SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 1

Weak and Semi-Contraction Theorywith Application to Network Systems

Saber Jafarpour , Member, IEEE, Pedro Cisneros-Velarde, Student Member, IEEE, andFrancesco Bullo , Fellow, IEEE

Abstract—We develop two generalizations of contraction the-ory: semi-contraction and weak-contraction theory. First, usingthe notion of semi-norm, we propose a geometric frameworkfor semi-contraction theory. We introduce matrix semi-measures,characterize their properties, and provide techniques to computethem. We also propose two optimization problems for the spectralabscissa of a given matrix using its semi-measures. For dynamicalsystems, we use the semi-measure of their Jacobian to studyconvergence of their trajectories to invariant subspaces. Second,we provide a comprehensive treatment of weakly-contractingsystems; we prove a dichotomy for asymptotic behavior oftheir trajectories and show that, for their equilibrium points,local asymptotic stability implies global asymptotic stability.Third, we introduce the class of doubly-contracting systemsand show that every trajectory of a doubly-contracting systemconverges to an equilibrium point. Finally, we apply our resultsto various important network systems including affine averagingand affine flow systems, continuous-time distributed primal-dualalgorithms, and networks of diffusively-coupled oscillators.

Index Terms—contraction theory, stability analysis, synchro-nization

I. INTRODUCTION

Problem description: Strict contractivity is a useful andclassical property of dynamical systems, which ensures globalexponential stability of a unique equilibrium. However, nu-merous example applications fail to satisfy this property andexhibit richer dynamic properties. In this paper, motivated byapplications in network systems, we study systems that satisfyrelaxed versions of the standard contractivity conditions. Wecharacterize the implications of these relaxed conditions onthe asymptotic behavior of the dynamical system. We aimto develop a generalized contractivity theory that can explainthe rich behavior of some classic example systems, includingaffine averaging and flow systems, distributed primal-dualdynamics, and networks of diffusively-coupled oscillators.

Literature review: Studying contractivity of dynamicalsystems using matrix measures has a long history that can betraced back to Lewis [22] and Demidovic [12]. In the controlcommunity, matrix measures were adopted by Desoer andVidyasagar [13], [42] and contraction theory was introducedby Lohmiller and Slotine [24]. We refer to [33] for a historicalreview and to the surveys [1], [14] for recent developments

This work was supported in part by the U.S. Defense Threat ReductionAgency under grant HDTRA1-19-1-0017.

Saber Jafarpour, Francesco Bullo, and Pedro Cisneros-Velarde arewith the Center of Control, Dynamical Systems and Computation, UCSanta Barbara, CA 93106-5070, USA. {saber,bullo}@ucsb.edu,pacisne@gmail.com.

and applications to consensus and synchronization in complexnetworks.

Several generalizations of contraction theory have been pro-posed in the literature. In [40], the notion of partial contractionis introduced to study convergence of system trajectories toa specific behavior or to a manifold. The idea is to imposecontractivity only on a part of the states of the system.Partial contraction with respect to the `2-norm has been furtherdeveloped in [45], [34] to study synchronization in complexnetworks. A similar notion of partial contraction is studiedby [14] in the context of convergence to invariant subspaces.Generalizations of contraction theory based on matrix mea-sures have also been studied in the literature. In [41], it isshown that if the sum of the two largest eigenvalues of thesymmetric part of the Jacobian of the system is negative,then every bounded trajectory of the system converges toan equilibrium point. In [21], a relaxed contraction propertyis defined based on the sign of the truncated trace of thesymmetric part of the Jacobian.

Extensions of contraction theory to non-Euclidean normsand metrics have been explored in the context of monotonedynamical systems. For compartmental systems, contractivitywith respect to the `1-norm has been used to study convergenceto the equilibrium point [25]. For monotone systems, [17]uses the contractivity of the so-called Hilbert metric to proposea nonlinear generalization of Perron–Frobenius theorem. Theconnection between monotonicity and contractivity of dynam-ical systems has been studied in detail in [9], [18]. Otherextensions of contraction theory include contraction theoryon Finsler manifolds [16], contraction theory on Riemannianmanifolds [39], transverse contraction for convergence to limitcycles [28], and contraction after transient [29].

Synchronization of diffusively-coupled oscillators has beenextensively studied using contraction theory, e.g., see [46],[45], [38], [11], [2]. Indeed the notion of partial contractionwas developed in [45] precisely to study this class of dy-namical systems. The early work [46] introduces what is nowknown as the QUAD condition for studying synchronization ofcoupled oscillators. In [26], local and global synchronizationof linearly-coupled oscillators over directed graphs have beenanalyzed using the QUAD condition. While explaining therelationship between QUAD condition and contractivity ofvector fields, [11] also studies diffusively-coupled identicalnonlinear oscillators on undirected graphs. [3] proposes an `2-norm condition for synchronization of diffusively-coupled os-cillators with time-invariant interconnections. For diffusively-coupled networks with time-varying interconnections, syn-

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chronization has been studied using non-Euclidean matrixmeasures in [6], [2]. In [36], the synchronization of complexnetworks is studied using a contraction-based hierarchicalapproach via mixed norms.

Primal-dual algorithms for optimization problems have beenwidely studied and adopted in several applications; we referto the survey [48] for a comprehensive review. In the lastdecade, there has been a renewed interest in convergenceanalysis [15] focusing on the convergence rate in centralizedoptimizations [35] and convergence guarantees in distributedoptimizations [44]. Recently, contraction theory has beenused to study convergence and robustness of continuous-timeprimal-dual algorithms [32], [8].

Contribution: In this paper, we develop two generaliza-tions of contraction theory that broaden its range of applicabil-ity and allow for richer and more complex dynamic behaviors.

The first generalization, called semi-contractivity, allows thedistance between trajectories to increase in certain directions,thus requiring the dynamical system to be contractive onlyon a certain subspace. Using the notion of semi-norm, wepresent a geometric framework for studying semi-contractingsystems. We introduce the semi-measure of a matrix, which isassociated with a semi-norm, and we study the linear algebraof semi-measures. We provide two optimization problems thatestablish a connection between semi-measures of a matrix andits spectral abscissa on certain subspaces. These optimizationproblems can be considered as extensions of [23, Proposition2.3] to semi-norms and semi-measures, and they play a crucialrole in studying the convergence rate of trajectories in ourframework. Next, we prove a generalization of the well-knownCoppel’s inequality [10], which provides upper and lowerbounds on the semi-norm of flows of linear time-varying sys-tems. These results generalize the classic treatment of matrixmeasures and Coppel’s inequality given for example in [42,Chapter 2]. Finally, for a given time-varying dynamical systemand semi-norm, we introduce the notion of semi-contractionof the system and two notions of invariance of the semi-norm, namely shifted- and infinitesimal invariance. We provevarious results. First, for a semi-contracting system with aninfinitesimally invariant semi-norm, the semi-distance betweenany two trajectories vanishes exponentially fast. Second, fora semi-contracting system with a shifted-invariant semi-norm,every trajectory converges to the shifted kernel of the semi-norm. Third and final, an infinitesimally invariant semi-normis also shifted-invariant for a time-invariant dynamical system.

The notion of semi-contraction is related to the notions of(i) partial contraction, as proposed in [40], elaborated in [45],[34], and surveyed in [14]; and (ii) horizontal contraction onFinsler manifolds, as proposed in [16] and elaborated in [47].Beside providing a comprehensive unifying framework, ourtreatment of semi-contraction differs from partial contractionsince it allows arbitrary semi-measures and from horizontalcontraction as it leads to sharper statements for a morerestricted class of systems. In Remark 14 we provide a moredetailed comparison.

The second generalization, called weak-contractivity, allowsthe distance between trajectories to be non-increasing. Ourfirst result is a dichotomy for qualitative behavior of weakly-

contracting systems; a trajectory of a weakly-contracting sys-tem is bounded if and only if the system has an equilib-rium point. One interesting feature of this dichotomy is theconnection between the topological properties of the dynam-ical system and the algebraic properties of its vector field.Moreover, we show that if a weakly-contracting system hasa locally asymptotically stable equilibrium point, then thisequilibrium point is globally asymptotically stable. This resultcan be considered as the generalization of [27, Theorem 2 andTheorem 3] and [25, Lemma 6] to weakly-contracting systems.Finally, for systems that are weakly-contracting with respectto a weighted `1-norm or a weighted `∞-norm, we showthat the dichotomy can be more elaborate; every trajectoryconverges to an equilibrium point if and only if the dynamicalsystem has an equilibrium point. We conclude by providingan example which shows that, for weakly-contracting systemswith respect to an arbitrary `p-norm, a bounded trajectory doesnot necessarily converge to an equilibrium point.

In contrast to time-invariant contractive systems whosetrajectories converge to their globally stable equilibriumpoints, trajectories of semi-contracting or weakly-contractingsystem do not necessarily exhibit a unique asymptotic be-havior. We illustrate this fact using examples of semi-and weakly-contracting systems. We introduce the class ofdoubly-contracting systems, i.e, systems which are both semi-contracting and weakly-contracting (possibly with respect totwo different semi-norms). We show that, for a doubly-contracting system whose equilibrium points form a subspaceof Rn, every trajectory converges exponentially to an equilib-rium point. Moreover, we provide a convergence rate that isindependent of the norm and the semi-norm and depends onlyon the limiting equilibrium point.

We provide several applications of our results to networksystems: affine averaging and flow systems, continuous-timeprimal-dual dynamics, and networks of coupled oscillators.First we show that affine averaging (affine flow) systems areweakly-contracting with respect to the `∞-norm (`1-norm).Using our dichotomy result for weakly-contracting systems,we show that trajectories of both affine averaging and affineflow systems converge to an equilibrium point if and only if thesystem has an equilibrium point. We also show that both affineaveraging and affine flow systems are doubly-contracting, thusrecovering the exact rate of convergence of trajectories to theequilibrium points.

Second, we study a distributed implementation of thecontinuous-time primal-dual algorithm for optimizing a func-tion over a connected network. We show that the resultingdynamical system is weakly-contracting and every trajectoryof the system converges to an equilibrium point. As recentlystated in [8], compared to the analysis in [35], [32], weshow that the distributed primal-dual algorithm is weakly-contracting with respect to the `2-norm and prove its globalconvergence to the solution of the optimization problem whenthe cost function is weakly convex.

Finally, for networks of diffusively-coupled identical os-cillators, we use our semi-contraction framework to proposenovel sufficient conditions for global synchronization. A keystep in obtaining these synchronization conditions is the in-

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 3

troduction of a new class of norms called (2, p)-tensor norms,with various useful properties. Compared to the contraction-based approaches using the `2-norm [46], [45], [11], oursynchronization conditions (i) are based on the weighted `p-matrix measure of the Jacobian of oscillator dynamics forevery p ∈ [1,∞], (ii) provide an explicit rate of convergenceto the synchronized trajectories, and (iii) can recover the exactthreshold of synchronization for diffusively-coupled linearoscillators. Compared to the results in [6], [2], our synchro-nization conditions (i) are applicable to arbitrary undirectednetwork topology, (ii) allow for a arbitrary class of weightedp-norms, for every p ∈ [1,∞], and (iii) demarcate the roles ofinternal dynamics and the network connectivity.

Paper Organization: Section II introduces the notation.Section III introduces the matrix semi-measures and use themto study the semi-contracting systems. Sections IV and V studyweakly-contracting and doubly-contracting systems, respec-tively. Finally, Section VI analyzes three applications of oursemi-and weak-contraction theory to network systems. Thisdocument is an ArXiv technical report and, compared withits journal version, it additionally contains in Appendix E acontractivity-based analysis of the cooperative Lotka–Volterrapopulation model.

II. NOTATION

For a set S ⊆ Rn, its interior, closure, and diameter aredenoted by int(S), cl(S), and diam(S), respectively. The n×nidentity matrix is In and the all-ones and all-zeros columnvectors of length n are 1n and 0n, respectively. For A ∈Cn×m, the conjugate transpose of A is AH , the real part ofA is <(A), the range of A is Img(A) and the kernel of Ais Ker(A). The Moore–Penrose inverse of A is the uniquematrix A† ∈ Cm×n such that AA†A = A, A†AA† = A† andAA† and A†A are Hermitian matrices. It can be shown thatAA† is the orthogonal projection onto Img(A) and A†A is theorthogonal projection onto Img(AH). Let λ1(A), . . . , λn(A)and spec(A) denote the eigenvalues and the spectrum of A ∈Cn×n. Given a vector subspace S ⊆ Cn, a vector v ∈ Cn,and a matrix A ∈ Cn×n, the orthogonal complement of S isS⊥ and we define v + S = {v + u | u ∈ S} and AS :={Au | u ∈ S}. The vector subspace S ⊆ Cn is invariantunder A ∈ Cn×n if AS ⊆ S. Given A ∈ Cn×n and a vectorsubspace S ⊆ Cn, S is invariant under A if and only if S⊥ isinvariant under AH . Given A ∈ Cn×n and a vector subspaceS ⊆ Cn invariant under A, define

specS(A) := {λ ∈ spec(A) | ∃v ∈ S s.t. Av = λv}.

Note spec(A) = specS(A)∪specS⊥(AH). Define the spectralabscissa of A restricted to S by:

αS(A) := max{<(λ) | λ ∈ specS(A)}.

For A ∈ Cn×n such that <(λ) ≤ 0, for every λ ∈ spec(A),the essential spectral abscissa of A is defined by

αess(A) := max{<(λ) | λ ∈ spec(A)− {0}}.

The absolute value of z ∈ C is denoted by |z|. A norm ‖ · ‖on Cn is (i) absolute if ‖x‖ = ‖y‖ for every x, y ∈ Cn such

that |xi| = |yi|, i ∈ {1, . . . , n}, (ii) monotone if ‖x‖ ≤ ‖y‖for every x, y ∈ Cn such that |xi| ≤ |yi|, i ∈ {1, . . . , n}. Letx ∈ Rn, r > 0, and ‖·‖ be a norm on Rn. Then the open ball of‖·‖ centered at x with radius r is B‖·‖(x, r) = {y ∈ Rn | ‖y−x‖ < r} and the closed ball of ‖·‖ centered at x with radius ris B‖·‖(x, r) = {y ∈ Rn | ‖y−x‖ ≤ r}. A norm ‖·‖ on Rn ispolyhedral if its unit ball {x ∈ Rn | ‖x‖ = 1} is a polyhedron.For p ∈ [1,∞], denote the `p-norm on Rn by ‖ · ‖p. All `p-norm are absolute and monotone. The `p-norm is polyhedralif and only if p ∈ {1,∞}. For any two complex matrices Aand B, their Kronecker product is denoted by A ⊗ B. Giventwo vector spaces V and W , the tensor product space V ⊗Wis given by V ⊗W = span{v⊗w | v ∈ V, w ∈W}. Considerthe time-varying dynamical system:

x = f(t, x), t ∈ R≥0, x ∈ Rn. (1)

We assume that (t, x) 7→ f(t, x) is twice-differentiable in xand essentially bounded in t. Denote the flow of (1) startingfrom x0 by t 7→ φ(t, x0). A set S ⊆ Rn is invariant withrespect to (1) if φ(t,S) ⊆ S for every t ∈ R≥0. A vector fieldX : Rn → Rn is piecewise real analytic if there exist closedsets {Σi}mi=1 which partition Rn and X is real analytic onint(Σi), for every i ∈ {1, . . . ,m}.

III. SEMI-CONTRACTING SYSTEMS

In this section we generalize the contracting theory byproviding a geometric framework for semi-contracting systemsvia semi-norms.

A. Linear algebra and matrix semi-measures

We start with semi-norms and associated semi-measures.

Definition 1 (Semi-norms). A function |||·||| : Rn → R≥0 is asemi-norm on Rn, if it satisfies the following properties:

(i) |||cv||| = |c||||v|||, for every v ∈ Rn and c ∈ R;(ii) |||v + w||| ≤ |||v|||+ |||w|||, for every v, w ∈ Rn.

For a semi-norm |||·|||, its kernel is defined by

Ker |||·||| = {v ∈ Rn | |||v||| = 0}.

It is easy to see that Ker |||·||| is a subspace of Rn and |||·||| is anorm on the vector space Ker |||·|||⊥. Semi-norms can naturallyarise from norms. Let k ≤ n, and ‖ ·‖ : Rk → R≥0 be a normon Rk and R ∈ Rk×n. Then the R-weighted semi-norm onRn associated with the norm ‖ · ‖ on Rk is defined by

‖v‖R = ‖Rv‖, for all v ∈ Rn. (2)

It is easy to see that the R-weighted semi-norm ‖ · ‖R is anorm if and only if k = n and R is invertible.

A semi-norm |||·||| on Rn naturally induces a semi-norm onthe space of real-valued matrices Rn×n.

Definition 2 (Induced semi-norm). Let |||·||| : Rn → R≥0 bea semi-norm on Rn, the induced semi-norm on Rn×n (whichwithout any confusion, we denote again by |||·|||) is defined by

|||A||| = sup{|||Av||| | |||v||| = 1, v ⊥ Ker |||·|||}.

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 4

The following properties of induced semi-norms areknown [19] and we omit the proof in the interest of brevity.

Proposition 3 (Properties of induced semi-norms). Let |||·||| :Rn → R≥0 be a semi-norm on Rn and assume that we denotethe induced semi-norm on Rn×n again by |||·|||. Then, for everyA,B ∈ Rn×n and c ∈ R,

(i) |||In||| = 1, |||A||| ≥ 0, and |||cA||| = |c||||A|||;(ii) |||A+B||| ≤ |||A|||+ |||B|||;

(iii) |||Av||| ≤ |||A||||||v|||, for every v ⊥ Ker |||·|||.

Definition 4 (Matrix semi-measures). Let |||·||| : Rn → R≥0 bea semi-norm on Rn and we denote the induced semi-norm onRn×n again by |||·|||. Then the matrix semi-measure associatedwith |||·||| is defined by

µ|||·|||(A) = limh→0+

|||In + hA||| − 1

h. (3)

Theorem 5 (Properties of matrix semi-measures). Let |||·||| :Rn → R≥0 be a semi-norm on Rn and let µ|||·||| be the as-sociated matrix semi-measure. Then, for every A,B ∈ Rn×n,the followings hold:

(i) µ|||·|||(A) is well-defined;(ii) µ|||·|||(A+B) ≤ µ|||·|||(A) + µ|||·|||(B);

(iii) |µ|||·|||(A)− µ|||·|||(B)| ≤ |||A−B|||.Moreover, if Ker |||·||| is invariant under A, then(iv) αKer |||·|||⊥(AT) ≤ µ|||·|||(A).

Proof. Regarding part (i), define the function f : R≥0 → Rby f(h) := |||In+hA|||−1

h . It suffices to show that the limitlimh→0+ f(h) exists. First note that, for every h ∈ R≥0, wehave |||In + hA||| ≥ |||In||| − |||hA||| = 1−h|||A|||. This impliesthat f(h) ≥ −|||A|||, for every h ∈ R≥0. Thus f is boundedbelow. Moreover, for every 0 < h1 < h2, we have

|||(1/h1)In +A||| − |||(1/h2)In +A|||≤ |||(1/h1 − 1/h2)In||| = 1/h1 − 1/h2.

As a result, we get

f(h1) = |||(1/h1)In +A||| − (1/h1)

≤ |||(1/h2)In +A||| − (1/h2) = f(h2).

Therefore f : R≥0 → R is a strictly increasing func-tion which is bounded below. Thus, limh→0+ f(h) exists.Regarding part (ii), the result is straightforward using thetriangle inequality (Theorem 3(ii)) for the induced semi-norm. Regarding part (iii), the proof is straightforward usingthe triangle inequality proved in part (ii). Finally, regardingpart (iv), suppose that |||·|||C is the complexification of |||·|||and λ ∈ C is an eigenvalue of AT with the right eigenvectorv ∈ Ker |||·|||⊥ ⊂ Cn normalized such that |||v|||C = 1. Thenwe have

<(λ) = limh→0+

|||v + hλv|||C − 1

h= limh→0+

∣∣∣∣∣∣v + hATv∣∣∣∣∣∣

C − 1

h

≤ limh→0+

∣∣∣∣∣∣In + hAT∣∣∣∣∣∣− 1

h= µ|||·|||(A

T).

Given the R-weighted semi-norm ‖ · ‖R as defined in (2),we denote its induced semi-norm on Rn×n by ‖ · ‖R and its

matrix semi-measure by µR. Specifically, for the `p-norm onRk, we let ‖ ·‖p,R and µp,R denote the associated R-weightedsemi-norm on Rn and matrix semi-measure, respectively.

Lemma 6 (Computation of semi-measures). Let ‖ · ‖ be anorm with the associated matrix measure µ and let k ≤ nand R ∈ Rk×n be full rank and ξ ∈ Rn≥0. Then, for everyA ∈ Rn×n,

(i) ‖A‖R = ‖RAR†‖,(ii) µR(A) = µ(RAR†),

(iii) µ1,diag(ξ)(A) = maxj:ξj 6=0

{ajj + ξj

∑i:ξi 6=0

|aij |ξi

},

(iv) µ∞,diag(ξ)(A) = maxi:ξi 6=0

{aii + ξi

∑j:ξj 6=0

|aij |ξj

}.

Moreover, if Ker(R) is invariant under A, then(iv) µ2,R(A) = 1

2αKerR⊥(A+ P †ATP

), with P = RTR.

Proof. First note that Ker ‖ · ‖R = {v ∈ Rn | ‖v‖R = 0}.Moreover, we have ‖v‖R = ‖Rv‖ and since ‖ · ‖ is a norm,we get ‖v‖R = 0 if and only if Rv = 0k. This implies thatKer ‖ · ‖R = KerR. Regarding part (i), by definition of theinduced-norm we have

‖A‖R = {‖RAv‖ | ‖v‖ = 1, v ⊥ KerR}.

Note that R†R is an orthogonal projection and Ker(R†R) =Ker(R). Thus, for every v ⊥ Ker(R), there exists x ∈ Rnsuch that v = R†Rx. As a result,

‖A‖R = {‖RAR†Rx‖ | ‖RR†Rx‖ = 1}= {‖RAR†Rx‖ | ‖Rx‖ = 1},

where for the second equality, we used the fact that RR†R =R. Since R is full rank, for every y ∈ Rk, there exists x ∈ Rnsuch that y = Rx. This means that

‖A‖R = sup{‖RAR†Rx‖ | ‖Rx‖ = 1}= sup{‖RAR†y‖ | ‖y‖ = 1} = ‖RAR†‖

Regarding part (ii), note that

µR(A) = limh→0+

‖In + hA‖R − 1

h

= limh→0+

‖RR† + hRAR†‖ − 1

h

= limh→0+

‖Ik + hRAR†‖ − 1

h= µ(RAR†),

where in the third equality, we used the fact that R has fullrow rank and therefore RR† = Ik. Regarding part (iii), defineξ ∈ Rn≥0 by ξi = ξ−1i if ξi > 0 and ξi = 0 if ξi = 0. Assumethat ξ has r ≤ n non-zero entries and define Qξ ∈ Rr×n(Qξ ∈ Rr×n) as the matrix whose ith row is the ith non-zerorow of diag(ξ) (diag(ξ)). Then it is easy to see that Qξ is fullrank and Q†ξ = QT

ξ. Note that ‖v‖1,diag(ξ) = ‖ diag(ξ)v‖1 =

‖Qξv‖1, for every v ∈ Rn. Therefore, by part (ii),

µ1,diag(ξ)(A) = µ1(QξAQ†ξ) = µ1(QξAQ

Tξ

).

The result follows using the formula for `1-norm matrixmeasure. The proof of part (iv) is similar to (iii) and weomit it. Regarding part (iv), note that by part (ii), we have

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 5

µ2,R(A) = µ2(RAR†). Moreover, using the formula for the`2-norm matrix measure,

µ2(RAR†) = 12 max{λ | λ ∈ spec(RAR† + (R†)TATRT)}.

Note that Ker(R) is invariant under A. Therefore, usingLemma 33, we get specKerR⊥(AT) = spec(RATR†). Notingthe fact that R is full rank and RR† = Ik and RP † =RR†(R†)T = (R†)T and PR† = RT, we get

µ2(RAR†) = 12 max{λ | λ ∈ specKerR⊥(A+ P †ATP )}.

B. Spectral abscissa as an optimal matrix measure

Theorem 5(iv) shows that the semi-measures of a matrix islower bounded by its spectral abscissa. In the next theoremwe study this gap and we show that on the space of all semi-measures this lower bound is tight. In this part, we use thegeneralization of the results in Section III-A to Cn. Specifi-cally, we use Lemma 6(i) and (ii) and and Theorem 5(iv) formatrices, norms, and matrix measures defined on Cn.

Theorem 7 (Optimal matrix measures and spectral abscissa).Let A ∈ Cn×n and let S ⊆ Cn be a (n − k)-dimensionalsubspace which is invariant under A. Then

(i) αS⊥(AH) = inf{µ|||·|||(A) | |||·||| a semi-norm with kernel S};(ii) let ‖ · ‖ be an absolute norm with its associated ma-

trix measure µ, then αS⊥(AH) = inf{µR(A) | R ∈Ck×n,Ker(R) = S}.

Proof. First note that, by Theorem 5(iv), for every semi-norm|||·||| with Ker |||·||| = S , we have αS⊥(AH) ≤ µ|||·|||(A). Thisimplies that

αS⊥(AH) ≤ inf{µ|||·|||(A) | |||·||| a semi-norm ,Ker |||·||| = S}≤ inf{µR(A) | R ∈ Ck×n,Ker(R) = S}. (4)

Therefore, in order to prove statements (i) and (ii), we needto show that

αS⊥(AH) ≥ inf{µR(A) | R ∈ Ck×n,Ker(R) = S}≥ inf{µ|||·|||(A) | |||·||| a semi-norm ,Ker |||·||| = S}.

We start by proving statement (ii). Consider the case that Ais diagonalizable. Let specS⊥(AH) = {λ1, . . . , λk}. Since Ais diagonalizable, there exists a set of linearly independentvectors {v1, . . . , vk} ⊂ S⊥ such that AHvi = λivi. Definethe matrix R ∈ Ck×n, where Ri (the ith row of matrix R) isequal to vHi , for every i ∈ {1, . . . , k}. Note that {v1, . . . , vk}is linearly independent and therefore R is full rank. Moreover,it is easy to see that KerR = S. On the other hand, wehave RA = ΛR, where Λ = diag{λH1 , . . . , λHk } ∈ Ck×k.This implies that RAR† = Λ. As a result, we have µR(A) =µ(RAR†) = µ(Λ) and therefore

µR(A) = µ(Λ) = limh→0+

‖Ik + hΛ‖ − 1

h

= maxi∈{1,...,k}

{<(λi)} = αS⊥(AH),

where the third equality holds because ‖ · ‖ is absolute andso ‖In +hΛ‖ = maxi∈{1,...,k}{|1 +hλi|}. Thus αS⊥(AH) ≥inf{µR(A) | R ∈ Ck×n,Ker(R) = S}. Now, consider the

case when A is not diagonalizable. Note that, in the complexfield C, the set of diagonalizable matrices are dense in Cn×n.Therefore, for every ε > 0, there exists a diagonalizable Aεsuch that ‖A − Aε‖ ≤ ε. Theorem 5(iii) implies |µR(A) −µR(Aε)| ≤ ε‖R‖‖R†‖. As a result

µR(A) ≤ µR(Aε) + ε‖R‖‖R†‖.

By taking the infimum over the set of R ∈ Ck×n withKer(R) = S and noting the fact that sup{‖R‖‖R†‖ | R ∈Ck×n, Ker(R) = S} ≤M , for some M ∈ R≥0, we get

inf{µR(A) | R ∈ Ck×n,Ker(R) = S}≤ inf{µR(Aε) | R ∈ Ck×n,Ker(R) = S}+Mε

= αS⊥(AHε ) +Mε,

where the last equality holds because Aε is diagonalizable.By continuity of eigenvalues, we get limε→0+ αS⊥(AHε ) =αS⊥(AH). This implies that

inf{µR(A) | R ∈ Ck×n,Ker(R) = S} ≤ αS⊥(AH).

This completes the proof of statement (ii). Statement (i) is aconsequence of (4) and statement (ii).

Remark 8. (i) Statement (i) is a generalization to semi-norms and semi-measures of [23, Proposition 2.3], whichstates α(A) = inf{µ‖·‖(A) | ‖ · ‖ a norm}.

(ii) Statement (ii) is a generalization to absolute normsof [43], which essentially proves the result for the `2case. For the special case of `p-norm on Rn, Theorem 7shows that the infimum of the weighted `p-measure of amatrix with respect to the weights recovers the spectralabscissa of the matrix, that is, for any p ∈ [1,∞],

α(A) = infR invertible

µp,R(A). 4

Next we present an application to algebraic graph theory.

Lemma 9 (Semi-measures of Laplacian matrices). Let G bea weighted digraph with a globally reachable vertex and withLaplacian matrix L (with eigenvalue λ1(L) = 0). Let ‖ · ‖ bea norm with associated matrix measure µ. Then

inf{µR(−L) | R ∈ C(n−1)×n,Ker(R) = span(1n)}= α1⊥n

(−LT) = αess(−L) < 0. (5)

Moreover, if G is undirected, then

min{µR(−L) | R ∈ C(n−1)×n,Ker(R) = span(1n)}= µRV (−L) = α1⊥n

(−LT) = −λ2(L) < 0, (6)

where V = {v2, . . . , vn} are orthonormal eigenvectors of theLaplacian L and

RV =[v2 . . . vn

]T ∈ R(n−1)×n. (7)

Proof. Theorem 7(ii) with A = −L and S = span(1n) gives

infR∈C(n−1)×n

Ker(R)=span(1n)

µR(−L) = α1⊥n(−L)

Since G has a globally reachable node, λ1 = 0 is a simpleeigenvalue of L with associated right eigenvector 1n and all

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 6

the other eigenvalues have positive real parts. This means thatα1⊥n

(−LT) = αess(−L) < 0.If G is undirected, then L is symmetric and all its eigen-

values are real. Let 0 = λ1 < λ2 ≤ λ3 ≤ . . . ≤λn denote the eigenvalues of L. Then it is clear thatα1⊥n

(−LT) = αess(−L) = −λ2 < 0. Moreover, we haveRVLR

TV = diag(λ2, . . . , λn). Additionally, we know that

R†V = RTV(RVR

TV)−1 = RT

V [30, E4.5.20]. As a result,

µRV (−L) = µ(−RVLR†V) = µ(−RVLRTV) = −λ2.

Remark 10. Equation (5) implies that, for any ε > 0, thereexists a matrix Rε ∈ C(n−1)×n with Ker(R) = span(1n)satisfying µRε(−L) ≤ αess(−L) + ε. 4

C. Solution estimates for dynamical systems

Consider a continuous map t 7→ A(t) ∈ Rn×n and thedynamical system

x(t) = A(t)x(t), for t ≥ t0 ∈ R (8)

with the initial condition x(t0) = x0.

Theorem 11 (Generalized Coppel’s inequality). Let |||·||| be asemi-norm on Rn and let µ|||·||| be the associated matrix semi-measure. Assume that, for every t ≥ t0, Ker |||·||| is invariantunder A(t). Then, for every t ≥ t0, we have

exp

(∫ t

t0

µ|||·|||(−A(τ))dτ

)|||x(0)||| ≤ |||x(t)|||

≤ exp

(∫ t

t0

µ|||·|||(A(τ))dτ

)|||x(0)|||.

Moreover, if A is time-invariant, we have

exp(tµ|||·|||(−A))|||x(0)||| ≤ |||x(t)||| ≤ exp(tµ|||·|||(A))|||x(0)|||.

Proof. Note that t 7→ A(t) is continuous. Therefore thesolutions t 7→ x(t) of the time-varying dynamical system (8)are differentiable. Thus, for small enough h, we can writex(t+ h) = x(t) + hA(t)x(t) + o(h). Let P be the orthogonalprojection onto Ker |||·|||⊥. Then we have

Px(t+ h) = Px(t) + hPA(t)x(t) + o(h)

= Px(t) + hPA(t)Px(t) + o(h).

where the last equality holds because A(t) Ker |||·||| ⊆ Ker |||·|||.Therefore, |||Pv||| = |||v||| and Theorem 3 together imply

|||x(t+ h)||| − |||x(t)|||h

≤ |||In + hA||| − 1

h|||x(t)|||+ o(h)

h.

Taking the limit as h → 0+, we obtain ddt |||x(t)||| ≤

µ|||·|||(A)|||x(t)|||. The result then follows from theGronwall–Bellman inequality.

Definition 12 (Semi-contracting systems). Let C ⊆ Rn be aconvex set, c > 0 be a positive number, |||·||| be a semi-normon Rn, and µ|||·||| be its associated matrix semi-measure. Thetime-varying dynamical system (1) is

(i) semi-contracting on C with respect to |||·||| with rate c, if

µ|||·|||(Df(t, x)) ≤ −c, for all t ∈ R≥0, x ∈ C

Moreover, the subspace Ker |||·||| is(i) infinitesimally invariant under the system (1), if

Df(t, x) Ker |||·||| ⊆ Ker |||·|||, for all t ∈ R≥0, x ∈ C;(P1)

(ii) shifted-invariant under the system (1), if there exists x∗ ∈C, such that

f(t, x∗ + Ker |||·|||) ⊆ Ker |||·|||, for all t ∈ R≥0. (P2)

Now, we study the asymptotic behavior of trajectories ofthe dynamical system (1) when it is semi-contracting.

Theorem 13 (Trajectories of semi-contracting systems). Con-sider the time-varying dynamical system (1) with a convex andinvariant set C. Assume the system (1) is semi-contracting onC with respect to a semi-norm |||·||| with rate c > 0. Thefollowing statements hold:

(i) if (P1) holds, then for every x0, y0 ∈ Rn and t ∈ R≥0,

|||φ(t, x0)− φ(t, y0)||| ≤ e−ct|||x0 − y0|||;

(ii) if (P2) holds for some x∗ ∈ Rn, then for every x0 ∈ Rnand t ∈ R≥0,

|||φ(t, x0)− x∗||| ≤ e−ct|||x0 − x∗|||,

and t 7→ φ(t, x0) converges to x∗ + Ker |||·||| withexponential rate c.

Moreover, assuming the system (1) is time-invariant,(iii) if (P1) holds, then for every x0 ∈ Rn and every t ∈ R≥0,

|||f(φ(t, x0))||| ≤ e−ct|||f(x0)|||;

(iv) if (P1) holds and C = Rn, then (P2) holds for somex∗ ∈ Rn.

Proof. Regarding part (i), for α ∈ [0, 1], define ψ(t, α) =φ(t, αx0 + (1 − α)y0

)and note ψ(0, α) = αx0 + (1 − α)y0

and ∂ψ∂α (0, α) = x0 − y0. We then compute:

∂

∂t

∂

∂αψ(t, α) =

∂

∂α

∂

∂tψ(t, α)

=∂

∂αf(t, ψ(t, α)) =

∂f

∂x(t, ψ(t, α))

∂

∂αψ(t, α).

Therefore, ∂ψ∂α (t, α) satisfies the linear time-varying differen-

tial equation ∂∂t∂ψ∂α = Df(t, ψ)∂ψ∂α . Since the system (1) has

property (P1), Theorem 11 implies∣∣∣∣∣∣∣∣∣∂ψ∂α (t, α)∣∣∣∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣∣∂ψ∂α (0, α)

∣∣∣∣∣∣∣∣∣ exp(∫ t

0

µ(Df(t, ψ(τ, α))

)dτ)

≤ e−ct|||x0 − y0|||, (9)

where we used µ|||·|||(Df(t, x)) ≤ −c, for every t ∈ R≥0 andx ∈ Rn. In turn, inequality (9) implies

|||φ(t, x0)− φ(t, y0)||| = |||ψ(t, 1)− ψ(t, 0)|||

=

∣∣∣∣∣∣∣∣∣∣∣∣∫ 1

0

∂ψ(t,α)∂α dα

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ ∫ 1

0

∣∣∣∣∣∣∣∣∣∂ψ(t,α)∂α

∣∣∣∣∣∣∣∣∣dα ≤ e−ct|||x0 − y0|||.This completes the proof of part (i).

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 7

Regarding part (ii), let P be the orthogonal projection ontoKer |||·|||⊥ and consider the dynamical system on Ker |||·|||⊥:

y = fP(t, y) := Pf(t, y + (In − P)φ(t, x0)). (10)

Note that, for every v ∈ Rn, we have |||Pv||| = |||v|||. Thisimplies that, for A ∈ Rn×n,

µ|||·|||(PA) = limh→0+

|||In + hPA||| − 1

h

= limh→0+

∣∣∣∣∣∣P + hP2A∣∣∣∣∣∣− 1

h

= limh→0+

|||P + hPA||| − 1

h

= limh→0+

|||In + hA||| − 1

h= µ|||·|||(A).

As a result, for every (t, y) ∈ R≥0 ×Ker |||·|||⊥,

µ|||·|||(DfP(t, y)) = µ|||·|||(PDf(t, y + (In − P)φ(t, x0))

= µ|||·|||(Df(t, y + (In − P)φ(t, x0))) ≤ −c.

where for the last inequality, we used the fact thatµ|||·|||(Df(t, x)) ≤ −c, for every (t, x) ∈ R≥0×Rn. Recall that|||·||| is a norm on Ker |||·|||⊥. Thus, the dynamical system (10)is contractive with respect to |||·||| on Ker |||·|||⊥. Additionally,(In − P) is the orthogonal projection onto Ker |||·||| andtherefore, for every t ≥ 0,

x∗ + (In − P)φ(t, x0) ∈ x∗ + Ker |||·|||.

Using (P2), for every t ≥ 0, we obtain

f(t, x∗ + (In − P)φ(t, x0)) ⊆ Ker |||·|||.

This implies that fP(t, x∗) = Pf(t, x∗+ (In−P)φ(t, x0)) =0n. As a result, x∗ ∈ Ker |||·|||⊥ is an equilibrium point ofthe dynamical system (10). Moreover, one can see that t 7→Pφ(t, x0) is another trajectory of the dynamical system (10).Since the dynamical system (10) is contractive on Ker |||·|||⊥,

|||Pφ(t, x0)− x∗||| ≤ e−ct|||Pφ(0, x0)− x∗|||= e−ct|||x0 − x∗|||,

where the last equality hold because Ker |||·||| = Img(In−P).Moreover,

limt→∞

|||Pφ(t, x0)− x∗||| ≤ limt→∞

e−ct|||x0 − x∗||| = 0.

Note that Pφ(t, x0) − x∗ ∈ Ker |||·|||⊥ and |||·||| is a normon Ker |||·|||⊥. This implies that limt→∞ Pφ(t, x0) = x∗, orequivalently, limt→∞ φ(t, x0) ∈ x∗ + Ker |||·||| with exponen-tial convergence rate c.

Regarding part (iii), note that the map t 7→ f(φ(t, x0))satisfies d

dtf(φ(t, x0)) = Df(φ(t, x0))f(φ(t, x0)). Since (P1)holds, Theorem 11 implies

|||f(φ(t, x0))|||

≤ exp

(∫ t

0

µ|||·|||(Df(f(φ(τ, x0)))

)dτ

)|||f(φ(0, x0))|||

≤ e−ct|||f(x0)|||,

where we used µ|||·|||(Df(t, x)) ≤ −c, for every t ∈ R≥0 andx ∈ Rn.

Regarding part (iv), consider the following dynamical onKer |||·|||⊥ system

y = Pf(y) (11)

Note that µ|||·|||(PDf(y)) = µ|||·|||(Df(y)) ≤ −c, for everyy ∈ Ker |||·|||⊥ and |||·||| is a norm on Ker |||·|||⊥. Therefore,the dynamical system (11) is contracting on Ker |||·|||⊥ withrespect to |||·|||. Since (11) is time-invariant, it has a uniqueglobally stable equilibrium point x∗ ∈ Ker |||·|||⊥. Thus, wehave Pf(x∗) = 0n. On the other hand, for every x0 ∈ Rn,recall that t 7→ φ(t, x0) is the trajectory of the dynamicalsystem (1) starting form x0. Therefore, by part (iii),

|||f(φ(t, x∗))||| ≤ e−ct|||f(x∗)||| = 0.

This implies that the trajectory t 7→ φ(t, x∗) remains in x∗ +Ker |||·|||, for every t ≥ 0. Using part (i), for every x0 ∈ x∗ +Ker |||·||| and every t ∈ R≥0,

|||φ(t, x0)− φ(t, x∗)||| ≤ e−ct|||x0 − x∗||| = 0,

where the last equality holds because x0 − x∗ ∈ Ker |||·||| andthus |||x0 − x∗||| = 0. This means that, for every x0 ∈ x∗ +Ker |||·||| and every t ∈ R≥0, we have φ(t, x0) ∈ x∗+Ker |||·|||.As a consequence, property (P2) holds for x∗.

Remark 14 (Comparison with literature). Theorem 13(i)is related to horizontal contraction theory on Finsler man-ifolds [16]. In short, semi-contraction theory offers a morecomprehensive treatment with a smaller domain of applicabil-ity. Specifically, semi-contraction theory studies convergenceto vector subspaces, whereas horizontal contraction theorycan handle more general submanifolds. On the other hand,the notion of semi-measure leads to a generalized Coppel’sinequality, to a less restrictive invariance condition for semi-contractivity, and to sharp results about convergence rates; wepostpone a comprehesive comparison to Appendix A.

Theorem 13(ii) is related to partial contraction theory [34],[14]. Specifically, Theorem 13(ii) is more general than [34,Theorem 1], whereby partial contraction is restricted to the `2-norm, and [14], whereby partial contraction is defined using anorthonormal set of vectors. In contrast, our statement and proofof Theorem 13(ii) is geometric and coordinate independent.

To the best of our knowledge, the results Theorem 13(iii)and Theorem 13(iv) are novel. 4

According to Theorem 13, if the affine space x∗+Ker |||·||| isinvariant for a semi-contracting system, then every trajectoryconverges to x∗ + Ker |||·|||. However, the theorem does notstate that the asymptotic behavior of generic trajectories isthe same as the asymptotic behavior of the trajectories on theinvariant affine subspace x∗+Ker |||·|||. The following exampleelaborates on this aspect.

Example 15. The dynamical system on R2

x1 = −x1,x2 = x1x

22

(12)

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 8

is semi-contracting with respect to the semi-norm |||·||| definedby∣∣∣∣∣∣(x1, x2)T

∣∣∣∣∣∣ = |x1|, for every (x1, x2)T ∈ R2. Addi-tionally, Ker |||·||| = {(0, x2) | x2 ∈ R} is shifted-invariantunder (12). Thus, by Theorem 13(ii), every trajectory of thesystem (12) converges to the set Ker |||·||| = {(0, x2) | x2 ∈R}. Moreover, S is the set of equilibrium points for (12). Onthe other hand, the curve t 7→ (x1(t), x2(t)) := (e−t, et) isa trajectory of (12) such that limt→∞ x2(t) = ∞. Therefore,this trajectory does not converge to any equilibrium point.

IV. WEAKLY-CONTRACTING SYSTEMS

We here generalize contraction theory by providing a com-prehensive treatment to weakly-contracting systems.

Definition 16 (Weakly-contracting systems). Let C ⊆ Rn be aconvex set, ‖·‖ be a norm on Rn, and µ be its associated matrixmeasure. The time-varying dynamical system (1) is weakly-contracting on C with respect to ‖ · ‖, if

µ(Df(t, x)) ≤ 0, for all t ∈ R≥0, x ∈ C. (13)

We aim to study the qualitative behaviors of weakly-contracting systems. For the rest of this section, we restrictourselves to the time-invariant dynamical systems:

x = f(x), (14)

on convex invariant sets and we assume that x 7→ f(x) istwice-differentiable. We start with a useful and novel lemmawhich shows a weakly contracting system with a boundedtrajectory has a compact convex invariant set.

Lemma 17 (Weak contraction and bounded trajectory implyinvariant set). Consider the time-invariant dynamical sys-tem (14) with a closed convex invariant set C. Suppose that ‖·‖is a norm on Rn, f is continuously differentiable and weaklycontracting with respect to the norm ‖ · ‖, and t 7→ x(t) is abounded trajectory of the system in C. Then the system (14)has a compact convex invariant set W ⊆ C.

Our first result is a dichotomy for qualitative behavior oftrajectories of weakly contracting systems.

Theorem 18 (Dichotomy for weakly contracting systems).Consider the time-invariant dynamical system (14) with aconvex invariant set C ⊆ Rn. Let ‖ · ‖ be a norm on Rn withthe associated matrix measure µ. Suppose that the system (14)is weakly contracting on C with respect to ‖ · ‖. Then eitherof the following exclusive conditions hold:

(i) the dynamical system (14) has at least one stable equilib-rium point x∗ ∈ C and every trajectory of (14) startingin C is bounded; or

(ii) the dynamical system (14) has no equilibrium point in Cand every trajectory of (14) starting in C is unbounded.

Moreover, the following statements hold:(iii) if x∗ is locally asymptotically stable, then x∗ is globally

asymptotically stable in C,(iv) if additionally Df(x∗) is Hurwitz then x∗ is the unique

globally asymptotically stable equilibrium point in C,(v) if additionally µ(Df(x∗)) < 0, then x∗ is the unique

globally asymptotically stable equilibrium point in C and

x 7→ ‖x − x∗‖ is a global Lyapunov function and x 7→‖f(x)‖ is a local Lyapunov function.

Proof. Suppose that the equation f(x) = 0n has at least onesolution x∗ in C. Let t 7→ x(t) be a trajectory of the system.Since the system (14) is weakly-contracting, we have

‖x(t)− x∗‖ ≤ ‖x(0)− x∗‖, ∀t ≥ 0.

By setting M = ‖x(0) − x∗‖ and using triangle inequality,we see that ‖x(t)‖ ≤ ‖x∗‖ + M , for every t ≥ 0. Thisimplies that t 7→ x(t) is bounded and thus statement (i) holds.Now suppose that the algebraic equation f(x) = 0n does nothave any solution in C. We now need to show that everytrajectory of (14) is unbounded. By contradiction, assumet 7→ x(t) is a bounded trajectory of (14). In Lemma 17we establish that there exists a compact convex invariantset W ⊆ C for the dynamical system (14). Therefore, byYorke Theorem [20, Lemma 4.1], the system (14) has anequilibrium point inside W ⊆ C. This is in contradictionwith the assumption that f(x) = 0n has no solution insideC. Therefore, every trajectory of (14) is unbounded. Thiscompletes the proof of statement (ii). Regarding part (iii), sincex∗ is a locally exponentially stable equilibrium point for thedynamical system, there exists ε, T > 0 such that B‖·‖(x∗, ε)is in the region of attraction of the equilibrium point x∗ and,for every z ∈ B‖·‖(x∗, ε), we have φ(T, z) ∈ B‖·‖(x∗, ε/2).Let t 7→ x(t) denote a trajectory of the dynamical system.Assume that y ∈ ∂B‖·‖(x∗, ε) is a point on the straight lineconnecting x(0) to the unique equilibrium point x∗. Then wehave

‖x(T )− x∗‖ ≤ ‖x(T )− φ(T, y)‖+ ‖φ(T, y)− x∗‖≤ ‖x(0)− y‖+ ε/2 = ‖x(0)− x∗‖ − ε/2,

where the last equality holds because x∗, y, and x(0) are on thesame straight line. Therefore, after time T , t 7→ ‖x(t)− x∗‖decreases by ε/2. As a result, there exists a finite time Tinfsuch that, for every t ≥ Tinf , we have x(t) ∈ B‖·‖(x

∗, ε).Since B‖·‖(x∗, ε) is in the region of attraction of x∗ the tra-jectory t 7→ x(t) converges to x∗. Regarding part (iv), Df(x∗)being Hurwitz implies that x∗ is a locally exponentiallystable equilibrium point for (14) and therefore it is a globallyexponentially stable equilibrium point by part (iii). Regardingpart (v), µ(Df(x∗)) < 0 implies that Df(x∗) is Hurwitz andthe global exponentially stability of the equilibrium followsfrom part (iv). Now we show that x 7→ ‖x − x∗‖ is aglobal Lyapunov function. It is positive definite and radiallyunbounded. Since µ(Df(x∗)) < 0 and f is continuouslydifferentiable, there exists a closed ball B‖·‖(x∗, r) and c > 0such that

µ(Df(x)) ≤ −c, ∀x ∈ B‖·‖(x∗, r).

Let t 7→ x(t) denote a trajectory of the system and, for everyt ∈ R≥0, assume that y(t) is the point on the intersection of

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 9

the line connecting x(t) to x∗ and ∂B‖·‖(x∗, r). Then, for

every s > t, we have

‖x(s)− x∗‖ ≤ ‖x(s)− φ(s− t, y)‖+ ‖φ(s− t, y)− x∗‖≤ ‖x(t)− y‖+ e−c(s−t)‖y − x∗‖< ‖x(t)− x∗‖.

This implies that t 7→ ‖x(t)−x∗‖ is strictly decreasing. Sincex 7→ ‖x − x∗‖ is radially unbounded, we can deduce thatx 7→ ‖x− x∗‖ is a global Lyapunov function for the system.On the other hand, for every t ∈ R≥0,

d

dtf(x(t)) = Df(x(t))f(x(t)).

Note that the dynamical system (14) is weakly contractingwith respect to the norm ‖ · ‖. Therefore, for every x0 ∈B‖·‖(x

∗, r), the trajectory t 7→ x(t) starting at x0 remainsinside B‖·‖(x∗, r). This implies that, for every t ∈ R≥0, wehave µ(Df(x(t))) ≤ −c. Using Theorem 11(iii), we get

‖f(x(t))‖ ≤ e−ct‖f(x(0))‖, for all t ∈ R≥0.

As a result x 7→ ‖f(x)‖ is a local Lyapunov function for thedynamical system (14).

The dichotomy in Theorem (18) completely characterizesthe qualitative behavior of the dynamical system when systemhas no equilibrium point. However, for weakly contractingsystems with at least one equilibrium point, this theoremcan only guarantee that the trajectories are bounded. Thefollowing theorem shows that, for some classes of norms,weak contractivity implies convergence of every trajectory toan equilibrium point.

Theorem 19 (Weak contraction and convergence to equilibria).Consider the time-invariant dynamical system (14) with aconvex invariant set C ⊆ Rn. Suppose that the vector field f isdifferentiable and piecewise real analytic with an equilibriumpoint x∗ ∈ C. Suppose that there exists p ∈ {1,∞} and aninvertible matrix Q ∈ Rn×n such that µp,Q(Df(x)) ≤ 0, forevery x ∈ C, i.e., the system is weakly contracting with respectto ‖ · ‖p,Q on C. Then every trajectory of (14) starting in Cconverges to an equilibrium point.

Proof. We prove the theorem for p = 1. The proof forp = ∞ is similar and we omit it. Suppose that the vectorfield f is piecewise real analytic and the system (14) isweakly contracting with respect to Q-weighted `1-norm. ByTheorem 18(i) every trajectory of the system (14) is bounded.Now we use the LaSalle Invariance Principle [5, Theorem15.7] for the function V (x) = ‖f(x)‖1,Q on the invariantconvex set C. We first show that, for every c > 0, V −1(c) isa closed invariant set for the dynamical system (14). Since Vis continuous, V −1(c) is closed. Now we show that V −1(c) isinvariant. First note that, for every trajectory t 7→ x(t) startinginside V −1(c), we have d

dtf(x(t)) = Df(x(t))f(x(t)). UsingTheorem 11,

‖f(x(t))‖1,Q ≤ exp

(∫ t

0

µ1,Q(Df(x(τ)))dτ

)‖f(x(0))‖1,Q

≤ ‖f(x(0))‖1,Q ≤ c,

where the second inequality holds because system (14) isweakly contracting with respect to Q-weighted `1-norm andthus µ1,Q(Df(x)) ≤ 0, for every x ∈ C and the last inequalityholds because x(0) ∈ V −1(c). Thus, V −1(c) is an invariantset for (14). Therefore, by the LaSalle Invariance Principle,the largest invariant set M inside the set

{x ∈ C | LfV (x) = 0} ∩ V −1(c)

is nonempty and every trajectory of (14) converges to M .Let t 7→ y(t) denote a trajectory in the set M (and thereforein the set C). Our goal is to show that t 7→ y(t) is anequilibrium point. Since the vector field f is piecewise realanalytic, there exists a partition {Σj}mj=1 of C such that fis real analytic on int(Σj), for every j ∈ {1, . . . ,m}. Nowconsider the trajectory t 7→ y(t). It is clear that, there existsk ∈ {1, . . . ,m} and T > 0 such that y(t) ∈ cl(Σk), for everyt ∈ [0, T ]. Since f is real analytic on int(Σk), there exists areal analytic vector field g : C → Rn such that g(x) = f(x),for every x ∈ cl(Σk). This implies that, for every t ∈ [0, T ],the curve t 7→ y(t) is a solution of the dynamical system

y(t) = f(y(t)) = g(y(t)).

Therefore, the curve t 7→ y(t) is real analytic for every t ∈[0, T ]. Note that, for every t ∈ [0, T ],

0 =d

dtV (y(t)) =

n∑i=1

sign(

(Qf)i(y(t)))

(Qf)i(y(t))

=

n∑i=1

sign(

(Qg)i(y(t)))

(Qg)i(y(t)).

Since g is real analytic and t 7→ y(t) is real analytic on [0, T ],then t 7→ (Qg)i(y(t)) is real analytic, for i ∈ {1, . . . , n}.Therefore, either of the following conditions hold:

(i) (Qg)i(y(t)) 6= 0, for every t ∈ [0, T ], or(ii) (Qg)i(y(t)) = 0, for every t ∈ [0, T ].

Since t 7→ (Qg)i(y(t)) is continuous, for every i ∈ {1, . . . , n},this implies that sign((Qg)i(y(t))) = sign((Qg)i(y(0))), forevery t ∈ [0, T ] and every i ∈ {1, . . . , n}. Define w ∈ Rn by

w = sign(Qg(y(0))).

Thus, we have LfV (y(t)) = wTQg(y(t)) = 0, for everyt ∈ [0, T ]. By integrating this condition, we get

wTQg(y(t)) = β, ∀t ∈ [0, T ],

for some constant β ≥ 0. We first show that η = 0. Note thatg(y(t)) = f(y(t)) = y(t). Therefore, we have wTQy(t) = β,for every t ∈ [t1, t2]. Integrating with respect to time, we get

wTQy(t) = βt+ η, ∀t ∈ [0, T ].

for some constant η ∈ R. Since every trajectory of (14) startingin C is bounded, we have β = 0. Now, note that, for everyt ∈ [0, T ],

‖f(y(t))‖1,Q =

n∑i=1

sign((Qf)i(y(t))(Qf)i(y(t))

= wTQg(y(t)) = 0.

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 10

This implies that f(y(t)) = 0n, for every t ∈ [0, T ]. Sincet 7→ y(t) is a trajectory of (14) and it is continuous, wehave f(y(t)) = 0n, for every t ≥ 0. This implies thatevery trajectory inside M is an equilibrium point. Therefore,every trajectory of (14) starting in C converges to the set ofequilibrium points.

Remark 20. (i) Theorem 19(iii) is a generalization of [25,Lemma 6] to weakly contracting systems. Theorem 18(v)is due to [9, Corollary 8].

(ii) We believe that Theorem 19 can be generalized to poly-hedral norms; we omit this generalization in the interestof brevity.

(iii) Theorem 19 does not hold in general for systems thatare weakly contracting with respect to `p-norms for p 6∈{1,∞}, as the following example illustrates. 4

Example 21. The dynamical system on R2

x1 = x2,

x2 = −x1,(15)

has a unique equilibrium point 02 and its vector field f ispiecewise real analytic. Let µ2 denote the matrix measure withrespect to the `2-norm on R2. Then

µ2(Df(x)) = λmax

(Df(x) +DfT(x)

2

)= λmax (02×2) = 0,

so that system (15) is weakly contracting with respect to ‖·‖2.However, t 7→ (sin(t), cos(t)) is a periodic trajectory of (15)which does not converge to the equilibrium point. 4

V. DOUBLY-CONTRACTING SYSTEMS

Examples 15 and 21 illustrate that, for time-invariant sys-tems, semi-contractivity or weak-contractivity alone do notguarantee the convergence of trajectories to equilibrium points.Here we show that, for time-invariant systems, a combinationof these two properties can ensure that every trajectory con-verges to an equilibrium point.

Theorem 22 (Double contraction and convergence to equilib-ria). Consider the time-invariant dynamical system (14) witha convex invariant set C ⊆ Rn. Suppose that S ⊆ Rn is avector subspace consists of equilibrium points of (14). Assumethat there exist

(A1) a norm ‖ · ‖ such that (14) is weakly-contracting withrespect to ‖ · ‖ on C; and

(A2) a semi-norm |||·||| with Ker |||·||| = S such that (14) issemi-contracting with respect to |||·||| on C.

Then, for every trajectory t 7→ x(t) of (14) starting in C, thereexists x∗ ∈ S such that limt→∞ x(t) = x∗ with exponentialconvergence rate |αS⊥(Df(x∗))|.

We refer to systems satisfying (A1) and (A2) as doubly-contracting in the sense that their trajectories satisfy a weakcontractivity and a semi-contractivity property:

‖φ(t, x0)− φ(t, y0)‖ ≤ ‖x0 − y0‖,|||φ(t, x0)||| ≤ e−ct|||x0|||,

for every x0, y0 ∈ C, every t ∈ R≥0, and some c ∈ R≥0.

Proof. Let t 7→ x(t) denote a trajectory of (14) startingfrom x(0) ∈ C. Let P be the orthogonal projection onto thesubspace S⊥ = Ker |||·|||⊥. Note that, for every t ∈ R≥0, theorthogonal projection of x(t) onto S = Ker |||·||| is given by(In−P)x(t) and it is an equilibrium points for (14). Moreover,by Assumption (A1), system (14) is weakly-contracting withrespect to the norm ‖ · ‖. This implies that, for every t ∈ R≥0and every s ≥ t, the point x(s) remains inside the closed ballB‖·‖((In −P)x(t), ‖Px(t)‖). Therefore, for every t ≥ 0, thepoint x(t) is inside the set Dt defined by

Dt = cl( ⋂τ∈[0,t]

B‖·‖((In − P)x(τ), ‖Px(τ)‖)).

It is easy to see that, for s ≥ t, we have Ds ⊆ Dt. Thisimplies that the family {Dt}t∈[0,∞) is a nested family ofclosed subsets of Rn such that diam(Dt) ≤ ‖Px(t)‖, forevery t ∈ [0,∞). On the other hand, for every t ∈ [0,∞),the set Dt is a closed convex invariant set for the system (14).Moreover, Ker |||·||| = S consists of equilibrium points of (14).This implies that, Ker |||·||| is shifted-invariant under the sys-tem (14). Using Assumption (A2) and Theorem 13(ii), thetrajectory t 7→ x(t) converges to the subspace S. This meansthat limt→∞ diam(Dt) = limt→∞ ‖Px(t)‖ = 0. Thus, bythe Cantor Intersection Theorem [31, Lemma 48.3], thereexists x∗ ∈ C such that

⋂t∈[0,∞)Dt = {x∗}. We show

that limt→∞ x(t) = x∗. Note that x∗, x(t) ∈ Dt, for everyt ∈ R≥0. This implies that ‖x(t) − x∗‖ ≤ diam(Dt). Thisin turn means that limt→∞ ‖x(t) − x∗‖ = 0 and t 7→ x(t)converges to x∗. On the other hand, the trajectory t 7→ x(t)converges to the subspace S. Therefore, x∗ ∈ S and x∗

is an equilibrium point. Regarding the convergence rate, byTheorem 7, we have

αS⊥(Df(x∗))

= inf{µ|||·|||(Df(x∗)) | |||·||| a semi-norm, Ker |||·||| = S}.

First note that system (14) is semi-contracting with respect to|||·||| on C. This implies that αS⊥(Df(x∗)) ≤ µ|||·|||(Df(x∗)) <0. Let ε > 0 be such that ε ≤ 1

3 |αS⊥(Df(x∗))|. There exists asemi-norm |||·|||ε such that µ|||·|||ε(Df(x∗)) ≤ αS⊥(Df(x∗)) +ε < 0. Consider the family of closed convex invariant sets{Dt}t≥0. Since limt→∞ diam(Dt) = 0, and f is twicedifferentiable, there exists tε ∈ R≥0 such that µ|||·|||ε(Df(x)) ≤αS⊥(Df(x∗)) + 2ε < 0, for every x ∈ Dtε . Therefore, usingTheorem 13(ii), the trajectory t 7→ x(t) converges to thesubspace S with the convergence rate |αS⊥(Df(x∗)) + 2ε|.Since ε can be chosen arbitrarily small, the trajectory t 7→ x(t)converges to the subspace S with the rate |αS⊥(Df(x∗))|.

Remark 23. Theorem 22 provides a convergence rate for tra-jectories of the system which depends only on the convergingpoint and is independent of both the norm ‖ · ‖ in Assump-tion (A1) and the semi-norm |||·||| in Assumption (A2). Thistheorem also generalizes the classical results in contractiontheory where the convergence rate depends on the norm. 4

VI. APPLICATION TO NETWORK SYSTEMS

In this section, we apply our results to example systems. Weshow that (i) affine averaging systems and affine flow systems

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 11

are doubly-contracting, (ii) distributed primal-dual dynamics isweakly-contracting, and (iii) networks of diffusively-coupledoscillators with strong coupling are semi-contracting.

A. Affine averaging and flow systems are doubly-contracting

Theorem 24 (Affine averaging system). Let L be the Lapla-cian matrix of a weighted digraph with a globally reachablenode. Let v denote the dominant left eigenvector of L satisfying1Tnv = 1. Let b ∈ Rn. Then the affine averaging system

x = −Lx+ b (16)

(i) is weakly-contracting with respect to ‖ · ‖∞ and semi-contracting with respect to ‖·‖∞,Rε , where Rε is definedin Remark 10 as function of a sufficiently small ε,

(ii) if vTb 6= 0, then every trajectory is unbounded,(iii) if vTb = 0, then the trajectory starting from x(0) = x0

converges to the equilibrium point L†b+ (vTx0)1n withexponential rate −αess(−L).

Proof. Regarding part (i), for f(x) := −Lx + b, it is easyto see that µ∞(Df(x)) = µ∞(−L) = 0. Therefore, thedynamical system (16) is weakly-contracting with respectto `∞-norm. Regarding part (ii), if vTb 6= 0, then theredoes not exist x ∈ Rn such that −Lx + b = 0. Thus,Theorem 18(i) implies every trajectory of (16) is unbounded.Regarding part (iii), since G has a reachable node, 0 is a simpleeigenvalue of L and the other eigenvalues of L have positivereal parts. Thus, if vTb = 0, then the algebraic equation−Lx + b = 0n has solutions x = L†b + β1n, for β ∈ R.Therefore, by Theorem 19, every trajectory of (16) convergesto an equilibrium point in span(1n). Let t 7→ x(t) be atrajectory of (16). Since vTL = 0, we have vTx(t) = vTx0.Thus, the trajectory t 7→ x(t) converges to the equilibriumpoint L†b+(vTx0)1n. In order to find the rate of convergence,we use the change of variable z = x+ L†b to get

z = −Lz. (17)

Note the span(1n) is an invariant subspace for (17) consistingof only equilibrium points. Moreover, for f(z) := −Lz wehave µ∞(Df(z)) = µ∞(−L) = 0. Thus the dynamicalsystem (17) is weakly-contracting with respect to `∞-norm.Lemma 9 now implies

inf{µ∞,R(−L) | R ∈ C(n−1)×n,Ker(R) = span(1n)}= α1⊥n

(−LT) = αess(−L) < 0.

This implies that there exists R ∈ C(n−1)×n such thatKerR = span(1n) and µ∞,R(−L) ≤ −c for some positivec > 0. Thus the assumptions of Theorem 22 hold for thenorm ‖ · ‖∞ and the semi-norm |||·|||∞,R. This implies thatevery trajectory of the system (17) converges to span(1n) withconvergence rate −αess(−L) > 0.

Next, we state an analogous theorem for affine flow systems,whose proof is omitted in the interest of brevity.

Theorem 25 (Affine flow system). Under the same assump-tions on L, v, and b as in Theorem 24, the affine flow system

x = −LTx+ b (18)

(i) is weakly-contracting with respect to ‖ · ‖1 and semi-contracting with respect to ‖ · ‖1,Rε , where Rε is definedin Remark 10 as function of a sufficiently small ε,

(ii) if 1Tnb 6= 0, then every trajectory is unbounded,

(iii) if 1Tnb = 0, then the trajectory starting from x(0) = x0

converges to the equilibrium point (LT)†b+(1Tnx0)v with

exponential rate −αess(−L).

B. Distributed primal-dual dynamics is weakly-contracting

In the second application, we study a well-known distributedimplementation of unconstrained optimization problems. Sup-pose that we have n agents connected though an undirectedweighted connected graph with Laplacian L. We want tominimize an objective function f : Rk → R which can berepresented as f(x) =

∑ni=1 fi(x). This optimization problem

can be written as

min

n∑i=1

fi(x), (19)

When agent i has access to only the objective function fi, theoptimization problem (19) can be implemented in a distributedfashion as:

min

n∑i=1

fi(xi),

xi = xj , for every edge i, j.

(20)

The primal-dual algorithm associated with the distributedoptimization problem (20) is given by

xi = −∇fi(xi)−n∑i=1

aij(νi − νj),

νi =

n∑j=1

aij(xi − xj).(21)

The following theorem characterizes the global convergenceof the dynamics (21); for a more detailed treatment we referto [8].

Theorem 26 (Distributed primal-dual algorithm). Considerthe distributed optimization problem (20) and the primal-dual dynamics (21) over a connected undirected weightedgraph with Laplacian matrix L. Assume that the optimizationproblem (19) has a solution x∗ ∈ Rk and that, for eachi ∈ {1, . . . , n}, fi is convex and ∇2fi(x

∗) � 0. Then(i) system (21) is weakly contracting with respect to ‖ · ‖2,

(ii) each trajectory (x(t), ν(t)) of (21) converges exponen-tially to (1n ⊗ x∗,1n ⊗ ν∗), where ν∗ =

∑ni=1 νi(0).

Proof. We set x = (x1, . . . , xn)T and ν = (ν1, . . . , νn)T,and define h(x) =

∑ni=1 fi(xi). Then algorithm (21) can be

written as

x = −∇h(x)− (L⊗ Ik)ν,

ν = (L⊗ Ik)x.(22)

Regarding (i), let (x, ν) = FPD(x, ν). Then DFPD(x, ν) =[−∇2h(x) −L⊗ IkL⊗ Ik 0

]and µ2(DFPD(x, ν)) = 0. Therefore,

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 12

the system (21) is weakly-contracting with respect to the `2-norm. Regarding part (ii), let t 7→ (x(t), ν(t)) be a trajectoryof the system (21). Note that, for every u ∈ Rk, we have (1n⊗u)Tν(t) = (1n ⊗ u)T(L ⊗ Ik)x(t) = 0nk. This implies that∑ni=1 νi(t) =

∑ni=1 νi(0) and the subspace V ⊆ Rnk × Rnk

defined by

V ={

(x, ν) ∈ R2nk∣∣ n∑i=1

νi = 0}

is an invariant subspace of the system (21). For the Laplacianmatrix L, define RV by equation (7). Define new coordinates(x, v) on V by x = x and ν = (RV ⊗ Ik)ν. Thus, thedynamical system (21) restricted to V can be written in thenew coordinate (x, v) as

˙x = −∇h(x)− (LRTV ⊗ Ik)ν,

˙ν = (RVL⊗ Ik)x.(23)

Let ( ˙x, ˙ν) := FPD(x, ν) Note that (x, v) = (1n⊗x∗,0(n−1)k)is an equilibrium point of the dynamical system (23) and

DFPD(x, ν) =

[−∇2h(x) −LRT

V ⊗ IkRVL⊗ Ik 0

].

Again we note that µ2(DFPD(x, ν)) = 0, for every (x, ν) ∈V , and therefore, the system (23) is weakly-contracting withrespect to the `2-norm. Moreover, −∇2h(1n ⊗ x∗) ≺ 0and Ker(LRT

V ⊗ Ik) = ∅. Thus, by [7, Lemma 5.3], thematrix DFPD(1n ⊗ x∗, 0) is Hurwitz and the equilibriumpoint (1n ⊗ x∗,0(n−1)k) is locally asymptotically stable forthe dynamical system (23). Theorem 18(iii) applied to theweakly-contracting system (23) implies that (1n⊗x∗,0(n−1)k)is a globally asymptotically stable equilibrium point, i.e.,limt→∞(x(t), ν(t)) = (1n ⊗ x∗,0(n−1)k). Therefore, we getlimt→∞ x(t) = 1n ⊗ x∗ and

0nk = (RTV ⊗ Ik) lim

t→∞ν = lim

t→∞(RTVRV ⊗ Ik)ν(t)

= limt→∞

((In − 1

n1n1Tn)⊗ Ik

)ν(t)

= limt→∞

(ν(t)− 1n ⊗ ν∗) ,

where the last equality holds because 1n

∑ni=1 νi(t) = ν∗. As

a result, we get limt→∞ ν(t) = 1n ⊗ ν∗.

C. Strongly coupled oscillators are semi-contractingAs third application, we study synchronization phenomena

in networks of identical diffusively-coupled oscillators. Con-sider n agents connected through a weighted undirected graphwith Laplacian matrix L. Suppose the agents have identicalinternal dynamics described by the time-varying vector fieldf : R≥0 × Rk → Rk. Then the overall network dynamics is

xi = f(t, xi)−n∑j=1

aij(xi − xj), i ∈ {1, . . . , n}. (24)

Next, we introduce a novel useful norm. For p ∈ [1,∞],define the (2, p)-tensor norm ‖ · ‖(2,p) on Rnk ' Rn⊗Rk by:

‖u‖(2,p) = inf{( r∑

i=1

‖vi‖22‖wi‖2p) 1

2∣∣ u =

r∑i=1

vi ⊗ wi}.

(25)

The well-definiteness and properties of the (2, p)-tensor normare studied in Lemma 34 in the Appendix. The (2, p)-tensornorm is closely related to, but different from, the well-known projective tensor product norm (see [37, Chapter 2]for definition and properties of this well-known norm). The(2, p)-tensor norm is also different from the mixed globalnorm introduced in [36] for hierarchical analysis of networksystems. For the symmetric Laplacian L, let RV be definedby equation (7).

Theorem 27 (Networks of diffusively-coupled oscillators).Consider the network of diffusively-coupled identical oscil-lators (24) over a connected weighted undirected graph withLaplacian matrix L. Let Q ∈ Rk×k be an invertible matrixand p ∈ [1,∞]. Suppose there exists a positive c such that,for every (t, x) ∈ R≥0 × Rk,

µp,Q(Df(t, x)) ≤ λ2(L)− c. (26)

Then,(i) system (24) is semi-contracting with rate c with respect

to ‖ · ‖(2,p),RV⊗Q,(ii) system (24) achieves global exponential synchronization

with rate c, i.e., for each trajectory (x1(t), . . . , xn(t)) andfor each pair i, j, the distance ‖xi(t)−xj(t)‖2 vanishesexponentially fast with rate c.

Proof. Set x = (xT1 , . . . , xTn)T ∈ Rnk and define xave :=

1n

∑ni=1 xi. The dynamics (24) can be written as:

x = F (t, x)− (L⊗ Ik)x, (27)

where F (t, x) = [fT(t, x1), . . . , fT(t, xn)]T. Moreover,

DF (t, x) =

Df(t, x1) . . . 0k×k...

. . ....

0k×k . . . Df(t, xn)

∈ Rnk×nk.

Next, we show that µ(2,p),RV⊗Q(DF (t, x) − L ⊗ Ik) ≤ −c.First note that

µ(2,p),RV⊗Q(DF (t, x)− L⊗ Ik)

≤ µ(2,p),RV⊗Q(DF (t, x)) + µ(2,p),RV⊗Q(−L⊗ Ik).

From Lemma 9, recall that R†V = RTV(RVR

TV)−1 = RT

V andRVLR

TV = Λ = diag(λ2(L), . . . , λn(L)). Using Lemma 6(ii),

this implies that

µ(2,p),RV⊗Q(−L⊗ Ik) = µ(2,p)(−RVLRTV ⊗QQ−1)

= µ(2,p)(−Λ⊗ Ik) ≤ −λ2(L). (28)

where the last inequality holds by Lemma 34(v). Moreover,we have RVRT

V = In−1 and RV ⊗Q = (RV ⊗ Ik)(In ⊗Q).Thus, Lemma 6(ii) and Lemma 34(iv) imply that

µ(2,p),RV⊗Q(DF (t, x))

= µ(2,p),RV⊗Ik((In ⊗Q)DF (t, x)(In ⊗Q−1))

≤ µ(2,p)((In ⊗Q)DF (t, x)(In ⊗Q−1))

Note that

Γ := (In ⊗Q)DF (t, x)(In ⊗Q−1) =

Γ1 . . . 0k×k...

. . ....

0k×k . . . Γn

,

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 13

where Γi = QDf(t, xi)Q−1 ∈ Rk×k, for every i ∈

{1, . . . , n}. In turn, using Lemma 34(v),

µ(2,p),RV⊗Q(DF (t, x)) ≤ µ(2,p)(Γ)

≤ maxi∈{1,...,n}

{µp(Γi)} = maxi∈{1,...,n}

µp,Q(Df(t, xi)). (29)

Thus, combining (28) and (29), we get

µ(2,p),RV⊗Q(DF (t, x)− L⊗ Ik)

≤ maxi∈{1,...,n}

µp,Q(Df(t, xi))− λ2(L).

Using (26), for every t ∈ R≥0 and every x ∈ Rnk,

µ(2,p),RV⊗Q(DF (t, x)− L⊗ Ik) ≤ −c,

This implies that the system (24) is semi-contracting withrespect to ‖ · ‖(2,p),RV⊗Q with rate c. It is easy to see that

Ker ‖ · ‖(2,p),RV⊗Q = Ker(RV ⊗Q) = S

where S = span{1n ⊗ u | u ∈ Rk} is shifted-invariant underthe dynamical system (37). Thus, using Theorem 13(ii), everytrajectory of (37) converges exponentially with rate c to S.Now, define P = In − 1

n1n1Tn and note that P = PT is

the orthogonal projection onto 1⊥n and KerP = span(1n).Note that P ⊗ Ik is the orthogonal projection onto S⊥. Thus,Theorem 13(ii) implies that limt→∞ ‖(P⊗Ik)x(t)‖2 = 0 withrate c. Moreover, we know that

(P ⊗ Ik)x(t) = x(t)− 1n (1n1T

n ⊗ Ik)x(t)

= x(t)− (1n ⊗ xave(t)).

As a result, limt→∞ ‖x(t)− (1n⊗xave(t))‖2 = 0 with rate c,which in turn means that limt→∞ ‖xi(t) − xj(t)‖2 = 0 withrate c, for every i, j ∈ {1, . . . , n}.

Remark 28 (Comparison with literature). For networks ofdiffusively-coupled identical oscillators, condition (26) hastwo unique features: (i) it is based on matrix measures inducedby weighted `p-norm, for p ∈ [1,∞], thus extending theQUAD-based global synchronization conditions in [26], [11]and the `2-norm-based synchronization condition in [3]; and(ii) it is applicable to networks with arbitrary undirectedtopology, thus generalizing the synchronization conditionsdeveloped for specific network topologies in [2]. 4

Remark 29 (Diffusively-coupled linear systems). GivenA ∈ Rk×k and a weighted undirected graph, consider thediffusively-coupled linear identical systems:

xi = Axi −n∑j=1

aij(xi − xj), i ∈ {1, . . . , n}. (30)

It is known that the system (30) achieves synchronization ifand only if A − λ2(L)Ik is Hurwitz, e.g., see [5, Theorem8.4(ii)]. For a fixed p ∈ [1,∞], condition (26) reads

µp,Q(A) < λ2(L).

Using Lemma 7(ii), we get that infQ invertible µp,Q(A) =α(A). Thus, per Theorem 27(ii), if α(A) < λ2(L), thenthe diffusively-coupled linear identical systems (30) achievesynchronization with exponential rate λ2(L) − µp,Q(A). In

other words, condition (26) recovers the exact threshold ofsynchronization for diffusively-coupled linear systems. Wenote that the contraction-based synchronization conditions fordiffusively-coupled identical oscillators proposed in [3], [11],[2] do not recover this threshold for the linear systems. 4

VII. CONCLUSION

In this paper we have provided multiple analytic extensionsof the basic ideas in contraction theory, including advancedresults on semi-contracting, weakly-contracting and doubly-contracting systems. We have also illustrated how to applyour results to various network systems. Possible directionsfor future work include extensions to discrete-time and hybridsystems, to differential-geometric treatments, and to infinite-dimensional systems.

VIII. ACKNOWLEDGMENTS

The authors are very thankful to Zahra Aminzare for helpfulsuggestions about the proof of Theorem 27 and to SamuelCoogan for helpful suggestions about Theorem 13. The thirdauthor wishes to also thank Mario Di Bernardo, GiovanniRusso, Rodolphe Sepulchre, and John W. Simpson-Porco forinspiring discussions about contraction theory.

APPENDIX ASEMI-CONTRACTION VS. HORIZONTAL CONTRACTION

In this appendix, we study the connection between thesemi-contraction setting in Section III and the horizontalcontraction framework developed in [16]. We consider theEuclidean space Rn equipped with a semi-norm |||·||| and definea Finsler structure on Rn compatible with the semi-norm |||·|||.Specifically, assume H is a vector subspace of Rn such that

Rn = Ker |||·||| ⊕ H. (31)

Let πH : Rn → Rn denote the oblique projection onto Hparallel to Ker |||·||| and consider a norm ‖ · ‖ on Rn such that

‖πH(v)‖ = |||v|||, for all v ∈ Rn.

It is easy to see that the manifold Rn with the norm ‖ · ‖ isa Finsler manifold. Considering the decomposition (31) as avertical/horizontal decomposition of the tangent space Rn [16,Definition 4], the associated horizontal projection is given bythe oblique projection πH [16, Theorem 4] and the associatedpseudo-distance is given by d(x1, x2) = |||x1 − x2||| [16].Thus, we can restate [16, Theorem 4], for the Finsler manifold(Rn, ‖ · ‖) with the vertical/horizontal decomposition (31) asfollows.

Theorem 30 (Horizontal contraction on Euclidean spaces[16, Theorem 4]). Consider the dynamical system (1) on theFinsler manifold (Rn, ‖ · ‖) and assume that there existsc > 0 such that the following conditions hold, for every(t, x) ∈ R≥0 × Rn:

µ|||·|||(Df(t, x)) ≤ −c, (32)πHDf(t, x) = Df(t, x)πH, (33)

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 14

Then, there exists K ≥ 1 such that

|||φ(t, x0)− φ(t, y0)||| ≤ Ke−ct|||x0 − y0|||, for all t ≥ 0.

Now, we compare Theorem 13(i) with Theorem 30.

Remark 31 (Comparison). (i) The commutativity condi-tion (33) in Theorem 30 depends on the choice of thehorizontal subspace H in the decomposition (31). On theother hand, the infinitesimally invariant condition (P1)in Theorem 13(i) only depends on Ker |||·||| and is inde-pendent of this decomposition. Moreover, one can showthat condition (33) is a stronger requirement than theinfinitesimally invariance condition (P1). Assume thatcondition (33) holds, i.e.,

πHDf(t, x) = Df(t, x)πH, for all (t, x) ∈ Rn.

Then, for every v ∈ Ker |||·||| and every (t, x) ∈ R≥0×Rn,

πHDf(t, x)v = Df(t, x)πHv = Df(t, x)0n = 0n.

This implies that Df(t, x)v ∈ Ker |||·|||, for every (t, x) ∈R≥0 × Rn. As a result, if condition (33) holds then wehave Df(t, x) Ker |||·||| ⊆ Ker |||·|||, for every (t, x) ∈R≥0 × Rn, which means that Ker |||·||| is infinitesimallyinvariant under the dynamics (1). Example 32 belowshows that, in general, the converse is not true andinfinitesimal invariance of Ker |||·||| does not imply thecommutativity condition (33).

(ii) Theorem 30 provides a weaker contractivity result com-pared to Theorem 13(i). In fact, Theorem 30 guaranteesthat there exists K ≥ 1 such that

|||φ(t, x0)− φ(t, y0)||| ≤ Ke−ct|||x0 − y0|||, for all t ≥ 0.

On the other hand, Theorem 13(i) ensures semi-contraction of every two trajectories with respect to |||·|||,i.e.,

|||φ(t, x0)− φ(t, y0)||| ≤ e−ct|||x0 − y0|||, for all t ≥ 0.

Example 32. Consider the following linear system on R2:

x1 = −x1 := f1(x1, x2),

x2 = x1 − 2x2 := f2(x1, x2)(34)

with the semi-norm defined by∣∣∣∣∣∣(x1, x2)T

∣∣∣∣∣∣ = |x1|, for every(x1, x2)T ∈ R2. Note that Ker |||·||| = span{(0, 1)T}. Weconsider the vertical/horizontal decomposition (31) with H =Ker |||·|||⊥ = span{(1, 0)T}. Then, the horizontal projection is

given by πKer |||·|||⊥ =

[1 00 0

]. Note that Df(t, x)

[0 1

]T=[

0 −2]T

and therefore Ker |||·||| is infinitesimally invariantunder the system (34). However, the system (34) does notsatisfy condition (33), since

πKer |||·|||⊥Df(x1, x2) =

[−1 00 0

]6=[−1 01 0

]= Df(x1, x2)πKer |||·|||⊥

APPENDIX BUSEFUL LEMMAS

Lemma 33. Let A ∈ Cn×n and R ∈ Ck×n is a full rankmatrix such that Ker(R) is invariant under A. Then

specKerR⊥(AH) = spec(RAHR†).

Proof. Let λ ∈ specKerR⊥(AH). This means that there existv ⊥ Ker(R) and λ ∈ C such that AHv = λv. Since R†R is anorthogonal projection and v ⊥ Ker(R), there exists x ∈ Cnsuch that v = R†Rx. Thus, AHR†Rx = λR†Rx. Multi-plying both side by R, we get RAHR†Rx = λRR†Rx =λRx. This means that λ ∈ spec(RAHR†). Thus, we havespecKerR⊥(AH) ⊆ spec(RAHR†). On the other hand, sinceAKer(R) ⊆ Ker(R) and R is a full-rank matrix, we have|specKerR⊥(AH)| = k. It is easy to see that since RAHR† ∈Ck×k, we have

∣∣spec(RAHR†)∣∣ =

∣∣specKerR⊥(AH)∣∣ = k.

As a result specKerR⊥(AH) = spec(RAHR†).

Lemma 34 (Properties of (2, p)-tensor norms). Consider theidentification Rnk ' Rn ⊗Rk and the (2, p)-tensor norm de-fined in (25). Let A ∈ Rnk×nk, Λ = blkdg(Λ1, . . . ,Λn) suchthat Λi ∈ Rk×k, for every i ∈ {1, . . . , n} and U ∈ R(n−1)×n

is such that UUT = In−1, then

(i) (2, p)-tensor norm is well-defined;(ii) ‖U ⊗ Ik‖(2,p) = ‖UT ⊗ Ik‖(2,p) = 1;

(iii) ‖Λ‖(2,p) ≤ maxi ‖Λi‖p;(iv) µ(2,p),U⊗Ik(A) ≤ µ(2,p)(A);(v) µ(2,p)(Λ) ≤ maxi µp(Λi).

Proof. Regarding part (i), we follow the argument in [37,Proposition 2.1] for projective tensor product norm to showthat (2, p)-tensor norm is a norm. We first show that, for everyc ∈ R, we have ‖cu‖(2,p) = |c|‖u‖(2,p). The proof is trivial forc = 0. Let c 6= 0 and u =

∑ri=1 v

i⊗wi be a representation ofu. Then we have cu =

∑ri=1 cv

i ⊗ wi. Using definition (25),this implies that ‖cu‖(2,p) ≤ |c|‖u‖(2,p). Since c 6= 0, wecan deduce that ‖u‖(2,p) ≤ (1/|c|) ‖cu‖(2,p). Thus, we get‖cu‖(2,p) = |c|‖u‖(2,p).

Now we show that ‖ · ‖(2,p) satisfies triangle inequality. Letu, z ∈ Rn ⊗ Rk and let ε > 0. Then by definition (25), thereexist representations u =

∑ri=1 v

i⊗wi and z =∑si=1 x

i⊗yisuch that ( r∑

i=1

‖vi‖22‖wi‖2p) 1

2 ≤ ‖u‖(2,p) +ε

2,

( s∑i=1

‖xi‖22‖yi‖2p) 1

2 ≤ ‖z‖(2,p) +ε

2.

Note that∑si=1 x

i⊗ yi +∑ri=1 v

i⊗wi is a representation ofu+ z. As a result, we have

‖u+ z‖(2,p) ≤( s∑i=1

‖xi‖22‖yi‖2p +

r∑i=1

‖vi‖22‖wi‖2p) 1

2

≤( s∑i=1

‖xi‖22‖yi‖2p) 1

2 +( r∑i=1

‖vi‖22‖wi‖2p) 1

2

≤ ‖z‖(2,p) + ‖u‖(2,p) + ε

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 15

Since ε can be chosen arbitrarily small, then we have ‖u +z‖(2,p) ≤ ‖z‖(2,p) + ‖u‖(2,p).

Finally, we show that if ‖u‖(2,p) = 0, then we have u =0nk. Since ‖u‖(2,p) = 0, there exists a representation u =∑ri=1 v

i⊗wi such that∑ri=1 ‖vi‖22‖wi‖2p ≤ ε. Let φ : Rn →

R and ψ : Rk → R be two linear functionals. Then we have∑ri=1(φ(vi))2(ψ(wi))2 ≤ ε‖φ‖22‖ψ‖2p. Since this inequality

holds for every ε > 0, then we have∑ri=1(φ(vi))2(ψ(wi))2 =

0. This implies that, for every i ∈ {1, . . . , r}, either φ(vi) = 0or ψ(wi) = 0. Thus,

∑ri=1 φ(vi)ψ(wi) = 0. Now using [37,

Proposition 1.2], we get u = 0nk.Regarding part (ii), note that, for u ∈ Rn ⊗ Rk with a

representation u =∑ri=1 v

i ⊗ wi,

‖(U ⊗ Ik)u‖2(2,p) =∥∥ r∑i=1

(Uvi)⊗ wi∥∥2(2,p)

≤r∑i=1

‖Uvi‖22‖wi‖2p ≤r∑i=1

‖vi‖22‖wi‖2p,

where the first inequality holds by definition of (2, p)-tensor norm and the second inequality holds because, forevery i ∈ {1, . . . , r}, we have ‖Uvi‖2 ≤ ‖U‖2‖vi‖2and ‖U‖2 = maxi∈{1,...,n−1} λi(UU

T) = 1. Since theabove inequality holds for every representation of u, then‖(U ⊗ Ik)u‖(2,p) ≤ ‖u‖(2,p), for every u ∈ Rn ⊗ Rk. Thisimplies that ‖U ⊗ Ik‖(2,p) ≤ 1. Similarly, one can show that‖UT ⊗ Ik‖(2,p) ≤ 1. However, (U ⊗ Ik)(UT ⊗ Ik) = Ink,this implies that ‖U ⊗ Ik‖(2,p)‖UT⊗ Ik‖(2,p) ≥ 1 and in turn‖U ⊗ Ik‖(2,p) = ‖UT ⊗ Ik‖(2,p) = 1.

Regarding part (iii), for u ∈ Rn⊗Rk with a representationu =

∑ri=1 v

i ⊗ wi, we have Λu =∑ri=1

∑nj=1 v

ijej ⊗ Λjw

i,where ek is the kth standard basis in Rn, for every k ∈{1, . . . , n}. This implies that

‖Λu‖2(2,p) =∥∥ r∑i=1

n∑j=1

vijej ⊗ Λjwi∥∥2(2,p)

≤r∑i=1

n∑j=1

|vij |22‖Λjwi‖2p

≤r∑i=1

n∑j=1

|vij |22‖Λj‖2p‖wi‖2p

≤ maxi‖Λi‖2p

r∑i=1

n∑j=1

|vij |22‖wi‖2p

= maxi‖Λi‖2p

r∑i=1

‖vi‖22‖wi‖2p,

where the first inequality holds by the definition of (2, p)-tensor norm. For the second inequality, we used ‖Λjwi‖p ≤‖Λj‖p‖wi‖p, for every i ∈ {1, . . . , r} and every j ∈{1, . . . , n}. For the third inequality, we used the fact that‖Λj‖p ≤ maxi ‖Λi‖p. Finally, for the last equality, weused the fact that

∑nj=1 |vij |2 = ‖vi‖22. Since the above

inequality holds for every representation of u, then we have‖Λu‖(2,p) ≤ maxi ‖Λi‖p‖u‖(2,p), for every u ∈ Rn ⊗ Rk.This implies that ‖Λ‖(2,p) ≤ maxi ‖Λi‖p.

Regarding part (iv), note that U† = UT. As a result, usingLemma 6(ii), we have

µ(2,p),U⊗Ik(A) = µ(2,p)((U ⊗ Ik)A(UT ⊗ Ik))

= limh→0+

‖I(n−1)k + h(U ⊗ I)A(UT ⊗ Ik)‖(2,p) − 1

h.

Note that, for every h ∈ R≥0,

‖I(n−1)k − h(U ⊗ I)A(UT ⊗ Ik)‖(2,p)= ‖(U ⊗ Ik)(UT ⊗ Ik) + h(U ⊗ Ik)A(UT ⊗ Ik)‖(2,p)≤ ‖(U ⊗ Ik)‖(2,p)‖Ink + hA‖(2,p)‖(UT ⊗ Ik)‖(2,p)= ‖Ink + hA‖(2,p),

where for the first equality, we used the fact that UUT = In,and the last equality holds by part (ii). As a result, we getµ(2,p),U⊗Ik(A) ≤ µ(2,p)(A). Regrading part (v), note that

µ(2,p)(Λ) = limh→0+

‖Ink + hΛ‖(2,p) − 1

h.

On the other hand, we have Ink + hΛ = blkdg(Ik +hΛ1, . . . , Ik + hΛn). Thus, by part (iii), we have

µ(2,p)(Λ) = limh→0+

‖Ink + hΛ‖(2,p) − 1

h

≤ limh→0+

maxi ‖Ik + hΛi‖p − 1

h

= maxi

{limh→0+

‖Ik + hΛi‖p − 1

h

}= max

iµp(Λi).

APPENDIX CPROOF OF LEMMA 17

Proof of Lemma 17. Since the trajectory t 7→ x(t) is boundedin Rn, there exists r > 0 such that t 7→ x(t) is containedin B‖·‖(x(0), r) ∩ C. For every s ≥ 0, we define the setUs =

⋂t≥s(B‖·‖(x(t), r) ∩ C

). It is easy to show the family

of sets {Us}s≥0 satisfies the following monotonicity property:Us1 ⊆ Us2 , for s1 < s2. Additionally, we define the sets Uand V by

U =⋃

s≥0Us V =

⋃t≥0

(B‖·‖(x(t), r) ∩ C

).

Since t 7→ x(t) is bounded, the set V is bounded and thereforecl(V ) is a compact in Rn. Moreover, it is easy to check thatU ⊆ cl(V ). We set W = cl(U) and show that W is a non-empty compact convex bounded set with the property thatφ(t,W ) ⊆W , for every t ≥ 0.

(Step 1: W is non-empty). Consider 0 = t0 ≤ t1 < . . . < tnand define the set S =

⋂ki=0

(B‖·‖(x(ti), r) ∩ C

)Let t ≥ tk.

For every i ∈ {1, . . . , k}, we have

‖x(t)− x(ti)‖ ≤ ‖x(t− ti)− x(0)‖ ≤ r, (35)

where the first inequality holds because the system (14)is weakly contracting and the last inequality holds becauseB‖·‖(x(0), r) contains the trajectory t 7→ x(t). Therefore,the inequalities in (35) implies that x(t) ∈ S, for everyt ≥ 0. Thus S is non-empty. Now consider the family of sets

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 16

{(B‖·‖(x(t), r) ∩ C

)}t≥0. By the above argument, it is clear

that this family is inside the compact set cl(V ) and, for everyfinite set J ∈ R≥0, the intersection

⋂ti∈J

(B‖·‖(x(t), r) ∩ C

)is non-empty. Therefore, by [31, Theorem 26.9], for everys ≥ 0, the set Us is non-empty and, as a result, W is non-empty.

(Step 2: W is convex and compact). We start by showingthat U is convex. Note that intersection of any collection ofconvex sets is convex [4]. Therefore, Us is convex, for everys ≥ 0. Pick x1, x2 ∈ U and α ∈ [0, 1]. We show that αx1 +(1 − α)x2 ∈ U . By definition of U , there exists s1, s2 ≥ 0such that x1 ∈ Us1 and x2 ∈ Us2 . Using the monotonicityof the family {Us}s≥0, we have that x1, x2 ∈ U(s1+s2). SinceU(s1+s2) is convex, we have αx1+(1−α)x2 ∈ U(s1+s2) ⊆ U .This means that U is convex. Moreover, the closure of anyconvex set is again convex, thus W = cl(U) is convex. SinceU ⊆ V and V is compact, we have W = cl(U) ⊆ V . Thus,W is a closed subset of a compact set and it is compact.

(Step 3: W is invariant under (14)). Let y ∈ U . Bydefinition, there exists s ≥ 0 such that y ∈ Us. For everyη > 0 and every t ≥ s, we have

‖φ(η, y)− x(η + t)‖ ≤ ‖y − x(t)‖ ≤ r,

where the first inequality holds because the system (14) isweakly contracting and the last inequality holds because y ∈U . This implies that, for every η ≥ 0, we have φ(η, y) ∈ Uη+s.This implies that φ(η, y) ∈ U . So we have φ(η, U) ⊆ U , forevery η > 0. By a simple continuity argument and using thefact that W is closed, we get φ(η,W ) ⊆W .

APPENDIX DNETWORKS OF DIFFUSIVELY-COUPLED OSCILLATORS

In this appendix, we provide a slight generalization ofTheorem 27 for diffusively-coupled oscillators (24).

Theorem 35 (Networks of diffusively-coupled oscilla-tors–extended). Consider the network of diffusively-coupledidentical oscillators (24) over a connected weighted undirectedgraph with Laplacian matrix L. Let Q ∈ Rk×k be an invertiblematrix and p ∈ [1,∞]. Suppose there exists a positive c suchthat, for every (t, x) ∈ R≥0 × Rk,

µp,Q(Df(t, x)) ≤ λ2(L)− c. (36)

Then,(i) system (24) is semi-contracting with exponential rate c

with respect to ‖ · ‖(2,p),RV⊗Q,(ii) if t 7→ x(t) and t 7→ y(t) are trajectories of (24) starting

from x0 ∈ Rnk and y0 ∈ Rnk, respectively, then for everyt ≥ 0,

‖y(t)− x(t)‖(2,p),RV⊗Q ≤ e−ct‖y0 − x0‖(2,p),RV⊗Q

(iii) system (24) achieves global exponential synchronizationwith rate c, more specifically, for each trajectory x(t) =(x1(t), . . . , xn(t)), we have

limt→∞

‖x(t)− 1n ⊗ xave(t)‖2 = 0,

where xave(t) = 1n

∑ni=1 xi(t).

Proof. Set x = (xT1 , . . . , xTn)T ∈ Rnk and define P ∈

R(n+1)×n by

P =

n−1n − 1

n . . . − 1n

− 1n

n−1n . . . − 1

n...

.... . .

...− 1n − 1

n . . . − 1n

It is easy to see that P† = [In,−1n]. Now we define the newvariable z := (P ⊗ Ik)x ∈ R(n+1)k. Since P†P = In, we getx = (P† ⊗ Ik)z. As a result, in the new coordinate chart, thedynamical system (24) can be written as

z = (P ⊗ Ik)F (t, (P† ⊗ Ik)z)− (PLP† ⊗ Ik)z := G(t, z).(37)

where F (t, x) = [fT(t, x1), . . . , fT(t, xn)]T. It is clear that

DG(t, z) = (P ⊗ Ik)DF (t, (P† ⊗ Ik)z)(P† ⊗ Ik)

+ (PLP† ⊗ Ik), (38)

where

DF (t, x) =

Df(t, x1) . . . 0k×k...

. . ....

0k×k . . . Df(t, xn)

∈ Rnk×nk.

Note that (P† ⊗ Ik)(1n ⊗ u) = 0nk, for every u ∈ Rk. Thus,it is easy to check that

Ker ‖ · ‖(2,p),RVP†⊗Q = Ker(RVP† ⊗Q) = S

where S = span{1n ⊗ u | u ∈ Rk}. Moreover, fromequation (38), it is clear that DG(t, z)S = 0nk, for every(t, z) ∈ R≥0 × Rn+1. This means that S is infinitesimallyinvariant under the dynamical system (37). Now, we show thatthe dynamical system (37) is semi-contracting with respect tothe semi-norm ‖ · ‖(2,p),RVP†⊗Q. Note that,

µ(2,p),RVP†⊗Q(DG(t, x)− PLP† ⊗ Ik)

= µ(2,p),RV⊗Q(DF (t, x)− L⊗ Ik) ≤ −c,

where the last inequality follows from Theorem 27(i). There-fore, the dynamical system (37) is semi-contracting withrespect to ‖ · ‖(2,p),RVP†⊗Q with rate c and the subspaceS is infinitesimally invariant under (37). Moreover, we haveP†P = In. This implies that, for every t ≥ 0,

‖x(t)− y(t)‖(2,p),RV⊗Q = ‖(P ⊗ Ik)(x(t)− y(t))‖(2,p),RVP†⊗Q.

Thus, using Theorem 13(i),

‖x(t)− y(t)‖(2,p),RV⊗Q = ‖(P ⊗ Ik)(x(t)− y(t))‖(2,p),RVP†⊗Q≤ e−ct‖(P ⊗ Ik)(x0 − y0)‖(2,p),RVP†⊗Q= e−ct‖x0 − y0‖(2,p),RV⊗Q.

This completes the proof of part (ii).Regarding part (iii), define t 7→ y(t) by the following

differential equation

y(t) = f(t, y(t)),

y(0) = 1n

n∑i=1

xi(0).

SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, MAY 18, 2020 17

Then it is clear that t 7→ 1n ⊗ y(t) is a solution of thedynamical system (24). By part (ii),

‖x(t)‖(2,p),RV⊗Q = ‖x(t)− 1n ⊗ y(t)‖(2,p),RV⊗Q

≤ e−ct∥∥∥∥∥x0 − 1n ⊗ 1

n

n∑i=1

xi(0)

∥∥∥∥∥(2,p),RV⊗Q

, (39)

where the first equality holds because 1n ⊗ y(t) ∈ S. Now,define P = In − 1

n1n1Tn and note that P = PT is the

orthogonal projection onto 1⊥n and KerP = span(1n). Notethat P ⊗ Ik is the orthogonal projection onto S⊥. Thus,equation (39) implies that limt→∞ ‖(P ⊗ Ik)x(t)‖2 = 0 withrate c. Moreover, we know that

(P ⊗ Ik)x(t) = x(t)− 1n (1n1T

n ⊗ Ik)x(t)

= x(t)− (1n ⊗ xave(t)).

As a result, limt→∞ ‖x(t)− (1n⊗xave(t))‖2 = 0 with rate c,which in turn means that limt→∞ ‖xi(t) − xj(t)‖2 = 0 withrate c, for every i, j ∈ {1, . . . , n}.

APPENDIX ETHE LOTKA–VOLTERRA MODEL

In this appendix, we study the Lotka–Volterra model withmutualistic interactions and prove that it is weakly contracting.The Lotka–Volterra model is given by

x = diag(x)(Ax+ r), (40)

where x = (x1, . . . , xn)T ∈ Rn≥0 is the vector of populations,A ∈ Rn×n is the mutual interaction matrix, and r > 0n isthe intrinsic growth rate. We assume A is Metzler, i.e., theinteractions between any two species are mutualistic.

Theorem 36 (Global asymptotic stability and Lyapunov func-tions). Consider the Lotka–Volterra model (40) with a Metzlerand Hurwitz interaction matrix A. Let the positive vectorv ∈ Rn>0 satisfy vTA < 0n. Then

(i) the open positive orthant Rn>0 is invariant,(ii) x∗ = −A−1r > 0n is the unique globally asymptotically

stable equilibrium point of (40) restricted to Rn>0,(iii) the following distance between any two trajectories x(t)

and z(t) is decreasing:

dLV(x(t), z(t)) =∑n

i=1vi| ln

(xi(t)/zi(t)

)|,

(iv) the following functions are global Lyapunov functions:

x 7→∑n

i=1vi| ln(xi/x

∗i )|, x 7→

∑n

i=1vi|(Ax+ r)i|.

Proof. We omit the proof of statement (i) in the interest ofbrevity. For x ∈ Rn>0, let yi = ln(xi) ∈ R, i ∈ {1, . . . , n} andwrite the Lotka–Volterra model (40) as

y = A exp(y) + r := fLVe(y), (41)

where y and its entry-wise exponential exp(y) are vectors inRn. Note that DfLVe(y) = A diag(exp(y)) is Metzler sinceexp(y) > 0n. Since v ∈ Rn>0 satisfies vTA < 0n, there existsc > 0 such that vTA ≤ −cvT. Therefore, for every y ∈ Rn,

vTDfLVe(y) = vTAdiag(exp(y)) < −cvT diag(exp(y)) ≤ 0.

We now recall [9] that, for a Metzler matrix M , a positivevector v and a scalar b,

vTM ≤ bvT ⇐⇒ µ1,diag(v)−1(M) ≤ b.

This equivalence implies that fLVe is weakly contracting inits domain. After a change of coordinates, this establishesstatement (iii). At the equilibrium point x∗ = −A−1r > 0n,that is, at exp(y∗) = −A−1r ∈ Rn>0, we know

vTDfLVe(y∗) ≤ −cvT diag(exp(y∗))

= −cvT diag(−A−1r)

≤ −c(

mini∈{1,...,n}

(−A−1r)i)vT,

=⇒ µ1,diag(v)

(DfLVe(y

∗))≤ −c min

i∈{1,...,n}(−A−1r)i.

We now invoke Theorem 18(v): fLVe is weakly contractingover the entire Rn and has strictly negative matrix measure atthe equilibrium y∗. Therefore, y∗ is the unique globally expo-nentially stable equilibrium with global Lyapunov functions

y 7→ ‖y − y∗‖1,diag(v), and y 7→ ‖fLVe(y)‖1,diag(v).

Facts (ii) and (iv) follow from a change of coordinates.

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