  An optimal control strategy is developed to construct nanostructures of
desired geometry along line segments by means of directed self-assembly of
charged particles. Such a control strategy determines the electric potentials
of a set of electrodes located at fixed points in the line segment. The
particles move under the electric forces generated by these electrodes and by
the interactions between the particles themselves to form a desired pattern
eventually. Due to technology limitations, the particle positions cannot be
measured during the course of control, so that the control is open-loop in
nature. Such an open-loop control optimally changes the electrode potentials in
time in order to create a desired pattern with the highest probability, despite
the inherent uncertainty in the initial positions and the dynamical behaviors
of the particles. Two models are proposed to describe the uncertain dynamics of
the particles: a continuous model relying on a set of nonlinear stochastic
differential equations, and a discrete Ising model consisting of a large
dimensional continuous-time Markov chain. While the first model is more
mathematically tractable, the second one more precisely describes particles at
the nanometer scale. The control design procedure begins with the continuous
model and identifies the structure of its stable equilibria, which is used
later to propose a piecewise constant structure for the control and to
demonstrate that the optimal value of each piece is independently obtained from
a certain static optimization problem. It is shown next that the design
procedure can be applied to the discrete model with only minor modifications. A
numerical example of control design is presented.
