Optimal Boundary Control of Nonlinear Hyperbolic Conservation Laws with Switched Boundary Data

We consider the optimal control of initial-boundary value problems for entropy solutions of scalar hyperbolic conservation laws. In particular, we consider initial-boundary value problems where the initial and boundary data switch between different C-functions at certain switching points and both, the functions and the switching points, are controlled. We show that the control-tostate mapping is differentiable in a certain generalized sense, which implies Fréchet-differentiability with respect to the control functions and the switching points for the composition with a tracking type functional, even in the presence of shocks. We also present an adjoint-based formula for the gradient of the reduced objective functional.