Two Biological Applications of Optimal Control to Hybrid Differential Equations and Elliptic Partial Differential Equations

In this dissertation, we investigate optimal control of hybrid differential equations and elliptic partial differential equations with two biological applications. 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 opti- mality 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 two prob- lems: one is a type of hybrid system involving ordinary differential equations and a discrete time feature. We apply our approach to a tick-transmitted disease model in which the tick dynamics changes seasonally while hosts have continuous dynam- ics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. The other is a semilinear elliptic partial differential equation model for fishery harvesting. We consider two objective functionals: maximizing the yield and minimizing the cost or variation in the fishing effort (control). Existence, necessary conditions and uniqueness for the optimal control for both problems are established. Numerical examples are given to illustrate the results.