A quasi-sequential algorithm for PDE-constrained optimization based on space–time orthogonal collocation on finite elements

Abstract

Orthogonal collocation on finite elements (OCFE) has been used universally to approximate ODEs to date. For PDEs, this contribution presents a novel discretization scheme applying the methodology of OCFE to discretize both space and time domain simultaneously, named as space–time orthogonal collocation on finite elements (ST-OCFE). Due to the existence of boundary conditions, the selection of discrete points and the constitution of discretized algebraic equations are different in space and time domain. Furthermore, for solving optimal control problems constrained by PDEs, a discretize-then-optimize algorithm based on ST-OCFE and quasi-sequential approach is proposed. The formulation of discretized optimization problems and the procedure of sensitivity computation are deduced. In the algorithm, diverse types of PDEs, state constraints, and general control parameterization are considered. The proposed method has the advantages of generality, higher numerical accuracy, and easy handling of state constraints, as demonstrated by three examples.