Nonexpansive Piecewise Constant Hybrid Systems are Conservative

TL;DR

This paper proves that nonexpansive hybrid systems with piecewise constant dynamics are necessarily conservative, with trajectories corresponding to convex potential functions that are linear on each region.

Contribution

It establishes a fundamental link between nonexpansiveness and conservativeness in hybrid systems, showing such systems are characterized by convex potential functions.

Findings

Nonexpansive hybrid systems are conservative.

Trajectories correspond to convex potential functions.

Potential functions are linear on each region.

Abstract

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.