An Adaptive Method for Optimal Control Problems Constrained by Parabolic Differential Equations

Abstract

An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the spatial domain is discretized using the hp-Galerkin finite element method. To address nonlinearities in the variational form, a Kirchhoff-like integral transformation is applied to linearize the dynamics. In the temporal dimension, an orthogonal collocation scheme, the hp-flipped Legendre-Gauss-Radau method, is employed to fully discretize the problem, yielding a large, sparse nonlinear programming problem. Upon solving the nonlinear programming problem, solution accuracy is assessed through an implicit residual estimation procedure. This approach evaluates the local error by solving auxiliary residual problems over selected subdomains, providing a novel means of error estimation within an orthogonal collocation framework for optimal control. Based on the computed error estimate, the mesh is adaptively refined or coarsened to meet a prescribed error tolerance. Mesh refinement is guided by the estimated regularity of the solution which is determined via the decay rate of the coefficients of a Legendre polynomial expansion. In overcollocated regions, a mesh reduction strategy is adapted from orthogonal collocation methods for application within the finite element framework. Numerical examples demonstrate that the proposed method can reduce the error by up to five orders of magnitude in both spatial and temporal dimensions.

Figures (11)

• Figure 1: A depiction of the spatial discretization.
• Figure 2: A deptiction of the temporal differentiation matrix, $D_t$, and its organization.
• Figure 3: A depiction of the mesh for the local element residual problem with $p^{(k)} = 4$.
• Figure 4: A depiction of the mesh for the interval residual problem.
• Figure 5: Legendre polynomial coefficients of approximations of functions given in Eq. \ref{['eq:funcs']}.