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We show that a direct shooting method is mathematically equivalent to anindirect method in the sense of certain first-order conditions. Specificmathematical formulas pertaining to the equivalence of a direct shooting methodwith an indirect method are derived. We also show that a theoreticalequivalence does not necessarily translate to practical equivalence if theparameterized optimal control problem is simply patched to a nonlinearprogramming solver. A mathematical explanation is provided for the successesand failures of such patched nonlinear programming methods. In order togenerate the correct solution more consistently, the nonlinear programmingsolver used in a traditional direct method must be replaced or augmented by aHamiltonian programming method. The theoretical results derived in this paperfurther strengthen the connections between computational optimal control, deeplearning and automatic differentiation.

A Direct Shooting Method is Equivalent to an Indirect Method