  We are concerned with a control problem related to the vanishing viscosity
approximation to scalar conservation laws. We investigate the
$\Gamma$-convergence of the control cost functional, as the viscosity
coefficient tends to zero. A first order $\Gamma$-limit is established, which
characterizes the measure-valued solutions to the conservation laws as the
zeros of the $\Gamma$-limit. A second order $\Gamma$-limit is then
investigated, providing a characterization of entropic solutions to
conservation laws as the zeros of the $\Gamma$-limit.
