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

This note deals with some properties of particular bounded sets w.r.t. linear continuous-time systems described by x˙(t)=A(0)x(t)+c(t), where c(t)∈Ω⊂Rⁿ, Ω a compact set, and matrix e^tA(0) has the property of leaving a proper cone K positively invariant, that is e^tA(0)K⊂K. The considered bounded sets 𝒟(K; a, b) are described as the intersection of shifted cones. Necessary and sufficient conditions are given. They guarantee that such sets are positively invariant w.r.t. the considered system. The trajectories starting from x ₀∈Rⁿ/𝒟(K; a, b) (respectively to x₀ ∈Rⁿ) are studied in terms of attractivity and contractivity of the set 𝒟(K; a, b). The results are applied to the study of the constrained state feedback regulator problem