Zermelo Navigation Problem with State Constraints

The article uses the example of the Zermelo navigation problem to illustrate a simple way to address state constraints of a certain type. The problem of the horizontal distance maximization of an autonomous aircraft operating in a steady, homogeneous flow field is considered. Process time is fixed beforehand. The simple particle model of the aircraft is considered. The particle moves in a horizontal plane with a constant modulus velocity relative to the flow of the medium. The angular velocity of rotation of the particle velocity vector is considered as the control variable. The angle between the velocity vector and the horizontal axis is subjected to a phase constraint. The structure of the dynamic system allows to reduce the optimal problem to the problem of a smaller dimension. In reduced problem the state constraints transform to the constraints on the control variables. For the reduced problem, the optimal synthesis is designed. Next, for the original problem, the sequence and the number of the arcs with motion along state constraints are determined. The control law in the initial problem is established.