundary control of
various types of PDEs, systems that combine PDEs with ODEs, as well as systems
that involve multiple interacting PDEs [24, 38, 39]. While previous studies
focus on constant domain size in time, there are significant works on the
global results of moving domains in time [12, 31, 28, 5, 13, 20, 9]. This line
of work has been extended to derive local stability results for moving
boundary nonlinear hyperbolic PDEs [2, 4, 41]. Achieving stability for
nonlinear parabolic PDEs with moving boundaries, especially without using the
maximum principle, has been challenging until our recent study on axonal
growth [6, 7].

While the control methods discussed above operate continuously, certain
technologies necessitate interventions only when needed, driven by limitations
in energy, communication, and computation [17]. These constraints suggest
control actions that are executed only when needed and thereby enhance
actuation resource usage, which is called the event-triggering control. This
concept was initially developed for linear systems in [1, 18] and nonlinear
systems in [22]. Following these establishments, it was proposed for infinite
dimensional systems as event-triggered boundary control of PDE systems in
[14]. Static and dynamic triggering mechanisms have been developed for ODEs
and various classes of PDEs, specifically, for Stefan problem in [32, 33].
Periodic event-triggered control (PETC) and self-triggered control are
proposed in [35] and [36], respectively, for a class of reaction diffusion
PDEs. Moreover, this method finds application in ensuring the safety and
convergence to the set point of the Stefan problems in the presence of
actuator dynamics in [23].

In addressing the neuron growth problem, a dynamic event-triggering mechanism
for a coupled PDE-nonlinear ODE with a moving boundary is introduced in [8].
Motivated by the feasibility challenges encountered in real-time
implementation when applying feedback control to biological systems for
medical treatements, we convert the continuous time event-triggered control
(CETC) design by supplementing a periodic sampling rule. This introduces a
dynamic periodic event-triggering control (PETC) approach, where the
triggering function is only checked periodically while the control input being
updated aperiodically. The PETC improves the practical implementation of the
control law because it can be applied to standard time-sliced actuators (like
ChABC) for axon growth. The strategy involves deriving a novel triggering
condition and establishing an upper bound on the continuous-time event trigger
between two periodic examinations, explicitly derived as a sampling period in
our study. The problem statement in our current study diverges from that
addressed in [34], where observer-based PETC was utilized for the classical
one-phase Stefan problem. While the outcome in [34] pertains to a system
exhibiting _geometric nonlinearity,_ the neuron growth process involves both
_geometrical and analytical nonlinearities_ leading to a _local_ convergence
result. The PETC boundary controller guarantees $L^{2}$ local exponential
convergence in closed-loop, akin CETC.

The paper’s structure is as follows: Section II introduces the tubulin-driven
axon growth model, including the steady-state solution and reference error
analysis, and control law design. Section III presents the event-triggering
mechanism and results from prior work. Section IV introduces a novel periodic
event-triggering control for the axon growth problem. Finally, Section V
presents numerical simulation results.

##  II Tubulin-Driven Axon Growth Modeling and Control

This section introduces the tubulin-driven axonal growth model, a coupled
moving boundary PDE with ODEs, and a locally exponentially stabilizing
continuous-time boundary control law.

###  II-A Understanding axon growth

####  II-A1 A model with a moving boundary PDE

Tubulin, a collection of proteins, facilitates the development of a newly
formed axon. Under the assumption that unattached tubulin molecules along the
axon is insignificant, and considering that only tubulin molecules are
accountable for axon growth, the newborn evolution in time and space can be
modeled as follows [11, 10].

$\displaystyle c_{t}(x,t)=D$ $\displaystyle c_{xx}(x,t)-ac_{x}(x,t)-gc(x,t),$
(1) $\displaystyle c_{x}(0,t)+c(0,t)$ $\displaystyle=-q_{\rm s}(t),$ (2)
$\displaystyle c(l(t),t)=c_{\rm c}$ $\displaystyle(t),$ (3)
$\displaystyle\dot{c}_{\rm c}(t)=\tilde{a}$ $\displaystyle c_{\rm c}(t)-\beta
c_{x}(l(t),t)-\kappa c_{\rm c}^{2}(t)+c_{\infty}\tilde{r}_{\rm g},$ (4)
$\displaystyle\dot{l}(t)=r_{\rm g}$ $\displaystyle(c_{c}(t)-c_{\infty})$ (5)

where the constants in (4) are

$\displaystyle\tilde{a}_{1}=\frac{a-r_{\rm g}c_{\infty}}{l_{\rm
c}}-g-\tilde{r}_{\rm g},\quad\beta=\frac{D}{l_{\rm
c}},\quad\kappa=\frac{r_{\rm g}}{l_{\rm c}}.$ (6)

In this model, $c(x,t)$ is the tubulin concentration in the axon, varying with
the spatial variable $x$ over time $t$, while $q$ represents the combined
tubulin flux and concentration at soma. The axon length is denoted $l(t),$
which is the distance between the soma and the growth cone. The subscripts $s$
and $c$ are used for the soma and the growth cone, respectively. This notation
can be seen in Fig. 1. The parameters, $D$, $a$ and $g$ in (1) are tubulin
diffusivity, velocity, and degradation constants, respectively. The parameters
in (4) and (5) include $l_{\rm c}$ representing the growth ratio,
$\tilde{r}_{\rm g}$ denoting the reaction rate of the microtubules production
process, $c_{\infty}$ as the equilibrium of tubulin concentration in the cone,
and $r_{\rm g}$ serving as a lumped parameter. Detailed descriptions and
derivations of $r_{\rm g}$ and other parameters are available in [11].

Figure 1: Schematic of neuron and state variables

####  II-A2 The steady-state solution and reference error system

We derive a steady-state solution for the concentration, corresponding to a
desired axon length $l_{\rm s}$, by setting the time derivatives in (1), (4),
and (5) to zero. The steady-state spatially distributed steady-state tubulin
concentration is

$\displaystyle c_{\rm eq}(x)=c_{\infty}\left(K_{+}e^{\lambda_{+}(x-l_{\rm
s})}+K_{-}e^{\lambda_{-}(x-l_{\rm s})}\right),$ (7)

where

$\displaystyle\lambda_{\pm}=$
$\displaystyle\frac{a\pm\sqrt{a^{2}+4Dg}}{2D},\quad
K_{\pm}=\frac{1}{2}\pm\frac{a-2gl_{\rm c}}{2\sqrt{a^{2}+4Dg}},$ (8)

and the steady-state input for the combination of tubulin flux and
concentration in the soma is

$\displaystyle q_{\rm
s}^{*}=-c_{\infty}\left(K_{+}(1+\lambda_{+})e^{-\lambda_{+}l_{\rm
s}}+K_{-}(1+\lambda_{-})e^{-\lambda_{-}l_{\rm s}}\right).$ (9)

The reference error system relative to (1)-(5) is given by the dynamics of
errors given below

$\displaystyle u_{t}(x,t)=Du_{xx}(x,t)-au_{x}(x,t)-gu(x,t),$ (10)
$\displaystyle u_{x}(0,t)+u(0,t)=U(t),$ (11) $\displaystyle
u(l(t),t)=h(X(t)),$ (12)
$\displaystyle\dot{X}(t)=AX(t)+f(X(t))+Bu_{x}(l(t),t),$ (13)

where the reference error states, $u(x,t)$, $z_{1}(t)$ and $z_{2}(t)$, and the
reference error input $U(t)$ are defined as

$\displaystyle u(x,t)=$ $\displaystyle c(x,t)-c_{\rm eq}(x),\quad
U(t)=-(q_{\rm s}(t)-q_{\rm s}^{*})$ (14) $\displaystyle z_{1}(t)=$
$\displaystyle c_{\rm c}(t)-c_{\infty},\quad z_{2}(t)=l(t)-l_{\rm s}.$ (15)

Here, $X$ is a state vector in $\mathbb{R}^{2}$, given as

$\displaystyle X(t)=[z_{1}(t)\quad z_{2}(t)]^{\top}.$ (16)

The parameters and the functions in (10)-(13) are defined as follows:

$\displaystyle A$
$\displaystyle=\left[\begin{array}[]{cc}\tilde{a}_{1}&-\beta\tilde{a}_{2}\\\
r_{\rm g}&0\end{array}\right],\quad B=\left[\begin{array}[]{c}-\beta\\\
0\end{array}\right],$ (21) $\displaystyle f(X(t))$ $\displaystyle=-\kappa
z_{1}^{2}(t)+\beta f_{1}(z_{2}(t)),$ (22) $\displaystyle h(X(t))$
$\displaystyle=z_{1}(t)+\tilde{h}(z_{2}(t)),$ (23)
$\displaystyle\tilde{h}(z_{2}(t))$
$\displaystyle=c_{\infty}\left(1-K_{+}e^{\lambda_{+}z_{2}(t)}-K_{-}e^{\lambda_{-}z_{2}(t)}\right).$
(24) $\displaystyle\tilde{a}_{2}=$ $\displaystyle
c_{\infty}\left(\lambda_{+}^{2}K_{+}+\lambda_{-}^{2}K_{-}\right),$ (25)
$\displaystyle f_{1}(z_{2}(t))=$ $\displaystyle-
c_{\infty}\left(K_{+}\lambda_{+}e^{\lambda_{+}z_{2}(t)}+K_{-}\lambda_{-}e^{\lambda_{-}z_{2}(t)}\right)$
$\displaystyle+\tilde{a}_{2}z_{2}(t)+c_{\infty}\frac{a-gl_{\rm c}}{D}.$ (26)

###  II-B Control Law Design

####  II-B1 Linearization of the finite-dimensional part of the cascade system

We begin by linearizing the nonlinear ODEs defined in (13) around zero states:

$\displaystyle u_{t}(x,t)=$ $\displaystyle Du_{xx}(x,t)-au(x,t)-gu(x,t),$ (27)
$\displaystyle u_{x}(0,t)+u(0,t)=$ $\displaystyle U(t),$ (28) $\displaystyle
u(l(t),t)=$ $\displaystyle H^{\top}X(t),$ (29) $\displaystyle\dot{X}(t)=$
$\displaystyle A_{1}X(t)+Bu_{x}(l(t),t),$ (30)

where the vector $H\in\mathbb{R}^{2}$ is defined as

$\displaystyle A_{1}$
$\displaystyle=\left[\begin{array}[]{cc}\tilde{a}_{1}&\tilde{a}_{3}\\\ r_{\rm
g}&0\end{array}\right],\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}~{}H=\left[1\quad-\frac{(a-gl_{\rm
c})c_{\infty}}{D}\right]^{\top},$ (33)

where $\tilde{a}_{3}=\frac{a^{2}+Dg-agl_{\rm c}}{D^{2}}$. By applying the
following backstepping transformation

$\displaystyle w(x,t)=$ $\displaystyle u(x,t)-\int_{x}^{l(t)}k(x,y)u(y,t)dy$
$\displaystyle-\phi(x-l(t))^{\top}X(t),$ (34)

we can map the linearized reference error system to a desired target system
which is

$\displaystyle w_{t}(x,$ $\displaystyle t)=Dw_{xx}(x,t)-aw_{x}(x,t)-gw(x,t)$
$\displaystyle-\dot{l}(t)F(x,X(t)),$ (35) $\displaystyle w_{x}(0,t)+$
$\displaystyle w(0,t)=-\frac{1}{D}\left(H-\epsilon\right)^{\top}Bu(0,t),$ (36)
$\displaystyle w(l(t)$ $\displaystyle,t)=\epsilon^{\top}X(t),$ (37)
$\displaystyle\dot{X}$
$\displaystyle(t)=(A_{1}+BK^{\top})X(t)+Bw_{x}(l(t),t),$ (38)

with the redundant nonlinear term $F(x,X(t))\in R$ in (II-B1) is described as

$\displaystyle
F(x,X(t))=\left(\phi^{\prime}(x-l(t))^{T}-k(x,l(t))C^{T}\right)X(t),$ (39)

where $K\in\mathbb{R}^{2}$ is chosen as

$\displaystyle k_{1}>\frac{\tilde{a}_{1}}{\beta},\quad
k_{2}>\frac{\tilde{a}_{3}}{\beta}.$ (40)

to make $A_{1}+BK$ Hurwitz and $\epsilon\in\mathbb{R}^{2}$ will be chosen in
the stability analysis.

The approach for obtaining gain kernels in (II-B1), namely $k(x,y)$ and
$\phi(x)$, is detailed in [6]. Simply, $k(x,y)$ and $\phi(x)$ are obtained as:

$\displaystyle k(x,y)=-\frac{1}{D}\phi(x-y)^{\top}B,$ (41)
$\displaystyle\phi(x)^{\top}=\begin{bmatrix}(H-\epsilon)^{\top}&K^{\top}-\frac{1}{D}H^{\top}BH^{\top}\end{bmatrix}e^{N_{1}x}\begin{bmatrix}I\\\
0\end{bmatrix},$ (42)

where the matrix $N_{1}\in R^{4\times 4}$ is defined as

$\displaystyle
N_{1}=\begin{bmatrix}0&\frac{1}{D}\left(gI+A+\frac{a}{D}BH^{\top}\right)\\\
I&\frac{1}{D}\left(BH^{\top}+aI\right)\end{bmatrix}.$ (43)

The inverse transformation is presented as follows:

$\displaystyle u(x,t)=$ $\displaystyle w(x,t)+\int_{x}^{l(t)}q(x,y)w(y,t)dy$
$\displaystyle+\varphi(x-l(t))^{\top}X(t)$ (44)

Detailed solutions for the gain kernels are provided in [6].

####  II-B2 Continuous-time and sampled-data control law

By taking the spatial derivative of the transformation and substituting $x=0$
into both the backstepping transformation and its spatial derivative, and
setting boundary condition (36), the control law is derived as

$\displaystyle U(t)=-\frac{1}{D}\int_{0}^{l(t)}p(x)Bu(x,t)dx+p(l(t))X(t),$
(45)

where

$p(x)=\phi^{\prime}(-x)^{\top}+\phi(-x)^{\top}.$ (46)

The system outlined in (1)-(5), with the continuous-time controller input
(45), is locally exponentially stable in the $L^{2}$-norm sense, as
demonstrated in [8]. To develop the periodic event-triggered control
mechanism, the CTC input is sampled at discrete intervals, which holds it
constant between events. This approach yields the following sampled-data
control.

$\displaystyle U_{k}^{\omega}(t):=U(t_{k}^{\omega}),$ (47)

where is employed at

$\displaystyle u_{x}(0,t)+u(0,t)=U_{k}^{\omega}(t).$ (48)

for $\forall t\in[t_{k}^{\omega},t_{k+1}^{\omega})$, $k\in\mathbb{N}$ with the
increasing time sequence, $I^{\omega}=\\{t_{k}^{\omega}\\}_{k\in\mathbb{N}}$,
where $t_{0}^{\omega}=0$ and $\omega=\\{``c",``p"\\}$. The notations $``c"$
and $``p"$ represent CETC and PETC, respectively. It’s important to note that
at each sampling time, the control input is sampled from (45), in other word,
the sampled-data control law is the emulation of the continuous-time
controller that is to be implemented in a Zero-Order Hold fashion.

##  III Continuous-time Event Triggered control

In this section, we provide a summary of a CETC design detailed in [8].

###### Definition 1:

Continuous-time event-triggering consists of two stages: the occurrence of the
event and the application of the control signal when the event occurs. These
steps are

  1. 1.

Detection of the time that event occurs: The set
$I^{c}=\\{t^{c}_{0},t^{c}_{1},t^{c}_{2},\ldots\\}$ where $t^{c}_{0}=0$,
represents an increasing sequence of time instances where the events occur.
This set is generated using the following rules. (i) If
$S(t,t_{k}^{c})=\emptyset$, events occur at times
$\\{t_{0},\ldots,t_{k}^{c}\\}$. (ii) If $S(t,t_{k}^{c})\neq\emptyset$, the
subsequent event time is determined as
$t_{k+1}^{c}=\inf\left(S(t,t_{k}^{c})\right)$ where

$\displaystyle S(t,t_{k}^{c})=\\{t\in R_{+}|t>t_{k}^{c}\wedge d^{2}(t)>-\gamma
m(t)\\}$ (49)

for all $t\in[t_{k}^{c},t_{k+1}^{c})$, $d(t)$ is given as

$\displaystyle d(t)=U(t)-U_{k}^{c}(t)$ (50)

and $m(t)$ satisfies the ODE

$\displaystyle\dot{m}$ $\displaystyle(t)=-\eta m(t)+\rho
d(t)^{2}-\beta_{1}X(t)^{2}-\beta_{2}X(t)^{4}$
$\displaystyle-\beta_{3}X(t)^{6}-\beta_{4}|w(0,t)|^{2}-\beta_{5}||w(x,t))||^{2}.$
(51)

The event-triggering design parameters are $\sigma\in(0,1)$, $\gamma>0$,
$\eta>0$, and $\beta_{i}$ and $\rho$ are selected according to the
specifications outlined in [8] as

$\displaystyle\rho\geq\frac{d_{1}^{2}D}{\delta_{1}},\quad\beta_{i}=\frac{\alpha_{i}}{\gamma(1-\sigma)},$
(52)

where $\delta_{1}=2D^{2}$.

  2. 2.

Event-based control corresponding to the sampled-data control law (47) where
$p(x)$ is defined in (46).

###### Remark 1:

The triggering mechanism defined by (49) and (51) has the property of
$d^{2}(t)\leq\gamma m(t)$ and $m(t)>0,$ $\forall~{}t\in[0,\sup\\{I^{c}\\})$ as
detailed in [8].

###### Remark 2:

Considering the increasing set of event-times
$\\{t_{k}^{c}\\}_{k\in\mathbb{N}}$ with $t_{0}=0$, the following bound is
obtained for the time derivative of the input holding error

$\displaystyle\dot{d}^{2}(t)\leq$
$\displaystyle\rho_{1}d^{2}(t)+\alpha_{1}X(t)^{2}+\alpha_{2}X(t)^{4}+\alpha_{3}X(t)^{6}$
$\displaystyle+\alpha_{4}w(0,t)^{2}+\alpha_{5}||w(x,t)||^{2},$ (53)

where the parameters,
$\rho_{1},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}$ are given by

$\displaystyle\rho_{1}$ $\displaystyle=7|p(0)B|^{2},$ (54)
$\displaystyle\alpha_{1}$
$\displaystyle=\frac{21}{2}\left|\frac{1}{D}\zeta(y)B\int_{0}^{l(t)}\varphi(x-l(t))^{\top}dx\right|^{2}+28(p(0)Bp(l(t)))^{2}$
$\displaystyle+$ $\displaystyle
21\left(\left(p(0)\left(1-\frac{a}{D}\right)+\dot{p}(0)\right)B\right)^{2}(\varphi(0)^{\top})^{2}+28(p(l(t))A)^{2}$
$\displaystyle+$ $\displaystyle
14\left(\left|\dot{p}(l(t))+\frac{a}{D}p(l(t))+\frac{r_{\rm
g}}{D}e_{1}p(l(t))\right|BH^{\top}\right)^{2},$ (55) $\displaystyle\alpha_{2}$
$\displaystyle=7\left(r_{\rm
g}e_{1}\dot{p}(l(t))+2k_{n}\left|\dot{p}(l(t))+\frac{a}{D}p(l(t))\right|B+p(l(t))\kappa\right)^{2}$
(56) $\displaystyle\alpha_{3}$ $\displaystyle=7\left(\frac{2k_{n}}{D}r_{\rm
g}e_{1}p(l(t))B\right)^{2}+28\left(k_{m}p(l(t))\right)^{2},$ (57)
$\displaystyle\alpha_{4}$
$\displaystyle=21\left(\left(p(0)\left(1-\frac{a}{D}\right)+\dot{p}(0)\right)B\right)^{2},$
(58) $\displaystyle\alpha_{5}$
$\displaystyle=7\left|\frac{1}{D}\zeta(y)B\right|^{2}\left(\frac{9}{2}+\frac{9}{2}\left(\int_{0}^{l(t)}\int_{x}^{l(t)}q(x,y)^{2}dydx\right)\right)$
$\displaystyle+$ $\displaystyle
21\left(\left(p(0)\left(1-\frac{a}{D}\right)+\dot{p}(0)\right)B\right)^{2}\bar{G}(l(t))^{2},$
(59)
$\displaystyle\zeta(y):=\int_{0}^{l(t)}D\ddot{p}(y)-a\dot{p}(y)+gp(y)-p(0)Bp(y)dy,$
(60) $\displaystyle\bar{G}(l(t)):=\int_{0}^{l(t)}q(0,x)dx.$ (61)

###### Theorem 1:

[8] For the event-triggered mechanism described in (47)-(49), the set of
event-times $\\{t_{k}^{c}\\}_{k\in\mathbb{N}}$ ensures that the function
$\Gamma^{c}(t):=d(t)^{2}-\gamma m(t)$ remains non-positive for all
$t\in[t_{k}^{c},t_{k+1}^{c})$, where $k\in\mathbb{N}$.

The proof of this theorem and the following results are given in [8] and the
following hold:

  1. 1.

The set of event-times $\\{t^{c}_{k}\\}_{k\in\mathbb{N}}$ with triggering
mechanism (47)-(49) and with the design parameters specified in (54)-(59),
ensures that Zeno behavior does not occur. This is because there exists a
minimal dwell-time, $\tau>0$, between two execution times, given by

$\displaystyle\tau=\int_{0}^{1}\frac{1}{a_{1}s^{2}+a_{2}s+a_{3}}ds,$ (62)

where

$\displaystyle a_{1}$ $\displaystyle=\rho\sigma\gamma>0,$ (63) $\displaystyle
a_{2}$ $\displaystyle=1+2\rho_{1}+(1-\sigma)\rho+\eta>0,$ (64) $\displaystyle
a_{3}$
$\displaystyle=(1+\rho_{1}+\gamma(1-\sigma)\rho+\eta)\frac{1-\sigma}{\sigma}>0.$
(65)

  2. 2.

Given an initial condition $m(0)<0,$ the variable $m(t)$ governed by (51),
satisfies $m(t)<0$ for all $t>0$.

  3. 3.

The closed-loop system (1)-(5), along with the event-triggered mechanism (47),
locally exponentially converges to the desired axon length in the
$L^{2}$-sense.

In the next section, we propose a periodic event-triggering mechanism.

##  IV Periodic Event Triggering Mechanism

In this section, we propose a periodic event-triggering mechanism for axonal
growth.

###### Definition 2:

Consider the event-triggering function $\Gamma^{p}(t)$, which undergoes
periodic evaluation with a period of $h>0$. The PETC that generates the events
are characterized by two parts:

  1. 1.

The event-trigger mechanism: A periodic event-trigger that determines the
event times

$\displaystyle t_{k+1}^{p}=\inf\\{t\in\mathbb{R}_{+}|t>t_{k}^{p},$
$\displaystyle~{}\Gamma^{p}(t)>0,~{}t=nh,$
$\displaystyle~{}h>0,~{}n\in\mathbb{N}\\},$ (66)

with $t_{0}^{p}=0$ where $h$ is sampling period and

$\displaystyle\Gamma^{p}(t)=\upsilon_{1}d^{2}(t)-\upsilon_{2}m(t)$ (67)

where $\upsilon_{1}>0$ and $\upsilon_{2}>0$.

  2. 2.

The feedback control law that is derived as

$\displaystyle U_{k}^{p}(t)=$
$\displaystyle-\frac{1}{D}\int_{0}^{l(t_{k}^{p})}p(x)Bu(x,t_{k}^{p})dx+p(l(t_{k}^{p}))X(t_{k}^{p})$
(68)

for all $t\in[t_{k}^{p},t_{k+1}^{p})$ for $k\in\mathbb{N}$.

Note that periodicity in the triggering conditions (66), allows us to monitor
the triggering function _periodically_ and update the control laws
_aperiodically,_ removing the continuous monitoring of the PDE-ODE state
variables. Then, the boundary condition (11) becomes

$\displaystyle u_{x}(0,t)+u(0,t)=U(t_{k}^{p}).$ (69)

###  IV-A Design of the periodic event triggering function $\Gamma^{p}(t)$

Selection of the sampling period. The sampling period, denoted as $h$,
represents the unit of time during which the control input is updated. Let the
periodic event-triggered function given by (66), along with the boundary
condition in (69) and the plant dynamics from (1)-(5), satisfy the condition
$\Gamma^{p}(t)\leq 0$ for all $t$ within the interval
$t\in[t_{k}^{p},t_{k+1}^{p})$ for $k\in\mathbb{N}$. Hence, it follows that
$m(t)<0$ for all $t>0$. The parameter $h$ is selected to satisfy

$\displaystyle 0<h\leq\tau,$ (70)

where the upper bound, $\tau$, is the minimum inter-event time of the CETC
design defined in (62)-(65).

###### Proposition 1:

Under the definition of the periodic event-triggered boundary control (69),
with the sampling period $h<\tau$, it holds that

$\displaystyle\Gamma^{c}(t)\leq\frac{1}{q}\bigg{(}(a+\gamma\rho)d^{2}(nh)$
$\displaystyle e^{q(t-nh)}-\gamma\rho d^{2}(nh)$ $\displaystyle+q\gamma
m(nh)e^{-\eta(t-nh)}\bigg{)},$ (71)

for all $t\in[nh,(n+1)h)$ and any
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$, where $q=1+\eta+\rho_{1}$
and $\Gamma^{c}(t)=d^{2}(t)-\gamma m(t)$ for $\gamma>0$.

###### Proof.

Taking the time derivative of $\Gamma^{c}(t)$ in $t\in[nh,(n+1)h)$ and
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$, one can show that

$\displaystyle\dot{\Gamma}^{c}(t)$
$\displaystyle=2d(t)\dot{d}(t)-\gamma\dot{m}(t)$ (72) $\displaystyle\leq
d^{2}(t)+\dot{d}^{2}(t)-\gamma\dot{m}(t).$ (73)

By using Lemma 1, we get

$\displaystyle\dot{\Gamma}^{c}(t)\leq$
$\displaystyle(1+\rho_{1}+\gamma\rho)d^{2}(t)-\left(\gamma\beta_{1}-\alpha_{1}\right)X(t)^{2}$
$\displaystyle-\left(\gamma\beta_{2}-\alpha_{2}\right)X(t)^{4}-\left(\gamma\beta_{3}-\alpha_{3}\right)X(t)^{6}$
$\displaystyle-\left(\gamma\beta_{4}-\alpha_{4}\right)u(0,t)^{2}-\left(\gamma\beta_{5}-\alpha_{5}\right)||u(x,t)||^{2}$
$\displaystyle+\eta\gamma m(t).$ (74)

By using the definition of $\Gamma^{c}(t)$, we get

$\displaystyle\dot{\Gamma}^{c}(t)\leq$
$\displaystyle(1+\rho_{1}+\gamma\rho)\Gamma^{c}(t)-\left(\gamma\beta_{1}-\alpha_{1}\right)X(t)^{2}$
$\displaystyle-\left(\gamma\beta_{2}-\alpha_{2}\right)X(t)^{4}-\left(\gamma\beta_{3}-\alpha_{3}\right)X(t)^{6}$
$\displaystyle-\left(\gamma\beta_{4}-\alpha_{4}\right)u(0,t)^{2}-\left(\gamma\beta_{5}-\alpha_{5}\right)||u(x,t)||^{2}$
$\displaystyle+\left(\left(1+\rho_{1}+\gamma\rho\right)\gamma+\eta\gamma\right)m(t).$
(75)

Since $m(t)$ satisfies Remark 1 and (1)-(5) with the event-triggered control
law (47) is locally exponentially convergen, (75) exhibit smooth behavior in
the interval $t\in[nh,(n+1)h)$ and for any
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$. This establishes the
existence of a non-negative function $\iota(t)\in
C^{0}((t_{k}^{p},t_{k+1}^{p});\mathbb{R}_{+})$ such that:

$\displaystyle\dot{\Gamma}^{c}(t)=$
$\displaystyle(1+\rho_{1}+\gamma\rho)\Gamma(t)-\left(\gamma\beta_{1}-\alpha_{1}\right)X(t)^{2}$
$\displaystyle-\left(\gamma\beta_{2}-\alpha_{2}\right)X(t)^{4}-\left(\gamma\beta_{3}-\alpha_{3}\right)X(t)^{6}$
$\displaystyle-\left(\gamma\beta_{4}-\alpha_{4}\right)u(0,t)^{2}-\left(\gamma\beta_{5}-\alpha_{5}\right)||u(x,t)||^{2}$
$\displaystyle+\left(\left(1+\rho_{1}+\gamma\rho\right)\gamma+\eta\gamma\right)m(t)-\iota(t),$
(76)

for all $t\in[nh,(n+1)h)$ and for any
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$. Moreover, through the
substitution of $d^{2}(t)=\Gamma^{c}(t)+\gamma m(t)$, we can rephrase the
dynamics of $m(t)$ as follows:

$\displaystyle\dot{m}(t)=$
$\displaystyle-\rho\Gamma^{c}(t)-\left(\eta+\rho\gamma\right)m(t)+\beta_{1}X(t)^{2}+\beta_{2}X(t)^{4}$
$\displaystyle+\beta_{3}X(t)^{6}+\beta_{4}|u(0,t)|^{2}+\beta_{5}||u(x,t))||^{2},$
(77)

for all $t\in[nh,(n+1)h)$ and for any
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$. Subsequently, by combining
(76) with (77), we can derive the subsequent system of ODEs:

$\displaystyle\dot{r}(t)=A_{1}r(t)+v(t),$

where

$\displaystyle r(t)=\left[\begin{array}[]{c}\Gamma^{c}(t)\\\
m(t)\end{array}\right],A_{1}=\left[\begin{array}[]{cc}q-\eta+\gamma\rho&\gamma\left(q+\gamma\rho\right)\\\
-\rho&-\eta-\rho\gamma\end{array}\right],$ (82) $\displaystyle
v(t)=\left[\begin{array}[]{c}f_{1}(t)\\\ f_{2}(t)\end{array}\right],$ (85)

where

$\displaystyle f_{1}(t)=$
$\displaystyle-\left(\gamma\beta_{1}-\alpha_{1}\right)X(t)^{2}-\left(\gamma\beta_{2}-\alpha_{2}\right)X(t)^{4}$
$\displaystyle-\left(\gamma\beta_{3}-\alpha_{3}\right)X(t)^{6}-\left(\gamma\beta_{4}-\alpha_{4}\right)u(0,t)^{2}$
$\displaystyle-\left(\gamma\beta_{5}-\alpha_{5}\right)||u(x,t)||^{2}-\iota(t),$
(86) $\displaystyle f_{2}(t)=$
$\displaystyle\beta_{1}X(t)^{2}+\beta_{2}X(t)^{4}+\beta_{3}X(t)^{6}+\beta_{4}|u(0,t)|^{2}$
$\displaystyle+\beta_{5}||u(x,t))||^{2},$ (87)

and

$\displaystyle q=1+\eta+\rho_{1}.$ (88)

The solution to (85) for all $t\in[nh,(n+1)h)$ and for any
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$ can be expressed as:

$\displaystyle
r(t)=e^{A_{1}(t-nh)}r(nh)+\int_{nh}^{t}e^{A_{1}(t-\xi)}v(\xi)d\xi,$ (89)

which gives us

$\displaystyle\Gamma^{c}(t)=C_{1}e^{A_{1}(t-nh)}r(nh)+C_{1}\int_{nh}^{t}e^{A_{1}(t-\xi)}v(\xi)d\xi,$
(90)

where $C_{1}=\left[1\quad 0\right]$. Since matrix $A_{1}$ has two distinct
eigenvalues, we can diagonalize the matrix exponential $e^{A_{1}t}$ as it is
defined in [35]. Thus, we can derive the second part of (90) as

$\displaystyle
C_{1}e^{A_{1}(t-\xi)}v(\xi)=\left[\begin{array}[]{cc}g_{1}(t)&g_{2}(t)\end{array}\right]\left[\begin{array}[]{c}f_{1}(t)\\\
f_{2}(t)\end{array}\right],$ (94)

where

$\displaystyle g_{1}(t)$
$\displaystyle=\frac{q+\gamma\rho}{q}e^{(1+\rho_{1})t}-\frac{\rho\gamma}{q}e^{-\eta
t},$ (95) $\displaystyle g_{2}(t)$
$\displaystyle=\frac{\gamma\left(q+\gamma\rho\right)}{q}\left(e^{(1+\rho_{1})t}-e^{-\eta
t}\right).$ (96)

Since we have the following relationship

$\displaystyle 1+\eta+7|p(0)B|^{2}>0,$ (97)

we can get

$\displaystyle g_{1}(t)$
$\displaystyle=\frac{1}{q}\left(-\gamma\rho+(q+\rho\gamma)e^{qt}\right)e^{-\eta
t},$ (98) $\displaystyle g_{2}(t)$
$\displaystyle=\frac{\gamma(q+\gamma\rho)}{q}\left(-1+e^{qt}\right)e^{-\eta
t}.$ (99)

It’s apparent that $g_{1}(t)$ remains positive for $t>0$. Furthermore,
considering the relation (52), and using ascending order of triggering times
that is the solution of (62) is represented by

$\displaystyle\tau=\frac{1}{q}\ln{\left(1+\frac{\sigma
q}{(1-\sigma)(q+\gamma\rho)}\right)},$ (100)

one can show that

$\displaystyle C_{1}$ $\displaystyle
e^{A_{1}(t-\xi)}v(\xi)=\frac{\alpha_{1}(q+\gamma\rho)}{q}\left(e^{q\tau}-e^{q(t-\xi)}\right)e^{-\eta(t-\xi)}X(t)^{2}$
$\displaystyle+\frac{\alpha_{2}(q+\gamma\rho)}{q}\left(e^{q\tau}-e^{q(t-\xi)}\right)e^{-\eta(t-\xi)}X(t)^{4}$
$\displaystyle+\frac{\alpha_{3}(q+\gamma\rho)}{q}\left(e^{q\tau}-e^{q(t-\xi)}\right)e^{-\eta(t-\xi)}X(t)^{6}$
$\displaystyle+\frac{\alpha_{4}(q+\gamma\rho)}{q}\left(e^{q\tau}-e^{q(t-\xi)}\right)e^{-\eta(t-\xi)}u(0,t)^{2}$
$\displaystyle+\frac{\alpha_{5}(q+\gamma\rho)}{q}\left(e^{q\tau}-e^{q(t-\xi)}\right)e^{-\eta(t-\xi)}||u(x,t)||^{2}.$
(101)

Given the stipulated intervals $nh\leq\xi\leq t\leq(n+1)h$, and $h\leq\tau$,
upon thorough examination of (101), it emerges that the inequality
$(\gamma\beta_{i}-\alpha_{i})g_{1}(t-\xi)-\beta_{i}g_{2}(t-\xi)>0$ satisfied
for all $i=1,2,3,4,5$. This observation prompts us to establish
$C_{1}e^{A_{1}(t-\xi)}v(\xi)$ which holds for all $t$ and $\xi$ within the
range of $nh\leq\xi\leq t\leq(n+1)h$, and for
$n\in[t_{k}^{p}/h,t_{k+1}^{p}/h)\subset\mathbb{N}$. Taking this observation
into account alongside (90), we can derive the following expression for
$t\in[nh,(n+1)h)$:

$\displaystyle\Gamma^{c}(t)$
$\displaystyle\leq\frac{1}{q}\left(-\gamma(q+\gamma\rho)m(nh)-\gamma\rho\Gamma^{c}(nh)\right.$
$\displaystyle\left.+(q+\gamma\rho)\left(\Gamma^{c}(nh)+\gamma
m(nh)\right)e^{q(t-nh)}\right).$ (102)

Upon performing the substitution $\Gamma^{c}(nh)$ into (102), we are able to
derive the inequality (71) which is valid for all $t\in[nh,(n+1)h)$. This
concludes the proof. ∎

Building upon Lemma 2, the update time for the control input can be determined
by identifying when the subsequent condition is met for any $t\in[nh,(n+1)h)$,
thereby challenging the positive definiteness of $\Gamma^{c}(t)$.

$\displaystyle(q+\gamma\rho)d^{2}(nh)e^{q(t-nh)}-\gamma\rho d^{2}(nh)+q\gamma
m(nh)>0,$ (103)

Thus, one can choose this condition as $\Gamma^{p}(t)$ such that

$\displaystyle\Gamma^{p}(t)=(q+\gamma\rho)e^{qh}d^{2}(t)-\gamma\rho
d^{2}(t)+q\gamma m(t),$ (104)

which completes the design process.

###### Theorem 2:

Let the design parameters, $\rho$, $\rho_{1}$ and $\beta_{i}$ as defined in
(52), (54)-(59), set the sampling rate in accordance with (70), let
$\gamma,\eta>0$ and $\sigma\in(0,1)$. Let us consider the periodic event-
triggering mechanism (66)-(68) with the $\Gamma^{p}(t)$ as defined in (104)
which generates the increasing sequence of times
$\\{t_{k}^{p}\\}_{k\in\mathbb{N}}$ with $t_{0}^{p}=0$. Then, for
$\Gamma^{c}(t)$ and $m(t)$ with $m(t)>0$, it holds that $\Gamma^{c}(t)\leq 0$
and $m(t)>0$ for all $t>0$.

###### Proof.

Due to space constraints, we omit this proof, which can be stated following
the steps of the proof of Theorem 2 in [35]. ∎

###  IV-B Local exponential convergence under PETC

In order to prove that the closed-loop system (1)-(5) with the control law
(47) and the periodic event-triggering mechanism (66) and (104), is locally
exponentially convergent, we first obtain the following target system by
applying transformation (II-B1)

$\displaystyle
w_{t}(x,t)=Dw_{xx}(x,t)-aw_{x}(x,t)-gw(x,t)-\dot{l}(t)F(x,X(t))$
$\displaystyle\quad\quad\quad-\phi(x-l(t))^{\top}f(X(t))-G(x,l(t))h^{*}(X),$
(105) $\displaystyle
w_{x}(0,t)+w(0,t)=d(t)-\frac{1}{D}\left(H-\epsilon\right)^{\top}Bu(0,t),$
(106) $\displaystyle w(l(t),t)=h^{*}(X(t))+\epsilon^{\top}X(t),$ (107)
$\displaystyle\dot{X}(t)=(A+BK)X(t)+f(X(t))+Bw_{x}(l(t),t),$ (108)

where
$G(x,l(t)):=\left(\phi^{\prime}(x-l(t))^{\top}+\frac{a}{D}\phi(x-l(t))^{\top}\right)B.$
Using the transformation below

$\displaystyle\varpi(x,t)=w(x,t)-h^{*}(X(t))$ (109)

converts (IV-B)-(108) into

$\displaystyle\varpi_{t}(x,t)=D\varpi_{xx}(x,t)-a\varpi_{x}(x,t)-g\varpi(x,t)$
$\displaystyle\quad+gh^{*}(X(t))-\dot{l}(t)F(x,X(t))-\dot{h}^{*}(X(t))B\varpi_{x}(l(t),t)$
$\displaystyle\quad-\phi(x-l(t))^{\top}f(X(t))-G(x,l(t))h^{*}(X)$
$\displaystyle\quad-\dot{h}^{*}(X(t))\left((A+BK)X(t)+f(X(t))\right),$ (110)
$\displaystyle\varpi_{x}(0,t)+\varpi(0,t)=d(t)-\frac{1}{D}\left(H-\epsilon\right)^{\top}Bu(0,t)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+h^{*}(X(t)),$ (111)
$\displaystyle\varpi(l(t),t)=\epsilon^{\top}X(t),$ (112)
$\displaystyle\dot{X}(t)=(A+BK)X(t)+f(X(t))+B\varpi_{x}(l(t),t).$ (113)

Below, we state the convergence result.

###### Theorem 3:

Let the design parameters, $\rho$, $\rho_{1}$ and $\beta_{i}$ given as defined
in Theorem 2. Consider the periodic event-triggering rule (66)-(68) with the
periodic event-triggering function (104) and sampling rate $h$ defined in
(70), which generates an increasing event-times
$\\{t_{k}^{p}\\}_{k\in\mathbb{N}}$. Assuming the well-posedness, the closed-
loop system of (1)-(5) with the boundary control law (104) and (46) is locally
exponentially convergent in $L^{2}$-norm sense.

###### Proof.

To demonstrate the local convergence of the system, we initially establish the
system properties in a non-constant spatial interval as derived in [8],
outlined as follows:

$\displaystyle 0<l(t)\leq\bar{l},\quad|\dot{l}(t)|\leq\bar{v}$ (114)

for some $\bar{l}>l_{\rm s}>0$ and $\bar{v}=\frac{D}{16(D+1)}$. As
demonstrated in Theorem 2 of [8], $m(t)<0$ for all
$t\in[t_{k}^{p},t_{k+1}^{p})$ where $k\in\mathbb{N}$, implying
$\Gamma^{c}(t)\leq 0$ for $t\in[t_{k}^{p},t_{k+1}^{p})$. Assuming the well-
posedness of the closed-loop system and following the methodology outlined in
[8], the subsequent Lyapunov functional is considered

$\displaystyle V(t)=V_{1}(t)-m(t),$ (115)

where

$\displaystyle V_{1}(t)=$ $\displaystyle
d_{1}\frac{1}{2}\int_{0}^{l(t)}\varpi(x,t)^{2}dx+X(t)^{\top}\left(d_{2}P_{1}+\frac{1}{2}P_{2}\right)X(t)$
(116)

and $d_{1}>0$, $d_{2}>0$, $P_{1}\succ 0$ and $P_{2}\succeq 0$ are positive
definite and positive semidefinite matrices satisfying the Lyapunov equations:

$\displaystyle(A+BK^{\top})^{\top}P_{1}+P_{1}(A+BK^{\top})=-Q_{1},$
$\displaystyle(A+BK^{\top})^{\top}(P_{1}+P_{2})+(P_{1}+P_{2})(A+BK^{\top})=-Q_{2}$

where

$\displaystyle P_{1}=\begin{bmatrix}p_{1,1}&p_{1,2}\\\
p_{1,2}&p_{2,2}\end{bmatrix},\quad
P_{2}=\begin{bmatrix}\frac{D\epsilon_{1}}{\beta}-2p_{1,1}&0\\\
0&0\end{bmatrix}$ (117)

where we pick $\epsilon\in\mathbb{R}^{2}$ as $\epsilon_{1}\geq 2l_{\rm
c}p_{1,1}$ and $\epsilon_{2}=\frac{p_{1,2}}{l_{\rm c}d_{1}}$ for some positive
definite matrices $Q_{1}\succ 0$ and $Q_{2}\succ 0$. By taking the time
derivative of (116), applying Poincaré’s, Agmon’s, and Young’s inequalities,
we first derive the following expression:

$\displaystyle\dot{V}\leq$
$\displaystyle-\alpha^{*}V+\xi_{1}V^{3/2}+\xi_{2}V^{2}+\xi_{3}V^{5/2}+\xi_{4}V^{3}$
(118)

where

$\displaystyle\alpha^{*}$
$\displaystyle=\min\left\\{\frac{g}{2},\frac{1}{2\lambda_{\min}(P_{1}+P_{2})},\eta\right\\},$
(119) $\displaystyle\xi_{1}$
$\displaystyle=\frac{\left(Dd_{1}|\epsilon\bar{B}|+2d_{2}\left|P_{1}^{\top}B\bar{B}\right|\right)\kappa^{2}+\frac{d_{1}r_{\rm
g}}{2}(1+L_{1})+r_{\rm g}}{d_{2}^{3/2}\lambda_{\min}(P_{1}+P_{2})^{3/2}},$
(120) $\displaystyle\xi_{2}$
$\displaystyle=\frac{\Xi_{1}}{d_{2}^{2}\lambda_{\min}(P_{1}+P_{2})^{2}},\quad\xi_{3}=\frac{4d_{2}k_{m}|P_{1}|}{d_{2}^{5/2}\lambda_{\min}(P_{1}+P_{2})^{5/2}},$
(121) $\displaystyle\xi_{4}$
$\displaystyle=\frac{\Xi_{2}}{d_{2}^{3}\lambda_{\min}(P_{1}+P_{2})^{3}},$
(122)

where $d_{1}$ and $d_{2}$ are chosen to satisfy

$\displaystyle d_{1}$
$\displaystyle\geq\max\left\\{\frac{8\bar{l}\left(D+2\right)+16\bar{l}\beta_{4}}{D},\frac{4\beta_{5}+7}{g}\right\\},$
(123) $\displaystyle d_{2}$
$\displaystyle\geq\frac{4}{\lambda_{\min}(Q_{2})}\left(Dd_{1}\left|\epsilon\bar{B}|(A+BK)\right|+\beta_{1}\right)$
$\displaystyle+\frac{4}{\lambda_{\min}(Q_{2})}\left(\left(D+2+Dd_{1}+\frac{d_{1}a}{2}+2\beta_{4}\right)\frac{2}{\beta^{2}}\right).$
(124)

Note that the positive constant parameters are also defined as follows:

$\displaystyle F(0,X(t))^{2}\leq L_{1}X^{\top}X,$ (125)
$\displaystyle\int_{0}^{l(t)}\left(\phi(x-l(t))^{\top}\right)^{2}dx\leq
L_{n_{2}},$ (126)
$\displaystyle\int_{0}^{l(t)}\left(\phi^{\prime}(x-l(t))^{\top}B-ak(x,l(t))\right)^{2}dx\leq
L_{n_{3}}$ (127)
$\displaystyle\Xi_{1}=4Dd_{1}|\epsilon\bar{B}|k_{m}^{2}|P_{1}|^{2}+8d_{2}\left|P_{1}^{\top}B\bar{B}\right|k_{m}^{2}|P_{1}|^{2}+\beta_{2}$
$\displaystyle\quad+2d_{1}^{2}L_{n_{3}}k_{n}^{2}+\frac{d_{1}^{2}}{2}L_{n_{2}}\kappa^{2}+d_{1}^{2}c_{\infty}^{2}r_{\rm
g}^{2}k_{l}+8d_{2}\kappa|P|\beta_{5}k_{l}\ $
$\displaystyle\quad+2d_{2}\kappa|P|\left(d_{1}^{2}\left(\beta^{2}(1-\epsilon_{1})^{2}\left(1+\bar{G}(l(t))^{2}\right)+D\right)\right)k_{l},$
(128) $\displaystyle\Xi_{2}=d_{1}^{2}c_{\infty}^{2}r_{\rm
g}^{2}k_{l}+\frac{d_{1}^{2}}{2}L_{n_{2}}4k_{m}^{2}|P_{1}|^{2}+\beta_{3}$
$\displaystyle\quad+\left(d_{1}^{2}\left(\beta^{2}(1-\epsilon_{1})^{2}\left(1+\bar{G}(l(t))^{2}\right)+D\right)+4\beta_{5}\right)k_{l},$
(129) $\displaystyle
k_{l}=\max\left\\{\left|K_{+}\lambda_{+}\right|,\left|K_{-}\lambda_{-}\right|\right\\}^{2},\quad\bar{B}=[-\beta^{-1}~{}0],$
(130) $\displaystyle
k_{n}=c_{\infty}\max\\{K_{+}\lambda_{+}^{2},K_{-}\lambda_{-}^{2}\\},$ (131)
$\displaystyle
k_{m}=c_{\infty}\max\\{K_{+}\lambda_{+}^{3},K_{-}\lambda_{-}^{3}\\},$ (132)
$\displaystyle-e^{x}+x+1\leq x^{2}\quad\text{for}\quad x\leq 1.79.$ (133)

Given (118), we can demonstrate that within the region
$\Omega_{1}:=\\{(\varpi,X)\in L^{2}\times\mathbb{R}^{2}|V(t)<M_{0}\\}$ where
$t\in(t_{k}^{p},t_{k+1}^{p})$ for $k\in\mathbb{N}$, there exists a positive
constant $M_{0}>0$ ensuring the satisfaction of the system properties (114).
The existence of such $M_{0}>0$ is established in Lemma 2 of [6]. From this
result, we have $M_{0}=\frac{\lambda_{\rm min}(P_{1})}{d_{2}}r^{2}$ where
$r=\min\left\\{\frac{\bar{v}}{r_{\rm g}},l_{\rm s},\bar{l}-l_{\rm s}\right\\}$
for $t\in(t_{k}^{p},t_{k+1}^{p})$, $k\in\mathbb{N}$. Next, we analyze the
convergence within the time interval $t\in(t_{k}^{p},t_{k+1}^{p})$ for
$k\in\mathbb{N}$, and subsequently for $t\in(0,t)$. Then, a positive constant
$M$ exists such that when $V(t_{j})<M$, the following norm estimate is valid
for $t\in[t_{k}^{p},t_{k+1}^{p})$, where $k\in\mathbb{N}$:

$\displaystyle V(t_{k+1}^{p})\leq
V(t_{k}^{p})e^{-\frac{\alpha^{*}}{2}(t_{k+1}^{p}-t_{k}^{p})}.$ (134)

For $M>0$, we define the set $\Omega:=\\{(\varpi,X)\in
L^{2}\times\mathbb{R}^{2}|V(t)<M\\}$. From Lemma 2 in [6], it is clear that if
$M\leq M_{0}$, then $\Omega\subset\Omega_{1}$ which satisfy the system
properties (114) and the norm estimate defined in (134) for
$t\in[t_{k}^{p},t_{k+1}^{p})$, where $k\in\mathbb{N}$. Hence, we set $M\leq
p^{*}$, where $p^{*}$ is a non-zero root of the following polynomial

$\displaystyle-\alpha^{*}V+\xi_{1}V^{3/2}+\xi_{2}V^{2}+\xi_{3}V^{5/2}+\xi_{4}V^{3}=0$
(135)

for $V>0$. Given that all coefficients of this polynomial are positive, at
least one positive root $p^{*}$ exists. Thus, (118) implies

$\displaystyle\dot{V}\leq-\frac{\alpha^{*}}{2}V(t)$ (136)

for $t\in[t_{k}^{p},t_{k+1}^{p})$, where $k\in\mathbb{N}$ and
$M=\min\\{M_{0},p^{*}\\}$. The smoothness of $V(t)$ within this interval
ensures that $V(t_{k+1}^{p^{-}})=V(t)$ and $V(t_{k}^{p^{+}})=V(t_{k}^{p})$,
where $t_{k}^{p^{+}}$ and $t_{k}^{p^{-}}$ denote the right and left limits of
$t=t_{k}^{p}$, respectively. Thus, we can have the norm estimate in (134).
Then, for any $t\geq 0$ in $t\in[t_{k}^{p},t_{k+1}^{p})$ where
$k\in\mathbb{N}$, we have

$\displaystyle V(t)\leq e^{-\alpha^{*}(t-t_{k}^{p})}V(t_{k}^{p})\leq
e^{-\alpha^{*}t}V(0).$ (137)

Recalling $m(t)<0$ and (115), we can write

$\displaystyle V_{1}(t)-m(t)\leq e^{-\alpha^{*}t}V(0)$ (138)

by applying the comparison principle one can obtain the following norm
estimate for the target system $(\varpi,X)$:

$\displaystyle
d_{1}\frac{1}{2}||\varpi(x)||^{2}+d_{2}X(t)^{\top}\left(P_{1}+\frac{1}{d_{2}}P_{2}\right)X(t)$
$\displaystyle\leq$ $\displaystyle
e^{-\alpha^{*}t}\left(\frac{d_{1}}{2}||\varpi(0)||^{2}+d_{2}X(0)^{\top}\left(P_{1}+\frac{1}{d_{2}}P_{2}\right)X(0)\right)$
$\displaystyle-e^{-\alpha^{*}t}m(0)$ (139)

Utilizing the invertibility of the transformation (109), we subsequently prove
that the target system $(w,X)$ is also locally exponentially convergent. For
the original system $(u,X)$, we leverage the invertibility of the backstepping
transformation given in (II-B1). Consequently, we conclude that the closed-
loop system is also locally exponentially convergent. This completes the
proof. ∎

Figure 2: Comparison between periodic-event triggering control input
$U_{k}^{p}(t)$, continuous time event triggering control input $U_{k}^{c}(t)$
and the continuous control law $U(t)$

##  V Numerical Simulations

In this section, we conduct numerical simulations for the system represented
by equations (1)-(5), employing the control law (45) along with the designed
periodic event-triggering mechanism (66) utilizing the triggering function
(104). The model parameters are detailed in Table I. Initial conditions are
specified as $c_{0}(x)=1.5c_{\infty}$ for the tubulin concentration along the
axon and $l_{0}=1\mu m$ for the initial axon length. Control gain parameters
are set as $k_{1}=-0.001$ and $k_{2}=3\times 10^{13}$. The event-triggering
parameters are set as follows $m(0)=-0.5$, $\beta_{1}=2.5\times 10^{8}$,
$\beta_{2}=8\times 10^{9}$, $\beta_{3}=1\times 10^{11}$, $\beta_{4}=4\times
10^{11}$, $\beta_{5}=4.5\times 10^{11}$, $\rho=1.5\times 10^{-15}$,
$\gamma=1$, $\eta=2$ and $\sigma=0.8$. Moreover, the sampling period for the
periodic event-triggering mechanism is selected as $h=0.5~{}ms$ which is
smaller than the minimal dwell time $\tau\approx 0.54~{}ms$.

Fig. 2 illustrates the evolution of the continuous-time control input, $U(t)$,
the event-triggering control input, $U_{k}^{c}(t)$, as defined in (47) with
the triggering mechanism given by (49)-(51), and the periodic event-triggering
control input, $U_{k}^{p}(t)$, as defined in (68) with the triggering
condition in (66) and triggering function in (104). While PETC closely
emulates the CETC control input behavior, both PETC and CETC minimized the
necessity of control law updates by maintaining comparable performance. In
Fig. 3, tubulin concentration, $c(x,t)$, and axon length, $l(t)$, converge to
the steady-state solution of the tubulin concentration and the desired axon
length. Note that tubulin concentration exhibits smoother changes with the
PETC mechanism compared to the CETC mechanism which enhances the practical
applicability.

TABLE I: Biological constants and control parameters Parameter | Value | Parameter | Value
---|---|---|---
$D$ | $10\times 10^{-12}m^{2}/s$ | $\tilde{r}_{\rm g}$ | $0.053$
$a$ | $1\times 10^{-8}m/s$ | $\gamma$ | $10^{4}$
$g$ | $5\times 10^{-7}\ s^{-1}$ | $l_{\rm c}$ | $4\mu m$
$r_{\rm g}$ | $1.783\times 10^{-5}\ m^{4}/(mols)$ | $l_{s}$ | $12\mu m$
$c_{\infty}$ | $0.0119\ mol/m^{3}$ | $l_{0}$ | $1\mu m$

(a)

(b)

(c)

Figure 3: The tubulin concentration governed by (1)-(5), $c(x,t)$, converges
to the steady-state tubulin concentration, $c_{\rm eq}(t)$ by about $t=4.5$min
for both continuous control input, CETC and PETC. The axon length, $l(t)$,
also converges to the desired axon length, $l_{s}$, by about $t=4$min for all
sampling mechanisms.

##  VI Conclusion

This paper proposes a periodic-event triggering control of the axonal growth
problem which is modeled as coupled PDE and nonlinear ODE. The nature of the
actuation mechanism acting on the soma motivates the development of proof-
based sampled-and-hold control techniques for practicality. By virtue of
emulating the continuous-time feedback law and using recent techniques to
carefully refine a specific category of continuous-time dynamic event trigger,
we conceived a strategy that only requires periodic monitoring of the
triggering condition and aperiodic updates of the control action. Given the
vast number of neuron cells each with distinct parameters, future research
will consider the unknown model parameters by prioritizing the estimation of
these parameters and implementing adaptive control for this model. More
precisely, exploiting Batch Least Squares Identifiers (BaLSI) techniques
finite-time identification of the unknown parameters, and local exponential
convergence are challenging but promising.

## Acknowledgemts

The authors express appreciation to Bhathiya Rathnayake for engaging in
generous and selfless discussions on event-triggered control.

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