Analysis and Control of Dynamical Systems on Complex Networks of Unbounded Size
Abstract
Networks are ubiquitous in modern society and the need to analyse, design and control them is evident. However an equally obvious feature of many networks is their apparently unbounded growth over time. Examples are given by wireless sensor networks, the power grid, social, information and transport networks. An undesirable consequence of such growth is that, inevitably, any analysis or control method founded upon techniques whose effectiveness decreases with the size of the network will eventually be overwhelmed methodology for the analysis and control of dynamical systems distributed over networks of unbounded size. The methodology for the proposed research program involves the utilization of the theory of graphons, together with a fundamental extension of Mean Field Game theory, and it is this combination which permits us to construct a theory and methodology for the decentralized control of systems on unbounded networks. Graphon theory is a continuum approach to the stud of sequences of large, asymptotically infinite, graphs or networks; it gives a rigorous formulation of the notion of limits for infinite sequences of networks of increasing size (as measured by the number of nodes). One of the principal contributions of graphon theory is that it facilitates the application of the methods of mathematical analysis to the study of arbitrarily large, complex networks. The second principal methodology to be employed is Mean Field Game theory, of which the propose is a co-creator. Mean Field Game (MFG) theory establishes the existence of Nash equilibria in games involving a large number of asymptotically negligible agents modelled by controlled stochastic dynamical systems and provides an approximation theory for the application of the infinite population equilibrium strategies to the finite population case. This is achieved by exploiting the relationship between the finite and corresponding infinite limit population problems, where the solution to the infinite population problems is given via (i) the Hamilton-Jacobi-Bellman (HJB) equation of optimal control, and (ii) the Fokker-Planck-Kolmogorov (FPK) equation )or equivalent McKean-Vlasov stochastic differential equation) for a generic agent. These equations are linked by the distribution of the state of a generic agent, otherwise known as the systemÕs mean field. The significance of the proposed work is both theoretical and methodological. Two key theoretical innovations in the proposed project which must be highlighted are (i) the construction of stochastic dynamical systems distributed on the nodes of infinite graphon networks, and (ii) the corresponding generalization of the well known MFG equations to what are termed her the Graphon Ð Network MFG equations. The methodological significance is that tractable computational methods will be developed for the decentralized control of systems on arbitrarily large networks, a task which is inherently impossible by the use classical computation methods. As a result the work presented in the proposal constitutes a completely new contribution to the challenging problem area of systems control on large complex networks, it will permit, for the first time, the analysis and synthesis of decentralized control strategies for non-cooperative and cooperative systems distributed over complex networks of unbounded size. This is very likely to have a considerable impact on large scale systems and control theory, network theory, and game theory. The expected initial applications will be to control distributed wireless sensors, electric vehicles on smart grids and to residential power storage control for the integration of renewable energy sources.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 01, 2019
- Source ID
- W911NF1910110
Entities
People
- Peter E. Caines
Organizations
- Army Contracting Command
- McGill University
- United States Army