An Adaptive Orthogonal Collocation Method for the Optimal Control of Parabolic Partial Differential Equations

An adaptive flipped Legendre-Gauss-Radau collocation method is presented for the solution of optimal control problems constrained by parabolic partial differential equations. A temporal, local error estimate is presented that measures the satisfaction of the differential equations at points other than the original collocation points. In intervals that do not satisfy a requested error tolerance, the mesh is refined based on an approximation of the decay rate of coefficients used in a Legendre polynomial state approximation. Based on the size of the approximated decay rate, either the number of intervals or the degree of the state approximation is increased. To avoid an overcollocated mesh, an avenue for mesh reduction is provided that is extended from ordinary differential equation applications to parabolic systems. Lastly, numerical examples demonstrate that the use of an adaptive flipped Legendre-Gauss-Radau method can lead to a reduction in the required number of collocation points to compute accurate solutions in comparison to other methods.