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

这个项目提出了解决非光滑和奇异最优控制问题的计算方法，在该问题中，选择一个控制输入来最大化或最小化某个目标函数。当最优控制包含突然跳跃时，会出现非光滑问题。当控制输入不直接影响目标函数时，出现奇异问题。最优控制问题的解是使用建立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捉跳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来获得更高的精度。这一创新还有助于更好地划分问题的任何单一区域。该项目进一步改进了奇异区域问题的控制解决方案，只对这些区域的目标函数进行了修正。一系列具有国家重要性的问题可以表示为非光滑或奇异的最优控制问题。这些措施包括对高速车辆的控制、疾病的治疗和制造工艺的优化。该项目的成果将为这些问题提供更快、更准确的解决方案，从而造福于国家健康和繁荣。来自不同背景的研究生将在这项研究中发挥核心作用，其中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中实现，并将被广泛使用。本研究的重点是开发新的配置方法，称为惠普方法，并将这些配置方法用于求解具有非光滑解的最优控制问题。这种方法纯粹是计算性的，不需要任何关于最优解结构的先验知识。此外，该方法的目的是解决当最优控制是奇异时出现的挑战性问题。HP配置方法的开发方式使得能够准确地识别最优解是非光滑的点。本研究开发的方法可以非常有效地使用稀疏非线性优化技术，并将在不需要了解解结构的先验知识的情况下提供更高精度的解。在适当布置可变网格点的情况下，即使解可能是非光滑的，hp配置法也能以指数级的速度收敛。在这项研究中开发的方法可以导致非常快速的网格优化过程，其中将保持较小的网格，并且只需要很少的网格优化迭代就可以获得最优解的高精度近似值。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.