UC San Diego

This dissertation introduces a control mechanism for addressing neuronal growth problems, which can be applied to neurological disorders such as spinal cord injuries, Parkinson's disease, and Alzheimer's disease that limit neuronal functionality. We consider a recent medical therapy, Chondroitinase ABC (ChABC), as a control mechanism for these conditions. ChABC aims to treat these conditions by restoring neuron functionality through axon growth for damaged neurons. It manipulates the extracellular matrix (ECM), a network of macromolecules and minerals that surrounds neurons and regulates their activity. As a result, neurons produce tubulin proteins, which cause the axon to elongate. This process is modeled as a Partial Differential Equation (PDE), representing the behavior of tubulin concentration along the axon, with a moving boundary governed by Ordinary Differential Equations (ODE) consisting of the dynamics of the axon length and tubulin concentration in the growth cone. In this dissertation, we propose nonlinear design methods for a novel state feedback control law, an observer, and an output feedback control law for a one-dimensional model of axonal elongation. We demonstrate the robustness of the model to parameter changes of up to 40\% relative to the original design and analysis framework. We also address potential challenges, such as input delay, and propose a compensation mechanism to overcome these issues. In addition to theoretical challenges, we enhance the practical applicability of the proposed control law by introducing an event-triggered control mechanism that allows users to update the control law in a sample-based manner. We ensured local exponential stability and convergence of the closed-loop system, integrating the plant dynamics with the proposed control law across all these techniques. The performance of the designed control methods was validated through numerical simulations, demonstrating neuron elongation by up to three orders of magnitude. These advancements offer promising avenues for enhancing neural regeneration therapies and contribute significantly to the understanding of neural growth dynamics, while also advancing theoretical control of Stefan-type moving boundary PDE-ODE coupled systems.