Direct approaches for PDE-constrained dynamic optimization based on space–time orthogonal collocation on finite elements

Abstract

Dynamic optimization constrained by partial differential equations (PDE) involves the state distribution on space, and can generate improved operational policies compared to ODE-constrained dynamic optimization. With the help of space–time orthogonal collocation on finite elements (ST-OCFE) which can guarantee relatively high accuracy in both space and time, PDE models are transformed into a set of algebraic equations. For solving PDE-constrained optimization problems, three direct approaches, namely ST-OCFE based single shooting, ST-OCFE based multiple shooting, and ST-OCFE based simultaneous collocation are proposed. The similarities and differences among these approaches are analyzed, where the discretized nonlinear programming problems to be solved in each algorithm are described. Furthermore, the first-order and second-order sensitivity computation for ST-OCFE based shooting approaches is introduced with algorithmic differentiation to efficiently evaluate the exact Hessian of the Lagrangian. Three cases including different dynamics and constraints are investigated. The optimal solution given by these ST-OCFE based direct approaches is consistent, demonstrating the effectiveness of these algorithms.