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

应用中出现的状态空间约束最优控制问题分析

• 批准号: 0305965
• 负责人: Heinz Schaettler
• 金额: $ 10.25万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-15 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305965&HistoricalAwards=false
• 关键词: Analysis; Optimal; Control; Problems; State

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.

在该项目中，将开发具有状态空间约束的最优控制问题中受控轨迹最优性的综合类型充分条件，包括参考轨迹周围极值场形式的局部结果和常规综合形式的全局结果。 这项研究受到应用中出现的问题的强烈推动，并将用于证明极值族对于两个不同应用领域问题的最优性，即（a）电子学中半导体器件中基本渡越时间的最小化；（b）当包括明确的癌细胞数量上限或骨髓下限时，对癌症化疗的数学模型进行分析。 (a) 在电子学中，选择基极掺杂分布以最小化半导体器件中的基极渡越时间的问题通常是通过数值解决的。 在该项目中，该问题被表述为具有状态空间约束的最优控制问题，并分析了对应于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，将导出一般载流子扩散系数的显式解析解，并通过构建完整的综合来证明其最优性。 (b) 在许多癌症化疗的数学模型中，目的是在预定的固定治疗间隔结束时最大限度地减少癌细胞的数量，同时通过惩罚项限制药物的毒性。 然而，这并不总能阻止癌细胞数量在两者之间上升到不可接受的高水平。 因此，考虑将癌细胞的明确限制作为状态空间约束的模型更为现实。 将继续对此类模型的最优控制进行分析。在该项目中，将开发数学条件，以保证在存在严格限制的情况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。 在电子主题中，将为描述某些半导体器件速度的简化模型开发完整的分析解决方案。 在生物医学领域，癌症或其他疾病化疗的数学模型将在存在限制的情况下进行分析，例如，骨髓不得低于指定的最低水平，或者不允许癌细胞增加超过最高水平。 这些主题还将用于说明本科水平的最优控制方法，其中包含比当前大多数该主题教科书更相关和更现实的问题。