For our first lab involving a problem linear in the control, we examine a simple tree harvesting simulation. Consider a timber farm which, due to environmental regulations, can harvest at most a fixed percentage of its tree population, which must then be replanted. We assume the farm operates at this constant percentage of harvesting, producing raw timber in the amount of x(t) at time t. The growth and death rates of individual trees are not considered. The amount of timber is based solely on the size of the farm (or equivalently, the number of trees). We also assume the harvest percentage level is low enough so that tree age need not be considered and there will always be mature trees ready for harvest. Once the timber has been processed, it is immediately sold. The money can either be kept as profit or reinvested in the farm by purchasing land and labor for further tree growth. The owners of the farm wish to find the reinvestment schedule which maximizes profit over a fixed time interval. Let the control u(t) represent the percentage of timber revenue reinvested in the frame. Reinvestment will lead to the growth of more trees and the production of more timber, so we have x′(t) = kx(t)u(t), where k is the return constant, which takes into account the average cost of labor and land. Further, if p is the market price of a unit of timber, then the profit at time t is px(t)(1− u(t)), and the total profit is
x(t)[1− u(t)] dt.