  We compute the quasi-potential and determine minimizing paths for an action
functional related to scalar conservation laws on an interval with boundary
conditions in the sense of Bardos et al. (1979). Taking as input an
exclusion-like flux function, a strictly convex entropy, and boundary data, we
obtain a generalization of the functional derived by Derrida, Lebowtiz and
Speer (2003) for the stationary large deviations of the asymmetric exclusion
process.
