具有奇异势和趋化作用的相场肿瘤生长模型中最优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
AI总结：
本文研究含奇异势与趋化作用的Cahn-Hilliard肿瘤生长模型的分布最优控制，建立系统适定性、极小元存在性及一阶必要与二阶充分最优性条件。
摘要：
本文研究一个Cahn-Hilliard型肿瘤生长模型的分布最优控制问题，该模型包含趋化作用并可能具有奇异势，其中控制变量与状态变量非线性耦合。首先，我们在关于势的非常一般假设下讨论该系统的弱适定性，这些势可以是奇异且非光滑的。随后，我们在一个简化设定下建立系统的强适定性，然而该设定允许对数势：这一分析将为研究相应的最优控制问题奠定基础。关于优化问题，我们处理极小元的存在性，并同时建立最优性的一阶必要条件和二阶充分条件。在通过隐函数定理证明解映射在适当的Banach空间之间二次连续可微之后，这里完整完成了数学上具有挑战性的二阶分析。然后，我们完全确定控制到状态算子的二阶Fréchet导数，并对相关性质进行了透彻而详细的研究。
英文摘要：
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.
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