Optimal Control of Spatially Distributed Systems

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Lyapunov methods
Riccati equations
graph theory
linear systems
multidimensional systems
optimal control
state-space methods
Riccati equations
arbitrary graphs
distant-dependent coupling functions
heterogeneous linear control systems
infinite collection
infinite-horizon linear quadratic criteria
operator Lyapunov equation
optimal control
spatial structure analysis
spatially decaying
spatially distributed systems
Distributed control
infinite-dimensional systems
networked control
optimal control
spatially decaying systems
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Motee, Nader

In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant-dependent coupling functions over arbitrary graphs. We study the structural properties of optimal control problems with infinite-horizon linear quadratic criteria, by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD). These operators are a generalization of translation invariant operators used in the study of spatially invariant systems. We prove that given a control system with a state-space representation consisting of SD operators, the solution of operator Lyapunov and Riccati equations are SD. Furthermore, we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain, with the type of decay (e.g., exponential, polynomial or logarithmic) depending on the type of coupling between subsystems.

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Copyright 2009 IEEE. Reprinted from: Motee, N.; Jadbabaie, A., "Optimal Control of Spatially Distributed Systems," Automatic Control, IEEE Transactions on , vol.53, no.7, pp.1616-1629, Aug. 2008 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4623272&isnumber=4623249 This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.
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