Applications of optimal control theory to wave equations and a predator-prey model

In this dissertation, we investigate optimal control of partial differential equations. We prove the existence of optimal controls for which the objective functional are minimized or 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 studying a predator-prey parabolic system with Robin boundary condition, to control the effects of boundary habitat hostility. In a wave equation problem, a bilinear control is used to bring the state solution close to a desired profile under a quadratic cost control. We also establish the uniqueness of the solution of the optimality system and thus determine the unique optimal control. In the problem of a wave equation with viscous damping, we control the damping term. We seek to minimize the cost functional which measures the closeness of state solutions to a desired profile.