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In this work, stabilization of an axonal growth in a neuron associated withthe dynamics of tubulin concentration is proposed by designing a boundarycontrol. The dynamics are given by a parabolic Partial Differential Equation(PDE) of the tubulin concentration, with a spatial domain of the axon's lengthgoverned by an Ordinary Differential Equation (ODE) coupled with the tubulinconcentration in the growth cone. We propose a novel backstepping method forthe coupled PDE-ODE dynamics with a moving boundary, and design a control lawfor the tubulin concentration flux in the soma. Through employing the Lyapunovanalysis to a nonlinear target system, we prove a local exponential stabilityof the closed-loop system under the proposed control law in the spatial$H_1$-norm.

Neuron Growth Control by PDE Backstepping: Axon Length Regulation by Tubulin Flux Actuation in Soma