On the time-optimal 2D-trajectories in non-uniform mediums

This paper addresses two-dimensional trajectory optimization problems, where the mover''s speed monotonically decreases/increases in one of the space''s coordinates. For instance, it is well-known that the absolute value of the helicopter speed decreases in altitude because air pressure drops. Then, it is not trivial to find a time-minimal helicopter trajectory from one given point to another. We address such problems in different settings for the medium (atmosphere) and in the presence of obstacles. First, we consider the basic problem without any obstacles, where the mover''s speed decreases linearly in altitude. We show that the problem is reducible to the L''Hopital problem, one of the well-studied problems in geometrical optics. In this case, the time-optimal trajectory is a circular segment, and therefore can be expressed in a closed analytic form. Next, we adress the problem with linear speed decrease in presence of rectilinear obstacles. We show that this problem can be solved in polynomial time. Finally, we consider the case without obstacles, where the medium is non-uniform and the mover''s velocity is a piece-wise linear concave monotonically decreasing function. This is a widely accepted model in our motivating application of helicopter flights. In this case, we reduce the problem to the solution of a system of polynomial equations of fixed degree. Here, if the number of breakpoints of the velocity function is a constant, the algebraic elimination theory allows us to solve the problem in constant time with any given level of precision.