Fermat's Principle and Hamilton's Principle: Does a least action take a least time for happening?

We explore deeper and analyse in more detail Fermat's and Hamilton's principles. We try to address some questions: Is it possible to have S negative? Is Hamilton's principle always valid for entire path of the system? Is there a relation between Fermat's principle and Hamilton's principle? We assume analogy with Hamilton's principle, is Fermat's principle always valid for entire path of the system? Does a least action take a least time for happening?.