Optimal trajectories in a maximal height problem for a simplified version of the Goddard model in case of generalized media resistance function

We consider a family of the problems on maximization of the height of the vertical flight of a material point in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. Using the maximum principle we obtain classification of trajectories (w.r.t. problem parameters) which are suspected to be optimal and check that for some classical rocket systems optimal control in our model is the classical bang-bang or bang-singular-bang one. We obtain some new types of “potentially-optimal” trajectories which should be investigated.