Admitting the inadmissible: Adjoint formulation for incomplete cost functionals in aerodynamic optimization

The adjoint equations are derived for problems in aerodynamic optimization that are improperly considered as inadmissible. For example, a cost functional that depends on the density, rather than on the pressure, is considered inadmissible for an optimization problem governed by the Euler equations. We show that for such problems additional terms should be included in the Lagrangian functional when deriving the adjoint equations, These terms are obtained from the restriction of the interior partial differential equation to the control surface. Demonstrations of the explicit derivation of the adjoint equations for inadmissible cost functionals are given for the potential, Euler, and Navier-Stokes equations.