Optimal control of a stirred-tank reactor with rapidly-decaying catalyst

The singular perturbation solution to a general optimal control problem consisting of the optimization of a final-time function of the “slow” variables, subject to inequality control constraints in the inner control region, and a free-time terminal manifold for the fast-slow differential equations, is given. The example studied here is the optimal temperature control of a stirred-tank catalytic reactor in which the time constant of the catalyst decay is appreciable compared to the mean residence time. The first-order corrections to the usual quasi-steady state (zeroth-order outer) solution, consisting of the zeroth-order inner solutions at both ends of the interval of operation and the first-order outer solutions, are calculated. Boundary layers thus appear at start-up and shutdown. The second-order corrections can be calculated by the methods given here, and would entail interior boundary layers around the entry points to the control constraints. These entry points are not fixed in time, and a time-scaling procedure, developed previously for converting free-time to fixed-time problems, is employed. The solutions are thus matched at the edges of the boundary layers, and around the inequality-constraint entry and exit points. Some numerical solutions are given which show an appreciable increase in return on investment over the QSS solution.