Asymmetric invariants for a class of strictly hyperbolic systems including the Thimoshenko beam

We introduce a set of conserved quantities of energy-type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well-posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions.