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

在工程中，生物医学模型被制定并处理为无限视界最优控制问题。其中考虑了两类问题：1)化疗-抗血管生成肿瘤联合治疗的无限视界最优控制问题；2)非凸隔离费用的非药物流行病学干预。在我们之前的项目中，我们已经对所考虑的模型的无界规划地平线进行了研究，这是一个重要而具有挑战性的数学问题，已被证明是当今社会所渴望和目标的可持续性原则的适当理想化。导致无限视界公式的一个重要方面是，假设知道流行病或疾病（例如COVID-19或癌症的传播）的终点是不合理的。另一个重要的方面是从稳定受控动态系统的角度出发的，因为决策者试图将感染人数稳定在某个特定水平附近。优化目标和稳定目标在无限视界目标泛函中得到了合理的结合。所制定的模型，除了对特定背景下的长期最优策略有直接的兴趣外，还应该作为基准问题来检测和说明所观察到的数学现象。本项目的基本研究目标包括利用松弛技术解决非凸无限水平最优控制问题的理论方面，以及由于问题的非线性而迫切需要的这类问题的适当数值解方法的发展。第一部分是基于将Gamkrelidze的松弛思想应用于所考虑的问题类，以及将对偶概念Klötzler调整为加权功能空间中OCP的设置。充分最优性条件和庞特里亚金型极大原理的推导和证明是本课题的主要理论目标。第二部分结合对偶理论和松弛技术的概念，提出了一种拟谱方法来求解原问题和对偶问题的数值解。给出了收敛性证明，并与已有的双基直接伪谱法进行了比较。特别是在泛函分析方法的背景下，用松弛无限视界最优控制问题求解镇定问题，是一个新的具有挑战性的研究领域。从实际的角度来看，检测具有非凸目标的问题的所谓“抖动”解决方案并理解其在生物医学应用中的意义和适用性是特别有趣的。

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.