E. Asarin, O. Maler, A. Pnueli, **On the Analysis of
Dynamical Systems having Piecewise-Constant Derivatives**.

In this paper we consider a class of hybrid systems, namely
dynamical systems with piecewise-constant derivatives (PCD
systems). Such systems consist of a partition of the Euclidean
space into a finite set of polyhedral sets ({\it regions}).
Within each region the dynamics is defined by a constant vector
field, hence discrete transitions occur only on the boundaries
between regions where the trajectories change their direction.

With respect to such systems we investigate the reachability
question: {\it Given an effective description of the systems and
of two polyhedral subsets $P$ and $Q$ of the state-space, is
there a trajectory starting at some $\vx \in P$ and reaching some
point in $Q$? } Our main results are a decision procedure for two-dimensional
systems, and an undecidability result for three or more
dimensions. [Postscript]