From the electric power grid, to social networks, to the human brain, many systems of engineering and scientific interest are obtained by interconnecting simpler subsystems. This interconnection can be as simple as a feedback loop, or have a complicated network structure. However, in each case, the dynamic coupling results in complicated behaviors that cannot be explained simply by looking at the constituent components in isolation. This poses two major challenges: first, in terms of developing mathematical models to gain insight into the behavior of complex systems, and second, in terms of optimizing their operation or controlling them. The goal of this thesis is to develop a mathematical framework to tackle these challenges using tools from nonlinear dynamics, control theory, optimization, and network science.

This thesis is divided into two parts, each focusing on a specific class of interconnected system arising in real world applications. In the first part, we focus on developing a ``systems theory'' of optimization algorithms, in order to understand their properties and study their interconnection with physical processes. We demonstrate that tools from safety-critical control can be used to synthesize flows solving constrained nonlinear optimization problems, with safety, stability and robustness guarantees that make them ideal for online implementation when interconnected with physical processes. The second part of this thesis discusses mathematical modeling of interconnected neurological systems, in order to understand and control epileptic seizures. We model the epileptic brain with Linear Threshold Networks (LTNs), and analyze the interplay between the network structure and the dynamical properties they exhibit. We characterize conditions on the network structure under which oscillations spread in LTNs, and develop strategies to optimally modify networks to prevent the spread of epileptic seizures.