  The optimal time for the controllability of linear hyperbolic systems in one
dimensional space with one-side controls has been obtained recently for
time-independent coefficients in our previous works. In this paper, we consider
linear hyperbolic systems with time-varying zero-order terms. We show the
possibility that the optimal time for the null-controllability becomes
significantly larger than the one of the time-invariant setting even when the
zero-order term is indefinitely differentiable. When the analyticity with
respect to time is imposed for the zero-order term, we also establish that the
optimal time is the same as in the time-independent setting.
