This paper deals with an optimal control problem for a haptotaxis model which describes two types of cancer cell invading the surrounding healthy tissue. Cancer cells can be treated by therapeutic agents (radiotherapy or chemotherapy) considered as the control variable. Our primary goal is to characterize an optimal control which balances the therapeutic benefits with its side effects. Firstly, the global bounded weak solution of state system in spatial dimensions \(1\le N\le 5\) are obtained by using the Banach fixed-point theorem and a new iteration criterion of \(L^p\) estimates as well as the adapted Moser-type iteration technique of \(L^\infty \) estimates. Subsequently, the existence of optimal pair is proved by applying the technique of minimizing sequence. Furthermore, we derive the first-order necessary optimality condition by means of the Lipschitz continuity of the control-to-state mapping and the methods of convex perturbation. Finally, we present numerically the spatio-temporal dynamics of species, optimal control and optimal regimen of therapeutic agents administrated to illustrate the concrete realization of the theoretical results obtained in this work, and to validate some clinical findings.

本文研究了一种触觉模型的最优控制问题，该模型描述了两种类型的侵袭周围健康组织的癌细胞。癌细胞可以通过被视为控制变量的治疗剂（放射疗法或化学疗法）进行治疗。我们的主要目标是确定一种最佳控制措施，以兼顾治疗效果和副作用。首先，利用Banach不动点定理和新的迭代准则（L ^ p \）估计，得到空间系统\（1 \ le N \ le 5 \）中状态系统的全局有界弱解。作为\（L ^ \ infty \）的自适应Moser型迭代技术估计。随后，通过应用最小化序列技术证明了最优对的存在。此外，我们通过控制状态映射的Lipschitz连续性和凸摄动方法得出一阶必要最优条件。最后，我们从数字上介绍了物种的时空动态，所用治疗剂的最佳控制和最佳方案，以说明这项工作中获得的理论结果的具体实现，并验证一些临床发现。