A Unified Local Convergence Analysis of Differential Dynamic Programming, Direct Single Shooting, and Direct Multiple Shooting

We revisit three classical numerical methods for solving unconstrained optimal control problems - differential dynamic programming, direct single shooting, and direct multiple shooting - and examine their local convergence behaviour. In particular, we show that all three methods converge with the same linear rate if a Gauss-Newton (GN) - or Generalized Gauss-Newton (GGN) - Hessian approximation is used, which is the case in widely used implementations such as iLQR.