Optimal time for the controllability of linear hyperbolic systems in one dimensional space

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.