Logistic Keller-Segel 模型的趋化敏感性最优控制

中文摘要 AI 辅助

我们研究了一个在有界域上具有 Neumann 边界条件的双方程 Keller-Segel 系统的最优控制问题，其中趋化敏感性由作用于趋化通量 $\nabla\\!\cdot(f\\,u\\,\nabla v)$ 的控制函数 $f(x,t)$ 调制。我们在低正则性假设下证明了状态方程的适定性。该分析基于对 $\varepsilon$-正则化问题的研究、Schauder 不动点定理的应用、$\varepsilon$-无关性、正性和先验估计、当 $\varepsilon \to 0$ 时的极限过程以及唯一性论证。对于结合轨迹跟踪、趋化惩罚、控制正则化和终端目标的代价泛函，我们建立了最优控制的存在性，并通过 Lagrange 乘子框架导出了一阶最优性条件，从而得到伴随系统和最优控制的变分不等式。我们建立并实现了一种基于空间样条配点和 Runge-Kutta 时间步进的数值格式。数值实验在守恒和非守恒情形下模拟了自由和受控动力学。

英文摘要

We study an optimal control problem for a two-equations Keller-Segel system on a bounded domain with Neumann boundary conditions, where the chemotactic sensitivity is modulated by a control $f(x,t)$ acting on the taxis flux $\nabla\!\cdot(f\,u\,\nabla v)$. We prove the well-posedness of the state equation under low regularity assumptions. The analysis is based on the study of an $\varepsilon$-regularized problem, the application of Schauder's fixed point theorem, $\varepsilon$-independent, positivity and a priori estimates, the passage to the limit as $\varepsilon \to 0$, and a uniqueness argument. For a cost functional combining trajectory tracking, taxis penalization, control regularization, and terminal objectives, we establish the existence of optimal controls and derive the first-order optimality conditions through a Lagrange multiplier framework, leading to an adjoint system and a variational inequality for the optimal control. A numerical scheme based on spline collocation in space and Runge-Kutta time-stepping is set and implemented. Numerical experiments are performed to simulate the free and controlled dynamics in conservative and non-conservative cases.

发表机构

• Friedrich-Alexander-Universit¨at Erlangen–Nürnberg（埃尔朗根-纽伦堡大学）
• Université Yembila Abdoulaye TOGUYENI（Yembila Abdoulaye Toguyeni大学）
• George Mason University（乔治梅森大学）
• Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）

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