Optimizing a vehicle trans-atmospheric motion using Pontryagin’s maximum principle

The task of optimizing trans-atmospheric motion of a flight vehicle in order to maximize its final velocity with prescribed finite values of the height and flight path angle is considered. The angle of attack acts as control in passive motion of a vehicle. Previously, the sequential linearization method was used to solve this optimization task. It is shown that at great altitudes the control programs are slightly different depending on the chosen initial approximation.  Therefore, the aim of this work is to determine the optimum control program on the basis of a “strict” solution of the optimization task using the Pontryagin’s maximum principle. Solving the problem of optimizing trans-atmospheric motion of a flight vehicle is illustrated by passive climb of the sub-hypersonic vehicle MPV (the first stage of the aerospace system RASCAL designed in the USA). The coefficient of lift (angle of attack) increases in the greater part of the trajectory to provide the prescribed finite values of height and path inclination and then decreases to provide maximum final velocity. The correctness of the obtained solutions of the optimization task using the maximum principle is confirmed by the zero Hamiltonian value in the optimum trajectory. The results of vehicle motion simulation with optimal control and various initial conditions of motion and the vehicle mass are discussed. The results obtained show that the solutions of the optimization task under consideration using the maximum principle and the sequential linearization principle are in close agreement.