Regular presynthesis and synthesis, and optimality of families of extremals

The purpose of the theory of regular synthesis is to turn into a rigorous, precise theorem the vague assertion that “if a collection Γ of extremals covers the whole space in a smooth enough way, then all the members of Γ are optimal.” Naturally, a good definition should at the same time be stringent enough to lead to a correct proof of the sufficiency assertion, and broad enough to cover as large as possible a class of optimal control problems. In particular, it would be desirable for the theory to apply to well-known examples. The main goals of this paper are (a) to propose a new definition of regular synthesis, more general than those suggested by other authors, as well as an even more general notion of regular presynthesis, and (b) to give a detailed statement of the corresponding sufficiency theorem