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

无限视野最优控制问题及其在生物医学中的应用：模型、最优条件、数值解。

基本信息

批准号：
320486431
负责人：
Dr. Valeriya Lykina
金额：
--
依托单位：
Operations Research and Control Systems (ORCOS)
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/320486431?language=en
关键词：
Infinite horizon optimal control problems

项目摘要

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.

目前，具有固定有限时间范围的数学模型用于计算最佳癌症治疗策略。他们的目标主要是尽可能快地消除所有肿瘤细胞，而副作用和治疗成本应最小化（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，但其太高而不可接受。这些是免疫力低下，不必要地杀死或破坏健康组织以及肿瘤细胞抵抗力增加。相反，本研究项目涉及无限时域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。这些应该提供不太积极的稳定长期治疗。该项目的基本目标可分为两个部分。第一部分包括了所有属于基础研究范畴的研究，为第二部分奠定了理论基础。在这里，必要的最优性条件以及存在性定理应得出。问题的泛函分析表述和证明的泛函分析方法在此起着关键作用。另一个非常重要的问题是发展的数值方法，称为伪谱法，这是完全“定制”的最优控制问题的一般设置考虑。第二部分是关于生物医学应用的研究。在这一部分中，它是要找出哪些创新的癌症治疗策略，可以提供所考虑的控制问题的类，哪些具体的模型是足够的。肿瘤生长模型的最优控制问题的性能标准（目标函数）的选择对最优解以及因此对药物给药方案和治疗本身具有至关重要的影响。除了损伤功能，在这个项目中的稳定功能，这构成了偏离的过程（人体）从一个健康的平衡（“无肿瘤”或“肿瘤和正常细胞共存”）的动态系统，并同时稳定这一点，应交替考虑。应用该项目第一部分的结果，数值解应通过伪谱方法计算，并验证其最优性。客座机构的OCMat软件应用于癌症治疗的相同问题，应该能够比较最佳解决方案和定性分析的基础动力系统。有限视界与无限视界问题的最优解的结构比较对于由此产生的治疗是非常重要的。