Exact Controllability For Linear Hyperbolic Systems With Zero Eigenvalues

In this paper, we consider the exact controllability for first order linear hyperbolic systems with zero eigenvalues. We first give a necessary and sufficient condition on the initial and final conditions for the exact boundary controllability. By adding two-side control and one-side control, we realize the exact boundary controllability. Then, under the assumption that the necessary and sufficient condition is not satisfied, by adding suitable internal controls to those equations which correspond to the non-zero eigenvalues, we get the exact controllability for the systems with suitable boundary controls.