Optimal control of crystallization processes

In this paper an optimal control problem for polymer crystallization is investigated. The crystallization is described by a non-isothermal Avrami-Kolmogorov model and the temperature at the boundary of the domain serves as control variable. The cost functional takes into account the spatial variation of the crystallinity and the final degree of crystallization. This results in a boundary control problem for a parabolic equation coupled with two ordinary differential equations, which is treated by an adjoint variable approach. We prove the existence and uniqueness of solutions to the state system as well as the existence of a minimizer for the cost functional under consideration. The adjoint system is derived and we use a steepest descent algorithm to solve the problem numerically. Numerical simulations illustrate the applicability and performance of the optimization algorithm.