  Neurological studies show that injured neurons can regain their functionality
with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics
promote axon elongation by manipulating the injured neuron and its
intercellular space to modify tubulin protein concentration. This fundamental
protein is the source of axon elongation, and its spatial distribution is the
state of the axon growth dynamics. Such dynamics often contain time delays
because of biological processes. This work introduces an input delay
compensation with state-feedback control law for axon elongation by regulating
tubulin concentration. Axon growth dynamics with input delay is modeled as
coupled parabolic diffusion-reaction-advection Partial Differential Equations
(PDE) with a boundary governed by Ordinary Differential Equations (ODE),
associated with a transport PDE. A novel feedback law is proposed by using
backstepping method for input-delay compensation. The gain kernels are provided
after transforming the interconnected PDE-ODE-PDE system to a target system.
The stability analysis is presented by applying Lyapunov analysis to the target
system in the spatial H1-norm, thereby the local exponential stability of the
original error system is proved by using norm equivalence.
