Optimal Control of Distributed Systems with Break Points

We consider optimal control of distributed parameter systems, that are frequently used in chemical engineering to model for example tubular reactors. Break points are introduced to take also cleaning operations into account, where a reactor is shut down for a while and then restarted again. The goal is to minimize a cost function over a fixed time horizon, where the number of cleaning operations, the length of reactor operation between successive cleanings, and the reactor feed rates for each time interval are to be computed. We assume that product prices and consumer demands are time-dependent. It must be guaranteed that the decrease of the free cross-sectional area of the tube caused by coke deposition never exceeds a certain limit. Moreover, there are time and position dependent constraints for the state and control variables such as a maximum bound for the temperature. The general mathematical model and the applied discretization schemes are outlined in detail. We present two different approaches, one based on the method of lines, the other on full discretization where discretized state variables are treated as additional optimization variables. Numerical results are presented for a case study, where optimal input feeds and maintenance times of an acetylene reactor are computed.