Exponential stability for 2D systems: the linear case

This short paper deals with a 2D discrete linear Roesser model. The results introduced here are a follow-up of a paper we proposed recently and where we explained and motivated the reasons we need to adopt a new definition of exponential stability for 2D systems. However this previous result left aside a crucial point that we would like to asses here: is our new definition of exponential stability coherent with the existing stability criterion in the linear case? We hereby show that, in the linear case, 1. our new definition of exponential stability is equivalent to asymptotic stability and 2. the characteristic polynomial-based stability criterion is a sufficient and necessary condition for the exponential stability we have introduced.