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 state-constrained 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.