The Ritz–Trefftz Method for State and Control Constrained Optimal Control Problems

The Ritz–Trefftz method is analyzed for control problems with quadratic cost, linear dynamics and linear inequality state and control constraints. Although the dual function is only semi-definite, a solution to the Ritz–Trefftz problem is proved to exist. First order convergence of the Ritz–Trefftz method in piecewise polynomial subspaces is easily proved; however, higher order convergence requires not only that higher order spaces be used, but also that the grid points in a neighborhood of changes in binding constraints be left as free parameters in the maximization procedure. The convergence proofs are based on a theorem on the regularity of the solution to the control problem and results on the approximation of either nonnegative or monotone functions by nonnegative or monotone polynomials. The paper concludes with a bound on the error in the free boundary (or time when a constraint becomes binding) for the Ritz–Trefftz problem.