Minkowski and Lyapunov Functions in Contractive Sets Characterization for Discrete-Time Linear Systems

The paper constructs a mathematically rigorous support for flow-invariance analysis of discrete-time linear system dynamics, with respect to a large family of contractive sets, considered in the comprehensive form of proper C-sets, exponentially decreasing with a certain rate. The first part develops a general algebraic point of view for invariance characterization, based on an inequality built in terms of Minkowski functions (which incorporates the classical Stein-Lyapunov inequality, as a particular case). The second part shows that the inequality with a general form can be addressed by set-embedding techniques that offer scenarios with a better numerical tractability - stated as optimization problems. The role of these techniques is illustrated by the applications presented in the final part; the applications refer to two important classes of contractive sets (which were separately reported in literature, without any connection between the embedding principles), namely, polyhedral sets, and symmetrical sets with arbitrary shapes.