Burbano Lombana, D. A., Russo, G., & Di Bernardo, M. (2019).Pinning Controllability of Complex Network Systems with Noise. IEEETransactions on Control of Network Systems, 6(2), 874-883.[8528873]. https://doi.org/10.1109/TCNS.2018.2880300

Peer reviewed version

Link to published version (if available):10.1109/TCNS.2018.2880300

Link to publication record in Explore Bristol ResearchPDF-document

This is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IEEE at https://ieeexplore.ieee.org/document/8528873 . Please refer to any applicable terms of use of thepublisher.

University of Bristol - Explore Bristol ResearchGeneral rights

This document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/

1

Pinning Controllability of Complex NetworkSystems With Noise

Daniel A. Burbano-L. Member, IEEE, Giovanni Russo Member, IEEE and Mario di Bernardo, Fellow, IEEE

Abstract—The ability of a network of nonlinear systems tosynchronize onto the desired reference trajectory in the presenceof one or more leader nodes is known as the pinning control-lability problem. This paper studies the pinning controllabilityof multi-agent networks subject to three different types of noisediffusion processes; namely, noise affecting the node dynamics,the communication links, and the pinning control action itself.By using appropriate stochastic Lyapunov functions, sufficientpinning controllability conditions are derived depending on thenode dynamics, network structure, noise intensity, and controlparameters. Counterintuitively, it is found that under somespecific conditions noise may enhance the pinning controllabilityof the network making it easier to drive all agents towards thedesired collective behavior. The effectiveness of the theoreticalresults is illustrated via two application examples arising inthe context of gene regulatory networks and synchronization ofchaotic systems.

Index Terms—Pinning Control, Nonlinear Systems, Stochas-tic/Uncertain Systems, Stability, Networks of Autonomous Agents

I. INTRODUCTION

P INNING control has been shown to be an effective ap-proach for controlling network systems towards a desired

collective behavior. This strategy consists of injecting controlsignals only into a fraction of agents (or nodes) so that thewhole network converges to a desired target state. This makespinning control applicable for large networks and even in thosecases where only some of the nodes are accessible for control[1]. Pinning control strategies and their extensions have beenwidely used to steer networks toward a synchronous state.For instance in [2], the control gains self-tune their value forguaranteeing synchronization in networks of circuits, whilein [3], an impulsive control law is used for synchronizingfiring neurons. For a comprehensive review of pinning controlstrategies the reader is referred to e.g. [4], [5] and referencestherein.

When designing a pinning control strategy, it is crucial todecide the number and location of nodes to be controlledand find the value of the control parameters guaranteeingconvergence. This problem is known as pinning controllabilitywhich provides a qualitative measure of the propensity of a

Daniel A. Burbano-L. is with the Department of Electrical Engineeringand Computer Science, Northwestern University, Evanston, IL 60208-3118,United States (e-mail: danielburbano@northwestern.edu)

Giovanni Russo is with the School of Electrical and Electronic En-gineering, University College Dublin, Dublin 14, Ireland (e-mail: gio-vanni.russo1@ucd.ie). Work supported, in part, by Science Foundation Irelandgrant 13/RC/2094.

M. di Bernardo is with the Department of Electrical Engineering andInformation Technology, University of Naples Federico II, Naples 80125,Italy, and also with the Department of Engineering Mathematics, Universityof Bristol, U.K. (e-mail: mario.dibernardo@unina.it).

network to synchronize onto a reference trajectory, see e.g.[6], [7]. Pinning controllability was studied in [8], where localconvergence conditions were derived, while in [9] conditionsfor pinning controllability were given in the global case.Moreover, in [10], [11] the authors proposed different methodsfor finding the optimal location of pinned nodes (i.e. the nodeson which control is directly exerted) by optimizing pinningcontrollability.

Most of the results on pinning controllability are based onthe assumption that the dynamics of the network is deter-ministic and noise-free. Unfortunately, in many applications,networks are often subject to certain unavoidable disturbancesand noise. For instance, in gene regulatory networks determin-istic models are not able to capture cell-to-cell fluctuationsin genetic switching [12], and also in engineered networks,communication between nodes might occur through quantizedstate variables [13] or fading communication links [14].

In this paper, which is motivated by the above observations,we study pinning controllability of networks subject to noisediffusion processes. In particular, we consider networks ofdiffusively coupled nonlinear agents, and investigate threedifferent scenarios; namely, the cases where (i) noise affectsthe control action, or (ii) noise propagates through the in-terconnection links, or (iii) noise affects the intrinsic nodedynamics. For such scenarios, we devise a set of sufficientconditions, which explicitly relate pinning controllability tothe node dynamics, the network structure and the noise dif-fusion process. The results show that, as one would expect,pinning controllability becomes worse as the noise intensityincreases, i.e. more control interventions and/or higher controlgains are needed to guarantee the network reaches the targetstate. However, we have found that the pinning controllabilitycan counterinutitevely, be enhanced if noise propagates in ahomogeneous manner. The effectiveness of the results is alsoillustrated via two application examples arising in the contextof regulatory gene networks and chaotic systems.

A. Related work

The influence of noise on synchronization and consensushas recently gained much research attention, see e.g. [15], [16],[17], [18], [14], [19]. It has also been recently suggested [20],that noise can be beneficial for network coordination and thatsmall random fluctuations, if properly injected in the network,can improve the collective performance of a group of agents.

The problem of controlling a network with stochastic termshas been studied in [21], [22]; however, convergence of thecontrol strategies are proved assuming all nodes are controlled.In [23], noise has been used to implement a distributed speed

2

advisory system, where vehicles are modeled as simple inte-grators. In [24] pinning control is investigated, via linearizationof the network dynamics, in scenarios where the control actionis applied to the network in a stochastic manner, while in [25]linear diffusively coupled networks with multiple disturbancesand Lipschitz nonlinear terms were studied. More recently in[26] networks with nonlinear stochastic terms were analyzedassuming homogeneous noise, that is, the Brownian motion isassumed to be one-dimensional and identical for all the nodes.

When compared to previous literature, the work reportedin this paper expands the state of the art in a number ofdifferent ways. In particular, (i) we study networks of dif-fusively coupled nodes affected by different types of possiblyheterogeneous noise diffusion processes; (ii) our results, whichare based on the use of stochastic Lyapunov functions, providepinning controllability conditions in the case where only afraction of the nodes is controlled; (iii) the nodes dynamicscan be nonlinear, with their vector field satisfying the so-calledQUAD (quadratic) condition, a more general condition thanLipschitz [27]; (iv) the coupling functions between nodes inthe network can be nonlinear (preliminary results on networkswith linear diffusive couplings were reported in [28]); and (v)under certain condition noise might be beneficial for enhancingthe pinning controllability.

II. MATHEMATICAL PRELIMINARIES

A. Notation

We denote by IN the identity matrix of dimension N ×N ;by 1N a N × 1 vector with unitary elements, while 0 is amatrix of zeros of appropriate dimensions. The symbol ‖·‖denotes the Euclidean norm for vectors, and the spectral norminduced by the L2-norm for matrices. A diagonal matrix,say D, with diagonal elements d1, . . . , dN is denoted byD = diagd1, . . . , dN; λk(A) denotes the k-th eigenvalue ofa squared matrixA, and ⊗ denotes the Kronecker product. Thetriple (Σ,F ,P) denotes a complete probability space whereΣ is the sample space with elements denoted by ω, F is aσ-algebra and P : F → [0, 1] is a probability measure. Asusual, P(ω) denotes the probability of an event ω. In whatfollows, Cp×q represents the space of continuous functionsf : Ω0 ⊆ Rm × R+ ∪ 0 7→ Rm that can be differentiatedp times in x and q times in t. For instance C2×1 denotes thespace of functions f(x, t) that are twice differentiable in xand differentiable in t. We also simply denote by C the spaceC1×1.

Definition 2.1: A function ϕ(x, t) : Ω0 ⊆ Rn×R+∪0 7→Rn, is said to satisfy the Quadratic-Lipschitz condition if thereexist constants κ1 ∈ R+ and κ2 ∈ R with κ2 ≤ κ1, such that∀x,y ∈ Ω0 and ∀t ≥ 0

‖ϕ(x, t)−ϕ(y, t)‖ ≤ κ1 ‖x− y‖ (1)

(x− y)> (ϕ(x, t)−ϕ(y, t)) ≥ κ2(x− y)>(x− y) (2)

Remark 2.1: Note that a Quadratic-Lipschitz function isa generalization of the bi-Lipschitz condition. Indeed, usingthe triangle inequality, condition (2) can be upper-boundedby κ2‖x− y‖2≤ ‖x− y‖‖ϕ(x, t)−ϕ(y, t)‖ for κ2 > 0.

Hence, the quadratic-Lipschitz condition (1)-(2) reduces tothe bi-Lipschitz condition κ2‖x− y‖≤ ‖ϕ(x, t)−ϕ(y, t)‖≤κ1‖x− y‖, which naturally arises in a number of applica-tions, including filtering and state estimation [29], [30]. Inaddition, any linear function satisfies (1)-(2). For instance,let ϕ(xi) = Axi, with A being a square matrix. Then,the Quadratic-Lipschitz condition is easily verified by settingκ1 = (1/2)λmax(A> +A) and κ2 = (1/2)λmin(A> +A).

B. Algebraic Graph Theory

An undirected graph G is defined by G = (N , E) whereN = 1, 2, · · · , N is the finite set of N node indices; E ⊂N ×N is the set containing the E edges between the nodes(i, j) for any i, j ∈ N . The adjacency matrix A(G ) ∈ RN×N(or simply A in what follows) of a graph G represents thetopology of the network of interconnections and its elementsaij are defined as aij = 1 for i 6= j if there is an edge fromnode i to node j and zero otherwise. We assume that thereare no self loops in the network, i.e. aii = 0, ∀i = 1, . . . , Nand that the edges are undirected, that is aij = aji for alli, j ∈ N , i 6= j. The Laplacian matrix L(G ) ∈ RN×N (orsimply L in what follows) of a graph G is given by L =diagA1N−A, where the matrix diagA1N is often calledthe degree matrix. The elements of L are denoted by `ij ,i, j = 1, . . . , N . A multigraph, is the set of M graphs M :=G1, · · · ,GM called layers of M , where all the graphs inM share the same set of nodes, that is Gk = (N , Ek), fork ∈ 1, · · · ,M.

Lemma 2.1: [31] Let G (N , E) be a connected undirectedand unweighted graph. Then, the eigenvalues of its corre-sponding Laplacian matrix L can be ordered as 0 = λ1 <λ2 ≤ · · · ≤ λN . Moreover, for any pair of vectors x =[x1, · · · , xN ],y = [y1, · · · , yN ] ∈ RN , the following relationhods x>Ly =

∑Ni,j∈E (xj − xi)(yj − yi).

C. Stability of Stochastic Differential Equations

Consider the m-dimensional stochastic differential equationof Ito type [32]

dx = f(x, t)dt+ g(x, t)dB, x(0) = x0 (3)

where x ∈ Ω ⊆ Rm is the state variable, x0 ∈ Ω, f ∈ C2×1,f : Ω×R+ ∪ 0 7→ Rm, g ∈ C, g : Ω×R+ ∪ 0 7→ Rm×pand B is a p-dimensional Brownian motion. Throughout thispaper we will assume that the open set Ω is a forward invariantset for the dynamics (3). That is, for any given initial conditionx0 ∈ Ω, a unique solution of (3) exists, say x(t) such that,x(t) ∈ Ω ∀t ≥ 0, see e.g. [33]. We will also assume thatf(0, t) = g(0, t) = 0 and the solution x = 0 is termed asthe trivial solution of (3). Clearly, this implies by constructionthat x = 0 ∈ Ω.

In this paper we are interested in exponential almost surestability. Before giving the formal definition of stability, it isimportant to recall that a sequence of random variables, sayX1, X2, . . ., converges almost surely (a.s.) to the randomvariable X if P (limn→+∞Xn(w) = X(w)) = 1; that is,convergence is attained with probability 1 (P = 1). We arenow ready to give the following definition.

3

Definition 2.2: [32] The trivial solution of (3) is said to bealmost surely exponentially stable if for all x0 ∈ Ω,

limt→+∞

sup1

tlog (‖x‖) < 0, a.s. (4)

The quantity whose limit is taken in (4) is called the sampleLyapunov exponent (see p. 63 [32]); therefore, its value canbe used as an estimate of the rate of convergence towards thetrivial solution of (3).

Definition 2.3: [32] Consider a non-negative functionV (x, t) : Ω × R+ ∪ 0 7→ R+ belonging to C2×1. Thedifferential operator L associated to the stochastic Ito equation(3) is defined as

LV (x, t) := Vt(x, t) + Vx(x, t)f(x, t) + Vgx(x, t) (5)

where Vt(x, t) := ∂V (x, t)/∂t, Vx(x, t) := [Vx1, . . . , Vxm

]with Vxk

= ∂V (x, t)/∂xk for k ∈ 1, · · ·m, and Vgx :=(1/2)trace

g(x, t)>Vxxg(x, t)

where Vxx is an m × m

matrix with entries [Vxixj ] := ∂2V (x, t)/∂xj∂xi for alli, j ∈ 1, · · ·m

Theorem 2.1: Consider a non-negative function V (x, t) :Ω × R+ ∪ 0 7→ R+ satisfying V (x, t) ∈ C2×1. Assumethere exist arbitrary constants ρ > 0, c1 > 0, c2 ∈ R,c3 ≥ 0, such that ∀x ∈ Ω − 0, and ∀t ≥ 0 thefollowing conditions are fulfilled: (i) c1‖x‖ρ≤ V (x, t); (ii)LV (x, t) ≤ c2V (x, t); (iii) ‖Vx(x, t)g(x, t)‖2≥ c3V

2(x, t).Then, limt→+∞ sup(1/t) log (‖x(t)‖) ≤ −(c3 − 2c2)/ρ, a.s.In particular, if c3 > 2c2; then, the trivial solution of (3) isalmost surely exponentially stable.

Proof: This is a straightforward extension of Theorem3.3, page 121 in [32] (where Ω ≡ Rm). The proof followsidentical steps as those used in [32] and hence it is omittedhere for the sake of brevity.

III. PROBLEM FORMULATION

A. Network Model

Consider a network of N > 1 diffusively coupled identicalnodes represented by an undirected and connected graphG = (N , E) associated to the Laplacian matrix L= [`ij ],where each node is described by a set of nonlinear stochasticdifferential equations of Ito type

dxi =

f(xi, t)− σN∑j=1

`ijh(xj , t) + ui

dt+ φi(x)dBi

(6)with xi ∈ Ω0 ⊆ Rn, xi(0) = xio, i ∈ N being the initialconditions for the ith node. We assume that both f (i.e. thenode dynamics) and h (i.e. the coupling function) belong toC2×1 and that f ,h : Ω0 ⊆ Rn×R+∪0 7→ Rn. It is implicitin the notation that the set Ω0 is an open, forward-invariantsubset of Rn. In certain applications, the set Ω0 is not an openset (for example, for a biochemical system, this is given bynon-negativity constraints on the state variables). For a non-open set Ω0, the fact that f(·, ·) and h(·, ·) belong to C2×1means that the function f(·, t) and h(·, t) can be both extendedas a twice differentiable function to some open set whichincludes Ω0, and that f(x, ·) and h(x, ·) are differentiable

on this open set. The non-negative constant σ representsthe coupling strength, and ui is an exogenous control input.Finally, φi(x) is a possibly nonlinear function modeling thediffusion of noise through the network, x := [x>1 , · · · ,x>N ]>,while Bi is a p-dimensional Brownian motion defined on theprobability space (Σ,F ,P).

Assumption 3.1: There exists a positive constant, say kf ,such that for all x,y ∈ Ω0 and for all t ≥ 0

(x− y)> (f(x, t)− f(y, t)) ≤ kf (x− y)>(x− y) (7)

Assumption 3.2: The coupling function h(·, ·) satisfies thequadratic-Lipschitz condition for some κ1 = kl, κ2 = kh, withkl, kh > 0 (see Definition 2.1).

Remark 3.1: The quadratic condition (7) is well known asthe QUAD or one-side Lipschitz condition, which is widelyused within the synchronization literature and is closely relatedto the Lipchitz condition and the notion of contractive vector-fields [27]. We also note that Assumption 3.2 implies that thepossibly nonlinear function h(·, ·) is required to be Lipschitzand strongly monotone. Indeed, the monotonicity conditionis crucial to guarantee convergence [34], [35]. We wish toemphasize that Assumption 3.2 encompasses a broader classof coupling functions that are not necessarily linear as oftenassumed in the existing literature [21], [22], [26].

The goal of this paper is that of designing the control law,ui, so that all the states of the network asymptotically convergeonto a common desired state. To that aim we use the wellknown pinning control strategy, where the control action isonly exerted on a fraction of nodes [4].

B. Pinning Control

We consider the control action ui(t) to be given by aproportional feedback controller of the form

ui(t) := −piα (h(xi, t)− h(xr, t)) (8)

where: (i) α > 0 represents the control strength; (ii) xr ∈ Ω0

is the reference (or target) state generated by a master (orpinner) node external to the network; (iii) pi is a constant valueequal to 1 if the i-th node is being controlled or 0 otherwise.We denote the set of pinned (or leader) nodes as a subset ofnode indices Np ⊂ N such that pi = 1.

Definition 3.1: Under the control action (8), the stochasticnetwork (6) is said to reach complete stochastic synchroniza-tion onto xr(t) if

limt→+∞

sup1

tlog (‖xi(t)− xr(t)‖) < 0, a.s., i ∈ N (9)

Definition 3.2: We say that network (6) controlled by (8)is pinning controllable in a stochastic sense if there exist avalue of the constant parameter α and a set of pinned nodesNp for which complete stochastic synchronization is achievedonto the reference trajectory xr(t).

4

(a) (b)

Fig. 1. (a) Graph of a network with noisy communication links, (b) Multiplexrepresentation of the noisy network

C. Noise on Networks

Next, we model different sources of noise in a network ofnonlinear agents such as (6) by making different choices forthe term φi(x)dBi therein. We first consider the case wherenoise enters the network through the control link between thepinner node and the nodes in the network which are directlycontrolled (or pinned). To model this scenario, we set

φi(x)dBi = −σ∗pi (h(xi, t)− h(xr, t)) dbi (10)

where σ∗ > 0 is a positive constant representing the intensityof noise and bi ∈ R are independent Brownian motions. Itis important to highlight that this type of perturbations mightbe used to model more realistic scenarios considering non-ideal sensors and actuators. For instance, consider the specialcase when h(xi, t) = xi, and each pinned node has access toquantized state variables for computing the control action ui.Then, as pointed out in [13], [14], the quantized measurementof xi−xr is given by xi−xr+σ∗(xi−xr)wi(t), with wi(t) =dbi/dt, i ∈ Np being independent white noises. Therefore, wecan rewrite the whole network dynamics as in (6)-(10).

As a second type of noise diffusion, we consider the casewhere the communication links are noisy. In this case we set

φi(x)dBi = σ∗N∑j=1

a∗ij (h(xj , t)−h(xi, t)) dbij (11)

where dbij are independent one-dimensional Brownian mo-tions, and a∗ij are the elements of an adjacency matrix A∗representing the structure of the network with noisy com-munication links. We denote such network structure by agraph G ∗ = (N , E∗). We wish to emphasize that, in general,A 6= A∗ and this situation may arise in the case wherenoise is only present (or the noise intensity is so small thatcan be neglected) on a subset of links of the network (6).See, for example, Figure 1(a), where G ∗ ⊂ G . Indeed, thepresence of two different types of link (noisy and noise-freelinks) between the same nodes is known as multiplexity [36];therefore, the structure of the network in (6) together withthe noisy communication links (10) can be modeled as amultigraph of two layers M = G ,G ∗ as shown in Figure1(b). As noted in [14], [37] and references therein, the typeof noise diffusion processes in (10) and (11) naturally ariseswhen modeling communications among agents that are subjectto quantization and/or the communication channel experiencesfading.

Finally, we model the case of uncertain or noisy nodedynamics by choosing

φi(x)dBi = σ∗g(xi, t)db (12)

where b is a one-dimensional Brownian motion and g : Ω0 ⊆Rn × R+ ∪ 0 7→ Rn is a nonlinear function that representsuncertainty on the vector-field f(xi, t), for instance wheresome parameters are subject to random environmental effects[32] (see application example in Section V-A).

Assumption 3.3: The function g(xi, t) in (12) satisfies thequadratic-Lipschitz condition (1) for some κ1 = klg, κ2 = kgwith klg ∈ R+ and kg ∈ R (see Definition 2.1).Note that klg can also take negative values.

IV. MAIN RESULTS

We start by providing a sufficient condition for stochasticpinning controllability of network (6) under pinning controland subject to noise on the control links.

A. Noise on the control action

Theorem 4.1: Consider the stochastic network (6) controlledby (8) and assume noise is acting on the feedback controlaction, as given by (10). Let Assumptions 3.1, and 3.2 hold,and assume that: (i) there is at least one pinned node, i.e.pi 6= 0, for some i ∈ N , and G is undirected and connected;(ii) xr(t) is a solution of the uncoupled dynamics, i.e. dxr =f(xr, t)dt, with initial condition xr(0) ∈ Ω0. Then, the closedloop network is stochastic pinning controllable if

λ >2kfkh

+(σ∗kl)

2

kh, (13)

where λ := λmin(L) is the smallest eigenvalue of the matrixL := σL+ αP with P := diagp1, · · · , pN.

Proof: For the sake of clarity we divide the proof in twosteps:

Step 1: We first define the error at each node as the differ-ence between the ith node state and the reference trajectory asei := xi−xr. Letting x := [x>1 , . . . ,x

>N ]>, and r := 1N⊗xr

be the stack vectors of the node states and reference trajectoryrespectively, the overall error dynamics e := [e>1 , . . . , e

>N ]>

can be recast in compact form as

de = F (e, t)dt+ G(e, t)dB, (14)

where,

F (e, t) = F (e+ r, t)− F (r, t) + α(P ⊗ In)H(r, t)

− (σ(L⊗ In) + α(P ⊗ In))H(e+ r, t)(15)

with F (e + r, t) := [f(e1 + xr, t)>, . . . ,f(eN + xr, t)

>]>,H(e + r, t) := [h(e1 + xr, t)

>, . . . ,h(eN + xr, t)>]>. The

diffusion term G(e, t) = −σ∗G(e, t), where G(e, t) is annN ×N matrix given by p1 (h(x1, t)− h(xr, t)) · · · 0nN×1

.... . .

...0nN×1 · · · pN (h(xN , t)− h(xr, t))

(16)

5

and B = [b1, · · · , bN ]>. Note that: (i) xr is a solution ofthe uncoupled dynamics with initial conditions in Ω0; then,xr(t) ∈ Ω0, ∀t ≥ 0; (ii) e = 0 is the trivial solution of(14). Therefore, Theorem 2.1 can be used to prove almostsure exponential stability of e = 0. To that aim, consider theLyapunov candidate function

V (e, t) = V (e) =1

2e>e (17)

It is easy to verify that V is a positive definite function andsatisfies condition (i) of Theorem 2.1 (where we assume ρ =2).

Step 2: Next we calculate LV (e) as defined in (5). It is easyto see that the first term is null, i.e. Vt(e) = 0 and Ve = e>.Calculating Ve(e)F (e, t) yields

Ve(e)F (e, t) = e> (F (e+ r, t)− F (r, t))

+ αe>(P ⊗ In)H(r, t)− e>(L⊗ In)H(x, t)

From Assumption 3.1 we have that e> (F (x, t)− F (r, t)) ≤kfe>e, therefore

Ve(e)F (e, t) ≤ kfe>e− e>(L⊗ In)H(x, t)

+ αe>(P ⊗ In)H(r, t)

Next, adding and subtracting the term e>(L⊗ In)H(r, t) tothe right-hand side of the last inequality, we obtain

Ve(e)F (e, t) ≤ kfe>e− e>(L⊗ In)(H(x, t)−H(r, t))

− e>(L⊗ In)H(r, t) + αe>(P ⊗ In)H(r, t)

and from the fact that −(L⊗In) +α(P ⊗In) = −(L⊗In),and (L ⊗ In)H(r, t) = (L ⊗ In)(1N ⊗ h(xr, t)) = 0 onehas

Ve(e)F (e, t) ≤ kfe>e− e>(L⊗ In) (H(x, t)−H(r, t))(18)

Next, defining ζ = −e>(L⊗In) (H(x, t)−H(r, t)) we get:

ζ = −σN∑

i,j∈E(xj − xi)>(h(xj , t)− h(xi, t))

−αN∑i=1

pi(xi − xr)>(h(xi, t)− h(xr, t)) (19)

where we used Lemma 2.1 [with x = e and y = H(x, t) −H(r, t)] and the fact that the matrix P is a diagonal matrix.Moreover, by means of Assumption 3.2 one has

ζ ≤ −khσN∑

i,j∈E(xj − xi)>(xj − xi)

−khαN∑i=1

pi(xi − xr)>(xi − xr)

= −khe> (σL⊗ IN ) e− khe> (αP ⊗ IN ) e

= −khe>(L⊗ In)e ≤ −khλe>e (20)

where λ is the smallest eigenvalue of matrix L. Therefore wecan rewrite (18) as

Ve(e)F (e, t) ≤ (kf − khλ)e>e (21)

Now, we calculate the last term of LV (e), Vge =

(1/2)traceG(e, t)>VeeG(e, t). Since Vee = INn one has

Vge =(σ∗)2

2

N∑i=1

pi(h(xi, t)− h(xr, t))>(h(xi, t)− h(xr, t))

from Assumption 3.1, we have that h(·, ·) satisfies the Lips-chitz condition (1), yielding

Vge ≤ (klσ∗)2

2

N∑i=1

pi(xi − xr)>(xi − xr) (22)

≤ (klσ∗)2

2e>Pe ≤ (klσ

∗)2

2e>e (23)

Next, using the upper bounds (21) and (23) we find that

LV (e) ≤ c2V (e) (24)

where c2 = 2(kf − khλ) + (klσ∗)2. Then conditions (ii)

and (iii) of Theorem 2.1 are fulfilled provided (13) holds andc3 = 0. Therefore, the closed-loop network reaches stochasticsynchronization onto the reference trajectory and the proof iscomplete.

B. Noise on the communication links

Next, we provide a result for pinning controllability ofnetwork (6), when this is controlled by (8) and the noisediffusion processes are given by (11).

Theorem 4.2: Consider the stochastic network (6) controlledby (8) where noise is acting on the communication links, i.e.φi(x)dBi is given by (11). Let Assumptions 3.1, and 3.2hold and assume that: (i) pi 6= 0, for some i ∈ N , and G isundirected and connected; (ii) xr(t) is a solution of dxr =f(xr, t)dt, with xr(0) ∈ Ω0. Then, the closed loop multiplexnetwork with M = G,G∗ is pinning controllable if

λ >kfkh

+(σ∗klλ

∗N )2

2kh(25)

Proof: Considering the error ei = xi−xr and followingsimilar steps to those already presented in the proof ofTheorem 4.1 yields (14) where F (e, t) is given in (15) andG(e, t) = σ∗G(x, t) with G(x, t) being a nN ×N2 matrixgiven by

G(x, t) =

Z1 01×N · · · 01×N

01×N Z2 · · · 01×N...

.... . .

...01×N 01×N · · · ZN

(26)

where Zi ∈ is a n×N matrix given by Zi := [a∗i1(h(x1, t)−h(xi, t)), · · · , a∗iN (h(xN , t)−h(xi, t))]. a∗ij are the elementsof the adjacency matrix A∗ denoting the location of noisycommunication links. Moreover, B = [b>1 , · · · , b>N ] with bi =[bi1, bi2, · · · , biN ]>. Next, consider the candidate Lyapunovfunction (17) and calculating LV (e) along the trajectories of(14)-(15)-(26) yields LV (e) = Ve(e)F (e, t) + Vge, wherean upper-bound for Ve(e)F (e, t) is given in (21), whileVge = (1/2)traceG>G. In order to find an upper-boundfor Vge we start by noticing [from (26)] that G(r, t) = 0, so

6

that we can rewrite G(e, t) = G(x)−G(r, t). Then Vge canbe rewritten as Vge = (σ∗)2

2 trace M with

M = (G(x, t)− G(r, t))>(G(x, t)− G(r, t))

Since M is a block-diagonal matrix we have thattrace M = (σ∗)2

2

∑Ni=1 trace

Z>i Zi

where Zi :=

[a∗i1(h1 − hi), · · · , a∗iN (hN − hi)] with hi := h(xi, t) −h(xr, t) for i ∈ N . Then, letting H = H(x, t)−H(r, t) =[h1, · · · , hN ]> one has

Vge =(σ∗)2

2

N∑i=1

traceZ>i Zi

=

(σ∗)2

2

N∑i=1

N∑j=1

(a∗ij)2(hj − hi)>(hj − hi)

=(σ∗)2

2H>((L∗)2 ⊗ In)H ≤ (σ∗λ∗N )2

2H>H

≤ cN∑i=1

(h(xi, t)− h(xr, t))>(h(xi, t)− h(xr, t))

(27)

with c = (σ∗λ∗N )2/2. Now, from Assumption 3.2, we find that

(h(xi, t)−h(xr, t))>(h(xi, t)−h(xr, t)) ≤ k2l (x−r)>(x−r)

so that Vge ≤ (σ∗λ∗Nkl)2V (e), and LV (e) ≤ c2V (e) where

c2 = 2(kf − khλ) + (λ∗Nσ∗kl)

2. Therefore, if condition (25)holds, setting c3 = 0 in Theorem 2.1 guarantees completestochastic synchronization of the closed-loop network onto xrwhich completes the proof.

Remark 4.1: Theorems 4.1 and 4.2 provide sufficient al-gebraic conditions guaranteeing pinning controllability of agiven network of interest. Such conditions depend on thenode dynamics (via kf ), the structure of the network and thelocation of the pinned nodes (via λ), the coupling functions(via kh and kl), and on the intensity of the noise (via σ∗ andσ∗λ∗N for noisy control and communication links respectively).Intuitively, the conditions indicate that, for a given networkstructure, noise can be compensated by either adding morecontrol interventions or by increasing the feedback controlgain α. We shall see that this is not the case for the nextscenario –the case where noise affects the node dynamicsitself– Indeed, it is shown that under certain conditions, noisecan be useful for enhancing pinning controllability.We also remark that the conditions in Theorems 4.1 and4.2 depend on λ. Analytical estimates of λ as a function ofthe number of pinned nodes, control strength α and networkstructure L can be obtained as shown in e.g. [9], [38]. Inaddition, λ can also be computed in a distributed manner byusing the distributed power iteration method [39].

C. Noise on the node dynamics

The following result gives a sufficient condition for pin-ning controllability of network (6)-(8) when noise diffusionprocesses are modeled as in (12).

Theorem 4.3: Consider the stochastic network (6) controlledby (8) and with noise propagating according to (12). Let

Assumptions 3.1, 3.2, and 3.3 hold, and assume that: (i)pi 6= 0, for some i ∈ N , and G is undirected and connected;(ii) xr is a solution of dxr = f(xr, t)dt+σ

∗g(xr, t)db. Then,the closed-loop network is pinning controllable if

λ >kfkh− (σ∗)2

2kh(2k2g − k2lg) (28)

Proof: Let ei := xi − xr be the error at each node. Asin the proof of Theorem 4.1, we have that the overall errordynamics can be written as (14) where F (e, t) is given by(15) and G(e, t) = σ∗[(g(x1, t)−g(xr, t))

>, . . . , (g(xN , t)−g(xr, t))

>]>. Considering the candidate Lyapunov function(17) and calculating LV (e) yields LV (e) = Ve(e, t)F (e, t)+Vge where Ve(e)F (e, t) is given by (21) while

Vge =1

2traceG>(e, t)G(e, t)

=(σ∗)2

2

N∑i=1

(g(xi, t)− g(xr, t))>(g(xi, t)− g(xr, t))

≤ (σ∗klg)2

2

N∑i=1

(xi − xr)>(xi − xr)

≤ (σ∗klg)2

2e>e = (σ∗klg)

2V (e). (29)

Therefore, we have that LV (e) ≤ c2V (e) with c2 =2(kf − khλ) + (σ∗kl)

2. Next, we have that e>G(e, t) =

σ∗∑Ni=1 (xi − xr)>(g(xi, t)− g(xr, t)), and from Assump-

tion 3.3 yields e>G(e, t) ≥ kge>e so that ‖e>G(e, t)‖2≥

4(σ∗kg)2V (e). If condition (28) holds; then, condition (ii)-(iii)

of Theorem 2.1 are fulfilled with c3 > 2c2 so that the closed-loop network reaches complete stochastic synchronization ontoxr and the proof is complete.

Remark 4.2: Condition (28) reveals that noise can bebeneficial to enhance the ability of a network to be controlled.Indeed, if kg/klg > (

√2/2); then, pinning controllability

can be improved for lower values of λ by adding noise.This will be illustrated by some representative examples inSection V. We wish to highlight that noise can be potentiallyexploited for control purposes by injecting random signals onthe nodes using noise generators. Interestingly, this observationis consistent with the recent findings in [40], [20], and alsowith previous studies in chaos control [41].

Remark 4.3: Note that the reference signal xr is assumedto be the solution of an isolated node perturbed by noise. Thisis crucial to guarantee exact convergence of all nodes towardsthe noisy target evolution. If the reference signal does nothave any stochastic term, i.e. it is the solution of xr = f(xr),then convergence is still achieved but a residual error will bepresent at steady state. Indeed, as for standard proportionalcontrol, the mean and variance of the steady state error mightbe attenuated as the control gain increases (see the examplein the Applications Section V-A). Finding explicit bounds ofthe expectation and variance of the steady state error when thereference trajectory is noise-free is an open problem which isleft for future work.

7

D. Control design

The location of pinned nodes and the control gain α can beproperly designed according to the simple algebraic conditionsobtained in the previous results. Indeed, the control design isbased on the relation λ > ck, where

ck =

2c0 + (σ∗kl)

2/kh, k = 1 Controlc0 + (σ∗klλ

∗N )2/2kh, k = 2 Links

c0 − (σ∗)2/2kh(2k2g − k2lg) k = 3 Nodes(30)

with c0 = kf/kh. Note that given a certain network topologyL and noise intensity σ∗, the control design consist on findingthe matrix P and a positive gain α such that (30) is satisfiedaccording to on of the three different scenarios. Moreover,in the third case (k = 3), noise might be beneficial and canbe also used as an extra degree of freedom for designing thepinning control strategy (see Remark 4.2). We will illustratein next Section the use of condition (30) for designing thepinning controller

V. APPLICATION EXAMPLES

A. Gene regulatory networks

As a first application example we study the pinning con-trollability of the genetic Toggle Switch, originally introduced(and engineered in Escherichia coli) in [42]. Such a biochem-ical circuit consists of two genes mutually inhibiting eachothers’ promoters, see Figure 2(a).

1) Node dynamics: Following [42], the dynamics of asingle Toggle Switch can be modeled with the following setof differential equations

dU

dt=

α1

1 + V β− U, dV

dt=

α2

1 + Uγ− V (31)

where U ∈ R+ and V ∈ R+ are the state variablesrepresenting the concentration of repressor 1 and repressor 2respectively. The parameters: (i) α1 and α2 are two positiveconstants denoting the effective rates of synthesis of repressor1 and 2 respectively; (ii) β, γ are two nonnegative constantsrepresenting the cooperativity of repression of promoter 2and 1, respectively. We consider the same parameter valuesas reported in [42], α1 = 50, α2 = 20, β = 2.5, andγ = 1. With this choice of parameters, the set onto which(31) evolves (i.e. Ω0) is the positive orthant of R2. Thebasins of attraction of the toggle switch stable equilibria(U, V ) = (0.0301, 19.4161) (or (U, V )=(OFF,ON)), (U, V ) =(44.2548, 0.4420) (or (U, V )=(ON,OFF)) are also depicted inFigure 2(b).

2) Stochastic network dynamics: Different experimentalresults have confirmed that gene expressions are stochasticprocesses [43], where noise plays a very important role in theswitching of such bistable systems [12], [44], and also it canbe exploited for control [45]. We consider a network of N > 1diffusively coupled toggle switches; that is, network (6) withn = 2,

f(xi, t) = f(xi) =

[α1/(1 + V βi )− Uiα2/(1 + Uβi )− Vi

], i ∈ N , (32)

(a)

5 10 15 20 25 30 35 40

0

5

10

15

20

V = ON

U = OFF

V = OFF

U = ON

(b)

Fig. 2. (a) Schematic of the toggle switch design [42], (b) Basins of attractionof the toggle switch model for α1 = 50, α2 = 20, β = 2.5, and γ = 1.

where xi := [Ui, Vi]> with Ui, Vi ∈ R+ representing the

concentrations of the i-th node. The coupling functions areassumed to be linear i.e., h(xi, t) = h(xi) = xi. Wewish to emphasize that such couplings can be realized by anadditional reaction xj xi with reaction rate aij = aji [46],where aij are the entries of the adjacency matrix A (withassociated Laplacian matrix L) representing the network ofinterconnection between the toggle switches.

In this example we examine the scenario where nodes areaffected by noise. Specifically, we consider the case wherethe effective rates of synthesis are subject to some randomenvironmental effect, that is, α1 = α1 + σ∗w(t) and α2 =α2 + σ∗w(t), where α1 = 50 and α2 = 20 are the nominalvalues, w(t) = db/dt is a white noise and σ∗ represent theintensity of noise. Then the nonlinear stochastic term in (6) canbe written as in (12) with g(xi) = [1/(1+V βi ), 1/(1+Uγi )]>.

3) Control design: Our control target is to drive the con-centrations of all N units to the desired state (Ui, Vi) =(44.2548, 0.4420) (or Ui=ON, Vi=OFF) for all i ∈ N . To thisaim, we use the pinning controller (8) where the reference signalxr := [Ur, Vr]

> is given by the solution of an isolated Toggle switch(31) with initial conditions (30, 1) (green region of Figure 2(b)).

To find the control gain α and the number and locations of pinnednodes we use condition (30) for k = 3 (or Theorem 4.3).

QUAD condition: In order to use condition (30) we need first toshow that the vector-field f(xi) in (32) satisfies the QUAD condition(7). To that aim let us first define ϕ(θ) = f(y + θ(x−y)), for θ ∈[0, 1] and x,y ∈ Ω0. Noticing that ϕ(0) = f(y) and ϕ(1) = f(x),from the Fundamental Theorem of calculus one has that f(x) −f(y) = ϕ(1) − ϕ(0) =

∫ 1

0(dϕ(θ)/dθ)dθ. Then, f(x) − f(y) =∫ 1

0Df(y + θ(x− y))dθ(x−y) where Df is the Jacobian matrix

given by

Df =

[−1 Df12

Df21 −1

](33)

Df12 =−125(y2 + θ(x2 − y2))3/2((y2 + θ(x2 − y2))5/2 + 1

)2 (34)

Df21 =−20(

(y1 + θ(x1 − y1))5/2 + 1)2 (35)

Next, we have that

‖f(x)− f(y)‖ ≤ supθ∈[0,1]

‖Df(y + θ(x− y))‖ ‖x− y‖

from the triangle inequality we have ‖Df(y + θ(x− y))‖ ≤2 + ‖Df12‖+‖Df21‖ yielding ‖f(x)− f(y)‖ ≤ L ‖x− y‖ withL = 59.5, and from the fact that a Lipschitz function is also QUAD[27], kf = L. The analysis above proves that the model of interest isdescribed by a QUAD vector field but provides a conservative esti-mate of the Lipschitz constant kf . Indeed using Matlab optimization

8

0 5 10 15 20 25 30

0

20

40

60

0.1 0.2 0.3

3.5

4

4.5

5

5.5

Reference

(a)

0 5 10 15 20 25 30

0

10

20

30

0.01 0.02 0.03 0.04 0.05 0.06

18

20

22

Reference

(b)

Fig. 3. Open-loop dynamics (ui = 0) for an all-to-all network of five toggleswitches with noise acting on the nodes.

toolbox, we obtained numerically a better estimate for kf which isequal to 21. Similarly, we find that the coupling function satisfiesAssumption 3.2 with kh = 1 and that the nonlinear function g(xi)satisfies Assumption 3.3 with klg = 0.6 and kg = −0.868.

We consider the network topology L to be an all-to-all networkof five nodes (N = 5, N = 1, · · · , 5) (note that in this caseλ2 = 5) and σ = 14. Then, we have from (30) with k = 3 thatλ > 21 − 0.5734σ∗. Then in the absence of noise, i.e. σ∗ = 0we obtain the classic deterministic condition λ > 21 [9]. Then thepinning controllability condition is satisfied by setting α = 73, andpinning three out of five nodes. If two nodes are pinned instead, thepinning controllability condition is not fulfilled.

Moreover, considering the noisy term with σ∗ = 3, we find thatpinning two nodes yields λ = 16.1082 > 21−0.5734σ∗ = 15.8392so that the condition is fulfilled and pinning controllability is guar-anteed from Theorem 4.3. Note that increasing the noise intensityσ∗ implies the network can be controlled for lower values of λ sothat less control interventions, and even lower values of the controlstrength α can guarantee the network reaches stochastic pinningsynchronization. However, for large values of σ∗ the variance of thereference trajectory increases.

4) Numerical simulations: In order to illustrate our results wefirs simulate the network dynamics in the absence of control, i.e.ui = 0 for all i ∈ N . The time response of the open-loop stochasticnetwork is shown in Figure 3 where the initial conditions of eachnode were randomly chosen on the yellow region of Figure 2(b).Note that all nodes reach the equilibrium U = OFF, V = ON andthe control target is not fulfilled. Then we apply the control feedback(8) pining two nodes with α = 71 and converge to the desired targettrajectory is achieved as shown in Figure 3.

For the sake of completeness we further test the pinning controlstrategy when the reference signal does not have a stochastic term,i.e. xr = f(xr). In this case, perfect agreement cannot be achieved(see remark 4.3), yet the error e(t) remain bounded at steady state.Indeed, we calculate the mean and variance of ‖e(t)‖ at steady statefor different values of the control gain α and over one thousand trialsstarting from random initial conditions. The result is shown in Figure5. Note that the mean of the error decreases as α increases.

B. Synchronization of chaotic oscillatorsAs an additional example, we now turn our attention to study

pinning controllability of diffusively coupled chaotic systems. In

0 5 10 15 20 25 30

0

20

40

60

0.05 0.1 0.15 0.2 0.25 0.3 0.35

5

10

15

20

25

Reference

(a)

0 5 10 15 20 25 30

0

10

20

30

0.01 0.02 0.03 0.04 0.05 0.06

5

10

15

20

Reference

(b)

Fig. 4. Time response of an all-to-all network of five toggle switches withnoise acting on the nodes and pinning control being exerted at one singlenode.

60 80 100 120 140 160 180 200 220

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 5. Mean and variance of the error dynamics ‖e(t)‖ at steady state fordifferent values of the control gain α.

particular we consider networks of chaotic Lorenz systems wherethe intrinsic nonlinear dynamics are given by

f(xi, t) = f(xi) =

µ(qi − pi)pi(ρ− ri)− qipiqi − ωri

, i ∈ N (36)

with xi = [pi, qi, ri]> and the parameters are set as µ = 10, ρ = 28

and ω = 2 (for such parameters the system exhibits a chaotic behavior[47]). The network topology is assumed to be given by the one shownin Figure 1(a) where N = 6 and σ = 30. In addition the couplingfunctions are assumed to be nonlinear and given by h(xi) = xi +tan−1(xi). Next we illustrate the effectiveness of our approach intwo different scenarios; namely, noise on the communication andcontrol links

1) Noise on the communication links: We first consider thecase where noise is propagating thorough the communication links.Particularly we assume σ∗ = 1 and noise propagating over a subsetof links of the original network topology (illustrated in Figure 1(b))for which λ∗N = 4.2143. Next, we use Theorem 4.2 to design thepinning controller. To that aim, we first recall the fact that f(xi)satisfies the QUAD assumption (7) with kf = 14 [47]. Next, we caneasily verify that the nonlinear function h(xi) has bounded Jacobianand hence it can be shown to satisfy Assumption 3.2 with kl = 2and kh = 1, respectively. From (30) with k = 2 we can concludethat the network of chaotic Lorenz systems is pinning controllable ifλ > 49.5206.

9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-20

0

20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-50

0

50

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

50

100

Fig. 6. Time response of a multiplex network (Figure 1) of six chaotic Lorenzoscillators pinning the nodes 1, 4 and 5. The black dashed-line represents thedesired trajectory xr .

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

10

20

30

1 2 3 4 5 6 7

10-3

16

18

20

22

24

26

Fig. 7. Time evolution of the error dynamics for a network of chaotic Lorenzwith noise acting on the control links.

We find that the network is pinning controllable if nodes 1, 4, and 5are pinned with α = 350. Hence, the closed-loop network convergesto the desired target trajectory (see Figure V-B1) as theoreticallypredicted by Theorem 4.2.

2) Noise on the control links: We consider again the networkof chaotic Lorenz systems presented above but, this time, we supposethe noise is acting on the control action via equation (10) withσ∗ = 2.2. Now, from (30) for k = 1 we find the network is pinningcontrollable if λ > 47.41. This condition is fulfilled by using thesame set of pinned nodes as in the previous example and by settingα = 350. Indeed, as shown in Figure 7, with this choice of the pinnednodes and control strength, the closed-loop network converges to thedesired target trajectory provided all conditions of Theorem 4.1 arefulfilled.

VI. CONCLUSIONS

In this paper pinning controllability has been investigated innetworks affected by different types of noise diffusion processes.The results leverage the use of appropriate stochastic Lyapunovfunctions and the notion of almost sure exponential stability toobtain sufficient condition guaranteeing convergence of the closed-loop network onto the desired target state. We find that, surprisingly,noise can enhance pinning controllability when it propagates acrossthe network in an homogeneous fashion. The effectiveness of ourtheoretical results was illustrated via two representative applicationsarising in the context of gene regulatory networks and ensembleschaotic oscillators. In the former, we confirmed that noise can bebeneficial for enhancing pinning controllability, while in the latter weshowed that more control interventions and higher gains are neededto cope with the noisy term. Ongoing work is aimed at extending ourapproach to directed network topologies. Also, the effect of varyingthe topology of the noisy layer should be studied more in detail, sincepreliminary numerical results showed that this could enhance pinningcontrollability [28].

REFERENCES

[1] P. DeLellis, F. Garofalo, and F. L. Iudice, “The partial pinning controlstrategy for large complex networks,” Automatica, vol. 89, pp. 111–116,2018.

[2] P. DeLellis, M. Di Bernardo, and F. Garofalo, “Adaptive pinning controlof networks of circuits and systems in lur’e form,” IEEE Transactions onCircuits and Systems I: Regular Papers, vol. 60, no. 11, pp. 3033–3042,2013.

[3] J. Zhou, Q. Wu, and L. Xiang, “Impulsive pinning complex dynamicalnetworks and applications to firing neuronal synchronization,” Nonlineardynamics, vol. 69, no. 3, pp. 1393–1403, 2012.

[4] H. Su and X. Wang, Pinning control of complex networked systems: Syn-chronization, consensus and flocking of networked systems via pinning.Springer, 2013.

[5] X. Wang and H. Su, “Pinning control of complex networked systems:A decade after and beyond,” Annual Reviews in Control, vol. 38, no. 1,pp. 103–111, 2014.

[6] X. F. Wang and G. Chen, “Pinning control of scale-free dynamicalnetworks,” Physica A: Statistical Mechanics and its Applications, vol.310, no. 3-4, pp. 521–531, 2002.

[7] X. Li, X. Wang, and G. Chen, “Pinning a complex dynamical network toits equilibrium,” IEEE Transactions on Circuits and Systems I: RegularPapers, vol. 51, no. 10, pp. 2074–2087, 2004.

[8] F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, “Controllabilityof complex networks via pinning,” Physical Review E, vol. 75, no. 4, p.046103, 2007.

[9] M. Porfiri and M. di Bernardo, “Criteria for global pinning-controllability of complex networks,” Automatica, vol. 44, no. 12, pp.3100–3106, 2008.

[10] C. W. Wu, “On the relationship between pinning control effectivenessand graph topology in complex networks of dynamical systems,” Chaos:An Interdisciplinary Journal of Nonlinear Science, vol. 18, no. 3, p.037103, 2008.

[11] Y. Orouskhani, M. Jalili, and X. Yu, “Optimizing dynamical networkstructure for pinning control,” Scientific reports, vol. 6, p. 24252, 2016.

[12] T. Tian and K. Burrage, “Stochastic models for regulatory networksof the genetic toggle switch,” Proceedings of the National Academy ofSciences, vol. 103, no. 22, pp. 8372–8377, 2006.

[13] D. V. Dimarogonas and K. H. Johansson, “Stability analysis for multi-agent systems using the incidence matrix: Quantized communication andformation control,” Automatica, vol. 46, no. 4, pp. 695–700, 2010.

[14] T. Li, F. Wu, and J.-F. Zhang, “Multi-agent consensus with relative-state-dependent measurement noises,” IEEE Transactions on AutomaticControl, vol. 59, no. 9, pp. 2463–2468, 2014.

[15] W. Li, H. Su, and K. Wang, “Global stability analysis for stochasticcoupled systems on networks,” Automatica, vol. 47, no. 1, pp. 215–220,2011.

[16] G. Russo, F. Wirth, and R. Shorten, “On synchronization in continuous-time networks of nonlinear nodes with state dependent and degeneratenoise diffusion,” IEEE Transactions on Automatic Control, pp. 1–1,2018.

[17] G. Russo and R. Shorten, “On common noise-induced synchronizationin complex networks with state-dependent noise diffusion processes,”Physica D: Nonlinear Phenomena, vol. 369, pp. 47–54, 2018.

[18] T. Li and J.-F. Zhang, “Mean square average-consensus under measure-ment noises and fixed topologies: Necessary and sufficient conditions,”Automatica, vol. 45, no. 8, pp. 1929–1936, 2009.

[19] Z. Li and J. Chen, “Robust consensus of multi-agent systems withstochastic uncertain channels,” American Control Conference, pp. 3722–3727, 2016.

[20] H. Shirado and N. Christakis, “Locally noisy autonomous agents im-prove global coordination in network experiments,” Nature, vol. 545,pp. 370 – 374, 2017.

[21] X. Yang, J. Cao, and J. Lu, “Stochastic synchronization of complexnetworks with nonidentical nodes via hybrid adaptive and impulsivecontrol,” IEEE Transactions on Circuits and Systems I: Regular Papers,vol. 59, no. 2, pp. 371–384, 2012.

[22] Y. Tang, H. Gao, and J. Kurths, “Distributed robust synchronization ofdynamical networks with stochastic coupling,” IEEE Transactions onCircuits and Systems I: Regular Papers, vol. 61, no. 5, pp. 1508–1519,2014.

[23] W. Griggs, G. Russo, and R. Shorten, “Leader and leaderless multi-layerconsensus with state obfuscation: An application to distributed speedadvisory systems,” IEEE Transactions on Intelligent TransportationSystems, vol. 19, no. 3, pp. 711–721, 2018.

10

[24] V. Mwaffo, P. DeLellis, and M. Porfiri, “Criteria for stochastic pinningcontrol of networks of chaotic maps,” Chaos, vol. 24, no. 1, p. 013101,2014.

[25] Y. Tang, H. Gao, J. Lu, and J. Kurths, “Pinning distributed synchroniza-tion of stochastic dynamical networks: A mixed optimization approach,”IEEE transactions on neural networks and learning systems, vol. 25,no. 10, pp. 1804–1815, 2014.

[26] X.-J. Li and G.-H. Yang, “Graph theory-based pinning synchronizationof stochastic complex dynamical networks,” IEEE Transactions onNeural Networks and Learning Systems, vol. 28, no. 2, pp. 427–437,2017.

[27] P. DeLellis, M. di Bernardo, and G. Russo, “On QUAD, lipschitz, andcontracting vector fields for consensus and synchronization of networks,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 58,no. 3, pp. 576–583, March 2011.

[28] D. A. Burbano-L., G. Russo, and M. di Bernardo, “Pinning controllabil-ity of complex stochastic networks,” in IFAC World Congress, Toulouse,France, vol. 50, no. 1, 2017, pp. 8327–8332.

[29] R. Talmon and R. R. Coifman, “Empirical intrinsic geometry for non-linear modeling and time series filtering,” Proceedings of the NationalAcademy of Sciences, vol. 110, no. 31, pp. 12 535–12 540, 2013.

[30] J. Kim, C. Lee, H. Shim, Y. Eun, and J. H. Seo, “Detection of sensorattack and resilient state estimation for uniformly observable nonlinearsystems,” in 2016 IEEE 55th Conference on Decision and Control(CDC), Dec 2016, pp. 1297–1302.

[31] C. W. Wu and L. O. Chua, “Synchronization in an array of linearlycoupled dynamical systems,” IEEE Transactions on Circuits and SystemsI: Fundamental Theory and Applications, vol. 42, no. 8, pp. 430–447,1995.

[32] X. Mao, Stochastic Differential Equations and Applications, SecondEdition, 2nd ed. Woodhead Publishing, 2008.

[33] H. Tuckwell, Stochastic Processes in the Neurosciences. Society forIndustrial and Applied Mathematics, 1989. [Online]. Available:http://epubs.siam.org/doi/abs/10.1137/1.9781611970159

[34] X. Liu, T. Chen, and W. Lu, “Consensus problem in directed networksof multi-agents via nonlinear protocols,” Physics Letters A, vol. 373,no. 35, pp. 3122–3127, 2009.

[35] X. Liu and T. Chen, “Exponential synchronization of nonlinear coupleddynamical networks with a delayed coupling,” Physica A: StatisticalMechanics and its Applications, vol. 381, pp. 82–92, 2007.

[36] S. Boccaletti, G. Bianconi, R. Criado, C. I. Del Genio, J. Gomez-Gardenes, M. Romance, I. Sendina-Nadal, Z. Wang, and M. Zanin, “Thestructure and dynamics of multilayer networks,” Physics Reports, vol.544, no. 1, pp. 1–122, 2014.

[37] J. Wang and N. Elia, “Distributed averaging under constraints oninformation exchange: emergence of levy flights,” IEEE Transactionson Automatic Control, vol. 57, no. 10, pp. 2435–2449, 2012.

[38] S. Chretien, S. Darses, C. Guyeux, and P. Clarkson, “On the pinningcontrollability of complex networks using perturbation theory of ex-treme singular values. application to synchronisation in power grids,”Numerical Algebra, Control & Optimization, vol. 7, no. 3, pp. 289–299,2017.

[39] P. Yang, R. A. Freeman, G. J. Gordon, K. M. Lynch, S. S. Srinivasa,and R. Sukthankar, “Decentralized estimation and control of graphconnectivity for mobile sensor networks,” Automatica, vol. 46, no. 2,pp. 390–396, 2010.

[40] R. d. J. Leon-Montiel, M. A. Quiroz-Juarez, R. Quintero-Torres, J. L.Domınguez-Juarez, H. M. Moya-Cessa, J. P. Torres, and J. L. Aragon,“Noise-assisted energy transport in electrical oscillator networks withoff-diagonal dynamical disorder,” Scientific reports, vol. 5, 2015.

[41] S. Rajasekar and M. Lakshmanan, “Controlling of chaos in bonhoeffer-van der pol oscillators,” International Journal of Bifurcation and Chaos,vol. 2, no. 1, pp. 201 – 204, 1992.

[42] T. S. Gardner, C. R. Cantor, and J. J. Collins, “Construction of a genetictoggle switch in escherichia coli,” Nature, vol. 403, no. 6767, pp. 339–342, 2000.

[43] W. J. Blake, M. Kærn, C. R. Cantor, and J. J. Collins, “Noise ineukaryotic gene expression,” Nature, vol. 422, no. 6932, pp. 633–637,2003.

[44] G. Karlebach and R. Shamir, “Modelling and analysis of gene regulatorynetworks,” Nature Reviews Molecular Cell Biology, vol. 9, no. 10, pp.770–780, 2008.

[45] D. K. Wells, W. L. Kath, and A. E. Motter, “Control of stochastic andinduced switching in biophysical networks,” Physical Review X, vol. 5,no. 3, p. 031036, 2015.

[46] C. Kirst, M. Timme, and D. Battaglia, “Dynamic information routing incomplex networks,” Nature communications, vol. 7, p. 11061, 2016.

[47] Q. Song, F. Liu, J. Cao, and W. Yu, “Pinning-controllability analysisof complex networks: an M-matrix approach,” IEEE Transactions onCircuits and Systems I: Regular Papers, vol. 59, no. 11, pp. 2692–2701,2012.

Daniel Alberto Burbano Lombana received theB.S. (2010) in Electronic Engineering and M.S.(2012) in Industrial automation both from the Na-tional University of Colombia. In 2015, he obtaineda Ph.D. in control engineering from the University ofNaples Federico II, Italy, with a dissertation centeredon distributed PID control for networks. He was laterappointed at the same university as a postdoc forworking on stochastic networks till 2016. From 2017he has been working at Northwestern University,USA as a postdoctoral researcher. His research is

focused on the analysis and control of complex nonlinear systems andnetworks with particular attention to problems in network reconstruction,network control, consensus, and synchronization.

Giovanni Russo received his Ph.D. degree fromthe University of Naples Federico II in 2010. Hisfocus was on the stability of nonlinear dynamicalsystems with applications to networked control andsystems biology. From 2012 to 2015, he was the leadsystem engineer and integrator of the Honolulu RailTransit Project. In 2015, he joined IBM ResearchIreland and from September 2018 he has been aLecturer in Cyber-Physical Systems at the UniversityCollege Dublin. He is currently a member of theBoard of Editors of IEEE Transactions on Circuits

ans Systems and of the IEEE Transactions on Control of Network Systems.

Mario di Bernardo (SMIEEE ?06, FIEEE 2012)is currently Professor of Automatic Control at theUniversity of Naples Federico II, Italy. He is alsoProfessor (part-time) of Nonlinear Systems and Con-trol at the University of Bristol, UK. In January2012 he was elevated to the grade of Fellow of theIEEE for his contributions to the analysis, controland applications of nonlinear systems and complexnetworks. On 28th February 2007 he was bestowedthe title of ”Cavaliere” of the Order of Merit ofthe Italian Republic for scientific merits from the

President of Italy. In 2009, He was elected President of the Italian Society forChaos and Complexity for the term 2010-2013. He was re-elected in 2010 forthe term 2014-2017. In 2006 and again in 2009 he was elected to the Board ofGovernors of the IEEE Circuits and Systems Society. From 2011 to 2014 hewas Vice President for Financial Activities of the IEEE Circuits and SystemsSociety. In 2014 he was appointed as a member of the Board of Governorsof the IEEE Control Systems Society. He was Distinguished Lecturer of theIEEE Circuits and Systems Society for the two-year term 2016-2017. Hisresearch interests include the analysis, synchronization and control of complexnetwork systems; piecewise-smooth dynamical systems; nonlinear dynamicsand nonlinear control with applications to engineering and computationalbiology. He authored or co-authored more than 220 international scientificpublications including more than 110 papers in scientific journals, a researchmonograph and two edited books. According to the international databaseSCOPUS (March 2018), his h-index is 38 and his publications received over6000 citations by other authors (h-index = 48, citations = 11422 according toGoogle Scholar, March 2018). He is Senior Editor of the IEEE Transactionson Control of Network Systems; Associate Editor of Nonlinear Analysis:Hybrid Systems and the IEEE Control Systems Letters, the ConferenceEditorial Board of the IEEE Control System Society and the European ControlAssociation (EUCA). From 1st January 2014 till 31st December 2015 he wasDeputy Editor-in-Chief of the IEEE Transactions on Circuits and Systems:Regular Papers. He is regularly invited as Plenary Speaker in Italy and abroad.