I am reading some chapters of a book on quantum optimal control that introduces Hilbert spaces $H$, $Z$, $W$ and describes $W$ as being "a space of mappings $\left]0,T\right[\to H$ such that $\psi(t)\in H$ and $\partial_t\psi(t)$ exists in $Z$". Further, some admissible set of control functions $U$ is introduced.
The problem considered is minimizing some cost function of $\psi\in W$ and $u\in U$ while obeying to the "Schrödinger-type" equation $$\partial_t\psi = (A+uB)\psi$$ for some linear operators $A,B\in\mathcal{L}(W,Z)$. Also, they want "$\psi(\cdot,0)=\psi_0$ for some $\psi_0\in H$" and speak of some set $\Omega$ that $\psi$ could take values on for its first component.
I haven't any background in physics so I have trouble locating where things live and am confused by the notation:
1) I reckon $\psi$ is some wave-function that would be dependent on time and place: $\psi\colon (x,t)\mapsto \psi(x,t)$. So how do the authors fit $\psi$ in $H^{\left]0,T\right[}$ while considering $\psi(\cdot, t)$ on $\Omega$ for $t\in[0,T]$? I assume they just use notation loosely?
I could imagine them wanting $W$ to be a subset of functions $\Omega\times[0,T]\to H$ and $Z$ to actually just be the common range of some set of functions $Z'=\Omega\times [0,T]\to Z$. So, if $\partial_t\psi$ denotes the function of weak derivatives $x\mapsto\partial\psi(x,\cdot)(t)$ the differential equation would read $$\partial_t\psi(x) = ((A+uB)\psi)(x,t)$$ for some linear operators $A,B\in\mathcal{L}(W,Z')$. Does this make any sense?
2) To be formally correct they'd need $U$ to be a set of functions on $[0,T]$ with range in the underlying field of $H$ ($\mathbb{R}$ or $\mathbb{C}$), no?
3) What space would I need to assume $\Omega$ to be a subset of? Just some Banach $X$?
Thank you very much in advance!