A Flipped Radau Finite Element Framework for the Optimal Control of the Viscous Burgers’ Equation

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A computational framework for the solution of optimal control problems governed by a parabolic partial differential equation is presented. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem (NLP) through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau (fLGR) 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 nonlinearities in the variational description of the problem. Lastly, it is demonstrated on a numerical example that the use of a multi-interval fLGR 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.