  This article contributes to a framework for a computational indirect method
based on the Pontryagin maximum principle to efficiently solve a class of state
constrained time-optimal control problems in the presence of a time-dependent
flow field. Path-planning in a flow with tidal variations is an emblematic
real-life task that serves as an example falling in the class of problems
considered here. Under rather natural assumptions on the flow field, the
problem is regular and the measure Lagrange multiplier, associated with the
state constraint, is continuous. These properties (regularity and continuity)
play a critical role in computing the field of extremals by solving the
two-point boundary value problem given by the maximum principle. Moreover, some
sufficient conditions preventing the emergence of singular control processes
are given. A couple of examples of time-periodic fluid flows are studied and
the corresponding optimal solutions are found.
