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

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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.