Collaborative Research: Regular synthesis for multi-input optimal control problems with applications to biomedicine

在这个项目中，完全相互作用的非线性多输入控制系统将作为最优控制问题进行分析。这项研究的动机来自于对癌症治疗的数学模型的系统研究，这些模型结合了形成肿瘤微环境的各种结构。在现代肿瘤学中，肿瘤不仅被视为癌细胞，而且被视为一个相互作用的组成系统，它以各种方式帮助和教唆肿瘤（如肿瘤血管系统），但也与肿瘤作斗争（如免疫系统）。因此，目前的治疗方法是多靶向治疗，不仅可以杀死癌细胞，还包括抗血管生成治疗、免疫治疗和许多其他选择。寻找最好的方法来管理这些治疗剂自然导致配方作为多输入非线性最优控制问题。这项研究的目的是为这样的系统开发最优控制的局部综合，即使对于只有部分结果已知的单输入系统，这也是一项艰巨的任务。这些新结果将与描述(i)肿瘤微环境复杂背景下的癌症治疗和（ii）在流行病学中控制疾病传播的最佳策略的系统研究联系起来。在后一个领域，数学模型已成为分析潜在动力学的相关工具，而在本项目中，将探索对该问题的最佳控制方法。虽然一个目标是开发具有广泛适用性的一般方法和程序，但研究的第二个重要方面是为重要现实问题的最佳解决方案的结构提供定性和定量的见解。在生物医学的及时问题的激励下，如如何设计节拍化疗方案，将考虑最优控制理论中的挑战性问题，其解决方案需要开发新的工具和方法。由于其应用和跨学科的特点，该项目预计会引起数学和工程专业学生的浓厚兴趣，因此它包含了大量的教育内容。将继续努力吸引妇女和少数民族，并扩大工程系学生的参与，因为这些群体的参与度特别低。在现代肿瘤学中，肿瘤不仅被视为癌细胞，而且被视为一个相互作用的组成系统，它以各种方式帮助和教唆肿瘤（如肿瘤血管系统），但也与肿瘤作斗争（如免疫系统）。因此，目前的治疗方法是多靶向治疗，不仅可以杀死癌细胞，还包括抗血管生成治疗、免疫治疗和许多其他选择。在医学试验中测试复杂的多靶点方案是非常困难和昂贵的。因此，数学模型的分析就具有了内在的价值。医学界越来越意识到，不仅给药是什么，而且给药方式，即剂量、频率和顺序，都可能对治疗结果产生重大影响。这导致了最近发起的“Metronomics全球健康倡议”。提出的研究是由该倡议的生物医学思想驱动的，研究人员认为几何最优控制的工具最适合给出这些问题的数学答案，从而为如何设计节拍器协议提供见解。关于第二个主题，传染病仍然是全世界最重要的健康问题之一。在这个项目中，研究者寻求理论结果，可以为如何以最佳方式进行疫苗接种和治疗的共同努力提供实践指导，以最大化效果并最小化社会经济成本。由于其应用和跨学科的特点，该项目预计会引起数学和工程专业学生的浓厚兴趣，因此它包含了大量的教育内容。将继续努力吸引妇女和少数民族，并扩大工程系学生的参与，因为这些群体的参与度特别低。

In this project, fully interacting nonlinear multi-input control systems will be analyzed as optimal control problems. The motivation for this research comes from a systematic study of mathematical models for cancer treatment that combine various structures that form the tumor microenvironment. In modern oncology, a tumor is viewed as not just cancerous cells, but as a system of interacting components that in various ways aid and abet the tumor (e.g., the tumor vasculature), but also fight it (e.g., the immune system). Current treatments therefore are multi-targeted therapies that not only kill cancer cells, but also include antiangiogenic therapy, immunotherapy and many other options. The search for best ways of administering these therapeutic agents naturally leads to formulations as multi-input nonlinear optimal control problems. The aim of the research is to develop local syntheses of optimal controls for such systems, a difficult task even for single-input systems with only partial results known. These new results will be developed in connection with the study of systems that describe (i) cancer treatment within the complex context of the tumor microenvironment and (ii) optimal strategies for the control of the spread of diseases in epidemiology. In the latter field, mathematical models have been a relevant tool in analyzing the underlying dynamics, whereas in this project a much less explored optimal control approach to the problem will be pursued. While one aim is to develop general methods and procedures that have wide applicability, a second important aspect of the research is to provide qualitative and quantitative insights into the structure of optimal solutions for important real-life problems. Motivated by timely problems in biomedicine, like how to design metronomic chemotherapy protocols, challenging problems in optimal control theory will be considered whose solutions require developing new tools and methods. Because of its applied and interdisciplinary character, the project is expected to be of strong interest to students from mathematics and engineering and consequently it contains a substantial educational component. Efforts to attract women and minorities will be continued and expanded by reaching out to engineering students where participation of these groups is particularly low.In modern oncology, a tumor is viewed as not just cancerous cells, but as a system of interacting components that in various ways aid and abet the tumor (e.g., the tumor vasculature), but also fight it (e.g., the immune system). Current treatments therefore are multi-targeted therapies that not only kill cancer cells, but also include antiangiogenic therapy, immunotherapy and many other options. It is very difficult and expensive to test complex multi-target protocols in medical trials. For this reason, the analysis of mathematical models becomes of intrinsic value. The medical community has become more and more aware that not only what drug is given, but also how it is administered, i.e., dosage, frequency and sequencing, can have a major impact on the outcome of the treatment. This led to a recently launched "Metronomics Global Health Initiative". The proposed research is motivated by the biomedical ideas of this initiative and the investigators believe that the tools of geometric optimal control are best suited to give mathematical answers to these questions and thus provide insights into how a metronomic protocol should be designed. Regarding a second topic, infectious diseases continue to be one of the most important health problems worldwide. In this project, the investigators seek theoretical results that can give practical guidelines how to pursue joint efforts of vaccination and treatment in an optimal way to maximize the effectiveness and minimize the social economic cost. Because of its applied and interdisciplinary character, the project is expected to be of strong interest to students from mathematics and engineering and consequently it contains a substantial educational component. Efforts to attract women and minorities will be continued and expanded by reaching out to engineering students where participation of these groups is particularly low.