Finite element method for the solution of state-constrained optimal control problems

This paper presents an extension of a finite element formulation based on a weak form of the necessary conditions to solve optimal control problems. First, a general formulation for handling internal boundary conditions and discontinuities in the state equations is presented. Then, the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions with touch points and solutions with stateconstrained arcs are considered. After the formulations are developed, suitable shape and test functions are chosen for a finite element discretization. It is shown that all element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form, yielding a set of algebraic equations. To demonstrate and analyze the accuracy of the finite element method, a simple state-constrained problem is solved. Then, for a more practical application of the use of this method, a launch vehicle ascent problem subject to a dynamic pressure constraint is solved. The paper also demonstrates that the finite element results can be used to determine switching structures and initial guesses for a shooting code.