Applications of Optimal Control

In this dissertation, we investigate optimal control of partial and ordinary differential equations. We prove the existence of an optimal control for which the objective functional is maximized. The goal is to characterize the optimal control in terms of the solution of the optimality system. The optimality system consists of the state equations coupled with the adjoint equations. To obtain the optimality system we differentiate the objective functional with respect to the control. This process is applied to harvesting in a predator-prey parabolic system, to analyzing surface runoff in a parabolic problem, and to controlling the effect of the HIV virus on T cells in an AIDS patient. In the predator-prey problem, the profit associated with harvesting is shown to be positive under certain constraints. In the runoff problem, the concentration of contaminants being deposited into a major river flow is modeled as point sources. To explicitly characterize the optimal controls, two choices of the revenue function are used. One revenue function is a Michaelis-Menton function and the other is a quadratic function. In the HIV problem, we control the effect that HIV has on the T cells in the immune system. We seek to maximize the number of T cells, minimize the free virus, and minimize the systemic cost to the body.