I having difficulty with some notation relating to control theory. Given that $H(s)$ is a strictly proper, scalar transfer function (i.e. a ratio of polynomial functions with a higher degree in the denominator), the text I am following introduces: \begin{equation} H\left(\frac{d}{dt} \right) u(t) \end{equation} where $u(t)$ is the control function. Along with this, the text also gives the Fourier transform as: \begin{equation} \mathcal{F} \left( H \left( \frac{d}{dt} \right) u(t) \right) = H \left( i \omega \right) \hat{u}(i\omega) \end{equation} This seems to follow from $\mathcal{F} \left( \frac{d^n f}{dt^n} \right) = (i \omega)^{n} \hat{f}(i \omega) $.If the polynomial was, for example, $p(s)= s^2 + 1 $, I would interpret $p\left(\frac{d}{dt} \right) u(t) $ as $\frac{d^2 u}{dt^2} + u $, but I am not sure if this correct, and it is not clear how to handle say $H(s) = \frac{1}{s^2 +s +1}$.
The text is: Athanasios C Antoulas. Approximation of large-scale dynamical systems, volume 6. Siam, 2005.
We can think $s$ as a derivative operator in time domain, e.g. $(s^2 + s + 1)u(t) = \ddot{u}(t) + \dot{u}(t) + u(t)$. But it is not appropriate to use it like this because $s$ is a variable in frequency domain. I think that's why the authors chose the notation $\left( \left( \frac{d}{dt} \right)^2 + \frac{d}{dt} + 1 \right) u(t)$.