Receding Horizon Control of Spatially Distributed Systems over Arbitrary Graphs

In this paper, we study the problem of receding horizon control of spatially distributed systems with arbitrary interconnection topologies. The key idea is the introduction of spatially decaying operators (SD) which serve as the main ingredient in the cost function that couples the state and control of individual agents with those of others. It is shown that coupling between subsystems of many well-known spatially distributed systems such as some of the recently studied models of distributed motion coordination with nearest neighbor interactions as well as spatially invariant systems can be characterized using such operators. We prove that for spatially distributed systems with input and state constraints in which the coupling is through an SD operator in a finite horizon cost function, optimal receding horizon controllers are piecewise affine (represented as a convolution sum plus an offset). More importantly, we prove that the kernel of each convolution sum decays exponentially in the spatial domain, thereby providing evidence that even centralized solutions to the receding horizon control problems for such systems has an inherent spatial locality. Our theoretical results are verified by numerical simulations