State-constrained optimal spatial field control for controlled release in tissue engineering

Distributed parameter control problems involving manipulation within the spatial domain arise in a variety of applications including vibration control, active noise reduction, epidemiology, tissue engineering, and cancer treatment. A state-constrained spatial field control problem motivated by a biomedical application is solved in which the manipulation occurs over a spatial field and the state field is constrained both in spatial frequency and by a partial differential equation (PDE) that effects the manipulation. An optimization algorithm combines three-dimensional Fourier series, which are truncated to satisfy the spatial frequency constraints, with exploitation of structural characteristics of the PDEs. The computational efficiency and performance of the optimization algorithm are demonstrated in a numerical example, for which the spatial tracking error is almost entirely due to the limitation on the spatial frequency of the manipulated field. The numerical results suggest that optimal control approaches have promise for controlling the release of macromolecules in tissue engineering applications.