Adjoint IMEX‐based schemes for control problems governed by hyperbolic conservation laws

Starting from relaxation schemes for hyperbolic conservation laws we derive continuous and discrete schemes for optimization problems subject to nonlinear, scalar hyperbolic conservation laws. We discuss properties of first‐ and second‐order discrete schemes and show their relations to existing results. In particular, we introduce first and second‐order relaxation and relaxed schemes for both adjoint and forward equations. We give numerical results including tracking type problems with non‐smooth desired states.