  We investigate global bounded solutions of higher regularity to boundary
value problems for a general linear nonautonomous first order 1D hyperbolic
system in a strip. We establish the existence of such solutions under the
assumption of exponential stability and certain dissipativity (or non-resonant)
conditions. Our results demonstrate the existence and uniqueness of $C^1$- and
$C^2$-bounded solutions, in particular, periodic and almost periodic solutions.
The number of imposed dissipativity conditions is related to the regularity of
solutions. This connection arises due to the nonautonomous nature of the
hyperbolic system under consideration, in contrast to autonomous settings.
  In the general case of global bounded smooth solutions we assume exponential
stability in $H^1$. In the case of periodic solutions, the weaker assumption of
$L^2$-exponential stability proves to be sufficient.
