Derivatives and Control in the Presence of Shocks Claude Bardos and Olivier Pironneau ∗ August 7, 2002 Abstract Sensitivity of shocks to data is a key point for fluid-structure and flutter control, and even more for sonic boom reduction. The linearized equations of fluids have Dirac masses and so it is not clear that the standard tools of optimal control apply to these. We show here that indeed great care have to be applied to find the linearized equations but that there is no such difficulties for control problems which do not involve explicitly the position of the shock in the criteria. Keywords: Partial differential equations, Burger equation, Euler flow, sensitivity, tran- sonic equation. Introduction Sensitivity of the position of the shocks with respect to the parameters of the flow is the problem we would like to investigate here. There are many important applications such as the fluttering of wings and the sonic boom of supersonic airplanes. In a land mark paper, Godlewski et al [6] have studied a similar situation for the shock tube flow problem and solved it completely for Burger’s equation when the sensitivity is with respect to initial data; their proof is however somewhat dependent on the explicit form of the solution and we will give here what we believe to be a simpler proof based on the definition of derivatives of distributions. For control, Giles [3] showed that the adjoint equation of the time dependent Euler equation is well posed and continuous across the shock. Here too we confirm the result. Then we turn to stationary problems and apply the same method to a simple transonic nozzle flow. 1 Burger’s Equation Consider the one dimensional Burger’s equation, ∂ t u + ∂ x ( u 2 2 )=0 ∀{x, t}∈ Q := R × (0, +∞), * Laboratoire Jacques-Louis Lions, Universit´ e Paris VI - IUF (pironneau@ann.jussieu.fr) 1