Reachability and Holdability of Nonnegative States

Linear differential systems ẋ(t) = Ax(t) (A ∈ Rn×n, x0 = x(0) ∈ Rn, t ≥ 0) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of A furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some a ≥ 0, A+aI must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions x0 that result in nonnegative states x(t) in finite time are shown to form a convex cone that is related to the matrix exponential etA and its eventual nonnegativity.