A new optimal control algorithm for the Keller-Segel problem

TL;DR

This paper introduces a novel optimal control algorithm for the Keller-Segel system, utilizing discretization, adjoint methods, and gradient descent to efficiently manage chemical substances in chemo-attraction models.

Contribution

The paper presents a new optimal control algorithm tailored for the Keller-Segel system, combining discretization, adjoint-based gradient computation, and gradient descent optimization.

Findings

The algorithm effectively controls chemical substances in the Keller-Segel model.

Numerical results demonstrate the efficiency and accuracy of the proposed method.

The approach outperforms existing control strategies in simulation tests.

Abstract

In this work we introduce a new optimal control algorithm for the Keller-Segel chemo-attraction system, where both boundary and distributed controls are considered and both are associated with introducing/removing the amount of chemical substances in the system. The key idea of our approach is to design the optimal control algorithm after discretizing the state problem system, which is done using an upwind finite volume scheme in space and a semi-implicit finite difference in time. Then, the discrete optimal control is approximated identifying the gradient of the reduced discrete cost via the discrete adjoint scheme. Finally, to minimize the reduced cost functional, we use a gradient descent type method (Adam scheme). Moreover, several numerical results are presented to illustrate the efficiency of the proposed approach.