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.

翻译：本文研究了一个具有退化扩散的耦合非线性反应-扩散方程系统的最优控制问题，该系统由描述细胞密度和趋化剂浓度的两个偏微分方程构成。通过控制化学底物的浓度，本研究可指导细胞的最优生长。本工作的创新点在于直接模型与对偶模型均保持在弱解框架下，这在近期求解最优控制系统的文献中并不常见。此外，伴随问题为量化模型输入相关的不确定性提供了有力工具，但这类系统通常缺乏闭式解，使得弱解的获取颇具挑战。为此，我们首先严格保证了直接问题的适定性。随后，建立了最优控制的存在性及一阶最优性条件。最后，针对非线性退化直接模型的伴随系统，我们提出并研究了其弱解的存在性。