Classification of extremals in a simplified Goddard model on the maximal height of rocket flight

We consider a problem on maximizing the height of vertical flight of a material point (“meteorological rocket”) in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The original Goddard problem is simplified by removing the dependence on the rocket mass from the equations of motion. Using the maximum principle we find all possible types of Pontryagin extremals and classify them w.r.t. problem parameters. Since the velocity of the point can be negative, we obtain some new types of extremals with two or three switching points, which optimality should be further investigated.