  In this paper we study the right differentiability of a parametric infimum
function over a parametric set defined by equality constraints. We present a
new theorem with sufficient conditions for the right differentiability with
respect to the parameter. Target applications are nonconvex objective functions
with equality constraints arising in optimal control and shape optimisation.
The theorem makes use of the averaged adjoint approach in conjunction with the
variational approach of Kunisch, Ito and Peichl. We provide two examples of our
abstract result: (a) a shape optimisation problem involving a semilinear
partial differential equation which exhibits infinitely many solutions, (b) a
finite dimensional quadratic function subject to a nonlinear equation.
