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

TL;DR

The paper tackles the bilinear optimal control of a two-species chemotaxis system with two-sided degenerate diffusion and a volume-filling effect, formalized by the state equations $N_t - abla\cdot(a(N)\nabla N) + \nabla\cdot(\chi(N)\nabla C)=0$ and $C_t - \Delta C = \gamma N - \beta C + f C 1_{\Omega_c}$. It develops a rigorous weak-solution framework for the degenerate direct problem via a regularized diffusion $a_{\\varepsilon}(N)=a(N)+\\varepsilon$ and a semi-discretization in time, proving global existence, positivity, and stability; it then establishes the existence of an optimal control by compactness arguments and derives first-order optimality conditions through a Lagrangian that yields a coupled adjoint system. The adjoint analysis is addressed through regularized Faedo–Galerkin constructions, transforming the final-time adjoint problem into a backward-in-time parabolic problem and proving weak solvability under localized degeneracy and, more generally, under strengthened regularity assumptions. Overall, the work provides a solid theoretical foundation for controllability and optimization of degenerate chemotaxis models in 2D, paving the way for robust numerical schemes and applications to population management and medical contexts.

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.

Theorems & Definitions (16)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• Theorem 4.1
• proof
• Definition 4.2