We present a motivating idea of optimal control theory in a classic application from King and Roughgarden [104] on allocation between vegetative and reproductive growth for annual plants. This plant growth model formulated by Cohen [36] divides the plant into two parts: the vegetative part, consisting of leaves, stems, and roots, and the reproductive part. The products of photosynthesis (growth) are partitioned into these parts, and the rate of photosynthesis is assumed to be proportional to the weight of the vegetative part. Let x1(t) be the weight of the vegetative part at time t and x2(t) the weight of the reproductive part. Consider the following ordinary differential equation model:

x′1(t) = u(t)x1(t), x′2(t) = (1− u(t))x2(t),

0 ≤ u(t) ≤ 1, x1(0) > 0, x2(0) ≥ 0,

where the function u(t) is the fraction of the photosynthate partitioned to vegetative growth. The natural evolution of the plant should encourage maximal growth of the reproductive part in order to ensure effective reproduction. Therefore, the goal is to find a partitioning pattern control u(t) which maximizes the functional ∫ T

ln(x2(t)) dt.