Cell-mass maximization in fed-batch cultures: sufficient conditions for singular arc and optimal feed rate profiles.

Complete solutions are provided for cell-mass maximization for free and fixed final times and constant and variable yields. The optimal feed rate profile is a concatenation of maximum, minimum and singular feed rates. The exact sequence and duration of each feed rate depends primarily on the initial substrate concentration, and degenerate cases arise due to the magnitude constraint on the feed rate and the length of final time t (f). When the final time is free and not in the performance index, it is infinite for constant yield so that any form of feed rate leads to the same amount of cells, while for variable yield the singular feed rate is exponential and maximizes the yield. For fixed final time the singular feed rate for constant yield is exponential and maximizes the specific growth rate by maintaining the substrate concentration constant, while for variable yield, it is semi-exponential and the substrate concentration starts near the maximum specific growth rate and moves toward the maximum yield. A simple sufficient condition for existence of singular feed rate requires an existence of a region bounded by the maxima of specific growth and cellular yield. Otherwise, the optimal feed rate profile is a bang-bang type and the bioreactor operates in batch mode.