  In this research, we consider the commercial aircraft trajectory optimization
problem for a general cruise model with arbitrary spatial wind fields to be
solved using the Pontryagin maximum principle. The model features two
fundamental controls, namely, throttle setting (which appears as a singular
control) and heading angle (appearing as a regular control). For a constrained
problem with minimum time-fuel objective, we show that the optimal heading
angle is fully defined through the classic Zermelo navigation identity. We also
show that the optimal throttle setting can be characterized through a complete
feedback function. The switching-point algorithm is employed to solve a case
study where we inspect the optimality conditions and graph the optimal controls
together with the optimal state and co-state variables.
