Stabilization of inhomogeneous patterns in a diffusion-reaction system under structural and parametric uncertainties

Area of Science:

• Nonlinear dynamics
• Reaction-diffusion systems
• Control theory

Background:

• Spatiotemporal phenomena like neuron firing and heartbeats are modeled by reaction-diffusion systems.
• The FitzHugh-Nagumo (FHN) model simplifies these systems, capturing essential dynamic features.
• Disruptions in these spatiotemporal waves are linked to cardiac and neurological disorders.

Purpose of the Study:

• To develop nonlinear feedback controllers for stabilizing moving fronts in the FHN system.
• To address structural or parametric uncertainty within these systems.
• To provide a control framework for preventing disorders related to spatiotemporal wave disruptions.

Main Methods:

• Utilizing the dissipative nature of reaction-diffusion systems.
• Applying principles from finite-dimensional nonlinear control theory.
• Developing and proving the stability and robustness of novel feedback controllers.

Main Results:

• A class of nonlinear feedback controllers was successfully developed.
• The controllers ensure stabilization of moving fronts in the FHN model.
• Theoretical proofs and simulation experiments validated the controller's stability and robustness.

Conclusions:

• The developed controllers offer a method to stabilize moving fronts in FHN systems, even with uncertainties.
• This approach holds potential for preventing cardiac and neurological disorders.
• The feedback logic may also influence biochemical and cellular organization processes.