  This work collects some methodological insights for numerical solution of a
"minimum-dispersion" control problem for nonlinear stochastic differential
equations, a particular relaxation of the covariance steering task. The main
ingredient of our approach is the theoretical foundation called $\infty$-order
variational analysis. This framework consists in establishing an exact
representation of the increment ($\infty$-order variation) of the objective
functional using the duality, implied by the transformation of the nonlinear
stochastic control problem to a linear deterministic control of the
Fokker-Planck equation. The resulting formula for the cost increment
analytically represents a "law-feedback" control for the diffusion process.
This control mechanism enables us to learn time-dependent coefficients for a
predefined Markovian control structure using Monte Carlo simulations with a
modest population of samples. Numerical experiments prove the vitality of our
approach.
