Minimum time multiple

In this paper, the problem of minimum time multiple-burn optimization of an upper stage with a limited thrust, and engine restart capability for satellite injection into geostationary orbit are considered. The goals are to find thrust vector angle, times of the engine firings, and optimal duration of active phases of the upper stage to minimize fuel consumption and meet the desired boundary conditions. Various flight sequences with multiple burns, from two burns up to six burns, are considered. Also, the optimal trajectory for each sequence is derived. To solve the multi-point boundary value problem, an improved indirect shooting method with high performance is presented and used for an optimal solution. All in all, this novel method presented for multi-burn problem, not only with a very good accuracy, but also with a very fast convergence to the desired end conditions