  We are concerned about the controllability of a general linear hyperbolic
system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) +
\gamma C(x) w(t, x) $ ($\gamma \in \mR$) in one space dimension using boundary
controls on one side. More precisely, we establish the optimal time for the
null and exact controllability of the hyperbolic system for generic $\gamma$.
We also present examples which yield that the generic requirement is necessary.
In the case of constant $\Sigma$ and of two positive directions, we prove that
the null-controllability is attained for any time greater than the optimal time
for all $\gamma \in \mR$ and for all $C$ which is analytic if the slowest
negative direction can be alerted by {\it both} positive directions. We also
show that the null-controllability is attained at the optimal time by a
feedback law when $C \equiv 0$. Our approach is based on the backstepping
method paying a special attention on the construction of the kernel and the
selection of controls.

Optimization and Control;Analysis of PDEs