Singular optimal atmospheric rocket trajectories

Singular subarcs arise in quite a few problems of flight dynamics. The present study is devoted to the specific problem of ascent and acceleration of a vehicle in atmospheric flight in which a variable-thrust arc forms a part of the optimal trajectory. A two-parameter family of singular arcs was generated for time-range-fuel problems of an ascending rocket, using the modelling of Zlatskiy and Kiforenko. The short-term optimality of the singular subarcs has been checked in terms of certain necessary conditions: the classical Clebsch condition, the Kelley condition or the Generalized Legendre-Clebsch condition and the Goh condition. All these are found to be satisfied computationally for all the candidates. The calculations were repeated for simplified thrust-along-the-path modelling and similar results on optimality obtained.