Reachability and Holdability of Nonnegative States

Linear differential systems $\dot{x}(t)=Ax(t)$ ($A\in\mathbb{R}^{n\times n}$, $x_0=x(0)\in\mathbb{R}^n$, $t\geq0$) 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 $a\geq0$, $A+aI$ must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions $x_0$ that result in nonnegative states $x(t)$ in finite time are shown to form a convex cone that is related to the matrix exponential $e^{tA}$ and its eventual nonnegativity.