Solving optimal control problems using hp-version finite elements in time

A previously published temporal finite element method for optimal control problems is updated to include higher-order hierarchical shape functions. The accuracy and power of the analysis and corresponding code are illustrated on optimal control problems with nonlinear system dynamics, using interfaces based on symbolic mathematics to automatically generate the necessary derivative expressions and nonlinear algebraic equations to be solved. The class of problems includes those that can have free or fixed final time, multiple phases, nonlinear/periodic boundary conditions, and control constraints. This code has been tested on a variety of problems, culminating in a three-phase, seven-state, two-control missile problem with nonlinear system dynamics, where accuracy was found to be significantly improved over h-version results, with potential order-of-magnitude reductions in CPU time and numbers of parameters for a given level of error.