Infinite time regular synthesis

In this paper we provide a new suficiency theorem for regular syntheses. The concept of regular synthesis is discussed in [12], where a suficiency 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 necessarily bounded. This research is motivated by the fact that one of the main tools toward the construction of optimal syntheses is the proof of a strong suficiency theorem. The regularity assumptions of the main theorem in [12] are verified by every piecewise smooth feedback control generating extremal trajectories that reach the target in finite time with a finite number of switchings (indeed even by more complicate syntheses like the Fuller one presenting trajectories with an infinite number of switchings). In the case of this paper the situation is even more complicate, since we admit both trajectories with nite and innite time. It is important to notice that, in spite of its complexity, this situation is encountered in many simple cases like linear quadratic problems (see the example of the last section and [13]). We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function. However, in this paper we need the value function to be continuous at the origin (see Remark 3.3 for more details). The general case of synthesis generated by general piecewise smooth feedback deserves a further careful investigation.