For the second lab, we will explore an optimal control problem with biological applications. Let x(t) be a population concentration at time t, and suppose we wish to reduce the population over a fixed time period. We will assume x has a growth rate r and carrying capacity M . The application of a substance is known to decrease the rate of change of x, by decreasing the rate in proportion to the amount of u and x. Let u(t) be the amount of this substance added at time t. For example, the population could be an infestation of an insect, or a harmful microbe in the body. Here we view x(t) as the concentration of a mold and u(t) a fungicide known to kill it. The differential equation representing the mold is given by

x′(t) = r(M − x(t))− u(t)x(t), x(0) = x0, where x0 > 0 is the given initial population size. Note the term u(t)x(t) pulls down the rate of growth of the mold. The effects of both the mold and fungicide are negative for individuals around them, so we wish to minimize both. Further, while a small amount of either is acceptable, we wish to penalize for amounts too large, so quadratic terms for both will be analyzed. Hence, our problem is as follows

min u

Ax(t)2 + u(t)2 dt

subject to x′(t) = r(M − x(t))− u(t)x(t), x(0) = x0. The coefficient A is the weight parameter, balancing the relative importance of the two terms in the objective functional. As we saw in the last lab, one weight term can be divided out, so only the A parameter is needed here. The other parameter in front of the u2 is taken to be 1. To begin, type lab2 and press enter. Enter the values

r = 0.3 M = 10 A = 1 x0 = 1 T = 5 . (6.1)

Do not vary any parameters for now. The control initially increases, then levels off to become constant. The state is also constant here; we say the control and state are in equilibrium, meaning both stay at constant values. The control eventually begins decreasing again, going all the way to 0. The

growth at the beginning and end of the interval and constant in the middle. In application, though, we wanted to eliminate the state, or at least decrease it. Note, we entered the value A = 1, meaning lowering the level of mold is as important as keeping the levels of fungicide down. This generally would not be the case, however. We are much more interested in removing the mold. Therefore, we should use a higher weight parameter. Enter the values

r = 0.3 M = 10 A = 10 x0 = 1 T = 5 . (6.2)

Here, the level of fungicide used is much higher. Notice that the state and control still experience the long period of equilibrium. The control begins at its greatest point, decreasing slightly before becoming constant, then decreasing to 0. As desired, the state decreases from its initial amount to about 0.95 and becomes constant. However, at the end of the interval, when the fungicide use decreases, the level of mold rapidly increases. Seemingly, the best course of action would be to begin another 5-day regimen of a second fungicide on about day 4. For comparison, see Figure 6.1, which shows the optimal state with these values, versus a mold population where no fungicide is used (u ≡ 0).