CAREER: Constrained Optimal Control of Partial Differential Equations for Improving Energy Utilization in Transportation and in the Built Environment

该学院早期职业发展（Career）奖将导致创建一个新的数学框架，用于解决由偏微分方程描述的系统中具有输入和状态约束的最优控制问题。偏微分方程与电化学和热能储存、能量分配和转换等工程领域有关，经常用于交通运输中的交通流量预测。将最优控制应用于此类偏微分方程系统将显著降低能源利用率，并提高住宅和交通部门的可持续性，然而，这些好处取决于控制算法满足输入和状态约束的能力。这项工作将建立一个新兴的跨学科研究项目，跨越热流体科学、控制理论、建模和仿真。该奖项将通过启动针对大二和大三机械工程专业学生的新项目，支持指导和激励本科生追求研究经验的教育目标。这一举措将支持学生留在他们选择的专业，特别是来自代表性不足的群体，并将导致更多的国内学生攻读研究生教育。本研究将建立一种理论和方法来获得偏微分方程约束最优控制问题的近似解。建立的实践依赖于一个两步过程，其中首先将对象离散为有限维系统，然后在降阶模型上设计控制器。这项研究将克服传统方法的理论和实践局限性，导致(i)不需要校准来补偿模型近似的控制器；（ii）保证满足无限维系统的约束；（iii）无需开发精简的植物模型。实现这一目标的数学工具是通过值函数的参数化推导出Hamilton-Jacobi-Bellman方程的近似解，这是本研究的核心贡献。最终，该项目将(i)推进偏微分方程的约束最优控制理论，通过变换方法对Hamilton-Jacobi-Bellman方程的参数解进行约化；（ii）通过将其解释为无限维系统的有限阶近似，推进对大型系统约束控制的理解；（iii）为用偏微分方程描述的系统开发实用的、在线的状态反馈控制器实现方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This Faculty Early Career Development (CAREER) award will lead to the creation of a new mathematical framework for the solution of optimal control problems with input and state constraints in systems described by Partial Differential Equations. Partial Differential Equations are relevant to several engineering fields, such as electrochemical and thermal energy storage, energy distribution and conversion, and are frequently used to predict traffic flows in transportation. Application of optimal control to such Partial Differential Equations systems will lead to a significant reduction in the energy utilization and improve sustainability in the residential and transportation sectors, however such benefits are contingent upon the ability of the control algorithm to satisfy input and state constraints. This work will establish an emerging and interdisciplinary research program that bridges across thermal and fluid sciences, control theory, modeling and simulation. This award will support the educational goal of mentoring and motivating undergraduate students in pursuing a research experience by starting a new program targeting Sophomore and Junior Mechanical Engineering students. This initiative will support the retention of students in their program of choice, particularly from underrepresented groups, and will result in larger number of domestic students pursuing graduate education.This research will lead to the creation of a theory and methods to obtain approximate solutions of constrained optimal control problems for Partial Differential Equations. The established practice relies on a two-steps process, in which the plant is first discretized to a finite dimensional system, then the controller is designed on the reduced order model. This research will overcome the theoretical and practical limitations of conventional methods, leading to (i) controllers that do not require calibration to compensate for model approximations; (ii) guaranteed satisfaction of constraints on the infinite dimensional system; (iii) eliminating the need to develop reduced plant models. The enabling mathematical tool to achieve this goal is the derivation of an approximate solution of the Hamilton-Jacobi-Bellman equation via parametrization of the value function, which is a central contribution of this research. Ultimately, this project will (i) advance the theory of constrained optimal control of Partial Differential Equations, through transformative methods for the parametric solution of the Hamilton-Jacobi-Bellman equation via reduction; (ii) advance the understanding of constrained control of large scale system, by interpreting them as finite-order approximations of infinite dimensional systems; (iii) develop methods for practical, online implementation of state feedback controllers for systems described by Partial Differential Equations.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.