肿瘤生长模型中具有奇异势的相场系统的稀疏最优控制
Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth
摘要：
本文研究了一类非线性反应扩散方程系统的最优控制问题，该系统是文献[12]中提出的肿瘤生长的相容相场模型的简化和松弛形式.该模型考虑了趋化性的影响，但忽略了速度的贡献。控制状态方程的未知量是化学势、（归一化的）肿瘤分数和营养细胞外水浓度。控制肿瘤分数的演变的方程由双阱势的变分导数支配，该双阱势可以是奇异的（例如，对数）类型。与最近关于同一系统的论文[10]相反，我们在本文中考虑稀疏效应，这意味着成本泛函包含不可微（但凸）的贡献，如$L^1-$范数。对于这样的问题，我们推导出一阶必要的最优性条件和条件的方向稀疏性，无论是相对于空间和时间，后者的情况下是特别感兴趣的实际医疗应用中的控制变量是由管理细胞毒性药物或营养物质的供应。除了这些结果，我们证明了相应的控制到状态的运营商是两次连续可微适当的巴拿赫空间之间，使用隐函数定理。这一结果补充和加强了文[10]中的一个可微性结果，为以后推导二阶充分最优性条件提供了前提条件。
英文摘要：
In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [12]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the $L^1-$norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.
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