  Large-size populations consisting of a continuum of identical and
non-cooperative agents with stochastic dynamics are useful in modeling various
biological and engineered systems. This paper addresses the stochastic control
problem of designing optimal state-feedback controllers which guarantee the
closed-loop stability of the stationary density of such agents with nonlinear
Langevin dynamics, under the action of their individual steady state controls.
We represent the corresponding coupled forward-backward PDEs as decoupled
Schr\"odinger equations, by applying two variable transforms. Spectral
properties of the linear Schr\"odinger operator which underlie the stability
analysis are used to obtain explicit control design constraints. Our
interpretation of the Schr\"odinger potential as the cost function of a closely
related optimal control problem motivates a quadrature based algorithm to
compute the finite-time optimal control.
