ABSTRACT
This chapter wants to explain the basic concepts of optimal presynthesis and synthesis, and show how the theory of subanalytic sets can be used to prove existence theorems. It aims to establish the link with several other papers, by showing how the results on synthesis lead to the search for theorems on the structure of optimal trajectories. The existence problem for optimal presynthesis and synthesis is closely related to the question of existence of optimal trajectories. Regarding the characterization of optimal presyntheses and syntheses, the situation is quite remarkably different from what happens for optimal trajectories. The Pontryagin Maximum Principle gives a necessary condition for optimality, and many other conditions have been found, but it has been clear for a long time that, except for very special problems such as linear systems with a convex Lagrangian, the question of sufficient conditions is essentially hopeless.
