论文标题
稀疏对具有奇异电位的相位系统的最佳控制在肿瘤生长的建模中产生
Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth
论文作者
论文摘要
在本文中，我们研究了反应扩散方程的非线性系统的最佳控制问题，该系统构成了最初在[12]中引入的热力学一致相位场模型的简化和放松版本。该模型考虑了趋化性的作用，但忽略了速度贡献。管理状态方程的未知数是化学电位，（归一化）肿瘤分数和营养外细胞外水浓度。控制肿瘤分数演变的方程是由双孔电势的变异衍生物主导的，这可能是单数（例如，对数）类型的。与最近的同一系统上的论文[10]相反，我们在本文中考虑了稀疏效应，这意味着成本功能包含诸如$ l^1- $ norm的非不同（但凸）的贡献。对于此类问题，我们得出了一阶必要的最佳条件和定向稀疏性的条件，无论是在时空而言，后一种情况对于实用的医疗应用特别感兴趣，在这种医学应用中，通过使用细胞毒性药物给予控制变量或通过养分提供控制变量。除了这些结果外，我们还证明使用隐式函数定理，相应的控制对状态操作员在合适的Banach空间之间连续区分了两倍。该结果对[10]中得出的可不同性结果补充并匹配，构成了未来二阶足够最佳条件的未来推导的先决条件。
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.
