Analysis of Optimal Control Problems with State Space Constraints Arising in Applications

在本项目中，将给出具有状态空间约束的最优控制问题中受控轨迹最优的综合型充分条件，包括以参考轨迹周围极值场的形式的局部结果和以正则综合形式的全局结果。这项研究是由应用中出现的问题强烈驱动的，并将被用来证明极值族对于来自两个不同应用领域的问题的最优性，即(A)最小化电子半导体器件中的基态渡越时间和(B)当包括对癌细胞数量的显式上限或骨髓的下限时癌症化疗的数学模型的分析。(A)在电子学中，选择基区掺杂分布以最小化半导体器件中的基区渡越时间的问题通常是用数值方法解决的。将该问题描述为状态空间受限的最优控制问题，分析了同质结和异质结双极晶体管的不同模型。对于硅双极晶体管，我们将得到一般载流子扩散系数的显式解析解，并通过构造一个完整的综合体来证明它们的最优性。(B)在许多癌症化疗的数学模型中，目标是在预先确定的固定治疗间隔结束时尽量减少癌细胞的数量，同时通过惩罚期限限制药物的毒性。然而，这并不总是能阻止癌细胞数量在两者之间上升到令人无法接受的高水平。因此，考虑将对癌细胞的明确限制作为状态空间约束的模型更为现实。在这个项目中，将发展数学条件，以保证在严格的限制条件下受控过程的最优性。这项理论研究将以两个实际问题为指导，一个来自电子学，另一个来自生物医学。在电子学主题中，将为描述某些半导体器件的速度的简化模型开发完整的解析解。在生物医学领域，将在存在限制的情况下分析癌症或其他疾病的化学治疗的数学模型，例如，骨髓不得低于指定的最低水平，或癌细胞不允许超过最高水平。这些主题还将用来说明本科生水平上的最优控制方法，这些方法具有比目前大多数教科书中关于这一主题的问题更相关和现实的问题。

In this project, synthesis type sufficient conditions for optimality of controlled trajectories in optimal control problems with state-space constraints will be developed, including both local results in the form of a field of extremals around a reference trajectory and global results in the form of a regular synthesis. This research is strongly motivated by problems arising in applications and will be used to prove optimality of families of extremals for problems from two different application areas, namely (a) the minimization of the base transit time in semi-conductor devices in electronics and (b) the analysis of mathematical models for cancer chemotherapy when explicit upper limits on the number of cancer cells or lower limits for bone marrow are included. (a) In electronics, the problem of choosing the base-doping profile to minimize the base transit time in semi-conductor devices generally is solved numerically. In this project, the problem is formulated as an optimal control problem with state space constraints, and different models corresponding to both homojunction and heterojunction bipolar transistors are analyzed. For silicon bipolar transistors, explicit analytic solutions for general carrier diffusion coefficients will be derived, and their optimality will be proven by constructing a complete synthesis. (b) In many mathematical models for cancer chemotherapy the aim is to minimize the number of cancer cells at the end of a pre-determined fixed therapy interval while limiting the toxicity of the drugs through a penalty term. This, however, does not always prevent the number of cancer cells from rising to unacceptably high levels in between. It therefore is more realistic to consider models that incorporate explicit limits on the cancer cells as state-space constraints. The analysis of optimal controls for such models will be pursued.In this project, mathematical conditions will be developed that guarantee the optimality of controlled processes in the presence of strict limits. This theoretical research will be guided by two practical problems, one from electronics, the other biomedical. In the electronics topic, complete analytical solutions will be developed for a simplified model describing the speed of certain semiconductor devices. In the biomedical area, mathematical models for chemotherapy of cancer or other diseases will be analyzed in the presence of restrictions, for example, that bone marrow must not fall below a specified minimum level or that cancer cells are not allowed to increase beyond a maximum level. These topics also will be used to illustrate optimal control methods on an undergraduate level with more relevant and realistic problems than are currently available in most textbooks on the subject.