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In this article, we investigate the BV stability of $2\times 2$ hyperbolicsystems of conservation laws with strictly positive velocities underdissipative boundary conditions. More precisely, we derive sufficientconditions guaranteeing the exponential stability of the system underconsideration for entropy solutions in BV. Our proof is based on a fronttracking algorithm used to construct approximate piecewise constants solutionswhose BV norms are controlled through a Lyapunov functional. This Lyapunovfunctional 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 suitableweights 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].

Dissipative boundary conditions for $2\times 2$ hyperbolic systems of conservation laws for entropy solutions in BV