General Optimal Periodic Control Problem

In Figure 1.11 we see a general trend that the steam flow rate must follow to optimize the profit O. Therefore a control system is needed which will (1) compute the best steam flow rate for every time during the reaction period and (2) adjust the valve (inserted in the steam line) so that the steam flow rate takes its best value [as computed in (1)]. Such problems are known as optimal control problems. 

However the accurate treatment of state variable inequality constraints presents a few problems. Parameter optimization problems obtained by discretizing the control profile generally allow inequality constraints to be active only at a finite set of points, simply because a finite set of decisions cannot influence an infinite number of values (i.e., keeping the state fixed at every point in a finite time period). 

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