Design and control sensitivity analysis via min max differentiability

The object of this paper is twofold. We introduce theorems on the differentiability of a Min Max with respect to a parameter and to show how such theorems can be applied to compute the directional derivative of the cost for a control problem and the material derivative in Shape Sensitivity Analysis Problems. We consider the Min Max of a functional which is parametrized by t. We show that, under appropriate conditions, the derivative of the Min Max with respect to t is the Min Max with respect to the points solution of the Min Max problem of the derivative of the original functional with respect to t. To illustrate the use of this theorem, we apply it to the control of an elliptic equation with a nondifferentiable observation and to shape design problems.