Franck Cassez, Thomas A. Henzinger, and Jean-Francois Raskin
In the literature, we find several formulations of the control problem for timed and hybrid systems. We argue that formulations where a controller can cause an action at any point in dense (rational or real) time are problematic, by presenting an example where the controller must act faster and faster, yet causes no Zeno effects (say, the control actions are at times 0, 0.5, 1, 1.25, 2, 2.125, 3, 3.0625, ...). Such a controller is, of course, not implementable in software. Such controllers are avoided by formulations where the controller can cause actions only at discrete (integer) points in time. While the resulting control problem is well-understood if the time unit, or "sampling rate" of the controller, is fixed a priori, we define a novel, stronger formulation: the discrete-time control problem with unknown sampling rate asks if a sampling controller exists for some sampling rate. We prove that this problem is undecidable even in the special case of timed automata.
Proceedings of the Fifth International Workshop on Hybrid Systems: Computation and Control (HSCC), Lecture Notes in Computer Science 2289, Springer, 2002, pp. 134-148.