The study of feedback control is arguably the most influential of engineering disciplines. Autonomous driving, spacecraft pointing, indoor temperature and humidity control, and modern cancer radiation therapy all hinge on the ability of a control system to robustly and reliably regulate system behaviour. Despite its diverse areas of application, the desire to optimize performance and guarantee acceptable behavior in the face of inevitable uncertainty is pervasive throughout control theory. This creates a fundamental challenge since the necessity of stable yet robust control schemes often favors conservative designs, while the desire to optimize performance typically demands the opposite. Our research uses the tools of applied mathematics to find the ideal balance between these competing priorities.

Model predictive control (MPC) is a valuable tool for incorporating system constraints and performance objectives in controller design. Though traditionally employed in process control, MPC is garnering ever more attention from other industries. This research tackles the problem of applying MPC to systems with discrete changes in behavior, like docking spacecraft or networks with evolving communication lines. Very little literature exists on the use of MPC for switched systems and this research promises to both broaden the applicability of MPC, and lighten computational loads, permitting the use of MPC in systems with limited computational resources.

Despite the importance of input-output stability theory to robust and optimal control, its theoretical power has been obscured by an over-reliance on two popular results, the Small Gain and Passivity Theorems. This research focuses on how more general results, such as George Zames' Conic Sector Theorem and notions of dissipativity, can aid in solving the most challenging of modern control problems.