Consensus of A Class of Nonlinear Systems with Varying Topology: A Hilbert Metric Approach

Dongjun Wu This work was supported in part by the European Research Council (ERC) through the European Union’s Horizon 2020 Research and Innovation Program under Grant 834142 (ScalableControl). The author is with Department of Automatic Control, Lund University, 22100 Lund, Sweden (email: dongjun.wu@control.lth.se).

Abstract

In this technical note, we introduce a novel approach to studying consensus of continuous-time nonlinear systems with varying topology based on Hilbert metric. We demonstrate that this metric offers significant flexibility in analyzing consensus properties, while effectively handling nonlinearities and time dependencies. Notably, our approach relaxes key technical assumptions from some standard results while yielding stronger conclusions with shorter proofs. This framework provides new insights into nonlinear consensus under varying topology.

Index Terms:

Hilbert metric, nonlinear consensus, varying topology

I Introduction

Consensus under varying topology is a fundamental research question in multi-agent and network dynamical systems [1]. This topic has been extensively studied over the last three decades within the control community. In the seminal work [2], Vicsek et al. first observed that consensus (or coordination, as referred to in their study ) is closely related to the topology of multi-agent systems. Following this observation, lots of efforts have been made to establish the theoretical foundations for consensus with varying topology. Notable contributions include the works of Jadbabaie et. al. [3], Moreau [1], and Ren and Beard [4], among others. These early 21st century studies sparked a surge in research on consensus with varying topology in subsequent years. The theoretical developments in this area have found applications across various domains, including power grids [5], social networks [6], and formation control [7]. More recent advancements can be found in works such as [8], [9], and [10].

Most of the aforementioned works have primarily focused on linear systems. However, nonlinearities are pervasive in multi-agent systems, such as coupled oscillators [11], mobile robots [12] and social networks [6]. For these systems, the tools for analyzing linear consensus are not directly applicable, necessitating the use of nonlinear techniques. Various tools have been introduced to address the problem, including dissipative analysis [13, 14], monotone system theory [15, 16], and non-smooth techniques [17, 18]. For further reviews on this topic, the readers are referred to [19, 20, 21] and the references therein.

As far as we know, the result obtained in [17] stands among the most general ones regarding nonliner consensus with varying topology. Prior to this, Lin et al. [22] studied consensus of a class of nonlinear systems with fixed topology using non-smooth analysis techniques, with invariance principle being key of the proof. They later extended their result to nonlinear systems with switching topology [17], requiring the switching signal to be sufficiently regular (the definition will be provided later). When switching is present, invariance principle can no longer be used. To handle this, a technical result from [23] was employed in [17]. It is worth mentioning that, even though the system in consideration might possibly be smooth, non-smooth techniques had to be used [22, 17]. The result in [17] can also be seen as a nonlinear extension of the work [1], as both works relied on the quasi-strongly connectedness of multi-agent system as a key assumption.

The aim of this note is to relax some of the technical assumptions made in [17] and to provide new understandings for the problem. To achieve this goal, we adopt a novel approach, different from standard practice of utilizing Lyapunov or non-smooth analysis, by leveraging the so-called Hilbert metric to analyze the system dynamics, which proves to be quite successful.

Hilbert metric has already been used to study consensus. For instance, in [24] Hilbert metric was used to further relax and extend the results obtained in [25]; in [26], it was used for consensus in non-commutative spaces. However, to the best of our knowledge, it has not been introduced to study nonlinear consensus with varying topology. This might be related to the fact that the Hilbert metric was considered more suitable for linear systems; e.g., one of the most widely used results – Birkhoff’s Theorem [27] – is applicable for linear systems (or more generally, homogeneous systems). In this note, however, we show that Hilbert metric can also serve as a systematic tool for studying nonlinear consensus. In particular, it performs well in handling nonlinearities and time dependencies.

Before outlining the contributions of this note, it is necessary to briefly recall the main result obtained in [17]. Consider the following multi-agent system with switching topology controlled by the switching signal $\sigma:\mathbb{R}_{\geq 0}\to\{1,\cdots,N\}$ :

\begin{cases}\dot{x}_{1}=f_{\sigma(t)}^{1}(x_{1},\cdots,x_{n})\\ \vdots\\ \dot{x}_{n}=f_{\sigma(t)}^{n}(x_{1},\cdots,x_{n})\end{cases}

(1)

where $x_{i}\in\mathbb{R}^{m}$ . For each $p\in\{1,\cdots,N\}$ , we associate a digraph $\mathcal{G}_{p}$ with the system defined by vector fields $\{f_{p}^{i}\}_{i=1}^{n}$ . $\mathcal{G}_{p}$ has $n$ vertices denoted as $\{1,\cdots,n\}$ , and a link $(i,j)$ is in $\mathcal{G}_{p}$ if $f_{p}^{i}$ depends explicitly on $x_{j}$ . The digraph is called quasi strongly connected (QSC) if there exists a node $k$ such that for each node $j\neq k$ , there is a directed path joining node $k$ to node $j$ . Such a node is called a center. In other words, a digraph is QSC if there is a directed spanning tree, and the root of the tree is a center. For a switching graph corresponding to the system (1), we say the graph is uniformly quasi strongly connected (UQSC) if there exists a constant $T>0$ such that the union of the graph on the interval $[t,t+T]$ for any $t$ is QSC. Let $\mathcal{C}_{i}$ be the polytope formed by $x_{i}$ and its neighboring agents, and $T_{x_{i}}\mathcal{C}_{i}$ the tangent cone of $\mathcal{C}_{i}$ at $x_{i}$ in $\mathbb{R}^{m}$ , see [17] for more detailed explanations of these terminologies.

One of the main results in [17] can be restated as:

Theorem 1 (Lin et al. [17]).

Consider the system (1). Assume that for each $p\in\{1,\cdots,{N}\}$ : 1) $f_{p}^{i}$ is locally Lipschitz and $f_{p}^{i}$ is in the relative interior of the cone $T_{x_{i}}\mathcal{C}_{i}$ , i.e., $f_{p}^{i}\in{\rm ri}(T_{x_{i}}\mathcal{C}_{i})$ ; 2) the switching signal is piece-wise constant with minimum dwell time $\tau_{D}$ ; then the system achieves global consensus if and only if the system (1) is UQSC.

A switching signal satisfying the assumptions (having a minimum dwell time $\tau_{D}$ ) in Theorem 1 is said to be regular.

Note that under the assumption of Theorem 1, the vector field $f_{p}^{i}$ can be written as $f_{i}^{p}=\sum a_{ij}^{p}(x)(x_{j}-x_{i})$ for some non-negative scalar functions $a_{ij}^{p}$ . Equivalently, this means that the system can be written as $\dot{x}=A_{p}(x)x$ with $(A_{p}(x))_{ij}=a_{ij}^{p}(x)$ , in which the matrix $A_{p}(x)$ is Metzler, that is, $A_{ij}\geq 0$ for all $i\neq j$ , and each row sums to zero. System of the form $\dot{x}=A(x)x$ with $A(x)$ being Metzler has been recently considered by Kawano and Cao [28], where such systems were referred to as “virtually positive”. Due to the special structure $\dot{x}=A_{\sigma(t)}(x)x$ , consensus of such systems shares a lot in common with linear time varying multi-agent systems. A remarkable result concerning the latter was obtained by Moreau in [25]:

Theorem 2 (Moreau [25]).

Consider the LTV system $\dot{x}=A(t)x$ . Assume that $A(\cdot)$ is uniformly bounded and piecewise continuous. Assume that, for every time $t$ , $A(t)=(a_{ij}(t))\in\mathbb{R}^{n\times n}$ is Metzler with zero row sums. If there exists an index $k\in\{1,\cdots,n\}$ , a threshold value $\delta>0$ and an interval length $T>0$ such that for all $t\geq 0$ ,

\int_{t}^{T+t}a_{ik}(s)ds\geq\delta,\quad\forall i\in\{1,\cdots,n\}\backslash\{k\},

(2)

then the system achieves exponential consensus.

The proof of Theorem 2 provided in [25] was based on Lyapunov analysis, see also [3, 1] for discrete time versions. In particular, separable Lyapunov functions were used. This is a technique commonly used in monotone systems, see for example [29, 30, 31, 32].

Theorem 1 and Theorem 2 and share the same spirit. Nevertheless, their proof techniques were quite different: the former relies on on non-smooth analysis while the latter on Lyapunov analysis. It is then a question whether the two results can be understood in a unifying framework. This has remained an open question. In this note, we provide an affirmative answer to it.

We address the problem by a non-Lyapunov-based method, i.e., through the analysis of the evolution of the system dynamics under the Hilbert metric. The contributions of the note are twofold:

1.

Utilizing Hilbert metric to analyze nonlinear consensus is new. It does not rely on Lyapunov [25] or non-smooth analysis [17]. Our proof based on Hilbert metric for consensus is largely simplified compared to existing works. This has led to new understandings and insights of nonlinear consensus.
2.

Instead of regular switching topology considered in [17], we are able to analyze more general time varying topologies, requiring only measurability of the switching. This includes cases such as piece-wise continuous switching studied in [25]. Consequently, our results extend classical findings, including Theorem 1 and 2, by relaxing key technical assumptions. Furthermore, we obtain stronger results; for instance, while [17] establishes only asymptotic consensus, our analysis demonstrates exponential consensus under mild additional assumptions.

Organization of the paper: In Section II, we provide the problem setting and prove some technical results. In particular, we give some explicit formulas and new definitions which will be crucial for the next section. Section III contains the main results of this note and Section IV demonstrate a simulation result.

Notations: $|\cdot|$ stands for Euclidean $2$ -norm. For a dynamical system, use $\phi(t,t_{0},x_{0})$ to represent the solution at $t$ from initial state $(t_{0},x_{0})$ . The interior of a set $S$ is denoted ${\rm Int}S$ . Being $X$ , $Y$ some topological spaces, denote $C(X,Y)$ the space of continuous maps from $X$ to $Y$ . Given a set $S\subseteq X$ , the indicator function $1_{S}:X\to\{0,1\}$ is defined to be $1_{S}(x)=1$ if $x\in S$ and $0$ otherwise. For a Metzler matrix $A$ with row sum zero, $\mathcal{G}^{A}$ represents the graph associated with $A$ . Given $x,y$ , $d(x,y)$ stands for the Hilbert metric between $x$ and $y$ . $\mathbb{N}_{+}$ is the set of positive natural numbers. $\mathds{1}_{n}$ a column vector of dimension $n$ with all ones. A continuous function $\beta:[0,a)\times[0,\infty)\to[0,\infty)$ is called a class $\mathcal{KL}$ function if 1) $r\mapsto\beta(r,s)$ is strictly increasing and $\beta(0,s)\equiv 0$ for all $s\geq 0$ ; 2) $s\mapsto\beta(r,s)$ is decreases to $0$ as $s\to\infty$ .

II Preliminary Results

II-A Problem setting

We consider continuous time multi-agent nonlinear systems of the form

\dot{x}_{i}=\sum_{j=1}^{n}a_{ij}(t,x)(x_{j}-x_{i}),

(3)

for $i=1,\cdots,n$ , where the state of each agent is in $\mathbb{R}$ and $t\mapsto a_{ij}(t,\cdot)$ is measurable; in addition, $a_{ij}(t,x)\geq 0$ for all $t\geq 0$ , $x\in\mathbb{R}^{n}$ and $i,j\in\{1,\cdots,n\}$ , $i\neq j$ .

Remark 1.

We can also consider more general systems having the following form

\dot{x}_{i}=a(t)x_{i}+b(t)+\sum_{j=1}^{n}a_{ij}(t,x)(x_{j}-x_{i})

(4)

where $a(\cdot)$ and $b(\cdot)$ are bounded on $\mathbb{R}_{+}$ . Indeed, define $y(t)=e^{\int_{0}^{t}a(s){\rm d}s}x-\left(\int_{0}^{t}e^{\int_{0}^{\tau}a(s){\rm d}s}b(\tau){\rm d}\tau\right)\mathds{1}$ we have $\dot{y}=\tilde{A}(t,y)y$ where $\tilde{A}$ is Metzler and satisfies $\tilde{A}\mathds{1}=0$ .

We mention a few practical examples that can be written as (3).

•

Kuramoto oscillators: This is a widely studied nonlinear consensus model with applications in various engineering domains. Consider a network of Kuramoto oscillators with time-varying and state-dependent coupling:

$\dot{\theta}_{i}=\omega_{i}(t)+\sum_{j}k_{ij}(t,\theta)\sin(\theta_{i}-\theta_{j})$

where $\omega_{i}(t)$ are the natural frequencies, and $k_{ij}(t,\theta)$ are non-negative coupling functions. Phase synchronization occurs when $|\theta_{i}(t)-\theta_{j}(t)|\to 0$ as $t\to\infty$ . This can only be achieved if all natural frequencies are identical. In this case, the model can be transformed into the standard form (3) by defining $a_{ij}(t,\theta):=k_{ij}(t,\theta)\frac{\sin(\theta_{i}-\theta_{j})}{\theta_{i}-\theta_{j}}$ . Generalizations of the Kuramoto oscillator that can be expressed in the form of (3) can be found in [21].

•

Cucker-Smale model: This model is commonly used to describe flocking behavior [33]:

	$\displaystyle\dot{x}_{i}$	$\displaystyle=v_{i}$
	$\displaystyle\dot{v}_{i}$	$\displaystyle=\frac{\lambda}{N}\sum_{j=1}^{N}\psi_{ij}(x,v)(v_{j}-v_{i})$

where $N$ is the number of agents, $\psi_{ij}(x,v)\geq 0$ represents the interaction force, and $\lambda>0$ is a constant. The system achieves flocking if $v_{i}(t)-v_{j}(t)\to 0$ for all $i,j$ . The second equation can clearly be written in the form of (3) and thus analyzed using the proposed methods.

•

Hegselmann–Krause model: This model is widely used for opinion dynamics [6]. A time-varying version can be written as

$\dot{x}_{i}=\sum_{j=1}^{N}\phi_{ij}(t,x_{i},x_{j})(x_{i}-x_{j})$

where

$\phi_{ij}(t,x_{i},x_{j})=\begin{cases}1,&|x_{i}-x_{j}|\leq\epsilon(t)\\ 0,&\text{otherwise}\end{cases}$

and $\epsilon:\mathbb{R}_{+}\to[0,1]$ is a measurable function.

•

Animal group models: The following is widely used to simulate animal group behavior (see, e.g., [34]):

\dot{x}_{i}=\sum_{j\in A_{i}(t)}\frac{\phi_{a}(x_{i},x_{j})}{|x_{i}-x_{j}|}(x_{j}-x_{i})+\sum_{j\in R_{i}(t)}\frac{\phi_{r}(x_{i},x_{j})}{|x_{i}-x_{j}|}(x_{i}-x_{j})

where $A_{i}$ and $\phi_{a}$ represent attraction, and $R_{i}$ and $\phi_{r}$ represent repulsion. Both $\phi_{a}$ and $\phi_{r}$ are non-negative. This model also has the form (3).

The system (3) can also be written in matrix form as

\dot{x}=A(t,x)x

(5)

in which $A(t,x)_{ij}=a_{ij}(t,x)$ for $i\neq j$ and $A(t,x)_{ii}=-\sum_{j\neq i}a_{ij}(t,x)$ . Note that $A(t,x)$ is Metzler and has the property $A(t,x)\mathds{1}=0$ .

Associated with the system (5) is a varying digraph $\mathcal{G}^{A(t,x)}$ understood in the following sense: for $i\neq j$ , if $a_{ij}(t,x)>0$ , then $(i,j)$ is a directed link from node $i$ to $j$ and the weight on this link is $a_{ij}(t,x)$ ; if $a_{ij}(t,x)=0$ , then there is no link from $i$ to $j$ . Therefore, for any Metzler matrix $A$ , there is an associated graph. Note that, we do not consider self loop, i.e., the graph $\mathcal{G}^{A}$ is characterized only by the off-diagonal elements of $A$ . The following definition collects a few important notions that we will use frequently in the paper.

Definition 1.

Let $A,B,C(z)\in\mathbb{R}^{n\times n},\;z\in Z$ be some Metzler matrices with row sum zero, $Z$ some index set,

1.

We denote $\mathcal{G}^{A}$ the graph associated with the matrix $A$ .
2.

We say that $\mathcal{G}^{A}\geq\mathcal{G}^{B}$ if $A_{ij}\geq B_{ij}$ for all $i\neq j$ .
3.

We say that the graph valued function $z\mapsto\mathcal{G}^{C(z)}$ is continuous if $z\mapsto C(z)$ is continuous.
4.

The graph $\mathcal{G}^{A}$ is said to be quasi-strongly connected (QSC) if $\exists k\in\mathbb{N}_{+}$ , such that $A_{ik}>0$ for all $i\neq k$ ; it is called $\delta$ -connected for some $\delta>0$ , if $\exists k\in\mathbb{N}_{+}$ , such that $A_{ik}\geq\delta$ for all $i\neq k$ .

In this note, consensus is understood in the following sense:

Definition 2.

Given a forward invariant set $D\subseteq\mathbb{R}^{n}$ , the system (1) is said to achieve

•

asymptotic consensus on $D$ if there exists a class $\mathcal{KL}$ function $\beta$ , such that $|x_{i}(t)-x_{j}(t)|\leq\beta(|x_{i}(0)-x_{j}(0)|,t)$ for all $i,j\in\{1,\cdots,n\}$ , $t\geq 0$ and $x(0)\in D$ ;
•

exponential consensus on $D$ if there exist some constants $k,\lambda>0$ such that $|x_{i}(t)-x_{j}(t)|\leq ke^{-\lambda t}|x_{i}(0)-x_{j}(0)|$ for all $i,j\in\{1,\cdots,n\}$ , $t\geq 0$ and $x(0)\in D$ .

Remark 2.

For any $a<b$ , $[a,b]^{n}$ is invariant under the system flow (3). Thus we may assume $D$ in Definition 2 is compact.

The following regularity condition will be imposed on $A(t,x)$ throughout the paper unless otherwise stated.

Assumption A1.

Assume, for any compact set $D\subseteq\mathbb{R}^{n}$ ,

1.

the mapping $t\mapsto A(t,x)$ is measurable for every fixed $x\in D$ ;
2.

there is a locally integrable function $k_{D}(t)$ such that

$|A(t,x)x-A(t,y)y|\leq k_{D}(t)|x-y|$

for all $x,y\in D,\;t\geq 0$ .
3.

there exists a constant $C_{D}>0$ , such that $|A(t,x)|\leq C_{D}$ for every $x\in D,\;t\geq 0$ .

Under Assumption A1, the solution to system (5) exists and is unique through every point $(t_{0},x_{0})\in\mathbb{R}_{+}\times D$ [35, Theorem 5.1, Theorem 5.3].

II-B The Hilbert metric

In this paper, we use Hilbert metric to analyze consensus problems. Hilbert metric is a metric defined on cones. More precisely, in our setting, a cone is some closed subset $K\subseteq\mathbb{R}^{n}$ satisfying the following four properties 1) The interior of $K$ is non-empty; 2) For $v,w\in K$ , $v+w$ is also in $K$ . 3) For all $\lambda\geq 0$ , and $v\in K$ , $\lambda v$ is also in $K$ . 4) $K\cap-K=\{0\}$ , where $-K:=\{-x:x\in K\}$ . Given a cone, we can define a partial ordering as $x\leq y$ if $y-x\in K$ and $x<y$ if $y-x\in{\rm Int}\;K$ . For $x,y\in{\rm Int}K$ , define two numbers $M(x/y)=\inf\{\lambda:x\leq\lambda y\}$ or $\infty$ if the set is empty, and $m(x/y)=\sup\{\mu:\mu y\leq x\}$ . Then the Hilbert metric between $x$ , $y$ is defined as $d(x,y)=\ln\frac{M(x,y)}{m(x,y)}$ . Define the diameter of a set $S\subseteq K$ as ${\rm diam}(S)=\sup_{x,y\in S}d(x,y).$ For any cone $S\subseteq{\rm Int}\mathbb{R}_{+}^{n}\cup\{0\}$ , we have ${\rm diam}S<+\infty$ . In fact, $d(x,y)=\ln\frac{\max_{i}(x_{i}/y_{i})}{\min_{i}(x_{i}/y_{i})}$ , where $x=(x_{1},\cdots,x_{n})$ , $y=(y_{1},\cdots,y_{n})$ , which is bounded on $S$ . A mapping $A:K\to K$ on a cone is called non-negative. If in addition, $A$ maps the interior of $K$ into its interior, we call $A$ a positive mapping. For example, when $K=\mathbb{R}_{+}^{n}$ , then a non-negative matrix $A$ represents a non-negative mapping while a positive matrix represents a positive mapping.

In the literature, positive mapping is often studied in the positive orthant $\mathbb{R}_{+}^{n}$ . The following simple example shows that the positive orthant may not be the right cone for studying consensus. Consider the system $\dot{x}_{1}=0$ , $\dot{x}_{2}=x_{1}-x_{2}$ . It is then obvious that the system achieves exponential consensus. The vector plot of the system is shown in Fig. 1. We can see that the system does not contract the positive orthant into its interior since $x_{1}=0$ is invariant. However, the system does contract the smaller cone painted in gray. This turns out to be a key observation we need.

Refer to caption — Figure 1: The red arrows represent the vector fields of a system. The $x_{2}$ -axis is invariant and hence cannot be mapped into the interior of the positive orthant. However, the system contracts the smaller cone painted in gray.

To proceed, we need to justify that contraction in Hilbert metric is equivalent to consensus that we defined earlier in Definition 2.

Lemma 1.

Suppose that $\mathcal{K}$ is a proper cone in $\mathbb{R}_{+}^{n}$ satisfying $\mathcal{K}\subseteq\text{Int}\ \mathbb{R}_{+}^{n}\cup\{0\}$ . Then a system achieves asymptotic (resp. exponential) consensus on $\mathcal{K}$ if and only if there exists class $\mathcal{KL}$ function (resp. positive constants $K,\lambda$ ) such that

	$\displaystyle d(x{(t)},\mathds{1})$	$\displaystyle\leq\beta(d(x{(0)},\mathds{1}),t)\text{ (asymptotic)}$
	$\displaystyle d(x{(t)},\mathds{1})$	$\displaystyle\leq Ke^{-\lambda t}d(x{(0)},\mathds{1})\text{ (exponential)}$

where $d(x,y)$ stands for the Hilbert metric between $x$ and $y$ .

Proof.

We prove the asymptotic case – the exponential case is similar. Suppose that the system achieves asymptotic consensus on $\mathcal{K}$ , i.e., there exists a class $\mathcal{KL}$ function $\beta$ such that $|x_{i}(t)-x_{j}(t)|\leq\beta(|x_{i}(0)-x_{j}(0)|,t)$ for all $t$ and $x(0)\in\mathcal{K}$ and $x(t)\in\mathcal{K}$ for all $t\geq 0$ . Let

A_{n}(x)=\left|x-\frac{|x|}{\sqrt{n}}\mathds{1}\right|^{2},\quad B_{n}(x)=\sum_{i,j=1}^{n}(x_{i}-x_{j})^{2};

we estimate:

	$\displaystyle d(x(t),\mathds{1})^{2}$	$\displaystyle=d\left(x(t),\frac{\|x(t)\|}{\sqrt{n}}\mathds{1}\right)^{2}\leq\frac{1}{c_{1}}A_{n}(x(t))\text{ (Lemma \ref{lem:2norm-Hnorm}})$
		$\displaystyle\leq\frac{1}{c_{1}n}B_{n}(x(t))\text{ (Lemma \ref{lem:An-Bn})}$
		$\displaystyle=\frac{1}{c_{1}n}\sum_{i,j=1}^{n}(x_{i}(t)-x_{j}(t))^{2}$
		$\displaystyle\leq\frac{1}{c_{1}n}\sum_{i,j=1}^{n}\beta(t,(x_{i}(0)-x_{j}(0))^{2})$
		$\displaystyle\leq\frac{n}{c_{1}}\beta(t,B_{n}(x(0)))$
		$\displaystyle\leq\frac{n}{c_{1}}\tilde{\beta}(t,d(x(0),\mathds{1}))\text{ (Lemma \ref{lem:An-Bn},\ref{lem:2norm-Hnorm})}$

in which $\tilde{\beta}$ is some class $\mathcal{KL}$ function. The converse proceeds in a similar fashion. ∎

Remark 3.

Although in Lemma 1, the initial condition is restricted to a cone $\mathcal{K}$ , in practice this is sufficient for consensus on any compact set $D\subseteq\mathbb{R}^{n}$ . Indeed, let $y=x+\alpha\mathds{1}$ . Then $\dot{y}=A(t,y-\alpha\mathds{1})y$ and consensus of $y$ is equivalent to consensus of $x$ . By choosing $\alpha>0$ sufficiently large, we may assume $D$ is in the interior of $\mathbb{R}_{+}^{n}$ .

We propose to study the following type of small cones. For $\gamma\in[0,\frac{1}{\sqrt{n}})$ , define a family of cones

\mathcal{K}(\gamma):=\left\{x\in\mathbb{R}^{n}:\frac{x_{i}}{|x|}\geq\frac{1}{\sqrt{n}}-\gamma,\;\forall i=1,\cdots n\right\}.

(6)

See Fig. 2 for an illustration of such cones. The following lemma summarizes some properties of the diameter (in Hilbert metric) of these cones.

Lemma 2.

Given $\gamma\in[0,\frac{1}{\sqrt{n}})$ , the diameter of $\mathcal{K}(\gamma)$ is

{\rm diam}\mathcal{K}(\gamma)=\log\left(1-n+\frac{n}{(1-\sqrt{n}\gamma)^{2}}\right)

Denote $\alpha(\gamma)={\rm diam}\mathcal{K}(\gamma)$ , then

•

$\alpha$ is smooth, strictly increasing on $[0,1/\sqrt{n})$ , and $\alpha(0)=0$ , $\alpha(t)\to\infty$ as $t\to\frac{1}{\sqrt{n}}$ . The derivative of $\alpha$ is lower bounded away from zero.
•

For any $\epsilon_{0}\in[0,\frac{1}{\sqrt{n}})$ and $C\in(0,1)$ , there exist some positive constants $k_{1},k_{2}$ and $k\in(0,1)$ , such that $k_{1}\gamma\leq\alpha(\gamma)\leq k_{2}\gamma$ , and

$\alpha(C\gamma)\leq k\alpha(\gamma)$ (7)

for all $\gamma\in[0,\epsilon_{0}]$ .

Proof.

We show (7). Notice that

\displaystyle\alpha(r)-\alpha(Cr)

\displaystyle=\alpha^{\prime}(\xi)(1-C)r\geq cr\geq\frac{c}{k_{2}C}\alpha(Cr)

from which it follows that

\alpha(Cr)\leq\frac{1}{1+c/k_{2}C}\alpha(r).

The rest is straightforward. ∎

II-C Accumulated graph

We have noted that in both [25] and [17], consensus is related to the accumulation of the graphs over time, either the union of the switching graph in [17], or the integration of the system matrix in [25]. This motivates us to define the accumulated graph for a time-varying graph $\mathcal{G}(t)$ , $\forall t\geq 0$ .

Definition 3 (Accumulated graph).

Let $\mathcal{G}(\cdot)=(a_{ij}(\cdot))$ be a measurable time-varying graph, i.e., $t\mapsto a_{ij}(t)$ is a measurable function for all $i\neq j$ . The accumulating graph of $\mathcal{G}(\cdot)$ over the interval $[t_{1},t_{2}]$ is the graph $\mathcal{G}|_{t_{1}}^{t_{2}}$ defined by the Lebesgue integral

\mathcal{G}|_{t_{1}}^{t_{2}}=\int_{t_{1}}^{t_{2}}\mathcal{G}(t){\rm d}t.

in the sense that $(\mathcal{G}|_{t_{1}}^{t_{2}})_{ij}=\int_{t_{1}}^{t_{2}}a_{ij}(t){\rm d}t$ .

Example 1.

For a system (1) with switching topology, the graph associated with it can be written as $\mathcal{G}(t)=\sum_{i=1}^{N}1_{A_{i}}(t)\mathcal{G}_{i}$ where $1_{A_{i}}$ is the indicator function of some measurable sets $A_{i}$ and $\mathcal{G}_{i}$ the graph corresponding to $p=i$ . Now the union graph on $[t,t+T]$ used in [17] was nothing but the accumulated graph $\mathcal{G}|_{t}^{t+T}$ . This is quite similar to the construction of the Lebesgue integration – define first for simple functions and then extend to larger class of functions.

With these technical preparations, we are now ready to present the main results of this paper.

III Main results

We start with a lemma which explains how connectedness of the graph is related to the contraction property under the Hilbert metric. The proof can be found in the Appendix.

Lemma 3.

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative matrix and there exist a constant $\delta\in(0,1)$ , and an integer $k$ such that

•

$A\mathds{1}=\mathds{1}$ ;
•

$a_{ik}>\delta$ for all $i=1,\cdots,n$ ;

then for any $0<\epsilon<\frac{1}{\sqrt{n}}$ , there exists a constant $C\in(0,1)$ such that

A\mathcal{K}(\epsilon)\subseteq\mathcal{K}(C\epsilon)

(8)

where $\mathcal{K}(\epsilon)$ is defined as in (6). Moreover, $C$ can be taken as $\frac{1-\delta}{1-\sqrt{n}\epsilon\delta}$ . As a result, there exist a positive constant $c\in(0,1)$ such that

{\rm diam}(A^{m}\mathcal{K}(\epsilon))\leq c^{m}{\rm diam}(\mathcal{K}(\epsilon)),\quad\forall m\geq 1.

(9)

where $k$ can be taken as $c=\frac{1}{1+\eta C}$ and $\eta>0$ depends on $\epsilon$ , and $n$ .

Our first main result is the following theorem.

Theorem 3.

Let $D$ be a compact set in $\mathbb{R}^{n}$ . Consider the multi-agent system (5) satisfying Assumption A1. Let $\{t_{k}\}_{1}^{\infty}$ be an increasing sequence with $t_{k}\xrightarrow{k\to\infty}\infty$ and $\sup_{k}|t_{k+1}-t_{k}|<\infty$ . If there exists a graph $\mathcal{G}^{B(x)}$ such that $B(x)$ is quasi-strongly connected and

\int_{t_{k}}^{t_{k+1}}\mathcal{G}^{A(t,\phi(t,t_{k},x))}{\rm d}t\geq\mathcal{G}^{B(x)}

(10)

for all $x\in D$ , $k\geq 1$ , then the system achieves asymptotic consensus on $D$ . If in addition, the graph $\mathcal{G}^{B(x)}$ is continuous, the system achieves exponential consensus on $D$ .

Proof.

We prove the second part first. As remarked earlier, by defining the coordinate transform $y=x+\alpha\mathds{1}$ for $\alpha>0$ large, we may assume that $D$ lies in a sufficiently small cone in $\mathbb{R}_{+}^{n}$ . First, assume $t\mapsto A(t,x)$ is piece-wise continuous. This assumption will be removed later. Consider the Euler approximation scheme on the interval $[t_{k},t_{k+1}]$ :

x_{i+1}=x_{i}+hA(t_{k}+ih,x_{i})x_{i},\quad i=0,\cdots,N-1

with $h=\frac{t_{k+1}-t_{k}}{N}$ , $N\in\mathbb{Z}_{+}$ large and $x_{0}=x$ . Choose $\lambda$ large enough such that $\bar{A}(t,x):=A(t,x)+\lambda I\geq 0$ for all $t\geq 0$ and $x\in D$ . The following calculation is in order

$\displaystyle x_{0}$	$\displaystyle=x$
$\displaystyle x_{1}$	$\displaystyle=(1-h\lambda)x+h\bar{A}(t_{k},x)x$
$\displaystyle x_{2}$	$\displaystyle=[(1-h\lambda)^{2}I+h(1-h\lambda)(\bar{A}(t_{k},x)+\bar{A}(t_{k}+h,x_{1}))+*]x$
	$\displaystyle\vdots$
$\displaystyle x_{N}$	$\displaystyle=[(1-h\lambda)^{N}I+(1-h\lambda)^{N-1}\sum_{i=0}^{N-1}h\bar{A}(t_{k}+ih,x_{i})+*]x$	(11)

where $*$ stands for non-negative terms. Now let $P_{k}^{N}(x)$ be the matrix in the bracket on the right hand side of (11) and define $P_{k}(x)=\lim_{{N}\to\infty}P_{k}^{N}(x)$ . Note that

\phi(t_{k+1},t_{k},x)=P_{k}(x)x.

(12)

We claim that $P_{k}(x)$ has the following properties: 1) $P_{k}(x)$ is non-negative and $P_{k}(x)\mathds{1}=\mathds{1}$ ; 2) There exists a non-negative matrix $S$ defining a QSC graph ¹¹1Here we are slightly abusing the concept of a QSC graph. It was previously defined as the graph associated with a continuous-time system 5. Here, it should be understood in the sense that there exists some $k\in\mathbb{N}_{+}$ , such that $S_{ik}>0$ for all $i\neq k$ . , independent of $k,x$ such that $P_{k}(x)\geq S$ . Item 1) is obvious, we verify 2). From (11) we see

P_{k}(x)\geq e^{-\lambda(t_{k+1}-t_{k})}\left(I+\int_{t_{k}}^{t_{k+1}}\bar{A}(t,\phi(t,t_{k},x)){\rm d}t\right)\geq S

for some non-negative $S$ such that $\mathcal{G}^{S}$ is QSC thanks to (10), the continuity of $\mathcal{G}^{B(x)}$ and compactness of $D$ . Without loss of generality, we may assume that $\mathcal{G}^{S}$ is $\delta$ -connected (since otherwise, we can consider the matrix $P_{k+n}P_{k+n-1}\cdots P_{k}$ , which will be lower-bounded by $S^{n}$ that is $\delta$ -connected). Invoking Lemma 3, we conclude that the system achieves exponential consensus on a cone of the form (6) which includes $D$ in its interior.

To remove the assumption on piece-wise continuity of $t\mapsto A(t,x)$ on $[t_{k},t_{k+1}]$ , it suffices to replace Riemann integration by Lebesgue integration as follows. View $\varphi:t\mapsto A(t,\cdot)$ as a mapping from $[t_{k},t_{k+1}]$ to $C(D;\mathbb{R}^{n\times n})$ equipped with norm $||g||_{*}=\sup_{x\in D}||g(x)||$ . Then $\varphi\in L^{\infty}([t_{k},t_{k+1}];C(D;\mathbb{R}^{n\times n}))$ , which can be approximated by simple functions. Let $\eta>0$ be an arbitrarily small constant, and $\mathscr{A}(t,x)=\sum_{i=0}^{N-1}1_{[s_{i},s_{i+1})}(t)A_{i}(x)$ the simple function such that $\|\varphi-\mathscr{A}\|_{\infty}<\eta$ . Assume that the partition is uniform (otherwise we can always refine the partition to make it close to uniform and then use approximation arguments), i.e., $s_{i+1}-s_{i}=\frac{t_{k+1}-t_{k}}{N}$ for all $i$ . Let $\bar{A}_{i}(x)=A_{i}(x)+\lambda I\geq 0$ , $\bar{\mathscr{A}}(t,x)=\mathscr{A}(t,x)+\lambda I$ . As before, the Euler approximation scheme gives (11). Now

	$\displaystyle\sum_{i=0}^{N-1}h\bar{A}(t_{k}$	$\displaystyle+ih,x_{i})=\sum_{i=0}^{N-1}h\bar{A}_{i}(x_{i})$
		$\displaystyle=\int_{t_{k}}^{t_{k+1}}\bar{\mathscr{A}}(t,\phi(t,t_{k},x)){\rm d}t$
		$\displaystyle\geq\int_{t_{k}}^{t_{k+1}}\bar{A}(t,\phi(t,t_{k},x)){\rm d}t-\eta(t_{k+1}-t_{k})\mathds{1}_{n\times n}.$

Since $|t_{k+1}-t_{k}|$ is uniformly bounded, we can choose $\eta$ sufficiently small such that $P_{k}^{N}(x)$ is lower-bounded by a QSC graph which is independent of $k$ . Thus the system achieves exponential consensus on $D$ .

It remains to prove asymptotic consensus when $\mathcal{G}^{B(x)}$ is not necessarily continuous. As before, we can assume that $t\mapsto A(t,x)$ is piecewise continuous. The Euler approximation still converges due to local Lipschitz continuity of the system vector fields. Now, instead of having a uniform lower bound on $P_{k}(x)$ (see (12)), we only know that it is bounded by some QSC graph. But still, we know that for small $\epsilon_{0}>0$ , $P_{k}(x)\mathcal{K}(\epsilon_{0})\subseteq\mathcal{K}(\epsilon_{1})$ . Since $D$ is compact, we may assume that $D\subseteq\mathcal{K}(\epsilon_{0})$ . Define $D_{1}=\phi(t_{1},t_{0},D)$ , $D_{k+1}=\phi(t_{k+1},t_{k},D_{k})$ for $k\geq 1$ and $\eta_{k}={\rm diam}D_{k}$ which is a non-negative decreasing sequence. Set a decreasing sequence $\{\epsilon_{k}\}_{1}^{\infty}$ , such that $D_{k}\subseteq\mathcal{K}(\epsilon_{k})$ . Clearly, $\eta_{k}$ is strictly decreasing and is bounded from below by $0$ . Therefore $\eta_{k}$ converges to a limit $c\geq 0$ .

Suppose $c>0$ , and choose $\eta_{m}$ sufficiently close to $c$ . There must exist $y\in\mathcal{K}(\epsilon_{m})\backslash\mathcal{K}(c)$ , such that $\phi(t_{m+1},t_{m},y)\in\mathcal{K}(\epsilon_{m})\backslash\mathcal{K}(c)$ . Choose $z\in\mathcal{K}(c)$ close to $y$ , then due to the continuity of $y\mapsto\frac{\phi(t,t_{0},y)}{|\phi(t,t_{0},y)|}$ , the error between $\frac{\phi(t_{m+1},t_{m},y)}{|\phi(t_{m+1},t_{m},y)|}$ and $\frac{\phi(t_{m+1},t_{m},z)}{|\phi(t_{m+1},t_{m},z)|}$ is of order $O(|\epsilon_{m}-c|)$ which can be made sufficiently small by choosing $m$ large enough. But

\min_{w\in\mathcal{K}(c)\cap D}\frac{\phi(t_{m+1},t_{m},w))_{i}}{|\phi(t_{m+1},t_{m},w)|}=\frac{1}{\sqrt{n}}-c^{\prime}>\frac{1}{\sqrt{n}}-c

(since $\eta_{m+1}<\eta_{m}$ ) implying that $\frac{\phi(t_{m+1},t_{m},z)_{i}}{|\phi(t_{m+1},t_{m},z)|}$ is away from $\frac{1}{\sqrt{n}}-c$ for any $m$ . This is a contradiction since $\frac{\phi(t_{m+1},t_{m},y)}{|\phi(t_{m+1},t_{m},y)|}\in\mathcal{K}(\epsilon_{m})\backslash\mathcal{K}(c)$ and that $\frac{\phi(t_{m+1},t_{m},y)}{|\phi(t_{m+1},t_{m},y)|}$ and $\frac{\phi(t_{m+1},t_{m},z)}{|\phi(t_{m+1},t_{m},z)|}$ are sufficiently close. Thus we conclude that $c=0$ and asymptotic consensus is achieved. The proof strategy is shown in Fig. 3.

The proof is now complete. ∎

Theorem 3 needs the computation of the integral $\int_{t_{k}}^{t_{k+1}}A(t,\phi(t,t_{k},x)){\rm d}t$ , which is impossible in most cases – except that $A(t,x)$ does not dependent on $x$ . The following corollary is more convenient for practical use.

Corollary 1.

Consider the system (3) under Assumption A1. Let $D$ be a compact invariant set. Suppose that there exists a (continuous) QSC graph $\mathcal{G}^{B(x)}$ , continuous on $D$ , and an increasing sequence $\{t_{k}\}_{1}^{\infty}$ with $t_{k}\xrightarrow{k\to\infty}\infty$ and $\sup|t_{k+1}-t_{k}|<\infty$ , such that

\int_{t_{k}}^{t_{k+1}}\mathcal{G}^{A(t,x)}{\rm d}t\geq\mathcal{G}^{B(x)},\quad\forall x\in D,\;k\geq 1

then the system achieves (asymptotic) exponential consensus on $D$ .

Proof.

Assume $B(x)$ is continuous. We utilize Euler approximation as before. Note that for fixed $r\neq s\in\{1,\cdots,n\}$ , there exists $x_{rs}\in D$ such that

\sum_{i=1}^{N-1}ha_{rs}(t_{k}+hi,\phi(t_{k}+hi,t_{k},x))\geq\sum_{i=1}^{N-1}ha_{rs}(t_{k}+hi,x_{rs})

since for every $k$ and $i$ , $x\mapsto a_{rs}(t_{k}+hi,\phi(t_{k}+hi,t_{k},x))$ is continuous. But the right hand side is an approximation of $\int_{t_{k}}^{t_{k+1}}a_{rs}(t,x_{rs}){\rm d}t$ , which is lower bounded by $B_{rs}(x_{rs})$ . Define a matrix $\tilde{B}:=(B_{rs}(x_{rs}))$ . Due to the continuity of $x\mapsto B(x)$ , $B(x)$ and hence $\tilde{B}$ , are lower bounded by some matrix $S$ such that $\mathcal{G}^{S}$ is QSC. The conclusion follows invoking Theorem 3. ∎

Example 2.

1) Theorem 2 is now a corollary of Theorem 3. Our result is slightly stronger: the mapping $t\mapsto A(t)$ is only required to be bounded measurable while in Theorem 2, this mapping is assumed to be piecewise continuous. Note that the proof strategy is rather different for the two theorems.

2) For Theorem 1, the system (1) can be written as

\dot{x}=\left[\sum_{k=1}^{p}1_{k}(\sigma(t))A_{k}(x)\right]x.

Fix an interval $[t_{0},t_{0}+T]$ , then on $[t_{0}-\tau_{D},t_{0}+T+\tau_{D}]$ , we integrate

\int_{t_{0}-\tau_{D}}^{t_{0}+T+\tau_{D}}\sum_{k=1}^{p}1_{k}(\sigma(t))A_{k}(x){\rm d}t\geq\tau_{D}\sum_{k=1}^{p}A_{k}(x).

By assumption, $\sum_{k=1}^{p}A_{k}(x)$ is QSC and hence the system achieves asymptotic consensus.

If we assume further that $A_{k}(x)$ is continuous, then we get exponential consensus from Theorem 3. But as far as we know, it is not clear how to prove this using the techniques in [17]. In addition, Theorem 1 was proven only for switching multi-agent system with regular switching signal while Theorem 3 only requires the “switchings” to be measurable which is always satisfied in practice.

Example 3.

Consider a Kuramoto model with identical frequency

\dot{\theta}_{i}=\omega+\sum_{j\in\mathcal{N}_{t}(i)}a_{ij}(t)\sin(\theta_{j}-\theta_{i}),\quad i=1,\cdots n

(13)

where $\mathcal{N}_{t}(i)$ stands for the neighboring node of $i$ at time $t$ and $a_{ij}(\cdot)\in L^{\infty}(\mathbb{R}_{\geq 0},\mathbb{R}_{>0})$ . Let $a,b$ be real numbers such that $0\leq b-a<\pi$ and the graph $\mathcal{G}^{A(t)}$ satisfies the assumption of Theorem 3 where the lower bound of the accumulated graph is now state-independent.

We claim that the system (13) achieves exponential consensus on $[a,b]^{n}$ . In particular, if $\dot{\theta}_{i}=\omega+\sum_{j\in\mathcal{N}(i)}\sin(\theta_{j}-\theta_{i})$ and the graph associated with the system is QSC, then exponential consensus is achieved on $[a,b]^{n}$ . To see the claim, let $x_{i}=\theta_{i}-\omega$ , the above model can be written as

\dot{x}_{i}=\sum_{j\in\mathcal{N}_{t}(i)}a_{ij}(t)\sin(x_{j}-x_{i})

or in matrix form $\dot{x}=A(t,x)x$ where $(A(t,x))_{ij}=\frac{\sin(x_{j}-x_{i})}{x_{j}-x_{i}}a_{ij}(t)$ for $i\neq j$ and $j\in\mathcal{N}_{t}(i)$ . Then on $[a,b]^{n}$ , $(A(t,x))_{ij}\geq ca_{ij}(t)$ for some positive constant $c$ . Thus consensus is determined by the graph $\mathcal{G}^{A(t)}$ .

IV Simulation Result

We simulate Example 3 for $\omega=0$ . Consider a chain structure as in Fig. 4 of $N=10$ oscillators. The weights on the link $(x_{i}\to x_{i+1})$ is $a_{i,i+1}(t)$ . The weights are generated in the following manner. First, generate some random intervals $[t_{k},t_{k+1}]$ for $k=1,\cdots,M$ for some large $M$ . Then, divide each $[t_{k},t_{k+1}]$ into $N$ smaller pieces $[s_{i}^{k},s^{k}_{i+1}]$ randomly. After that, Euler scheme will run on each interval $[s_{i}^{k},s^{k}_{i+1}]$ . In each step of the Euler scheme, we choose three random numbers $p,q,r$ from $\{1,\cdots,N-1\}$ and generate three random positives numbers $a_{p,p+1}$ , $a_{q,q+1}$ , $a_{r,r+1}$ lower bounded by a threshold $\delta>0$ and the rest of $a_{i,i+1}$ are set to zero.

By construction, the weights $a_{i,i+1}$ are zero for most of the time and since the these weights are generated randomly, they are quite irregular. But still, we can see from Fig. 5 that the system achieves consensus.

Refer to caption

Figure 4: A chain of coupled oscillators.

Refer to caption

Figure 5: Simulation result. Left: the signal

a_{1,1+1}(t)

. Right: the state components.

V Conclusion

In this technical note, we have shown that the Hilbert metric can serve as alternative tool to study consensus properties. It is advantageous in dealing with nonlinearities and time dependencies, and requires very weak regularity assumptions. The results obtained in this note are somewhat preliminary and open the door for future research.

VI Appendix

Proof of Lemma 3 .

Consider the set

\mathcal{E}(\epsilon)=\left\{e:\frac{1}{\sqrt{n}}-\epsilon\leq e_{i}\leq\frac{1}{\sqrt{n}},\;\forall i=1,\cdots,n,\;|e|\leq 1\right\}.

Then $\mathcal{K}(\epsilon)=\{\alpha e:\forall\alpha\in\mathbb{R}_{>0},\;e\in\mathcal{E}(\epsilon)\}$ . Thus it is sufficient to prove

\frac{(Ae)_{i}}{|Ae|}\geq\frac{1}{\sqrt{n}}-C\epsilon,\quad\forall i\geq 1,\;e\in\mathcal{E}(\epsilon).

(14)

Write $e=\frac{1}{\sqrt{n}}-\tilde{e}$ , then $0\leq\tilde{e}_{i}\leq\epsilon$ and $(Ae)_{i}=\frac{1}{\sqrt{n}}-(A\tilde{e})_{i}$ . On the one hand, $(A\tilde{e})_{i}=\sum_{j\neq k}^{n}a_{ij}\tilde{e}_{j}+a_{ik}\tilde{e}_{k}\geq\delta\tilde{e}_{k}$ . On the other hand,

	$\displaystyle\sum_{j\neq k}^{n}a_{ij}\tilde{e}_{j}+a_{ik}\tilde{e}_{k}$	$\displaystyle\leq\epsilon\sum_{j\neq k}^{n}a_{ij}+a_{ik}\tilde{e}_{k}$
		$\displaystyle=\epsilon(1-a_{ik})+a_{ik}\tilde{e}_{k}$
		$\displaystyle\leq\delta(\tilde{e}_{k}-\epsilon)+\epsilon.$

Thus we obtain the inequality

\frac{1}{\sqrt{n}}-\epsilon(1-\delta)-\delta\tilde{e}_{k}\leq(Ae)_{i}\leq\frac{1}{\sqrt{n}}-\delta\tilde{e}_{k}.

As a result $|Ae|\leq 1-\sqrt{n}\delta\tilde{e}_{k}$ and

\frac{(Ae)_{i}}{|Ae|}\geq\frac{\frac{1}{\sqrt{n}}-\delta\tilde{e}_{k}-\epsilon(1-\delta)}{1-\sqrt{n}\delta\tilde{e}_{k}}=\frac{1}{\sqrt{n}}-\frac{1-\delta}{1-\sqrt{n}\tilde{e}_{k}\delta}\epsilon

Recall that $\tilde{e}_{k}\leq\epsilon<\frac{1}{\sqrt{n}}$ , we get $\frac{1-\delta}{1-\sqrt{n}\tilde{e}_{k}\delta}\leq\frac{1-\delta}{1-\sqrt{n}\epsilon\delta}:=C\in(0,1)$ and (14) follows. In other words, we have shown $A\mathcal{K}(\epsilon)\subseteq\mathcal{K}(C\epsilon)$ .

To show (9), following the same procedure, we can prove $A\mathcal{K}(C\epsilon)\subseteq\mathcal{K}(C^{\prime}C\epsilon)$ , where $C^{\prime}=\frac{1-\delta}{1-\sqrt{n}C\epsilon\delta}<C$ . Thus $A^{2}\mathcal{K}(\epsilon)\subseteq A\mathcal{K}(C\epsilon)\subseteq\mathcal{K}(C^{2}\epsilon)$ . By induction, we can show $A^{m}\mathcal{K}(\epsilon)\subseteq\mathcal{K}(C^{m}\epsilon)$ . Thus the inequality (9). ∎

Lemma 4.

For $x\geq 0$ , define two quantities as

A_{n}(x)=\left|x-\frac{|x|}{\sqrt{n}}\mathds{1}\right|^{2},\quad B_{n}(x)=\sum_{i,j=1}^{n}(x_{i}-x_{j})^{2}

then $nA_{n}\leq B_{n}\leq 2nA_{n}$ .

Proof.

It suffices to prove for $|x|=1$ . Define $X=\sum_{i=1}^{n}x_{i}$ , then

A_{n}=2-\frac{2}{\sqrt{n}}X,\quad B_{n}=2n-2X^{2}.

Since $|x|=1$ , we have $0\leq X\leq\sqrt{n}$ . Thus $B_{n}\geq 2n-2\sqrt{n}X=nA_{n}$ . On the other hand, $2nA_{n}-B_{n}=2(X-\sqrt{n})^{2}\geq 0$ . ∎

Lemma 5.

Consider the cone $\mathbb{R}_{+}^{n}$ . Suppose that $\mathcal{K}$ is a proper cone such that $\mathcal{K}\subseteq\text{Int}\ \mathbb{R}_{+}^{n}\cup\{0\}$ . Then there exist some constants $C_{2}>C_{1}>0$ such that $C_{1}d(x,w)\leq|x-w|\leq C_{2}d(x,w)$ for all $x\in\mathbb{R}_{+}^{n},$ $w\in\mathcal{K}$ with $|w|=|x|=1$ .

Proof.

Since $|w|=|x|=1$ , we can find a constant $C>0$ , such that

C\tanh\left(\frac{1}{2}d(x,w)\right)\leq|x-w|\leq\exp\left(d(x,w)\right)-1

invoking [36, Theorem 4.1], see [36, (4.1) and (4.3)]. The conclusion follows by noticing that $d(x,w)$ is bounded for all $x\in\mathbb{R}_{+}^{n}$ , $w\in\mathcal{K}$ with $|w|=|x|=1$ . ∎

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