We propose a definition of regular synthesis that is more general than thos
e suggested by other authors such as Boltyanskii [SIAM J. Control Optim, 4
(1966), pp. 326-361] and Brunovsky [Math. Slovaca, 28 (1978), pp. 81-100],
and an even more general notion of regular presynthesis. We give a complete
proof of the corresponding sufficiency theorem, a slightly weaker version
of which had been stated in an earlier article, with only a rough outline o
f the proof. We illustrate the strength of our result by showing that the o
ptimal synthesis for the famous Fuller problem satis es our hypotheses. We
also compare our concept of synthesis with the simpler notion of a "family
of solutions of the closed-loop equation arising from an optimal feedback l
aw, and show by means of examples why the latter is inadequate, and why the
difficulty cannot be resolved by using other concepts of solution such as
Filippov solutions, or the limits of sample-and-hold solutions recently pro
posed as feedback solutions by Clarke et al. [ IEEE Trans. Automat. Control
, 42 (1997), pp. 1394-1407] for equations with a non-Lipschitz and possibly
discontinuous right-hand side.