*Published Paper*

**Inserted:** 7 dec 2007

**Last Updated:** 10 nov 2018

**Journal:** Arch. Rat. Mech. Anal.

**Volume:** 195

**Number:** 1

**Pages:** 261-309

**Year:** 2010

**Abstract:**

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.