Optimization of Metabolite Production in Fed-Batch Cultures: Use of Sufficiency and Characteristics of Singular Arc and Properties of Adjoint Vector in Numerical Computation

Abstract

Fed-batch fermentation for maximization of metabolite production at a final time is normally a singular control problem. The singular optimal feed rate strategy is dependent on the relative position of peaks in the specific rates of cell growth (μ), substrate consumption (σ), and product formation (π). Analyses and application of Pontryagin's minimum principle, singular control theory, the generalized Legendre Clebsch conditions and the properties of adjoint variables that represent the gradients of the performance index, with respect to state variables λi(t) = ∂J/∂xi, yield sufficient conditions for the existence of singular arcs and their characteristics for processes with various forms of μ, σ, and π. From the analysis, we can infer the optimal feed rate structure, the region of singular arc, the sign of adjoint variables, and the value of state variables on a singular arc, and we can apply them in numerical optimization techniques such as the multiple shooting method, in which proper selection of the initial values at each switching time is essential and critical for effective and efficient computation.