‘Paradoxes’ of optimal solutions in problems of space vehicle injection and reentry

Different types of local extremals obtained in solving trajectory optimization problems related to space vehicle injection and reentry by applying the Pontryagin's maximum principle are demonstrated. Some solutions differ qualitatively from traditional ones. It is shown that almost each local extremal revealed provides a global optimum within a certain range of parameter values. The change in the type of the global extremal is, as a rule, of bifurcation nature. Multiplicity of local extremals and significantly nonlinear effects in the behavior of globally optimal solutions are basically caused by the presence of aerodynamic forces which can be sufficiently small as compared with the vehicle weight. Numerical results demonstrate the importance of the study of different types of local extremals at all aerospace vehicle design stages because the implementation of nontraditional optimal control laws can demand changes in the space vehicle configuration.