Optimal Control of Parabolic Differential Equations Using Radau Collocation

TL;DR

This work addresses PDE-constrained optimal control for parabolic systems by marrying a multi-interval flipped Legendre-Gauss-Radau temporal collocation with a Galerkin finite element spatial discretization. The method maps the continuous problem to a sparse NLP, enforcing dynamics and boundary conditions at the terminal time via fLGR points and linearizing nonlinear variational terms through a Kirchhoff-type transformation. Key contributions include the finite-element–based discretization that avoids redefinition of NLP constraints, the multi-interval fLGR scheme that reduces the required number of temporal points, and a detailed derivation of gradients and Jacobians to enable efficient NLP solution. Numerically, the approach delivers competitive objective values against existing methods on Burgers’ and nonlinear heat problems, while enabling fewer temporal points and preserved problem structure, with practical impact for scalable PDE-constrained optimization.

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. Lastly, 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.

Figures (8)

• Figure 1: A depiction of the spatial discretization with piecewise linear elements and $n_x = 3$ LGR points between each node.
• Figure 2: Structure of the fLGR temporal differentiation matrix for a mesh with $J$ time intervals.
• Figure 3: A comparison between the fLGR and LGR temporal discretizations and the locations of the applied boundary conditions (boxed).
• Figure 4: Resulting structure of the NLP constraint Jacobian with P-1 Lagrange elements.
• Figure 5: The NLP constraint Jacobian for the Burgers' equation tracking problem with P-1 Lagrange elements and $N_t^{(1)} = N_t^{(2)} = 3$, $J = 2$, $N_x = 5$.