A Weak Hamiltonian Finite Element Method for Optimal Guidance of an Advanced Launch Vehicle

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of nodal values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal of this work is to evaluate the feasibility of this approach for real-time guidance applications. A simplified model for an advanced launch vehicle application is presented which is suitable for finite element solution. Numerical results for this model will be presented in a later paper.