Second-order tests in optimization theories

There are two main kinds of second-order tests in optimization theories. The simplest tests (Refs. 1–13), such as the generalized Legendre-Clebsch condition, require a local study around points of the trajectory of interest (generally, on a singular arc). Tests of the second kind (Refs. 14–22 and 51–53) are more difficult to use but also more efficient: they can be applied under various assumptions of convexity or linearity and generally require some integrations along the trajectory of interest (usually, these integrations can only be done by numerical methods). In favorable cases, the better tests of this second kind lead to either the conclusion of nonoptimality or the conclusion of local optimality. This survey paper begins with a broader question, the question of sufficient conditions for absolute optimality. Some results of this study are used to define anadjoint matrix (extension of the notion ofadjoint vector of Pontryagin). Then, the different second-order tests can be unified and generalized even if the trajectory of interest has switches and singular arcs. On a given trajectory, theconjugate points are easily related to the evolution of the adjoint matrix. Finally, this generalized second-order test is applied to a singular arc of astrodynamics, the reversible arc: this arc is globally optimal from end to end.