Positively invariant sets for constrained continuous-time systems with cone properties

The authors study the properties of some particular bounded sets with respect to trajectories of a linear continuous-time system with constrained input described by x(t)=Ax(t )+c(t), where c(t) εΩ, a compact set, and matrix e^tA has the property of leaving a proper cone K positively invariant, that is, e^tA K⊂K. Necessary and sufficient conditions guaranteeing that some bounded sets D(K; a,b) obtained from the intersection of shifted cones are positively invariant with respect to the considered system are given. The external behavior of motions is studied in terms of attractivity and contractivity of the set D(K; a,b). The proposed results are applied to the study of the saturated state feedback regulator problem