AI 中文总结
研究基于Gompertz增长的肿瘤细胞群体空间分布稳态反应扩散模型的最优控制问题，通过理论分析得出内在增长率恒定时均匀分布是极小值等结论，数值模拟补充分析，揭示优化总体积对扩散系数的单调依赖这一新现象。
AI 中文摘要
我们研究了一个描述具有Gompertz增长的肿瘤细胞群体空间分布的稳态反应扩散模型的最优控制问题。控制项m(x)表示一个治疗项，其作用为密度依赖的清除率，且受L¹-L∞约束。当内在增长率恒定时，治疗的均匀分布是唯一的极小值。对于最大化问题，我们证明每个最优控制都是bang-bang型。此外，在一维情况下且扩散率足够大时，最优控制的正性集是一个附着于区域极值之一的区间。最后，数值模拟补充了理论分析，探索了未被论文结果完全覆盖的情况。计算结果证实了最大化器的bang-bang结构，并说明了最优控制的形状和相关状态如何受到增长率的空间异质性、局部可接受治疗区域和扩散系数的影响。此外，它们揭示了优化后的总体积对扩散系数的单调依赖性：这在逻辑斯谛模型中是一个新现象。
英文摘要
We study optimal control problems for a stationary reaction--diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.
