In this paper we provide a new sufficiency theorem for regular syntheses. The concept of regular synthesis is discussed in [12], wherea sufficiency theorem for finite time syntheses is proved. There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarilybounded. The regularity assumptions of the main theorem in [12] are verified by every piecewise smooth feedback control generatingextremal trajectories that reach the target in finite time with a finite number of switchings. In the case of this paper the situation iseven more complicate, since we admit both trajectories with finite and infinite time. We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function.
