Final Value Theorem

The Final Value Theorem in control states that if all poles of $sY(s)$ are strictly stable or lie in the open left half-plane, i.e., have $\text{Re}(s)<0$, then

$$y(\infty )=\underset{s\to \infty }{\lim }sY(s)$$

The more general mathematical form of the theorem states that if ${\lim }_{t\to \infty }f(t)$ exists (it has a finite limit), then

$$\underset{t\to \infty }{\lim }f(t)=\underset{s\to 0}{\lim }sF(s)$$

where $F(s)$ is the one-sided Laplace transform of $f(t)$.