  In graph-theoretical terms, an edge in a graph connects two vertices while a
hyperedge of a hypergraph connects any more than one vertices. If the
hypergraph's hyperedges further connect the same number of vertices, it is said
to be uniform. In algebraic graph theory, a graph can be characterized by an
adjacency matrix, and similarly, a uniform hypergraph can be characterized by
an adjacency tensor. This similarity enables us to extend existing tools of
matrix analysis for studying dynamical systems evolving on graphs to the study
of a class of polynomial dynamical systems evolving on hypergraphs utilizing
the properties of tensors. To be more precise, in this paper, we first extend
the concept of a Metzler matrix to a Metzler tensor and then describe some
useful properties of such tensors. Next, we focus on positive systems on
hypergraphs associated with Metzler tensors. More importantly, we design
control laws to stabilize the origin of this class of Metzler positive systems
on hypergraphs. In the end, we apply our findings to two classic dynamical
systems: a higher-order Lotka-Volterra population dynamics system and a
higher-order SIS epidemic dynamic process. The corresponding novel stability
results are accompanied by ample numerical examples.

Systems and Control;Systems and Control