  How can one change a system, in order to change its statistical properties in
a prescribed way? In this note we consider a control problem related to the
theory of linear response. Given an expanding map of the unit circle with an
associated invariant density we can consider the inverse problem of finding
which first order changes in the transformation can achieve a given first order
perturbation in the density. We show the general mathematical structure of the
problem, the existence of many solutions in the case of expanding maps of the
circle and the existence of optimal ones. We investigate in depth the example
of the doubling map, where we give a complete solution of the problem.
