Input Delay Compensation for Neuron Growth by PDE Backstepping

Abstract

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 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 modelled as coupled parabolic diffusion-reaction-advection Partial Differential Equations (PDE) with a boundary governed by a nonlinear Ordinary Differential Equations (ODE), associated with a transport PDE. A novel feedback law is proposed by using backstepping method for PDE and 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.