论文标题
在具有单数电位和趋化性的相位场肿瘤生长模型中对最佳控制问题的二阶分析
Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis
论文作者
论文摘要
本文涉及Cahn-Hilliard类型的肿瘤生长模型的分布式最佳控制问题，其中包括具有奇异电位的趋化性，其中控制和状态变量是非线性耦合的。首先，我们讨论了系统对电势的一般假设的较弱，这可能是单数和非平滑的。然后，我们在降低的设置中建立了系统的强大良好性，但是它承认对数电位：该分析将为研究相应的最佳控制问题的研究奠定基础。关于优化问题，我们解决了最小化器的存在，并建立了一阶必需和二阶足够条件以实现最佳性。在数学上具有挑战性的二阶分析在此表明解决方案映射是通过隐式函数定理在合适的BANACH空间之间连续区分的两倍之后，在此完全执行。然后，我们完全确定控制对状态操作员的二阶Fréchet导数，并对相关特性进行详尽而详细的研究。
This paper concerns a distributed optimal control problem for a tumor growth model of Cahn-Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.
