Optimal Control of Parabolic Differential Equations Using Radau Collocation

Abstract

A method is presented for the numerical solution of optimal boundary control problems governed by parabolic partial differential equations. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau collocation method is implemented for temporal discretization alongside a Galerkin finite element spatial discretization. The finite element discretization allows for a reduction in problem size and avoids the redefinition of constraints required under a previous method. Further, a generalization of a Kirchoff transformation is performed to handle variational form nonlinearities in the context of numerical optimization. Due to the correspondence between the collocation points and the applied boundary conditions, the multi-interval flipped Legendre-Gauss-Radau collocation method is demonstrated to be preferable over the standard Legendre-Gauss-Radau collocation method for optimal control problems governed by parabolic partial differential equations. The details of the resulting transcription of the optimal control problem into a nonlinear programming problem are provided. Numerical examples demonstrate that the use of a multi-interval flipped Legendre-Gauss-Radau temporal discretization can lead to a reduction in the required number of collocation points to compute accurate values of the optimal objective in comparison to other methods. Lastly, a self-convergence analysis on each test problem illustrates that the error decays exponentially as a function of the mesh size in both the temporal and spatial dimensions.

39 pages, 11 figures. Portions of this work were presented at the 2024 American Control Conference (ACC) in Toronto, Ontario, Canada, https://doi.org/10.23919/ACC60939.2024.10644743. Portions of the work were also presented at the 2025 AAS/AIAA Spaceflight Mechanics Meeting in Lihue, Hawaii. Other portions will be presented at the 2025 American Control Conference (ACC) in Denver, Colorado