Nonexpansive Piecewise Constant Hybrid Systems are Conservative
This theoretical result clarifies the fundamental structure of a class of hybrid dynamical systems, showing that nonexpansiveness implies conservatism, which is a strong constraint.
The authors prove that nonexpansive piecewise constant hybrid systems in Euclidean space are necessarily conservative, meaning their trajectories match the negative subgradient flow of a convex potential function that is linear on each polyhedral region.
Consider a partition of $R^n$ into finitely many polyhedral regions $D_i$ and associated drift vectors $μ_i\in R^n$. We study ``hybrid'' dynamical systems whose trajectories have a constant drift, $\dot x=μ_i$, 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.