  We propose formulating the finite-horizon stochastic optimal control problem
for colloidal self-assembly in the space of probability density functions
(PDFs) of the underlying state variables (namely, order parameters). The
control objective is formulated in terms of steering the state PDFs from a
prescribed initial probability measure towards a prescribed terminal
probability measure with minimum control effort. For specificity, we use a
univariate stochastic state model from the literature. Both the analysis and
the computational steps for control synthesis as developed in this paper
generalize for multivariate stochastic state dynamics given by generic
nonlinear in state and non-affine in control models. We derive the conditions
of optimality for the associated optimal control problem. This derivation
yields a system of three coupled partial differential equations together with
the boundary conditions at the initial and terminal times. The resulting system
is a generalized instance of the so-called Schr\"{o}dinger bridge problem. We
then determine the optimal control policy by training a physics-informed deep
neural network, where the "physics" are the derived conditions of optimality.
The performance of the proposed solution is demonstrated via numerical
simulations on a benchmark colloidal self-assembly problem.
