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On the optimality system for a 1-D Euler flow problem

In this paper, a problem of shape design for a duct with the flow governed by the one-dimensional Euler equations is analyzed. The flow is assumed to be transonic, in the sense that we have a shock embedded in the flow. The presence of the shock introduces analytical and numerical difficulties that are overcome by introducing the shock location as an explicit state variable. We justify the use of adjoint variables by establishing conditions to ensure that the linearized constraint is surjective. These adjoint variables are used to state necessary optimality conditions and to compute gradients. In addition, our characterization of the adjoint variables is of interest for comparison to adjoints associated with shock-capturing formulations.