Infinite horizon optimal control problems with applications in biomedicine: models, optimality conditions, numerical solutions.

目前，具有固定有限时间范围的数学模型被用于计算最优癌症治疗策略。它们的主要目的是尽可能快地消除所有的肿瘤细胞，而副作用和治疗成本应该最小化(损伤功能最小化)。这种方法的缺点在于对整个人体的副作用，但这种副作用太高，无法接受。这些都是免疫薄弱，不必要地杀死或损害健康组织，以及肿瘤细胞抵抗力的增强。相反，本研究项目涉及无限范围最优控制问题及其在生物医学中的应用，即最优癌症治疗模型。这些应该会提供不那么激进、稳定的长期疗法。该项目的基本目标可以细分为两个部分。第一部分包括了属于理论研究范畴的所有调查，为第二部分奠定了理论基础。在这里，必须得到最优性条件和存在定理。因此，问题的功能分析提法和证明的功能分析方法论应发挥关键作用。另一个非常重要的问题是发展了一种被称为伪谱方法的数值方法，它完全是为所考虑的最优控制问题的一般设置而“定制”的。第二部分包括与生物医学应用相关的研究。在这一部分中，它是为了找出所考虑的一类控制问题可以提供哪些创新的癌症治疗策略，以及哪些具体的模型是合适的。肿瘤生长模型最优控制问题的性能准则(目标泛函)的选择对最优解的选择有至关重要的影响，从而影响给药方案和治疗本身。在这个项目中，除了损伤泛函之外，还应该考虑稳定泛函，它构成了动力系统的过程(人体)偏离健康平衡(无肿瘤或肿瘤与正常细胞共存)并同时稳定这一平衡。应用项目第一部分的结果，应用拟谱方法计算数值解，并验证其最优性。将客户机构的OCMat软件应用于癌症治疗的相同问题，应该能够比较最优解决方案，并对基本动力系统进行定性分析。鉴于由此产生的治疗方法，有限和无限范围问题的最优解的结构比较是非常重要的。

At the present time, mathematical models with a fix finite time horizon are used for computation of optimal cancer treatment strategies. Their aim is mostly to eliminate all the tumor cells as fast as possible whereas the side-efects and the therapy costs should be minimized (Minimization of a damage functional). The drawback of this method lies in the side-effects onto the entire human body, which are nevertheless too high to be acceptable. These are the immune weakness, unnecessarilly killed or damaged healthy tissues as well as the increased resistance of tumor cells. On the contrary, the present research project deals with infinite horizon optimal control problems and their applications to bio-medicine, namely to the models of optimal cancer treatment. These should provide less aggressive stabilizing long term therapies. The essential aims of the project can be subdivided into two parts. The first part includes all the investigations which fall in the category of fundamental research and build the theoretical background for the second part. Here necessary optimality conditions as well as an existence theorem should be derived. The functional analytical formulation of the problem and the functional analytical methodology of the proof should play hereby the key role. Another very important issue is the development of a numerical method, called pseudospectral method, which is exactly "tailored" for the considered general setting of the optimal control problem. The second part includes the investigations which concern the biomedical applications. In this part, it is to find out which innovative cancer treatment strategies can be provided by the considered class of control problems and which concrete models are adequate. The choice of the perfomance criterium (functional in the objective) of the optimal control problem of a tumor growth model has a crucial influence on the optimal solution and, consequently, on the drug administration protocols and therapy itself. Besides the damage functional, in this project a stabilization functional, which constitutes the deviation of the process (human body) from a healthy equilibrium ("tumor-free" or "tumor- and normal cells coexist") of the dynamical system and stabilizes this simultaneously, should be alternatively considered. Applying the results from the first part of the project, numerical solutions should be computed by means of the pseudospectral method and their optimality has to be verified. The application of the OCMat software of the guest institution to the same problems of cancer treatment should enable the comparison of the optimal solutions and the qualitative analysis of the underlying dynamical systems. The structural comparison of optimal solutions to the problems with finite vs. infinite horizon is of great importance in view of the resulting therapies.