  Neurological injuries predominantly result in loss of functioning of neurons.
These neurons may regain function after particular medical therapeutics, such
as Chondroitinase ABC (ChABC), that promote axon elongation by manipulating the
extracellular matrix, the network of extracellular macromolecules, and minerals
that control the tubulin protein concentration, which is fundamental to axon
elongation. We introduce an observer for the concentration of unmeasured
tubulin along the axon, as well as in the growth cone, using the measurement of
the axon length and the tubulin flux at the growth cone. We employ this
observer in a boundary control law which actuates the tubulin concentration at
the soma (nucleus), i.e., at the end of the axon distal from the measurement
location. For this PDE system with a moving boundary, coupled with a two-state
ODE system, we establish global exponential convergence of the observer and
local exponential stabilization of the [axon, observer] system in the spatial
$\mathcal{H}_1$-norm. The results require that the axon growth speed be
bounded. For an open-loop observer, this is ensured by assumption (which
requires that tubulin influx at the soma be limited), whereas for the
output-feedback system the growth rate of the axon is ensured by assuming that
the initial conditions of all the states, including the axon length, be
sufficiently close to their setpoint values.
