Bridging the Gap Between Multigrid, Hierarchical, and Receding-Horizon Control

We analyze the structure of the Euler-Lagrange conditions of a lifted long-horizon optimal control problem. The analysis reveals that the conditions can be solved by using block Gauss-Seidel schemes and we prove that such schemes can be implemented by solving sequences of short-horizon problems. The analysis also reveals that a receding-horizon control scheme is equivalent to performing a single Gauss-Seidel sweep. We also derive a strategy that uses adjoint information from a coarse long-horizon problem to correct the receding-horizon scheme and we observe that this strategy can be interpreted as a hierarchical control architecture in which a high-level controller transfers long-horizon information to a low-level, short-horizon controller. Our results bridge the gap between multigrid, hierarchical, and receding-horizon control.