Optimal control of spatially distributed systems

Spatially distributed dynamical systems appear in several domains of engineering and science. Examples include multi-agent robotic systems, networks of embedded systems, arrays of mobile sensor networks for monitoring environment, platoon of vehicles in automated highways, deflection of beams and membranes, and the temperature distribution of thermally conductive materials. These systems can be described by a finite or infinite number of coupled subsystems that are possibly heterogeneous and of low dimensions. Even when each individual subsystem has a predictable behavior, the resulting spatially distributed dynamical system displays a rich and complex behavior when viewed as a whole. In this thesis, we present a mathematical framework to study the structural properties of optimal control of linear spatially distributed dynamical systems with arbitrary interconnection topologies. Our aim is to determine to what extent the optimal control law is localized in space and that how much information from far away subsystems is required. Specifically, we prove that global features of spatially distributed dynamical systems such as stability and optimal performance are inherently localized in space. In this thesis, we develop a mathematical framework using tools from functional analysis and operator theory to analyze the locality properties of optimal control problems involving infinite-horizon linear quadratic criteria and constrained receding horizon control. We prove that the kernels of the corresponding optimal controllers decay in the spatial domain and thus the controllers are spatially localized. It is important to stress that these locality features are inherent and, therefore, verify the feasibility of spatial truncation without too much loss of performance