  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.
