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 derivative(t)w(t,x) = Sigma(x)partial derivative(x)w(t,x) + gamma C(x)w(t, x) (gamma is an element of R) 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 is an element of R and for all C which is analytic if the slowest negative direction can be alerted by both positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when C 0. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.