In this paper we study the stabilization of systems with decentralized controllers, when the stability criterion of interest is "non-overshooting stability". This criterion is stronger than those which have typically been studied, particularly for decentralized control, and requires that the size of the state is always decreasing. We identify a key property which allows centralized results for this type of stability to be extended, and this property indeed holds for the most common classes of decentralized control problems. This enables us to determine that stabilizability with respect to static controllers is equivalent to stabilizability with respect to dynamic controllers, and to derive a linear matrix inequality (LMI) which either synthesizes a stabilizing controller or produces a certificate of non-stabilizability. We then compare these results with those for internal state stability, i.e., fixed modes.