Algebraic tests for the asymptotic stability of parametric linear systems

In this paper, algebraic tests are proposed to establish asymptotic stability of parametric linear systems assuming that the parameters lie in a given variety. In particular, by using tools borrowed from algebraic geometry, necessary and sufficient condition are proposed to test whether a continuous-time or a discrete-time linear parametric system is asymptotically stable for all the specializations belonging to a given variety. Both the cases of zero dimensional and non-zero dimensional varieties are considered.