State feedback synthesis for on-line optimization of batch reactors with a non-affine manipulated input and free terminal time

In this paper, state feedback laws are developed for an online optimization problem where the manipulated input appears as a non-affine function of the state model and the final time is free (unspecified). The first order necessary conditions for optimality as a function of adjoints states are derived using the classical Pontryagin´s principle. Then the optimal state feedback laws are derived by the elimination of adjoints states and it is found that, in general, the optimal state feedback law is dynamic in nature and requires the online solution of an ordinary differential equation. Finally, the application of this technique is illustrated through an example of a microbial fermentation.