Optimal control Solutions for crystal shape manipulation

摘要

Optimal control algorithms and strategies for crystal shape manipulation are explored in this paper. We focus on minimum time trajectories with the temperature as the control input. A challenge results from the inherent constraints in the particle growth vector, which is confined to lie within a cone in the model state-space. As a consequence, switching trajectories with a number of subsequent growth and dissolution phases may be indispensable. Such a strategy employs the unequal growth and dissolution rates in order to achieve crystal morphologies which do not result directly from a pure growth process only. For instantaneous switching between subsequent phases we need to extend the temperature solution profiles to piecewise-continuous. By a suitable specification of the switching manifolds, we are then able to avoid unnecessary long paths in the state-space. Diverse solution approaches to the optimal control problem are suggested using the minimum principle and efficient numerical techniques which utilize the invertibility property of the particle growth dynamics. In particular, we prove that minimum-time scenarios are composed of constant supersaturation trajectory sections.