  This paper aims to provide a Dynamic Programming (DP) approach to solve the
Mission-Wide Chance-Constrained Optimal Control Problems (MWCC-OCP). The
mission-wide chance constraint guarantees that the probability that the entire
state trajectory lies within a constraint/safe region is higher than a
prescribed level, and is different from the stage-wise chance constraints
imposed at individual time steps. The control objective is to find an optimal
policy sequence that achieves both (i) satisfaction of a mission-wide chance
constraint, and (ii) minimization of a cost function. By transforming the
stage-wise chance-constrained problem into an unconstrained counterpart via
Lagrangian method, standard DP can then be deployed. Yet, for MWCC-OCP, this
methods fails to apply, because the mission-wide chance constraint cannot be
easily formulated using stage-wise chance constraints due to the
time-correlation between the latter (individual states are coupled through the
system dynamics). To fill this gap, firstly, we detail the conditions required
for a classical DP solution to exist for this type of problem; secondly, we
propose a DP solution to the MWCC-OCP through state augmentation by introducing
an additional functional state variable.
