Regular Synthesis and Sufficiency Conditions for Optimality

We propose a deenition of \regular synthesis," more general than those suggested by other authors such as Boltyanskii and Brunovsk y, and an even more general notion of \regular presynthesis." We give a complete proof of the corresponding suuciency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisses 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 law," and show by means of examples why the latter is inadequate, and why the diiculty cannot be resolved by using other concepts of solution|such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke, Ledyaev, Subbotin and Sontag|for equations with a non-Lipschitz and possibly discontinuous right-hand side.