Decentralized control of discrete event systems with bounded or unbounded delay communication

We introduce problems of decentralized control with delayed communication, where delays are either unbounded or bounded by a given constant k. In the k-bounded-delay model, between the transmission of a message and its reception, the plant can execute at most k events. In the unbounded-delay model, the plant can execute any number of events between transmission and reception. We show that our framework yields an infinite hierarchy of control problems, /spl Cscr//spl Cscr/ =/spl Dscr//spl Cscr//spl Cscr//sub 0//spl sup//spl Dscr//spl Cscr//spl Cscr//sub 1//spl sup//spl Dscr//spl Cscr//spl Cscr//sub 2//spl sup//spl middot//spl middot//spl middot//spl sup//spl Dscr//spl Cscr//spl Uscr//spl Cscr//spl sup//spl Dscr//spl Cscr/ , where CC is the set of control problems solvable with a single controller (centralized case) and /spl Dscr//spl Cscr//spl Cscr//sub k/ (resp. /spl Dscr//spl Cscr//spl Uscr//spl Cscr/, /spl Dscr//spl Cscr/) is the set of problems solvable with two controllers in a k-bounded-delay network (resp. two controllers in an unbounded-delay network, two controllers without communication). The above containments are strict. We prove the undecidability of checking the existence of controllers in the unbounded-delay case, or in the case without any communication. Finally, we prove that a decentralized observation problem with bounded-delay communication is decidable.