  Consider a partition of $R^n$ into finitely many polyhedral regions $D_i$ and
associated drift vectors $\mu_i\in R^n$. We study ``hybrid'' dynamical systems
whose trajectories have a constant drift, $\dot x=\mu_i$, whenever $x$ is in
the interior of the $i$th region $D_i$, and behave consistently on the boundary
between different regions. Our main result asserts that if such a system is
nonexpansive (i.e., if the Euclidean distance between any pair of trajectories
is a nonincreasing function of time), then the system must be conservative,
i.e., its trajectories are the same as the trajectories of the negative
subgradient flow associated with a potential function. Furthermore, this
potential function is necessarily convex, and is linear on each of the regions
$D_i$. We actually establish a more general version of this result, by making
seemingly weaker assumptions on the dynamical system of interest.
