  We give a singular control approach to the problem of minimizing an energy
functional for measures with given total mass on a compact real interval, when
energy is defined in terms of a completely monotone kernel. This problem occurs
both in potential theory and when looking for optimal financial order execution
strategies under transient price impact. In our setup, measures or order
execution strategies are interpreted as singular controls, and the capacitary
measure is the unique optimal control. The minimal energy, or equivalently the
capacity of the underlying interval, is characterized by means of a nonstandard
infinite-dimensional Riccati differential equation, which is analyzed in some
detail. We then show that the capacitary measure has two Dirac components at
the endpoints of the interval and a continuous Lebesgue density in between.
This density can be obtained as the solution of a certain Volterra integral
equation of the second kind.
