ROMJIST Volume 24, No. 4, 2021, pp. 402-417
Mihaela MATCOVSCHI, Marius APETRII, Octavian PASTRAVANU, Mihail VOICU Invariant sets with arbitrary convex shapes in linear system dynamics
ABSTRACT: A mathematical framework dedicated to flow-invariance study is proposed for continuous-time linear dynamics, with respect to general-shape contractive sets - defined by the class of proper C-sets (convex and compact sets, including the origin as an interior point), for which a constant-rate exponential decrease is considered. The main result provides a general algebraic characterization of flow-invariance, stated in terms of functions extending the concept of matrix measure (by using Minkowski functions, instead of vector norms, in the set description). A unifying point of view is created - able to connect several approaches to flow-invariance, separately reported in literature. This result is further exploited by set-embedding procedures that yield reformulations as optimization tasks - more reliable from the numerical perspective. Two examples are considered to illustrate how the flow-invariance criteria known for non-symmetrical polyhedrons, and symmetrical sets defined by weighted p-vector-norms, respectively, can be obtained as particular cases from our main result.KEYWORDS: continuous-time linear systems, invariant sets, Minkowski function, Lyapunov functionRead full text (pdf)