Minimax Differentiability via the Averaged Adjoint for Control/Shape Sensitivity*

A standard approach to the minimization of a state constrained objective function in Control/Shape Optimization problems is to consider the minimax of the associated Lagrangian. In this paper, this construction is used to obtain the semidifferential (sensitivity analysis) with respect to the control/shape variable of a state constrained objective function. By using the new notion of averaged adjoint of Sturm (2014, 2015), the minimax problem need not be related to a saddle point: non-convex objective functions and non-linear state equations can be directly considered. We firstly provide a new version of the condition of Sturm and secondly an extension from the single valued case to the case where the solutions of the state/averaged adjoint state equations are not unique in which case a non-differentiability can occur.