A Variational Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws

We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable ($C^1$) states left and right of it. The control consists of the $C^1$ parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon $T>0$ to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small. Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to $C^0$. In physical coordinates, this implies that the shock curves in $C^1$ and the states between the shocks in the topology of $C^0$ depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals.