  We consider an optimal control problem for the steady-state Kirchhoff
equation, a prototype for nonlocal partial differential equations, different
from fractional powers of closed operators. Existence and uniqueness of
solutions of the state equation, existence of global optimal solutions,
differentiability of the control-to-state map and first-order necessary
optimality conditions are established. The aforementioned results require the
controls to be functions in $H^1$ and subject to pointwise upper and lower
bounds. In order to obtain the Newton differentiability of the optimality
conditions, we employ a Moreau-Yosida-type penalty approach to treat the
control constraints and study its convergence. The first-order optimality
conditions of the regularized problems are shown to be Newton diffentiable, and
a generalized Newton method is detailed. A discretization of the optimal
control problem by piecewise linear finite elements is proposed and numerical
results are presented.
