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

Consistent Approximations for the Optimal Control of Constrained Switched Systems

Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments