I have the following differential equation
$$M \dot y(t) = y(t) + K p(t)$$
where $M$ is a nilpotent matrix of degree $m$ and $K$ is some matrix of suitable order. Functions $p(t)$ are piecewise continuous.
If $y(0) \ge 0$ lies in the positive orthant and $p(t) \ge 0$ for all $t \in \mathbb{R}$, under what conditions do all trajectories emanating from any point from the positive orthant $\mathbb{R}^n_{+}$ (boundary included) lie inside $\mathbb{R}^n_{+}$ only?
Solution of the equation is I found using the fact $I-MD$ is invertible ($d$ is derivative operator),
$$y(t)=-\sum_{i=0}^{m-1} M^{i}Kp^{(i)}(t)$$