In a linear dynamical system, the components of the input are quantities that we can control, such as control surface deflections on an airplane.

We can always assume that the number of inputs (dimension of the input vector) is less than or equal to the number of states (dimension of the state vector).

Suppose $h: \mathbf{R}_+ \to \mathbf{R}^{p \times m}$ is the impulse response of a linear dynamical system.

We always have $h(t) \to 0$ as $t \to \infty$.

If $h_{23}(t)$ is small for $t \leq 1.3$, then

Suppose that $h_{ij}(t) \geq 0$ for all $i,j$ and $t\geq 0$, and $y$ and $\tilde y$ are the outputs corresponding to inputs $u$ and $\tilde u$, respectively (with the same initial state in each case). Then $u_j(t) \geq \tilde u_j(t)$ for all $j$ and $t \geq 0$ implies $y_i(t) \geq \tilde y_i(t)$ for all $i$ and $t \geq 0$.

The state of a general dynamical system