Rapid progresses in information and computer technology allow the development of more advanced optimal control algorithms dealing with real-world problems. In this paper, which is Part 1 of a two-part sequence, a multiple-subarc gradient-restoration algorithm (MSGRA) is developed. We note that the original version of the sequential gradient-restoration algorithm (SGRA) was developed by Miele et al. in single-subarc form (SSGRA) during the years 1968–86; it has been applied successfully to solve a large number of optimal control problems of atmospheric and space flight. MSGRA is an extension of SSGRA, the single-subarc gradient-restoration algorithm. The primary reason for MSGRA is to enhance the robustness of gradient-restoration algorithms and also to enlarge the field of applications. Indeed, MSGRA can be applied to optimal control problems involving multiple subsystems as well as discontinuities in the state and control variables at the interface between contiguous subsystems. Two features of MSGRA are increased automation and efficiency. The automation of MSGRA is enhanced via time normalization: the actual time domain is mapped into a normalized time domain such that the normalized time length of each subarc is 1. The efficiency of MSGRA is enhanced by using the method of particular solutions to solve the multipoint boundary-value problems associated with the gradient phase and the restoration phase of the algorithm. In a companion paper, MSGRA is applied to compute the optimal trajectory for a multistage launch vehicle design, specifically, a rocket-powered spacecraft ascending from the Earth surface to a low Earth orbit. Single-stage, double-stage, and triple-stage configurations are considered and compared.
