Stability of bounded subsets of Metzler sparse matrix cones

We study the existence of Hurwitz matrices in a class of Metzler sparse matrix cones, where the non-zero entries are constrained to be in a bounded set. In particular, we provide necessary and sufficient conditions for the existence of one or more Hurwitz matrices in this class of sparse matrix spaces, for scenarios with uniform and non-uniform bounds. Several examples illustrate our results.