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Improved Numerical Methods for Solving Optimal Control Problems with Nonsmooth and Singular Solutions

解决具有非光滑和奇异解的最优控制问题的改进数值方法

基本信息

批准号：
2031213
负责人：
Anil Rao
金额：
$ 60.9万
依托单位：
University of Florida
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-01-01 至 2024-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2031213&HistoricalAwards=false
关键词：
Improved Numerical Methods Solving Optimal

项目摘要

This project advances computational methods for solving nonsmooth and singular optimal control problems, in which a control input is selected to maximize or minimize some objective function. Non-smooth problems arise when the optimizing control includes sudden jumps. Singular problems arise when the control input does not directly influence the objective function. Solutions to optimal control problems are found using numerical approximations that are constructed on a pattern of grid points. This project improves upon existing approximation methods for nonsmooth problems by constructing grid patterns that better capture the jump points, resulting in higher accuracy using less computing power. This innovation also helps better demarcate any singular regions of the problem. The project further improves the control solution for problems with singular regions by using corrective modifications to the objective function only in those regions. A wide range of problems of national importance may be formulated as nonsmooth or singular optimal control problems. These include control of high-speed vehicles, treatment of diseases, and optimization of manufacturing processes. The results of this project will confer benefits to the national health and prosperity by offering faster and more accurate solutions to these problems. Graduate students from diverse backgrounds will play a central role in the research including one mathematician and one engineer. The methods that are developed will be implemented in high quality software that will be made widely available.This research focuses on the development of new collocation methods, called hp methods, and the use of these collocation methods in solving optimal control problems with nonsmooth solutions. The approach is purely computational and does not require any a priori knowledge of the structure of the optimal solution. In addition, the methodology is aimed at solving challenging problems that arise when an optimal control is singular. The hp collocation methods are developed in a manner that enables accurate identification of the points where the optimal solution is nonsmooth. The methods developed in this research can be employed extremely efficiently using sparse nonlinear optimization techniques and will provide much higher accuracy solutions without a priori knowledge of the solution structure. With a suitable placement of the variable mesh points, the hp collocation methods converge exponentially fast even though the solution may be nonsmoooth. The approach developed in this research could lead to a very rapid mesh refinement process where a small mesh will be maintained and few mesh refinement iterations would be required to obtain a high-accuracy approximation of the optimal solution.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本计画提出求解非光滑及奇异最优控制问题的计算方法，其中控制输入被选择以最大化或最小化某个目标函数。当优化控制包括突然跳变时，出现非光滑问题。当控制输入不直接影响目标函数时，就会出现奇异问题。最优控制问题的解决方案是使用构造在网格点模式上的数值近似来找到的。该项目通过构建更好地捕获跳跃点的网格模式来改进现有的非光滑问题的近似方法，从而使用更少的计算能力获得更高的精度。这种创新还有助于更好地划分问题的任何单一区域。该项目通过仅在这些区域中对目标函数进行校正修改，进一步改进了奇异区域问题的控制解决方案。一系列具有国家重要性的问题可以被表述为非光滑或奇异的最优控制问题。其中包括控制高速车辆、治疗疾病和优化制造工艺。该项目的成果将通过为这些问题提供更快、更准确的解决方案，为国家健康和繁荣带来好处。来自不同背景的研究生将在研究中发挥核心作用，包括一名数学家和一名工程师。所开发的方法将在高质量的软件，将被广泛使用。本研究的重点是新的配置方法，称为HP方法的发展，并使用这些配置方法在解决最优控制问题的非光滑解决方案。该方法是纯粹的计算，不需要任何先验知识的结构的最佳解决方案。此外，该方法的目的是解决具有挑战性的问题时出现的最优控制是奇异的。的HP配置方法的方式，使准确识别的最优解是非光滑的点。在这项研究中开发的方法可以非常有效地使用稀疏非线性优化技术，并提供更高的精度解决方案，而无需先验知识的解决方案结构。通过适当地布置可变网格点，即使解可能是非光滑的，hp配点法也能指数快速收敛。在这项研究中开发的方法可能会导致一个非常快速的网格细化过程中，一个小的网格将保持和网格细化迭代将需要获得高精度近似的最佳solution.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。