Numerical computation of optimal feed rates for a fed-batch fermentation model

In this paper, we consider a model for a fed-batch fermentation process which describes the biosynthesis of penicillin. First, we solve the problem numerically by using a direct shooting method. By discretization of the control variable, we transform the basic optimal control problem to a finite-dimensional nonlinear programming problem, which is solved numerically by a standard SQP method. Contrary to earlier investigations by Luus in 1993, we consider the problem as a free final time problem, thus obtaining an improved value of the penicillin output. The results indicate that the assumption of a continuous control that underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to the fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple shooting techniques. It turns out that this BVP is sensitive, which is due to the rigid behaviour of the specific growth rate functions. By relaxation of the characteristic parameters, we obtain a simpler BVP, which can be solved by using the predicted control structure. Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution that satisfies all first-order and second-order necessary conditions of optimal control theory. The solution is similar to the one obtained by direct methods, but in addition it contains certain very small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.