Funnel control for the monodomain equations with the FitzHugh-Nagumo model

TL;DR

This paper applies funnel control to the monodomain equations coupled with the FitzHugh-Nagumo model, enabling output trajectory tracking in cardiac defibrillation simulations with proven solution properties.

Contribution

It introduces a funnel control approach for nonlinear reaction diffusion systems modeling cardiac electrical activity, proving existence, uniqueness, and regularity of solutions.

Findings

Successful simulation of reentry wave termination

Proven existence and uniqueness of controlled system solutions

Demonstrated effective output tracking with funnel control

Abstract

We consider a nonlinear reaction diffusion system of parabolic type known as the monodomain equations, which model the interaction of the electric current in a cell. Together with the FitzHugh-Nagumo model for the nonlinearity they represent defibrillation processes of the human heart. We study a fairly general type with co-located inputs and outputs describing both boundary and distributed control and observation. The control objective is output trajectory tracking with prescribed performance. To achieve this we employ the funnel controller, which is model-free and of low complexity. The controller introduces a nonlinear and time-varying term in the closed-loop system, for which we prove existence and uniqueness of solutions. Additionally, exploiting the parabolic nature of the problem, we obtain H\"older continuity of the state, inputs and outputs. We illustrate our results by a simulation of a standard test example for the termination of reentry waves.