Optimal and Approximate Solutions to Linear Quadratic Regulation of a Class of Graphon Dynamical Systems

Gao, Shuang; Caines, Peter E.

doi:10.1109/cdc40024.2019.9030095

Cited by 10 publications

(14 citation statements)

References 29 publications

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“…Based on (26) and (27), we obtain that the convergence of a sequence of graphons in · 2 or · 2 implies its convergence in · op . Under certain extra conditions, the convergence of a sequence of graphons under the cut norm implies its convergence under the L 2 [0, 1] norm [32, Corollary 1.1].…”

Section: Limits Of Sequences Of Network Systemsmentioning

confidence: 98%

Graphon Control of Large-Scale Networks of Linear Systems

Gao

Caines

2020

IEEE Trans. Automat. Contr.

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To achieve control objectives for extremely largescale complex networks using standard methods is essentially intractable. In this work, a theory of the approximate control of complex network systems is proposed and developed by the use of graphon theory and the theory of infinite dimensional systems. First, the graphon dynamical system models are formulated in an appropriate infinite dimensional space in order to represent arbitrary-size networks of linear dynamical systems, and to define the convergence of sequences of network systems with limits in the space. The exact controllability and the approximate controllability of graphon dynamical systems are then investigated. Second, the minimum energy state-to-state control problem and the linear quadratic regulator problem for systems on complex networks are considered. The control problem for the graphon limit systems is solved in each case and an approximation is defined which yields control laws for the original control problems. Furthermore, the convergence properties of the approximation schemes are established. A systematic control design methodology is developed within this framework. Finally, numerical examples of networks with randomly sampled weightings are presented to illustrate the effectiveness of the graphon control methodology.

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Section: Limits Of Sequences Of Network Systemsmentioning

confidence: 98%

Graphon Control of Large-Scale Networks of Linear Systems

Gao

Caines

2020

IEEE Trans. Automat. Contr.

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“…From the L 2 -solution granted by the theorem we can extract one version that satisfies (16) for all x ∈ I by replicating the argument from Theorem 2.…”

Section: Existence and Uniqueness Of Solutions To The Hamiltonian Systemmentioning

confidence: 99%

“…Step 0: Uniqueness in the Fubini extension Assume that (X, p, q) and ( X, p, q) are solutions to the FBSDE (16) in the sense that ( 16) is satisfied P λ-a.s. and E…”

Section: A2 Proof Of Theoremmentioning

confidence: 99%

“…Deterministic linear-quadratic graphon games were studied in [19]. In a setting similar to that of graphon games, a graphon-based model for optimal network interaction was used in [14,15,18,16,17] to represent control problems on large networks. All these works assume that the graph underpinning the network of interactions is dense in the sense that the number of edges (non trivial interactions) is of the same order as the number of vertices (agents).…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Graphon Games: II. The Linear-Quadratic Case

Aurell¹,

Carmona²,

Laurière³

2021

Preprint

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In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.

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“…Among these applications, graphon spectral properties are very significant [14]. In fact, the spectral analysis of largescale systems has been studied since the late 1960s [15] and it plays a key role to the low-complexity control synthesis of such systems [15]- [17].…”

Section: Introductionmentioning

confidence: 99%

Spectral Representations of Graphons in Very Large Network Systems Control

Gao

Caines

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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Graphon-based control has recently been proposed and developed to solve control problems for dynamical systems on networks which are very large or growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper, spectral representations, eigenfunctions and approximations of graphons, and their applications to graphon-based control are studied. First, spectral properties of graphons are presented and then approximations based on Fourier approximated eigenfunctions are analyzed. Within this framework, two classes of graphons with simple spectral representations are given. Applications to graphon-based control analysis are next presented; in particular, the controllability of systems distributed over very large networks is expressed in terms of the properties of the corresponding graphon dynamical systems. Moreover, spectral analysis based upon real-world network data is presented, which demonstrates that low-dimensional spectral approximations of networks are possible. Finally, an initial, exploratory investigation of the utility of the spectral analysis methodology in graphon systems control to study the control of epidemic spread is presented.

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Optimal and Approximate Solutions to Linear Quadratic Regulation of a Class of Graphon Dynamical Systems

Cited by 10 publications

References 29 publications

Graphon Control of Large-Scale Networks of Linear Systems

Graphon Control of Large-Scale Networks of Linear Systems

Stochastic Graphon Games: II. The Linear-Quadratic Case

Spectral Representations of Graphons in Very Large Network Systems Control

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