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A Robust Stability Condition for Subhomogeneous Cooperative

Time-delay Systems*

Vahid S. Bokharaie1and Oliver Mason2

Abstract— In this manuscript, we present conditions for

delay-independent stability of subhomogeneous cooperative

time-delay systems. We consider the case where the time-delay

system has a unique equilibrium at the origin and the case

where it has a unique equilibrium in the interior of the positive

orthant and prove global delay-independent stability results in

both cases.

I. INTRODUCTION

A common hypothesis in the modelling of physical sys-

tems is to assume that the future behaviour of the system

depends only on the present value of the states of the

system. Such models, when the number of states is ﬁnite, fall

into the category of ordinary differential equations, ODEs.

Sometimes, we face situations in which the inﬂuence of

the past states should also be considered. For example, in

population dynamics, time-delay should be added to the

model to account for hatching and maturation periods [9].

A familiar example in control engineering is the delay in

measuring the states of a plant due to technological or

physical limitations. Such systems are called delayed or time-

delay systems. The great number of monographs written

on the subject, particularly in recent years, is evidence for

the continuing interest of mathematicians and engineers in

delayed systems. For example, look at [22] [1] [13] and

references therein.

While modelling a time-delay system, based on the infor-

mation we have on τ, the value of the delay, we are usually

faced with three situations:

1) τis ﬁxed and its exact value is known;

2) τis ﬁxed but its exact value is unknown;

3) τis time-variable, i.e., τ=τ(t)for t≥[−τ, ∞).

In this manuscript, we are interested in the second case, i.e.,

when we know τis ﬁxed, but we do not know its precise

value. The stated results hold for all positive (but ﬁxed)

values of time-delay. Such stability results are called delay-

independent stability conditions.

Time-delay in the context of cooperative or monotone

systems has been studied since the late 1970s and early

1980s. The ﬁrst papers that introduced the quasimonotone

condition and the basic properties of monotone functional

differential equations are [12], [14] and [21]. In the past

*This work was supported by the Irish Higher Educational Authority

(HEA) PRTLI 4 Network Mathematics Grant and Science Foundation

Ireland award 09/SRC/E1780.

1V. S. Bokharaie is with Mathematics Applications Consortium

for Science and Industry (MACSI), Univeristy of Limerick, Ireland.

vahid.bokharaie@ul.ie

2O. Mason is with Hamilton Institute, National University of Ireland

Maynooth, Ireland. oliver.mason@nuim.ie

two decades, different aspects of monotone and positive

time-delay systems have been investigated and developed

in different directions. In [6], a condition for asymptotic

stability of the equilibrium of a class of linear functional

differential equations is presented. Similar results are also

presented for neutral linear functional differential equations

in [19], for positive linear Volterra equations in [20] and

for Linear Volterra-Stieltjes Differential Systems in [18] and

most recently, for a class of positive linear integro-differential

equations with inﬁnite delay in [17]. Also, systems with

delay in inputs are considered in [5]. The results of [6] are

extended to cooperative time-delay systems which are homo-

geneous with respect to standard dilation map in [16], and

to cooperative time-delay systems which are homogeneous

with respect to general dilation maps in [3].

The layout of the paper is as follows. In Section II, we

introduce notation as well as deﬁnitions and results that are

needed throughout the paper and review the basic properties

of time-delay systems with more emphasis on the properties

of monotone time-delay systems. In Section III, we present

delay-independent stability condition for subhomogeneous

cooperative systems that have a unique equilibrium at the

origin. In Section IV, we extend the results presented in

Section III to the case where the time-delay system has a

unique equilibrium in the interior of the positive orthant. In

Section V, we state our conclusions.

II. BACKGROU ND

Throughout this manuscript, Rdenote the ﬁeld of real

numbers. Rn×ndenotes the space of n×nmatrices with

real entries. Rnis the space of column vectors of size n

with real entries. For x∈Rnand i= 1, . . . , n,xidenotes

the ith coordinate of x. We deﬁne Rn

+:= {x∈Rn:xi≥

0,1≤i≤n}.Rn

+is called the positive orthant of Rn.

The interior of Rn

+is deﬁned as int (Rn

+) := {x∈Rn:

xi>0,1≤i≤n}. The boundary of Rn

+is deﬁned as

bd (Rn

+) := Rn

+\int (Rn

+).

We use 0to refer to a vector or matrix of appropriate

dimensions with all entries equal to zero. A real n×n

matrix A= (aij )is Metzler if its off-diagonal entries are

nonnegative. For vectors x, y ∈Rn, we write: x≥yif

xi≥yi;x > y if x≥yand x6=y; and xyif xi> yi

for 1≤i≤n.

Let x0be a point in Rn. An open ball B(x0;)of radius

> 0is often called an -neighbourhood of x0. By a

neighbourhood of x0, we mean any subset of Rnwhich

contains an -neighbourhood x0[11, p. 19]. Let Uand W

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be subsets of Rn. The set Wis a neighbourhood of the set

U, if it contains a neighbourhood of every point of U.

A. Basic Properties of Time-delay Systems

Mathematical representations of time-delay systems fall

under the category of functional differential equations.

The states of the functional differential equations we con-

sider in this manuscript belong to C([−τ, 0],Rn), where

C([a, b],Rn)is the space of continuous functions mapping

the interval [a, b]into Rnfor a, b ∈Rwith b > a.

Let tf∈Rwith tf≥0. For x∈C([−τ, tf],Rn)and

t∈[0, tf], we deﬁne x(t)∈C([−τ, 0],Rn)as follows:

x(t)(θ) = x(t+θ),θ∈[−τ, 0]

Let Ωbe a subset of C([−τ, 0],Rn). Then a functional

differential equation can be represented by

˙x(t) = h(x(t))(1)

with h: Ω 7→ Rn. A function xis said to be a solution or

trajectory of (1) on [−τ, tf), if there is tf>0such that

x∈C([−τ, tf),Rn),x(t)∈Ωand x(t)satisﬁes (1) for all

t∈[0, tf)[7].

For a given φ∈C([−τ, 0],Rn), we say x(t, φ)is a

solution of (1) with initial condition φ, or simply a solution

through φ, if there is a tf>0, such that x(t, φ)is a solution

of (1) on t∈[−τ, tf)and x(0) (θ) = φ(θ)for all θ∈[−τ, 0]

[7].

Similar to ordinary differential equations, we are only

interested in those cases where the solution of a functional

differential equation exists and is unique. In this manuscript,

we always assume that the conditions for existence and

uniqueness of the solutions of the system (1) are satisﬁed

(look at [7, Section 2.2] for more details on these conditions).

B. Monotonicity

Of special interest for us in this manuscript are cooperative

systems and monotonicity. We start with the deﬁnition of

monotone systems.

Deﬁnition 2.1: Let Dbe an open subset of Rn. Suppose

D ⊂ Rnis a forward invariant set for the system

˙x(t) = H(x), x(0) = x0(2)

The system (2) is monotone in Dif and only if ∀x0, y0∈ D

with x0≤y0, it holds that x(t, x0)≤x(t, y0),∀t > 0.

There is an easy way to check monotonicity of a system

which is due to [10]. It is called the Kamke Condition.

Deﬁnition 2.2 (Kamke Condition): The vector ﬁeld f:

D 7→ Rnon an open subset Dof Rnis said to be of type K or

to satisfy the Kamke Condition, if for each i,Hi(a)≤Hi(b)

for any two points aand bin Dsatisfying a≤band ai=bi.

The following Proposition, which is a restatement of Propo-

sition 3.1.1 in [23], links the Kamke condition with mono-

tonicity.

Proposition 2.1: Let Hbe of type K in an open subset D

of Rn. Then system (2) is monotone.

In the following sections, we deal with monotone time-

delay systems. Monotonicity in functional differential equa-

tions is deﬁned as follows.

Deﬁnition 2.3 (Monotonicity): Let Ωbe a subset of

C([−τ, 0],Rn)and let x(t, φ)represent the trajectory of the

system (1) with respect to initial condition φ∈Ωat time

t. Then the system (1) is said to be monotone if for every

φ, ψ ∈Ω, satisfying

φ(θ)≤ψ(θ),for all −τ≤θ≤0

we have:

x(t, φ)≤x(t, ψ)for all t > 0.

It is known that the quasimonotone condition, provides a

sufﬁcient condition for monotonicity of the system (1) [23,

Section 5.1]. The quasimonotone condition can be stated as

follows.

Deﬁnition 2.4 (Quasimonotone Condition): Whenever

φ(θ)≤ψ(θ), for all −τ≤θ≤0and φi(0) = ψi(0) holds

for some i, then hi(φ)≤hi(ψ).

The next theorem, which is a restatement of Theorem 5.1.1

in [23], formally states the relation between the quasimono-

tone condition and monotonicity of the system (1).

Theorem 2.2: Let Ωbe a subset of C([−τ , 0],Rn). The

system (1) is monotone in Ω, if the quasimonotone condition

is satisﬁed for every φ, ψ ∈Ω.

¿From now on, for any p∈Rnwe deﬁne ˆp∈

C([−τ, 0],Rn)to be:

ˆp(θ)≡p, ∀θ∈[−τ, 0] (3)

Let Ωbe an open subset of C([−τ, 0],Rn). The equilibria

of (1) in Ωare those φ∈Ωsuch that

h(φ)=0 (4)

It can be proved easily that if φis an equilibrium of the

system (1), then φis a constant function [23, p. 78]. In other

words, the set of equilibria of (1) is given by

E={ˆp∈Ω : p∈Rnand h(ˆp)=0}(5)

If ˆp∈Ewith p= 0, then we say the system (1) has an

equilibrium at the origin. If p∈int (Rn

+), then we say the

system (1) has an equilibrium in the interior of the positive

orthant. If p∈bd (Rn

+), then we say the system (1) has an

equilibrium on the boundary of the positive orthant.

The following lemma, which plays a crucial role in the

proofs of the main results of the following sections, states

that there is a one-to-one correspondence between the equi-

libria of the system (1) and the equilibria of the following

system

˙y(t) = H(y(t)) (6)

where H:Rn7→ Rnand

H(y) = h(ˆy)(7)

with y∈Rnand ˆyas deﬁned in (3).

Lemma 2.3: Let h: Ω 7→ Rn, where Ωis an open subset

of C([−τ, 0],Rn). Let hsatisfy quasimonotone condition

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and let pbe an equilibrium of the system (6). Then ˆp, as

deﬁned in (3), is an equilibrium of the system (1).

Proof: The proof is based on discussions in Section

5.1 (pp. 77-78) of [23]. Let Ωbe an open subset of

C([−τ, 0],Rn). As already discussed, the set of equilibria

of the system (1) in Ωis:

E={ˆp∈Ω : p∈Rnand h(ˆp) = 0}

Since H(p) = h(ˆp),H(p) = 0 if h(ˆp) = 0. Therefore, we

can conclude that the equilibria of (1) consists of those ˆpfor

which pis an equilibrium of (6). This concludes the proof.

The following result follows immediately from Corollary

5.2.2 of [23].

Lemma 2.4: Let h: Ω 7→ Rnsatisfy the quasimonotone

condition in Ω, where Ωis an open subset of C([−τ, 0],Rn).

If v∈Rnis such that ˆv∈Ωand h(ˆv)≤0(h(ˆv)≥0), then

x(t, ˆv), solution of (1) through ˆv, is non-increasing (non-

decreasing) for all t≥0.

Note that since H(v) = h(ˆv), as deﬁned in (7), then H(v)≤

0implies h(ˆv)≤0and vice versa. Therefore, Lemma 2.4

can be also stated as follows.

Lemma 2.5: Let h: Ω 7→ Rnsatisfy the quasimonotone

condition in Ω, where Ωis an open subset of C([−τ, 0],Rn)

and let Hbe deﬁned as (7). Assume that there exists a vector

v≥0with H(v)≤0(H(v)≥0). Then the trajectory x(t, ˆv)

of the system (1) is non-increasing (non-decreasing) for all

t≥0.

We use Lemma 2.5 instead of Lemma 2.4 in the following

sections, because the statement of Lemma 2.5 is more

suitable for the framework we have chosen for our stability

conditions.

We also use the following theorem which is a simple

adaptation of the Theorem 1.2.1 in [23].

Theorem 2.6 (Convergence Criterion): Let h: Ω 7→ Rn

satisfy the quasimonotone condition in an open subset Ωof

C([−τ, 0],Rn). Let x(t, ˆx0), the trajectory of the system (1),

be bounded for all t≥0and for all ˆx0∈Ω. If x(t, ˆx0)≤

ˆx0(or x(t, ˆx0)≥ˆx0) for tbelonging to some non-empty

subinterval of (0,∞), then x(t, ˆx0)→ˆp∈Eas t→ ∞,

where Eis the set of all equilibria of the system (1).

C. Subhomogeneous Systems

In this manuscript, we deal with vector ﬁelds which are

subhomogeneous. Subhomogeneous vector ﬁelds are deﬁned

as follows [4].

Deﬁnition 2.1: Let Wbe a neighbourhood of Rn

+. A

vector ﬁeld H:W → Rnis subhomogeneous of degree

α > 0if H(λv)≤λαH(v), for all v∈Rn

+,λ∈R

with λ≥1. System (2) is subhomogeneous when His

subhomogeneous.

The class of subhomogeneous vector ﬁelds given above

includes different classes of vector ﬁelds. For example, it is

easy to check that linear and homogeneous vector ﬁelds with

respect to the standard dilation map are subhomogeneous.

The following lemma, shows another important class of

subhomogeneous vector ﬁelds.

Lemma 2.7: Let H:Rn7→ Rnbe a concave vector ﬁeld

such that H(0) ≥0. Then His subhomogeneous of degree

1.

Proof: Based on the deﬁnition of concave vector ﬁelds

[8, p. 534], we have

H(αx + (1 −α)y)≥αH(x) + (1 −α)H(y)(8)

for all 0< α < 1and for all x, y ∈Rn,x6=y. Considering

x6= 0 and y= 0, we have:

αH(x) + (1 −α)H(0) ≤H(αx)

Since H(0) ≥0and α < 1, we can conclude αH(x)≤

H(αx). Changing the variable to z=αx, we have:

αH(1

αz)≤H(z)⇒H(1

αz)≤1

αH(z)

We deﬁne λ= 1/α. Since α∈(0,1), we have λ > 1.

Therefore:

H(λz)≤λH (z)∀z∈Rn, λ ≥1

Note that the inequality is trivially true for λ= 1. This

means H(·)is a subhomogeneous vector ﬁeld of degree 1.

The deﬁnition of subhomogeneity can be easily extended

to time-delay systems.

Deﬁnition 2.2: Let Ωbe a subset of C([a, b],Rn). A

vector ﬁeld h: Ω 7→ Rnis subhomogeneous of degree α > 0

if h(λφ)≤λαh(φ), for all φ∈ω,λ∈Rwith λ≥1. System

(1) is subhomogeneous when his subhomogeneous.

III. SUBHOMOGENEOUS COO PE RATIV E TIME-D EL AY

SYS TE M S WI TH A UNIQUE EQUILIBRIUM AT THE ORIGIN

The nonlinear time-delay system we consider in this

manuscript, is as follows:

˙x(t) = f(x(t)) +

m

X

i=1

g(i)(x(t−τi)) τi≥0(9)

for all i= 1,· · · , m where f:W → Rnand g(i):W → Rn

are vector ﬁelds on a neighbourhood Wof Rn

+and τi>0

are delays for i= 1,· · · , m. We deﬁne:

τ:= max

1≤i≤mτi

We also deﬁne h:C([−τ, 0],Rn)7→ Rnto be

h(φ) = f(φ(0)) +

m

X

i=1

g(i)(φ(−τi)) (10)

In this section and the next, fand g(i), for i= 1,· · · , m,

satisfy the following assumption.

Assumption 3.1: For i= 1,· · · , m, we have:

•fand g(i)are C1on W;

•fis cooperative in Rn

+and g(i)is non-decreasing in

Rn

+.

Since fand g(i), for i= 1,· · · , m, are C1the conditions

for existence and uniqueness of the solutions are satisﬁed.

Therefore, the ﬁrst part of assumption 3.1, guarantees exis-

tence and uniqueness of the solutions of the system (9).

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The second part of the assumption 3.1, guarantees mono-

tonicity of the system (9), as proved in the following lemma,

which follows readily from the discussion in [23, p. 79] .

Lemma 3.1: Let fand g(i), for i= 1,· · · , m, satisfy

Assumption 3.1. Then the system (9) satisﬁes the quasimono-

tone condition in Rn

+.

Our aim is to relate the stability properties of the equilib-

rium of the system (9), to the following system:

˙x(t) = H(x) = f(x(t)) +

m

X

i=1

g(i)(x(t)) i= 1,· · · , m

(11)

It is easy to see that system (11) is generated from (9), with

τi= 0, for i= 1,· · · , m. Note that based on Lemma 2.3,

there is a one-to-one correspondence between the equilibria

of (9) and (11).

The following proposition, which is Proposition 4.1 in [4],

also plays a key role in proving later results.

Lemma 3.2: Let Wbe a neighbourhood of Rn

+and let

H:W → Rnbe a cooperative vector ﬁeld. If the system

(2) has a GAS equilibrium at the origin then there exists a

vector v0such that H(v)0.

Our use of a vector vsatisfying the conditions of the Lemma

for stability analysis of monotone systems echoes earlier

work on the so-called MO condition for Wazewski systems

in [15]. For other work taking a similar approach, consult

[2], [15] and the reference therein.

Now we can state and prove the following result.

Proposition 3.3: Consider the system (9) where vector

ﬁelds fand g(i), for i= 1,· · · , m, satisfy assumption 3.1.

If the system (11) has an asymptotically stable equilibrium

at the origin then the system (9) has an asymptotically stable

equilibrium at the origin for all τi≥0, for i= 1,· · · , m.

Proof: Note that since the system (9) is monotone and

(11) has a unique equilibrium at the origin, then system

(9) also has a unique equilibrium at the origin. Let h(·)be

deﬁned as in (10) and H(·)be deﬁned as in (11). Based on

the Proposition 3.2, there exists a vector vwith v0, such

that H(v)0. Since H(v) = h(ˆv), we can conclude that

h(ˆv)0with ˆvdeﬁned as in (3).

It now follows from monotonicity of the system (9) and

Lemma 2.5 that the solution x(t, ˆv)of (9) is non-increasing

and bounded. Hence, Convergence Criterion (Theorem 2.6)

implies that it converges to an equilibrium which is the

origin. Finally, the monotonicity of (9) implies that the

solution x(t, φ)of (9) also converges to the origin as t→ ∞

for every φ∈C([−τ, 0] that satisﬁes φ≤ˆv.

Note that Theorem 3.3, states only a local condition for

delay-independent stability of the system (9). It means that

the result holds for only a certain set of initial conditions.

We now prove that adding subhomogeneity assumptions will

lead to global delay-independent stability results for system

(9). To prove the main result of this section, we need the

following proposition.

Lemma 3.4: Let Wbe a neighbourhood of Rn

+and let H:

W → Rnbe subhomogeneous of degree αand cooperative.

If (2) has a GAS equilibrium at the origin then for every

x0∈Rn

+there exists vx0such that H(v)0.

Proof: Since His cooperative, then based on Propo-

sition 3.2, we know that there exists a w0such that

H(w)0. Therefore, for every x0∈Rn

+there exists

aβ > 1such that v=βw x0. On the other hand,

since His subhomogeneous of degree α, we know that

H(v) = H(βw)< βαH(w)0and this concludes the

proof.

Now we are ready to prove the main result of this section.

Theorem 3.5: Consider the system (9) where vector ﬁelds

fand g(i), for i= 1,· · · , m, are subhomogeneous of degree

αand satisfy assumption 3.1. If the system (11) has a GAS

equilibrium at the origin then the system (9) has a GAS

equilibrium at the origin for all τi≥0, for i= 1,· · · , m.

Proof: Since the origin is the only equilibrium of

system (11), then system (9) has also a unique equilibrium

at the origin.

Based on Proposition 3.4, we can conclude that for any

initial condition φ∈C([−τ, 0],Rn

+), we can ﬁnd a v0

such that φˆvand h(ˆv)0where h(·)is deﬁned as in

(10).

It now follows from Lemma 2.5 that the solution x(t, ˆv)

of (9) is non-increasing and bounded. Hence, Convergence

Criterion (Theorem 2.6) implies that it converges to an

equilibrium which is the origin. Finally, based on the mono-

tonicity of the system (9), we know that

x(t, ˆ

0) ≤x(t, φ)≤x(t, ˆv)

which implies x(t, φ)also converges to the origin as t→ ∞.

This concludes the proof.

Remark 3.1: Note that in Theorem 3.5, we used subho-

mogeneity only to show that for all x0∈Rn

+, there exists a

vx0such that H(v)0. The rest of the proofs, follows

from monotonicity of the systems. This means that we

can easily extend the delay-independent stability condition

for homogeneous cooperative systems with a single delay

presented in [3] to homogeneous cooperative systems with

multiple delays of the form (9).

IV. SUBHOMOGENEOUS COO PER ATI VE TIME-D ELAY

SYS TE M S WI TH A UNI QU E EQUILIBRIUM IN THE

INTERIOR OF POSITIVE ORTH AN T

In this section we also consider the nonlinear time-delay

system (9), where vector ﬁelds fand g(i), for i= 1,· · · , m,

satisfy assumption 3.1. The difference is that we focus on

the case where the time-delay system (9) has a unique

equilibrium in the interior of the positive orthant.

To prove the main theorem of this section, we need some

preliminary results. These results are in principle similar to

the ones stated in the previous section, but adapted for the

systems we study in this section. The following proposition

is Proposition 4.2 in [4].

Lemma 4.1: Let fand gi, for 1 = 1,· · · , m, be vector

ﬁelds on a neighbourhood Wof Rn

+and satisfy Assumption

3.1. Assume that (11) has an asymptotically stable equilib-

rium at p0and that the domain of attraction of pcontains

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int (Rn

+). Then there exists vectors v, u ∈Rn

+, such that

vpwith H(v)0and 0wpwith H(w)0.

Now we can state and prove the following result.

Proposition 4.2: Consider the system (9) where vector

ﬁelds fand g(i), for i= 1,· · · , m, satisfy assumption 3.1.

If the system (11) has an asymptotically stable equilibrium

p∈int (Rn

+), then the system (9) has an asymptotically

stable equilibrium at ˆp(as deﬁned in (3)) for all τi≥0, for

i= 1,· · · , m.

Proof: The proof is similar to the proof of Proposition

3.3. Based on Lemma 4.1, there exists a vector vwith vp,

such that H(v)0. Since H(v) = h(ˆv), we can conclude

that h(ˆv)0with ˆvdeﬁned as in (3). It now follows

from monotonicity of the system (9) and Lemma 2.5 that

the solution x(t, ˆv)of (9) is non-increasing and bounded

with x(t, ˆv)≥x(t, ˆp).

Hence, Convergence Criterion (Theorem 2.6) implies that

x(t, ˆv)converges to ˆp. For every vector x0with p≤x0≤v,

based on the monotonicity of (9) we can conclude that

x(t, ˆp)≤x(t, ˆx0)≤x(t, ˆv)

which means x(t, ˆx0)also converges to the equilibrium ˆp.

Similarly, based on Lemma 4.1, there exists a vector w

with wpsuch that H(w)0which means h( ˆw)0.

Using similar argument as above, we can conclude that for

every vector x0with w≤x0≤p,x(t, ˆx0)also converges

to the equilibrium ˆp. This concludes the proof.

Similar to Proposition 3.3, Proposition 4.2 states only a

local condition for delay-independent stability of the system

(9). In order to prove a global delay-independent stability

result for subhomogeneous cooperative systems with an

equilibrium in the interior of the positive orthant, we need

some preliminary results.

Lemma 4.3: Let Wbe a neighbourhood of Rn

+and let H:

W → Rnbe subhomogeneous of degree αand cooperative.

If (2) has a GAS equilibrium at p∈int (Rn

+)then for every

x0≥pthere exists vx0such that H(v)0and for

every x0≤pthere exists a wx0such that H(w)0.

Proof: Lemma 4.1 implies that there exists a zp

such that H(z)0. It follows from the subhomogeneity of

Hthat for any λ≥1,

H(λz)≤λαH(z)0

Choosing a large enough value for λ, we can ﬁnd a v=

λz x0with H(v)0.

Similarly, for any 0< µ ≤1, based on subhomogeneity

of H, we have:

H(µu)≥µαH(u)0

We can choose a small enough 0< µ ≤1such that w=

µu ≤x0≤λv with H(w)0. This concludes the proof.

Now we can state and prove the main result of this section.

Theorem 4.4: Consider the system (9) where vector ﬁelds

fand g(i), for i= 1,· · · , m, are subhomogeneous of degree

αand satisfy assumption 3.1. If the system (11) has a

globally asymptotically stable equilibrium at p∈int (Rn

+)

then the system (9) has a globally asymptotically stable

equilibrium at ˆpfor all τi≥0, for i= 1,· · · , m.

Proof: Since pis the only equilibrium of system (11)

in int (Rn

+), then system (9) has also a unique equilibrium

in the interior of the positive orthant.

Based on Lemma 4.3, we can conclude that for any initial

condition φ∈C([−τ, 0],Rn

+), we can ﬁnd a vpand

wpsuch that ˆwφˆvwith h(ˆv)0and h( ˆw)0.

It now follows from Lemma 2.5 that the solution x(t, ˆv)

of (9) is non-increasing and bounded. Hence, Convergence

Criterion (Theorem 2.6) implies that it converges to an

equilibrium which is ˆp. Similarly, x(t, ˆw)is non-decreasing

and bounded, hence converges to ˆp. Finally, based on the

monotonicity of the system (9), we have

x(t, ˆw)≤x(t, φ)≤x(t, ˆv)

which means x(t, φ)also converges to the equilibrium ˆp.

This concludes the proof.

Remark 4.1: Note that in this section, we did not make

any assumption on the equilibria of the system (9) (or

equivalently (11)) on the boundary of the positive orthant.

In other words, the arguments presented in this section hold,

whether or not the system (9) or (11) has an equilibrium on

the boundary of the positive orthant.

V. CONCLUSIONS

In this paper, we stated delay-independent stability condi-

tions for subhomogeneous cooperative time-delay systems.

Subhomogeneous systems include different classes of non-

linear systems. We considered distinct cases; time-delay

systems with a unique equilibrium at the origin, and time-

delay systems with a unique equilibrium in the interior of

the positive orthant. In both cases, we proved a global delay-

independent stability condition for the time-delay system.

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