Graphon Control of Large-Scale Networks of Linear Systems

Abstract

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.

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

“…Therefore the optimal control problems (17) and (22) are equivalent. These, together with the result in Lemma 4, imply that solving the optimal control problems defined by (21) and (22) locally is equivalent to solving the original optimal control problem defined by (2) and (5).…”

“…The trajectories of the graphon dynamical system in (4) correspond one-to-one to the trajectories of the network system in (3). Moreover, the system in (2) can represent the limit system for a sequence of systems represented in the form of (4) when the underlying step function graphon sequences convergence in the L 2 [0,1] 2 metric [21]. III.…”

“…Theorem 2 If Assumptions 1-4 are satisfied, then solving the optimal control problems defined by (21) and (22) locally is equivalent to solving the original optimal control problem defined by (2) and (5). Moreover, the localized optimal control law for the γ th subsystem with γ ∈ [γ, γ] ⊂ [0, 1] is given by…”

“…2 PROOF First, sincex t =x t f ,ū t =ū t f and f 2 = 1 for 1 ≤ l ≤ d, the optimal control problem in (16) can be equivalently solved by solving (21) and then recovering the pair (x t ,ū t ) in the f eigendirection, 1 ≤ l ≤ d. Second, notice that J(ȗ) = 1 0 J(ȗ t (γ))dγ and J(ȗ(γ)) are non-negative for any γ ∈ [γ, γ] ⊂ [0, 1]. Therefore the optimal control problems (17) and (22) are equivalent.…”

“…2) the projections of the state x t onto each eigenfunction direction, that is,x t = x t , f for all 1 ≤ ≤ d; 3) its own state x t (γ). Alternatively, 2) can be replaced by the projections of the initial state x 0 onto each eigenfunction direction, that is, x 0 = x 0 , f for all 1 ≤ ≤ d. Givenx 0 , each subsystem can locally precompute the state {x t , t ∈ (0, T ]} based on the dynamics in (21) and the optimal control law in the f eigendirection given by (7).…”