Optimal Trajectory Generation With State Inequality Constraints for a Bioreactor

The dynamics of microorganisms in a bioreactor are described by population systems. The tracking of the biomass of a desired trajectory is ensured by means of a nonlinear control law, as long as the reference trajectory complies with certain naturally given inequality constraints. To generate such a reference trajectory, an optimal control problem based on a double integrator with nonlinear state inequality constraints is solved. The necessary optimality criteria are extracted and the differential equations are solved. Using these solutions and parameter assumptions, which are shown to be non-limiting, it is proven that at most one constraint at a time is active. In addition, it is shown that the intervals in which no constraint is active and those in which one constraint is active alternate.