# On the One-Dimensional Nonlinear Monodomain Equations with Moving Controls

## TL;DR

This paper establishes local exact controllability for one-dimensional nonlinear monodomain equations with moving controls, using a new Carleman inequality and addressing models like FitzHugh-Nagumo and Rogers-McCulloch.

## Contribution

It introduces a novel Carleman inequality for the linearized adjoint equations with moving control support, enabling controllability results for nonlinear monodomain models.

## Key findings

- Proves null controllability at any positive time.
- Establishes local exact controllability to trajectories.
- Develops a new Carleman inequality for the adjoint system.

## Abstract

In this paper local exact controllability to the trajectories for the one-dimensional monodomain equations with the FitzHugh-Nagumo and Rogers-McCulloch ionic models using distributed controls with a moving support is investigated. In a first step a new Carleman inequality for the adjoint of the linearized monodomain equations, under assumptions on the movement of the control region, is presented. It leads to null controllability at any positive time. Subsequently, a local result concerning the exact controllability to the trajectories for the nonlinear monodomain equations is deduced.