Optimal Control of Convective FitzHugh-Nagumo Equation

TL;DR

This paper develops and analyzes optimal control strategies for the convective FitzHugh-Nagumo equation, focusing on traveling wave solutions, using advanced numerical methods and validating second order optimality conditions.

Contribution

It introduces a numerical framework for controlling convective FitzHugh-Nagumo equations with traveling waves, including sparse control and validation of optimality conditions.

Findings

Numerical results demonstrate effective control of traveling waves.

Second order optimality conditions are validated numerically.

Distance estimates between controls and local optima are provided.

Abstract

We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian.