Tail Reoptimization in Desensitized Optimal Control

The principle of optimality ensures that the tail segment of an optimal trajectory provides an optimal solution to the original trajectory optimization problem, provided that the problem is formulated in standard Mayer, Lagrange, or Bolza form and that the initial states of the reoptimized tail align with the optimal reference solution. In desensitized optimal control (DOC) problems, the final values of both physical and sensitivity states often appear on the right-hand side of the differential equations. Transforming such problems into standard form necessitates introducing additional states with trivial dynamics and free initial values to represent the final values of the physical and sensitivity states. Numerical tests reveal that tail reoptimization in DOC problems can enhance performance if the initial values of these artificial states are left unconstrained. However, by virtue of the principle of optimality, this reoptimized tail cannot yield performance improvements for the overall path-planning problem starting from the initial time of the optimal reference solution. This counterintuitive result leads to the distinction between the planning phase and the execution phase. Moreover, for broad classes of DOC and stochastic optimal control problems, it implies that optimal trajectories can become suboptimal once execution begins, necessitating continuous reoptimization of the tail segment during execution to maintain optimality.