Boundary controllability of linear symmetric hyperbolic systems

We consider the finite time boundary value controllability of a linear symmetric hyperbolic system subject to what will be known as "natural" boundary conditions. Such a system occurs frequently in mathematical physics and is, in a sense, the most general linear hyperbolic equation. It includes the equations of linear elasticity, electro-magnetic theory and acoustic wave motion. It will be shown that when the system differential operator satisfies a backward uniqueness property and is self adjoint, there exists a time T₁ > 0 such that the system is approximately boundary controllable in any time T > 2T₁. It is also shown that there exists a time T₂ < T₁ such that the system is not boundary controllable in any time T < 2T₂. For a certain class of boundary controls, a necessary and sufficient condition for strict boundary controllability is obtained.