  We present a novel direct transcription method to solve optimization problems
subject to nonlinear differential and inequality constraints. We prove
convergence of our numerical method under reasonably mild assumptions:
boundedness and Lipschitz-continuity of the problem-defining functions. We do
not require uniqueness, differentiability or constraint qualifications to hold
and we avoid the use of Lagrange multipliers. Our approach differs
fundamentally from well-known methods based on collocation; we follow a
penalty-barrier approach, where we compute integral quadratic penalties on the
equality path constraints and point constraints, and integral log-barriers on
the inequality path constraints. The resulting penalty-barrier functional can
be minimized numerically using finite elements and penalty-barrier
interior-point nonlinear programming solvers. Order of convergence results are
derived, even if components of the solution are discontinuous. We also present
numerical results to compare our method against collocation methods. The
numerical results show that for the same degree and mesh, the computational
cost is similar, but that the new method can achieve a smaller error and
converges in cases where collocation methods fail.
