On the Existence and Approximationof Solutions for the Optimal Control ofNonlinear Hyperbolic Conservation Laws

Optimal control problems for possibly discontinuous entropy solutions of nonlinear multidimensional conservation laws with controls in source term and initial condition are considered. The control-to-state-mapping is analyzed by using monotone diierence schemes and existence results for optimal controls are proven. Moreover, a result on the convergence of optimal solutions of nite dimensional approximations to solutions of the original problem is given. In the 1-D case the theory of compensated compactness is used to prove that the control-to-state-mapping is compact from L 1 to C ((0; T ]; L 1 loc) which ensures the existence of optimal controls under very weak assumptions.