In this talk, we study variational problems for regularized conservation laws with Lax`s entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are flux-gradient flows. We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers` equation, are presented. We successfully compute the control of conservation laws by incorporating both primal-dual algorithms and monotone schemes. This is based on joint work with Siting Liu and Stanley Osher.