Control of 2D scalar conservation laws in the presence of shocks

Abstract

We analyze a model optimal control problem for a 2D scalar conservation law: The so-called inverse design problem, the goal being to identify the initial datum leading to a given nal time con guration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and nite di erences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The e ciency of the method is illustrated by numerical experiments.