For the one dimensional case, with $a, b, c$ being real constants, $u$ being the system input, $x$ the state, what is the Laplace transfer function of:
$$\dot x = bu + c$$
Ideally I'm looking for an expression $X(s)/U(s)$, but I don't know if this can be done with that additive constant $c$.
You cannot get the expression you're looking for as this system has no transfer function because it is not a linear system, i.e. it does not satisfy the additivity property.
Assuming zero initial conditions for $x$, you have
$$sX(s)=bU(s)+\frac{c}{s}$$
from which you can't get an independent transfer function $H(s)=X(s)/U(s)$.