Hurwitzness of $A^{-}$ implies local stability in the on-demand agent system

Consider the fluid-limit dynamics of the on-demand agent service system, represented by switched linear systems with system matrices $A^{-}$ and $A^{+}$. A matrix is Hurwitz when all its eigenvalues have strictly negative real parts. Hurwitz-matrix stability conjecture. If $A^{-}$ is Hurwitz, then the system is locally stable. In this model, $A^{+}$ is always Hurwitz. This conjecture is motivated by numerical and simulation experiments; the paper presents it as a sufficient condition but does not prove it in general.