  We study problems of optimal boundary control with systems governed by linear
hyperbolic partial differential equations. The objective function is quadratic
and given by an integral over the finite time interval $(0,\, T)$ that depends
on the boundary traces of the solution. If the time horizon $T$ is sufficiently
large, the solution of the dynamic optimal boundary control problem can be
approximated by the solution of a steady state optimization problem. We show
that for $T\rightarrow\infty$ the approximation error converges to zero in the
sense of the norm in the Hilbert space
  $L^2(0,\,1)$ with the rate $1/T$, if the time interval $(0,\, T)$ is
transformed to the fixed interval $(0,\, 1)$. Moreover, we show that also for
optimal boundary control problems with integer constraints for the controls the
turnpike phenomenon occurs. In this case the steady state optimization problem
also has the integer constraints. If $T$ is sufficiently large, the integer
part of each solution of the dynamic optimal boundary control problem with
integer constraints is equal to the integer part of a solution of the static
problem. A numerical verification is given for a control problem in gas
pipeline operations.
