Weighted functional spaces approach to non-convex nonlinear infinite horizon optimal control problems: stabilization, numerical analysis, bio-medical applications.

加权函数空间方法解决非凸非线性无限视野最优控制问题：稳定性、数值分析、生物医学应用。

• 批准号: 387882680
• 负责人: Dr. Valeriya Lykina
• 金额: --
• 依托单位: Lehrstuhl Mathematik insbesondere Optimierung
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/387882680?language=en
• 关键词: Weighted; functional; spaces; approach; non

In the project biomedical models are formulated and treated as infinite horizon optimal control problems. Among them one considers two classes of problems: infinite horizon optimal control problems of 1) the combined chemotherapeutic-antiangiogenic cancer treatment; 2) the non-pharmaceutical epidemiological intervention with non-convex isolation costs. The incorporation of an unbounded planing horizon for the considered models has been investigated in our previous projects and represents an important and challenging mathematical issue which has been proved to be a proper idealization for sustainability principle so much desired and targeted by the today's society. One important aspect leading to an infinite horizon formulation is the fact that it is not reasonable to assume to know the end point of e.g. epidemics or the decease, cf. spread of COVID-19 or cancer. Another important aspect results from the point of view of stabilization the controlled dynamic system, since the decision maker try to stabilize the number of infected around some specific level. Both optimization and stabilization goals are properly to combine in the infinite horizon objective functional. The formulated models, besides the direct interest in the long-term optimal strategies in the particular background, should serve as benchmark problems to detect and illustrate the observed mathematical phenomena as well. Fundamental research goals of the project cover both theoretical aspects of solving nonconvex infinite horizon optimal control problems by means of relaxation techniques and development of suitable numerical solution methods for the mentioned class of problems which is strongly required due to nonlinearity of the problems. The first part is based on relaxation ideas of Gamkrelidze applied to the considered problem class as well on the duality concept of Klötzler adjusted to the setting of the OCP in weighted functional spaces. Derivation and proof of sufficient optimality conditions and Pontryagin Type Maximum Principle is the main theoretical goal of the project. The second part includes the development of a pseudospectral method for numerical solution of the primal as well as the dual problems which combines the concepts of duality theory and the relaxation techniques. The convergence proofs and the comparison to the previously developed dual-based direct pseudo-spectral method are to establish. Especially the solving of stabilization problems by means of relaxed infinite horizon optimal control problems in the context of functional analytical approach represents a new and challenging research area. From practical point of view it is particularly interesting to detect the so called "dithering" solutions for problems with nonconvex objectives and understand their meaning and applicability in case of biomedical applications.

在该项目中，生物医学模型被制定和处理为无限时域最优控制问题。其中考虑了两类问题：1）癌症化疗-抗血管生成联合治疗的无限时域最优控制问题; 2）隔离费用为非凸的非药物流行病学干预问题。在我们以前的项目中已经研究了所考虑的模型的无限规划视野的结合，并且代表了一个重要且具有挑战性的数学问题，该问题已被证明是当今社会所期望和针对的可持续性原则的适当理想化。导致无限视界公式化的一个重要方面是这样一个事实，即假设知道例如流行病或死亡的终点是不合理的。COVID-19或癌症的传播。另一个重要的方面是从稳定受控动态系统的角度来看，因为决策者试图将感染人数稳定在某个特定水平。在无限时域目标泛函中，最优化目标和稳定化目标适当地联合收割机结合。制定的模型，除了在特定背景下的长期最优策略的直接利益，应该作为基准问题，以检测和说明所观察到的数学现象。该项目的基本研究目标包括两个理论方面的解决非凸无限时域最优控制问题的松弛技术和适当的数值求解方法的发展所提到的一类问题，这是强烈要求由于非线性的问题。第一部分是基于放松的思想Gamkrelidze适用于所考虑的问题类，以及对偶概念Klötzler调整设置的OCP在加权函数空间。最优性充分条件和庞特里亚金型最大值原理的推导和证明是该项目的主要理论目标。第二部分包括一个伪谱方法的发展的原始数值解以及对偶问题，结合对偶理论和松弛技术的概念。收敛性证明和比较以前开发的双基直接伪谱方法建立。 特别是在泛函分析方法的背景下，通过松弛的无限时域最优控制问题来解决镇定问题，代表了一个新的和具有挑战性的研究领域。从实践的角度来看，检测非凸目标问题的所谓“抖动”解决方案并了解它们在生物医学应用中的意义和适用性是特别有趣的。