This chapter presents the theoretical base on optimization using the Pontryagin maximum principle (PMP). It contains the background information for the following chapters. In particular, the formulation of PMP and a variation form for derivation of the boundary conditions for conjugate system (transversality conditions) are given, which are convenient for adapting to various problem statements, including branching processes with finite and continuous sets of branches, which are considered here. The significant advantages of PMP for the purposes of the aerospace trajectory optimization are substantiated in detail. Attention is focused on the methodological issues of solving multi-point boundary value problems, to which PMP leads, based on the solution continuation method and selection of local extremals. There is emphasized the importance for further applications of the physical meaning of the conjugate variables and Lagrange variables as sensitivity functions of the functional to the phase vector, boundary conditions and constraints. There are described the approach to verify computer programs at various stages of their development: from writing the optimality conditions to debugging. It is principal that these techniques are developed on objective basis, following from mathematical properties of the invariants and conjugate system. In conclusion, a description of the developed ASTER software package is given, with the help of which all the calculations of the first six chapters were performed.