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An optimal control strategy is developed to construct nanostructures ofdesired geometry along line segments by means of directed self-assembly ofcharged particles. Such a control strategy determines the electric potentialsof a set of electrodes located at fixed points in the line segment. Theparticles move under the electric forces generated by these electrodes and bythe interactions between the particles themselves to form a desired patterneventually. Due to technology limitations, the particle positions cannot bemeasured during the course of control, so that the control is open-loop innature. Such an open-loop control optimally changes the electrode potentials intime in order to create a desired pattern with the highest probability, despitethe inherent uncertainty in the initial positions and the dynamical behaviorsof the particles. Two models are proposed to describe the uncertain dynamics ofthe particles: a continuous model relying on a set of nonlinear stochasticdifferential equations, and a discrete Ising model consisting of a largedimensional continuous-time Markov chain. While the first model is moremathematically tractable, the second one more precisely describes particles atthe nanometer scale. The control design procedure begins with the continuousmodel and identifies the structure of its stable equilibria, which is usedlater to propose a piecewise constant structure for the control and todemonstrate that the optimal value of each piece is independently obtained froma certain static optimization problem. It is shown next that the designprocedure can be applied to the discrete model with only minor modifications. Anumerical example of control design is presented.

Directed Self-Assembly of Linear Nanostructures by Optimal Control of External Electrical Fields