A Novel Regularization-Based Computational Framework for State-Constrained Optimal Control

该项目的研究计划是由广泛的实际应用的动机。一个相关的例子是癌症的局部热治疗，其目的是加热肿瘤细胞，但同时确保附近的健康细胞不被加热，因此不被损坏。这类目标称为具有逐点状态约束的最优控制问题。主要研究人员将开发新的计算模型来解决这些控制问题。他们的研究是基于数学和应用领域的不同学科的思想的交叉施肥，并在工程领域具有很强的影响潜力，例如优化生产热钢型材的过程而不产生裂纹。研究团队还将把他们的研究整合到数学科学的大学教育计划中，并将制作基本软件，用于解决属于同一建模框架的其他重要应用。主要研究人员的目标是开发一种新的正则化方法的逐点约束最优控制问题的解决方案。这些问题是最近大量研究的焦点，并对找到可靠的解决方案提出了严峻的挑战。其中一个主要问题是，相关的拉格朗日乘子是氡措施，使控制具有较低的规则性。这会导致不利的影响，在分析层面上获得最优条件的控制问题，并在数值层面上进行离散化时。 缺乏规律性可以归因于下面的有序锥有一个空的内部。因此，没有普遍的Karush-Kuhn-Tucker理论。事实上，Slater型约束条件的失效是应用数学的许多分支中的常见障碍，包括最优控制、反问题、非光滑优化和变分不等式。圆锥正则化提供了一个统一的框架来研究最优化问题，Slater型约束资格由于与不等式约束相关联的有序锥的空内部而无法保持。研究人员计划为具有逐点状态约束的偏微分方程和变分不等式的最优控制的圆锥正则化开发新的误差估计。该项目将测试纳什均衡问题，线性弹性和网络上的供应链的新理论成果。该项目还对研究生的培训产生了教育影响。研究人员将把教育与研究和设计课程结合起来，教授最先进的最优控制技术。

The research plan for this project is motivated by a broad range of practical applications. A relevant example is in the localized heat treatment of cancer in which the intent is to heat the tumor cells, but at the same time assure that nearby healthy cells are not heated, and hence not damaged. A goal of this kind is called an optimal control problem with pointwise state constraints. The principal investigators will develop novel computational models for the solution of these control problems. Their research is based on the cross-fertilization of ideas from diverse disciplines of mathematics and application domains, and has strong potential for impact in engineering domains such as the optimization of the process of producing hot steel profiles without the generation of cracks. The research team will also integrate their research in the university educational program in the mathematical sciences and will produce basic software that can be made available for solution of other significant applications that fall into the same modeling framework. The principal investigators aim to develop a novel regularization approach for the solution of pointwise constrained optimal control problems. Such problems are a focus of considerable recent research and pose serious challenges for finding reliable solutions. One of the main issues is that the associated Lagrange multipliers are Radon measures so that the control has low regularity. This causes adverse effects at the analytical level when obtaining optimality conditions for the control problem, and at the numerical level when performing discretization. The lack of regularity can be attributed to the fact that the underlying ordering cone has an empty interior. Consequently, no general Karush-Kuhn-Tucker theory is available. In fact, the failure of a Slater-type constraint qualification is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, non-smooth optimization, and variational inequalities. Conical regularization provides a unified framework to study optimization problems for which a Slater-type constraint qualification fails to hold due to the empty interior of the ordering cone associated with the inequality constraints. The investigators plan to develop new error estimates for the conical regularization for optimal control of partial differential equations and variational inequalities with pointwise state constraints. The project will test the new theoretical results for Nash equilibrium problems, linear elasticity, and supply chains on networks. The project also has an educational impact in the training of graduate students. The investigators will integrate education with research and design courses to teach state-of-the-art techniques on optimal control.