Gamma-entropy cost for scalar conservation laws

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