Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws

We propose a rigorous procedure to obtain the adjoint-based gradient representation of cost functionals for the optimal control of discontinuous solutions of conservation laws. Hereby, it is not necessary to introduce adjoint variables for the shock positions. Our approach is based on stability properties of the adjoint equation. We give a complete analysis for the case of convex scalar conservation laws. The adjoint equation is a transport equation with discontinuous coefficients and special reversible solutions must be considered to obtain the correct adjoint-based gradient formula. Reversible solutions of the adjoint transport equation and the required stability properties are analyzed in detail.