Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise With Heading and Throttle Controls Using Pontryagin’s Maximum Principle

In this letter, we consider the commercial aircraft trajectory optimization problem for a general cruise model with arbitrary spatial wind fields to be solved using the Pontryagin’s maximum principle. The model features two fundamental controls, namely “throttle setting” (appearing as a singular control) and “heading angle” (appearing as a regular control). For a problem with state-inequality constraints and minimum time-fuel objective, we show that the optimal “heading angle” can be described through the classic Zermelo’s navigation identity. We also demonstrate, by analyzing the switching function, that the singular “throttle setting” can be characterized through a feedback function that relies on both the optimal states and “heading angle”. 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.