Many problems require bounds on the control to achieve a realistic solution. Suppose, for instance, that our control is the amount of a chemical used in a system. Then, clearly we require this amount to be nonnegative, i.e., u ≥ 0. Often, the control must also be bounded above. Perhaps there are physical limitations on the amount of chemicals or environmental regulations which prohibit a certain level of use. We could also have a problem where the control is the percentage of some strength or use. Then 0 ≤ u ≤ 1 would be our bounds. Recall that in Labs 2 and 3, the controls were a fungicide and the concen-
tration of a chemical nutrient, respectively. Clearly, both of these quantities must remain non-negative. We did not, however, enforce this with bounds in the problem, as the resulting optimal controls met this requirement without restriction. However, this is not always the case. Consider the following fish harvesting example.