One dimensional trajectory optimization for stratospheric airship with varying thruster efficiency

This paper studies the minimal energy trajectory to drive the airship from one spot keeping position to a target spot keeping position. After determining and analyzing the three possible work modes of the airship by the Pontryagin´s minimum principle, two main problems are investigated. One is to show that the airship can not repeat any of the three modes. By this result, the possible switch sequence of work mode is reduced to a very limited number. The other is to determine the airspeed of the singular interval so that one may know how to drive an airship in an energy optimal trajectory. It is shown that, when the flight range is greater than a specified value, the energy optimal trajectory for a stratospheric airship to move forward in the wind is to accelerate in the first stage, after its airspeed reach a certain value, it then keep constant speed in the second stage and in the last stage it decelerated until the airship reach the target; when the flight range is too short, only acceleration and deceleration modes are needed.