The significance of Gabriel localization for stability and stabilization of multidimensional input/output behaviors To Peter Gabriel on his 77th birthday

Serre (1953) was the first who considered categories of groups up to negligible ones, negligibility being determined by the considered context. Gabriel in his thesis (1962) developed this idea into a complete theory of quotient categories, rings and modules which is called Gabriel localization. This theory was nicely exposed by Stenström (1975). The author has recently observed that this theory is a valuable tool for stability and stabilization of multidimensional behaviors where a finitely generated multivariate polynomial torsion module is considered negligible if its characteristic variety has points in a preselected stability region only or, equivalently, if its associated autonomous behavior is stable, i.e., has polynomial-exponential solutions with frequencies in this stability region only. Via the Integral Representation Formula of Ehrenpreis/Palamodov corresponding properties hold for all trajectories of the behavior. In the one-dimensional standard cases stability signifies asymptotic stability. In our approach to output feedback stabilization of multidimensional systems almost direct decompositions, i.e., direct sum decompositions up to negligible modules, are essential. Quadrat had observed the significance of direct sum decompositions in stabilization theory. These decompositions are usually hidden in coprime factorizations of transfer matrices which, however, do not always exist. Bisiacco, Valcher and Napp Avelli studied almost direct decompositions for two-dimensional polynomial modules and behaviors, but without using localization theory. This paper explains Gabriel’s localization and quotient modules and their use in multidimensional stabilization theory. It also contains a new algorithm for the computation of the (Gabriel) quotient module of a finitely generated torsionfree module over a multivariate polynomial ring. This algorithm can also be used for the computation of Willems closures of such modules and thus generalizes work of Shankar, Sasane, Napp Avelli, van der Put et al.. It is also useful for the computation of the purity filtration of a finitely generated polynomial torsion module which is the subject of Barakat’s talk at this conference where also the history of this filtration is discussed. The author gratefully acknowledges financial support of the Austrian FWF. 1