Bilinear optimal control for chemotaxis model: The case of two-sidedly degenerate diffusion with Volume-Filling Effect

TL;DR

This paper develops a novel optimal control framework for a complex chemotaxis model with degenerate diffusion, ensuring well-posedness and deriving first-order optimality conditions in a weak setting.

Contribution

It introduces a weak formulation approach for direct and dual models in a degenerate chemotaxis system, which is uncommon in existing literature.

Findings

Established well-posedness of the direct problem

Proved existence of an optimal control

Derived first-order optimality conditions

Abstract

In this paper, we study an optimal control problem for a coupled non-linear system of reaction-diffusion equations with degenerate diffusion, consisting of two partial differential equations representing the density of cells and the concentration of the chemotactic agent. By controlling the concentration of the chemical substrates, this study can guide the optimal growth of cells. The novelty of this work lies on the direct and dual models that remain in a weak setting, which is uncommon in the recent literature for solving optimal control systems. Moreover, it is known that the adjoint problems offer a powerful approach to quantifying the uncertainty associated with model inputs. However, these systems typically lack closed-form solutions, making it challenging to obtain weak solutions. For that, the well-posedness of the direct problem is first well guaranteed. Then, the existence of an optimal control and the first-order optimality conditions are established. Finally, weak solutions for the adjoint system to the non-linear degenerate direct model, are introduced and investigated.