In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn–Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fréchet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.

在本文中，我们解决了相场类型非局部模型的最优分布式控制问题，描述了肿瘤细胞在营养物存在下的演化。该模型将相位参数的非局部粘性 Cahn-Hilliard 方程与营养物的反应扩散方程耦合起来。最优控制问题旨在找到一种疗法，以放疗和化疗的形式编码为系统中的源项，这可能导致相位变量朝着所需的最终目标演化。首先，我们证明了非线性偏微分方程组的强适定性。特别是，由于粘性正则化的存在，我们还可以考虑与趋化作用相关的奇异类型的双井电势和交叉扩散项。此外，反应项的特殊结构使我们能够证明此类系统的新规律性结果。然后，转向最优控制问题，我们证明了最优治疗的存在性，并通过研究控制状态算子和相应的伴随系统的Fréchet可微性性质，得到了一阶必要的最优性条件。