A Direct Shooting Method is Equivalent to an Indirect Method

We show that a direct shooting method is mathematically equivalent to an indirect method in the sense of certain first-order conditions. Specific mathematical formulas pertaining to the equivalence of a direct shooting method with an indirect method are derived. We also show that a theoretical equivalence does not necessarily translate to practical equivalence if the parameterized optimal control problem is simply patched to a nonlinear programming solver. A mathematical explanation is provided for the successes and failures of such patched nonlinear programming methods. In order to generate the correct solution more consistently, the nonlinear programming solver used in a traditional direct method must be replaced or augmented by a Hamiltonian programming method. The theoretical results derived in this paper further strengthen the connections between computational optimal control, deep learning and automatic differentiation.