  In this article, we investigate the BV stability of $2\times 2$ hyperbolic
systems of conservation laws with strictly positive velocities under
dissipative boundary conditions. More precisely, we derive sufficient
conditions guaranteeing the exponential stability of the system under
consideration for entropy solutions in BV. Our proof is based on a front
tracking algorithm used to construct approximate piecewise constants solutions
whose BV norms are controlled through a Lyapunov functional. This Lyapunov
functional is inspired by the one proposed in J. Glimm's seminal work [J.
Glimm, Comm. Pure Appl. Math., 18:697--715, 1965], modified with some suitable
weights in the spirit of the previous works [J.-M. Coron, G. Bastin, and B.
d'Andr\'ea Novel, SIAM J. Control Optim., 47(3):1460--1498, 2008] and [J.-M.
Coron, B. d'Andr\'ea Novel, and G. Bastin, IEEE Trans. Automat. Control,
52(1):2--11, 2007].
