This article deals with stability and stabilization issues for linear two-dimensional (2D) discrete models. More precisely, we focus on repetitive processes and Roesser models. Within the algebraic analysis approach to linear systems theory, we first show how a given linear repetitive process can be transformed into an equivalent linear Roesser model. We then prove that the structural stability is preserved by this equivalence transformation. This enables us to design new approaches for stabilizing linear repetitive processes by means of existing methods for computing a state feedback control law stabilizing a linear 2D discrete Roesser model. We also show that we can interpret the stability along the pass of a linear repetitive process as its structural stability. This implies that one of our new approaches can be applied to stabilize along the pass a linear repetitive process which is only stable from pass to pass.