Continuous-time differential dynamic programming with terminal constraints

In this work, we revisit the continuous-time Differential Dynamic Programming (DDP) approach for solving optimal control problems with terminal state constraints. We derive two algorithms, each for different order of expansion of the system dynamics and we investigate their performance in terms of their convergence speed. Compared to previous work, we provide a set of backward differential equations for the value function expansion by relaxing the assumption that the initial nominal control must be very close to the optimal control solution. We apply the derived algorithms to two classical optimal control problems, namely, the inverted pendulum and the Dreyfus rocket problem and show the benefit of second order expansion.