Exact and optimal controllability for scalar conservation laws with   discontinuous flux
  This paper deals with an optimal control problem and describes the reachable set for the scalar 1-D conservation laws with discontinuous flux. Regarding the optimal control problem we first prove the existence of a minimizer and then we prescribe an algorithm to compute it. The same method also applies to compute the initial data control. The proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics. 
Adimurthi and Shyam Sundar Ghoshal and Pierangelo Marcati
math.AP