## A Comparison of Control Problems for Timed and Hybrid Systems

*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.

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