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 whi ch describes the biosynthesis of penicillin. First, we solve the problem nu merically by using a direct shooting method. By discretization of the contr ol variable, we transform the basic optimal control problem to a finite-dim ensional nonlinear programming problem, which is solved numerically by a st andard SQP method. Contrary to earlier investigations (Luus, 1993), we cons ider the problem as a free final time problem, thus obtaining an improved v alue of the penicillin output. The results indicate that the assumption of a continuous control which underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to t he fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple sho oting techniques. It turns out that this BVP is sensitive, which is due to the rigid behavior 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 (Lim et al., 1986). Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution which satisfies all first-order and second-orde r necessary conditions of optimal control theory. The solution is similar t o the one obtained by direct methods, but in addition it contains certain v ery small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.