Stability Analysis of Monotone Systems via
Max-separable Lyapunov Functions

arXiv:1607.07966v1 [] 27 Jul 2016

H.R. Feyzmahdavian, B. Besselink, M. Johansson

Abstract
We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability
of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a
compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of
D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems,
equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function,
D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and
general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and
sufficient conditions for delay-independent stability are derived. Examples show how the results extend
the state-of-the-art.

I. I NTRODUCTION
Monotone systems are dynamical systems whose trajectories preserve a partial order relationship on their initial states. Such systems appear naturally in, for example, chemical reaction
networks [2], consensus dynamics [3], systems biology [4], wireless networks [5]–[8], and as
comparison systems in stability analysis of large-scale interconnected systems [9]–[11]. Due to
The authors are with the ACCESS Linnaeus Centre and the Department of Automatic Control, School of Electrical Engineering,
KTH Royal Institute of Technology, Stockholm, Sweden. Email: hamidrez@kth.se, bart.besselink@ee.kth.se, mikaelj@kth.se.
A preliminary version of this work is submitted as the conference paper [1]. This manuscript significantly extends the work [1]
by providing additional technical results and illustrative examples. Namely, [1] only shows the equivalence of statements 2) and
3) in Theorem 1 (rather than the full Theorem 1), presents a notion of D-stability that is less general than the notion in the current
manuscript, and discusses delay-independent stability of monotone systems with constant delays (rather than time-varying and
potentially unbounded delays).

their wide applicability, monotone systems have attracted considerable attention from the control
community (see, e.g., [12]–[15]). Early references on the theory of monotone systems include
the papers [16]–[18] by Hirsch and the excellent monograph [19] by Smith.
For monotone linear systems (also called positive linear systems), it is known that asymptotic
stability of the origin implies further stability properties. First, asymptotically stable monotone
linear systems always admit a Lyapunov function that can be expressed as a weighted maxnorm [20]. Such Lyapunov functions can be written as a maximum of functions with onedimensional arguments, so they are a particular class of max-separable Lyapunov functions
[21]–[23]. Second, asymptotic stability of monotone linear systems is robust with respect to
scaling of the dynamics with a diagonal matrix, leading to the so-called D-stability property.
This notion appeared first in [24] and [25], with additional early results given in [26]. Third,
monotone linear systems possess strong robustness properties with respect to time-delays [27]–
[36]. Namely, for these systems, asymptotic stability of a time-delay system can be concluded
from stability of the corresponding delay-free system, simplifying the analysis.
For monotone nonlinear systems, it is in general unknown whether asymptotic stability of the
origin implies notions of D-stability or delay-independent stability, even though results exist for
certain classes of monotone systems, such as homogeneous and sub-homogeneous systems [37]–
[42]. Recent results in [43] and [44] show that for monotone nonlinear systems, asymptotic
stability of the origin implies the existence of a max-separable Lyapunov function on every
compact set in the domain of attraction. Motivated by this result and the strong robustness
properties of monotone linear systems, we study stability properties of monotone nonlinear
systems using max-separable Lyapunov functions. Here, monotone nonlinear systems are neither
restricted to be homogeneous nor sub-homogeneous.
The main contribution of this paper is to extend the stability properties of monotone linear
systems discussed above to monotone nonlinear systems. Specifically, we demonstrate that
asymptotic stability of the origin for a monotone nonlinear system leads to D-stability, and
asymptotic stability in the presence of bounded and unbounded time-varying delays. Furthermore,
this paper has the following four contributions:
First, we show that for monotone nonlinear systems, the existence of a max-separable Lyapunov function on a compact set is equivalent to the existence of a path in this compact set such
that, on this path, the vector field defining the system is negative in all its components. This

allows for a simple evaluation of asymptotic stability for such systems.
Second, we define a novel and natural notion of D-stability for monotone nonlinear systems
that extends earlier concepts in the literature [37]–[39]. Our notion of D-stability is based on the
composition of the components of the vector field with an arbitrary monotonically increasing
function that plays the role of a scaling. We then show that for monotone nonlinear systems,
this notion of D-stability is equivalent to the existence of a max-separable Lyapunov function
on a compact set.
Third, we demonstrate that for monotone nonlinear systems, asymptotic stability of the origin is
insensitive to a general class of time-delays which includes bounded and unbounded time-varying
delays. Again, this provides an extension of existing results for monotone linear systems to the
nonlinear case. In order to impose minimal restrictions on time-delays, our proof technique uses
the max-separable Lyapunov function that guarantees asymptotic stability of the origin without
delays as a Lyapunov-Razumikhin function.
Fourth, we derive a set of necessary and sufficient conditions for establishing delay-independent
stability of monotone nonlinear systems. These conditions can also provide an estimate of the
region of attraction for the origin. As in the case of D-stability, we extend several existing
results on analysis of monotone systems with time-delays, which often rely on homogeneity and
sub-homogeneity of the vector field (see, e.g., [38]–[42]), to general monotone systems.
The remainder of the paper is organized as follows. Section II reviews some preliminaries
on monotone nonlinear systems and max-separable Lyapunov functions, and discusses stability
properties of monotone linear systems. In Section III, our main results for stability properties of
monotone nonlinear systems are presented, whereas in Section IV, delay-independent stability
conditions for monotone systems with time-varying delays are derived. Section V demonstrates
through a number of examples how these results extend earlier work in the literature. Finally,
conclusions are stated in Section VI.
Notation. The set of real numbers is denoted by R, whereas R+ = [0, ∞) represents the set of

nonnegative real numbers. We let Rn+ denote the positive orthant in Rn . The associated partial
order is given as follows. For vectors x and y in Rn , x < y (x ≤ y) if and only if xi < yi

(xi ≤ yi ) for all i ∈ In , where xi ∈ R represents the ith component of x and In = {1, . . . , n}.

For a real interval [a, b], C [a, b], Rn denotes the space of all real-valued continuous functions
on [a, b] taking values in Rn . A continuous function ω : R+ → R+ is said to be of class K if

ω(0) = 0 and ω is strictly increasing. A function ω : Rn → R+ is called positive definite if

ω(0) = 0 and ω(x) > 0 for all x 6= 0. Finally, 1n ∈ Rn denotes the vector whose components
are all one.
II. P ROBLEM S TATEMENT

AND

P RELIMINARIES

Consider dynamical systems on the positive orthant Rn+ described by the ordinary differential
equation
ẋ = f (x).

(1)

Here, x is the system state, and the vector field f : Rn+ → Rn is locally Lipschitz so that local
existence and uniqueness of solutions is guaranteed [45]. Let x(t, x0 ) denote the solution to (1)
starting from the initial condition x0 ∈ Rn+ at the time t ∈ R+ . We further assume that (1) has
an equilibrium point at the origin, i.e., f (0) = 0.
A. Preliminaries on Monotone Systems
In this paper, monotone systems will be studied according to the following definition.
Definition 1: The system (1) is called monotone if the implication
x′0 ≤ x0 ⇒ x(t, x′0 ) ≤ x(t, x0 ),

∀t ∈ R+ ,

(2)

holds, for any initial conditions x0 , x′0 ∈ Rn+ .
The definition states that trajectories of monotone systems starting at ordered initial conditions
preserve the same ordering during the time evolution. By choosing x′0 = 0 in (2), since x(t, 0) = 0
for all t ∈ R+ , it is easy to see that
x0 ∈ Rn+ ⇒ x(t, x0 ) ∈ Rn+ ,

∀t ∈ R+ .

(3)

This shows that the positive orthant Rn+ is an invariant set for the monotone system (1). Thus,
monotone systems with an equilibrium point at the origin define positive systems1.
Monotonicity of dynamical systems is equivalently characterized by the so-called Kamke
condition, stated next.
1

A dynamical system given by (1) is called positive if any trajectory of (1) starting from nonnegative initial conditions remains

n
forever in the positive orthant, i.e., x(t) ∈ Rn
+ for all t ∈ R+ when x0 ∈ R+ .

Proposition 1 ([19]): The system (1) is monotone if and only if the following implication
holds for all x, x′ ∈ Rn+ and all i ∈ In :
x′ ≤ x and x′i = xi ⇒ fi (x′ ) ≤ fi (x).

(4)

Note that if f is continuously differentiable on Rn+ , then condition (4) is equivalent to the
requirement that f has a Jacobian matrix with nonnegative off-diagonal elements, i.e.,
∂fi
(x) ≥ 0,
∂xj

x ∈ Rn+ ,

(5)

holds for all i 6= j, i, j ∈ In [19, Remark 3.1.1]. A vector field satisfying (5) is called cooperative.
In this paper, we will consider stability properties of monotone nonlinear systems. To this
end, we use the following definition of (asymptotic) stability, tailored for monotone systems on
the positive orthant.
Definition 2: The equilibrium point x = 0 of the monotone system (1) is said to be stable if,
for each ε > 0, there exists a δ > 0 such that
0 ≤ x0 < δ1n ⇒ 0 ≤ x(t, x0 ) < ε1n ,

∀t ∈ R+ .

(6)

The origin is called asymptotically stable if it is stable and, in addition, δ can be chosen such
that
0 ≤ x0 < δ1n ⇒ lim x(t, x0 ) = 0.
t→∞

Note that, due to the equivalence of norms on Rn and forward invariance of the positive
orthant, Definition 2 is equivalent to the usual notion of Lyapunov stability [45].
B. Preliminaries on Max-separable Lyapunov Functions
We will characterize and study asymptotic stability of monotone nonlinear systems by means
of so-called max-separable Lyapunov functions
V (x) = max Vi (xi ),
i∈In

(7)

with scalar functions Vi : R+ → R+ . Since the Lyapunov function (7) is not necessarily
continuously differentiable, we consider its upper-right Dini derivative along solutions of (1)
(see, e.g., [46]) as


V
x
+
hf
(x)
−
V
x
.
D + V (x) = lim sup
h
h→0+

(8)

The following result shows that if the functions Vi in (7) are continuously differentiable, then (8)
admits an explicit expression.
Proposition 2 ([47]): Consider V : Rn+ → R+ in (7) and let Vi : R+ → R+ be continuously
differentiable for all i ∈ In . Then, the upper-right Dini derivative (8) is given by
∂Vj
(xj )fj (x),
j∈J (x) ∂xj

D + V (x) = max

(9)

where J (x) is the set of indices for which the maximum in (7) is attained, i.e.,

J (x) = j ∈ In | Vj (xj ) = V (x) .

(10)

C. Preliminaries on Monotone Linear Systems
Let f (x) = Ax with A ∈ Rn×n . Then, the nonlinear system (1) reduces to the linear system
ẋ = Ax.

(11)

It is well known that (11) is monotone in the sense of Definition 1 (and, hence, positive) if and
only if A is Metzler, i.e., all off-diagonal elements of A are nonnegative [20]. We summarize
some important stability properties of monotone linear systems in the next result.
Proposition 3 ([23], [48]): For the linear system (11), suppose that A is Metzler. Then, the
following statements are equivalent:
1) The monotone linear system (11) is asymptotically stable, i.e., A is Hurwitz.
2) There exists a max-separable Lyapunov function of the form
V (x) = max
i∈In

xi
,
vi

(12)

on Rn+ , with vi > 0 for each i ∈ In .
3) There exists a vector w > 0 such that Aw < 0.
4) For any diagonal matrix ∆ ∈ Rn×n with positive diagonal entries, the linear system
ẋ = ∆Ax,
is asymptotically stable, i.e., ∆A is Hurwitz,
In Proposition 3, the equivalence of statements 1) and 2) demonstrates that the existence of a
max-separable Lyapunov function is a necessary and sufficient condition for asymptotic stability
of monotone linear systems. The positive scalars vi in the max-separable Lyapunov function

(12) in the second item can be related to the positive vector w in the third item as vi = wi for
all i ∈ In . Statement 4) shows that stability of monotone linear systems is robust with respect
to scaling of the rows of matrix A. This property is known as D-stability [24]. Note that the
notions of asymptotic stability in Proposition 3 hold globally due to linearity.
Another well-known property of monotone linear systems is that their asymptotic stability
is insensitive to bounded and certain classes of unbounded time-delays. This property reads as
follows.
Proposition 4 ([49]): Consider the delay-free monotone (positive) system
ẋ(t) = (A + B)x(t),

(13)

with A Metzler and B having nonnegative elements. If (13) is asymptotically stable, then the
time-delay linear system
ẋ(t) = Ax(t) + Bx(t − τ (t))

(14)

is asymptotically stable for all time-varying and potentially unbounded delays satisfying
lim t − τ (t) = +∞.

t→+∞

This results shows that asymptotic stability of the delay-free monotone linear system (13)
implies that (14) is also asymptotically stable. This is a significant property of monotone
linear systems, since the introduction of time-delays may, in general, render a stable system
unstable [50].
D. Main Goals
The main objectives of this paper are (i) to derive a counterpart of Proposition 3 for monotone
nonlinear systems of the form (1); and (ii) to extend the delay-independent stability property of
monotone linear systems stated in Proposition 4 to monotone nonlinear systems with bounded
and unbounded time-varying delays.

III. S TABILITY

OF MONOTONE NONLINEAR SYSTEMS

The following theorem is our first key result, which establishes a set of necessary and sufficient
conditions for asymptotic stability of monotone nonlinear systems.
Theorem 1: Assume that the nonlinear system (1) is monotone. Then, the following statements
are equivalent:
1) The origin is asymptotically stable.
2) For some compact set of the form

X = x ∈ Rn+ | 0 ≤ x ≤ v ,

(15)

with v > 0, there exists a max-separable Lyapunov function V : X → R+ as in (7) with
Vi : [0, vi ] → R+ differentiable for each i ∈ In such that
ν1 (xi ) ≤ Vi (xi ) ≤ ν2 (xi ),

(16)

holds for all xi ∈ [0, vi ] and for some functions ν1 , ν2 of class K, and that
D + V (x) ≤ −µ(V (x)),

(17)

holds for all x ∈ X and some positive definite function µ.

3) For some positive constant s̄ > 0, there exists a function ρ : [0, s̄] → Rn+ with ρi of class
K, ρ−1
differentiable on [0, ρi (s̄)] and satisfying
i

dρ−1
i
(s) > 0,
ds

(18)


for all s ∈ 0, ρi (s̄) and all i ∈ In , such that

f ◦ ρ(s) ≤ −α(s),

(19)

holds for s ∈ [0, s̄] and some function α : [0, s̄] → Rn+ with αi positive definite for all
i ∈ In .

4) For any function ψ : Rn+ × Rn → Rn given by ψ(x, y) = ψ1 (x1 , y1 ), . . . , ψn (xn , yn )
where

T

•

ψi : R+ × R → R for i ∈ In ,

•

ψi (xi , 0) = 0 for any xi ∈ R+ and all i ∈ In ,

•

ψi (xi , yi ) is monotonically increasing in yi for each nonzero xi , i.e., the implication
yi′ < yi ⇒ ψi (xi , yi′ ) < ψi (xi , yi )

(20)

holds for any xi > 0 and all i ∈ In ,
the nonlinear system

ẋ = ψ x, f (x) ,

(21)

has an asymptotically stable equilibrium point at the origin.
Proof: The proof is given in Appendix A.
Theorem 1 can be regarded as a nonlinear counterpart of Proposition 3. Namely, choosing the
functions Vi in the second statement of Theorem 1 as Vi (xi ) = xi /vi , vi > 0, and the function
ρ in the third statement as ρ(s) = ws, w > 0, recovers statements 2) and 3) in Proposition 3,
respectively. Note that we can let vi = wi for each i ∈ In since, according to the proof of
Theorem 1, the relation between V and ρ is
−1
Vi (xi ) = ρ−1
i (xi ) and ρi (s) = Vi (s), i ∈ In .

Statement 4) of Theorem 1, which can be regarded as a notion of D-stability for nonlinear
monotone systems, is a counterpart of the fourth statement of Proposition 3. More precisely, the
choice ψ(x, y) = ∆y for some diagonal matrix ∆ with positive diagonal entries recovers the
corresponding result for monotone linear systems in Proposition 3.
Remark 1: According to the proof of Theorem 1, if there is a function ρ satisfying the third
statement for s ∈ [0, s̄], then the Lyapunov function (7) with components Vi (xi ) = ρ−1
i (xi )
guarantees asymptotic stability of the origin for any initial condition

x0 ∈ X = x ∈ Rn+ | 0 ≤ x ≤ ρ(s̄) .

This means that the set X is an estimate of the region of attraction for the origin.

⊳

We now present a simple example to illustrate the use of Theorem 1.
Example 1: Consider the nonlinear dynamical system


−5x1 + x1 x22
.
ẋ = f (x) = 
x1 − 2x22

(22)

This system has an equilibrium point at the origin. As the Jacobian matrix of f is Metzler for
all (x1 , x2 ) ∈ R2+ , f is cooperative. Thus, according to Proposition 1, (22) is monotone on R2+ .
√
First, we will show that the origin is asymptotically stable. Let ρ(s) = (s, s), s ∈ [0, 4]. For
each i ∈ {1, 2}, ρi is of class K. It is easy to verify that
−1
2
ρ−1
1 (s) = s, ρ2 (s) = s .

Thus, ρ−1
i , i ∈ {1, 2}, is continuously differentiable and satisfies (18). In addition,


 
2
−5s + s
s
 ≤ − ,
f ◦ ρ(s) = 
−s
s
for all s ∈ [0, 4], which implies that (19) holds. It follows from the equivalence of statements
1) and 3) in Theorem 1 that the origin is asymptotically stable.
Next, we will estimate the region of attraction of the origin by constructing a max-separable
Lyapunov function. According to Remark 1, the monotone system (22) admits the max-separable
Lyapunov function

V (x) = max x1 , x22
that guarantees the origin is asymptotically stable for

x0 ∈ x ∈ R2+ | 0 ≤ x ≤ (4, 2) .
Finally, we discuss D-stability of the monotone system (22). Consider ψ given by
T

x
3
2
1
.
ψ(x, y) = x2 +1 y1 , x2 y2
1

For any x ∈ R2+ , ψ(x, 0) = 0. Moreover, each component ψi (xi , yi ), i ∈ {1, 2}, is monotonically
increasing for any xi > 0. Since the origin is an asymptotically stable equilibrium point of (22),
by the equivalence of statements 1) and 4) in Theorem 1, the monotone nonlinear system


x1
2 3

(−5x1 + x1 x2 )
2

ẋ = ψ x, f (x) =  x1 +1
x22 (x1 − 2x22 )
has an asymptotically stable equilibrium point at the origin.
Remark 2: A consequence of the proof of Theorem 1 is that all statements 1)–4) are equivalent
to the existence of a vector w > 0 such that f (w) < 0 and
lim x(t, w) = 0.

t→∞

(23)

Contrary to statement 3) of Proposition 3 for monotone linear systems, the condition f (w) < 0
without the additional assumption (23) does not necessarily guarantee asymptotic stability of
the origin for monotone nonlinear systems. To illustrate the point, consider, for example, a
scalar monotone system described by (1) with f (x) = −x(x − 1), x ∈ R+ . This system has

two equilibrium points: x⋆ = 0 and x⋆ = 1. Although f (2) < 0, it is easy to verify that any

trajectory starting from the initial condition x0 > 0 converges to x⋆ = 1. Hence, the origin is
⊳

not stable.

Remark 3: Several implications in Theorem 1 are based on similar results in the literature.
Namely, the implication 1) ⇒ 2) was shown in [43] (see also [44]) by construction of a maxseparable Lyapunov function that is not necessarily differentiable or even continuous (see [44,
Example 2]). The implication 3) ⇒ 2) was proven before in [51, Theorem III.2] for the case of
global asymptotic stability of monotone nonlinear systems considering a max-separable Lyapunov
function with possibly non-smooth components Vi . The implication 1) ⇒ 4) was shown in [37]–
[39] for particular classes of scaling function ψ. For example, if we choose ψi (xi , yi ) = di (xi )yi
with di (xi ) > 0 for xi > 0, then statement 4) recovers the results in [37], [39]. However, contrary
to [37]–[39], neither homogeneity nor sub-homogeneity of f is required in Theorem 1.
IV. S TABILITY

⊳

OF MONOTONE SYSTEMS WITH DELAYS

In this section, delay-independent stability of nonlinear systems of the form

 ẋ(t) = g x(t), x(t − τ (t)) , t ≥ 0,
 x(t) = ϕ(t)
, t ∈ [−τmax , 0],

(24)

is considered. Here, g : Rn+ × Rn+ → Rn is locally Lipschitz continuous with g(0, 0) = 0,

ϕ ∈ C [−τmax , 0], Rn+ is the vector-valued function specifying the initial state of the system,

and τ is the time-varying delay which satisfies the following assumption:

Assumption 1: The delay τ : R+ → R+ is continuous with respect to time and satisfies
lim t − τ (t) = +∞.

t→+∞

(25)

Note that τ is not necessarily continuously differentiable and that no restriction on its derivative
(such as τ̇ (t) < 1) is imposed. Roughly speaking, condition (25) implies that as t increases,
the delay τ (t) grows slower than time itself. It is easy to verify that all bounded delays,
irrespectively of whether they are constant or time-varying, satisfy Assumption 1. Moreover,
delays satisfying (25) may be unbounded (take, for example, τ (t) = γt with γ ∈ (0, 1)).
Unlike the non-delayed system (1), the solution of the time-delay system (24) is not uniquely
determined by a point-wise initial condition x0 , but by the continuous function ϕ defined over

the interval [−τmax , 0]. Assumption 1 implies that there is a sufficiently large T > 0 such that
t − τ (t) > 0 for all t > T . Define
τmax = − inf

0≤t≤T





t − τ (t) .

Clearly, τmax ∈ R+ is bounded (τmax < +∞). Therefore, the initial condition ϕ is defined on a
bounded set [−τmax , 0] for any delay satisfying Assumption 1, even if it is unbounded. Since g
is Lipschitz continuous and τ is a continuous function of time, the existence and uniqueness of
solutions to (24) follow from [52, Theorem 2]. We denote the solution to (24) corresponding to
the initial condition ϕ by x(t, ϕ).
From this point on, it is assumed that the time-delay system (24) satisfies the next assumption:
Assumption 2: The following properties hold:
1) g(x, y) satisfies Kamke condition in x for each y, i.e.,
x′ ≤ x and x′i = xi ⇒ gi (x′ , y) ≤ gi (x, y),

(26)

holds for any y ∈ Rn+ and all i ∈ In .
2) g(x, y) is order-preserving in y for each x, i.e.,
y ′ ≤ y ⇒ g(x, y ′) ≤ g(x, y),

(27)

holds for any x ∈ Rn+ .

System (24) is called monotone if given two initial conditions ϕ, ϕ′ ∈ C [−τmax , 0], Rn+ with

ϕ′ (t) ≤ ϕ(t) for all t ∈ [−τmax , 0], then

x(t, ϕ′ ) ≤ x(t, ϕ),

∀t ∈ R+ .

(28)

It follows from [19, Theorem 5.1.1] that Assumption 2 ensures the monotonicity of (24).
Furthermore, as the origin is an equilibrium for (24), the positive orthant Rn+ is forward invariant,
i.e., x(t, ϕ) ∈ Rn+ for all t ∈ R+ when ϕ(t) ∈ Rn+ for t ∈ [−τmax , 0].
We are interested in stability of the time-delay system (24) under the assumption that the
delay-free system



ẋ t = g x(t), x(t) =: f x(t) ,

(29)

has an asymptotically stable equilibrium point at the origin. Since time-delays may, in general,
induce oscillations and even instability [53], the origin is not necessarily stable for the time-delay

system (24). However, the following theorem shows that asymptotic stability of the origin for
monotone nonlinear systems is insensitive to time-delays satisfying Assumption 1.
Theorem 2: Consider the time-delay system (24) under Assumption 2. Then, the following
statements are equivalent:
1) The time-delay monotone system (24) has an asymptotically stable equilibrium point at
the origin for all time varying-delays satisfying Assumption 1.
2) For the non-delayed monotone system (29), any of the equivalent conditions in the statement
of Theorem 1 hold.
Proof: The proof is given in Appendix B.
According to Theorem 2, local asymptotic stability of the origin for a delay-free monotone
system of the form (29) implies local asymptotic stability of the origin also for (24) with bounded
and unbounded time-varying delays. Theorem 2 does not explicitly give any estimate of the region
of attraction for the origin. However, its proof shows that the stability conditions presented
in Theorem 1 for non-delayed monotone systems can provide such estimates, leading to the
following practical tests.
T1. Assume that for the delay-free monotone system (29), we can characterize asymptotic
stability of the origin through a max-separable Lyapunov function V satisfying the second
statement of Theorem 1. Then, for the time-delay system (24), the origin is asymptotically
stable with respect to initial conditions satisfying

ϕ(t) ∈ x ∈ Rn+ | 0 ≤ xi ≤ Vi−1 (c), i ∈ In ,
for t ∈ [−τmax , 0] with
c = min Vi (vi ).
i∈In

T2. If we demonstrate the existence of a function ρ such that the non-delayed system (29) satisfies the third statement of Theorem 1, then (24) with time-delays satisfying Assumption 1
has an asymptotically stable equilibrium point at the origin for which the region of attraction
includes initial conditions ϕ that satisfy
0 ≤ ϕ(t) ≤ ρ(s̄),

t ∈ [−τmax , 0].

T3. If we find a vector w > 0 such that g(w, w) < 0 and that the solution x(t, w) to the
delay-free monotone system (29) converges to the origin, then the solution x(t, ϕ) to the
time-delay system (24) converges to the origin for any initial condition ϕ that satisfies
0 ≤ ϕ(t) ≤ w,

t ∈ [−τmax , 0].

The following example illustrates the results of Theorem 2.
Example 2: Consider the time-delay system





−5x1 t + x1 t x22 t − τ (t)
ẋ(t) = g x(t), x(t − τ (t)) = 

 .
2
x1 t − τ (t) − 2x2 t

(30)

One can verify that g satisfies Assumption 2. Thus, the system (30) is monotone on R2+ . According to Example 1, this system without time-delays has an asymptotically stable equilibrium
at the origin. Therefore, Theorem 2 guarantees that for the time-delay system (30), the origin
is still asymptotically stable for any bounded and unbounded time-varying delays satisfying
Assumption 1.
We now provide an estimate of the region of attraction for the origin. Example 1 shows that
√
for the system (30) without time-delays, the function ρ(s) = (s, s), s ∈ [0, 4], satisfies the
third statement of Theorem 1. It follows from stability test T2 that the solution x(t, ϕ) to (30)
starting from initial conditions
0 ≤ ϕ(t) ≤ ( 4, 2 )T ,

t ∈ [−τmax , 0].

converges to the origin.
Remark 4: Our results can be extended to monotone nonlinear systems with heterogeneous
delays of the form

ẋi (t) = gi x(t), xτi (t) ,

i ∈ In ,

(31)


T
where g x, y = g1 (x, y), . . . , gn (x, y) satisfies Assumption 2, and


T
xτi t := x1 (t − τi1 (t)), . . . , xn (t − τin (t)) .

If the delays τij , i, j ∈ In , satisfy Assumption 1, then asymptotic stability of the origin for the
delay-free monotone system (29) ensures that (31) with heterogeneous time-varying delays also
has an asymptotically stable equilibrium point at the origin

⊳

V. A PPLICATIONS

OF THE MAIN RESULTS

In this section, we will present several examples to illustrate how our main results recover
and generalize previous results on delay-independent stability of monotone nonlinear systems.
A. Homogeneous monotone systems
First, we consider a particular class of monotone nonlinear systems whose vector fields are
homogeneous in the sense of the following definition.
Definition 3: Given an n-tuple r = (r1 , . . . , rn ) of positive real numbers and λ > 0, the
dilation map δλr : Rn → Rn is defined as


δλr x := λr1 x1 , . . . , λrn xn .
When r = 1n , the dilation map is called the standard dilation map. A vector field f : Rn → Rn
is said to be homogeneous of degree p ∈ R with respect to the dilation map δλr if


f δλr (x) = λp δλr f (x) ,

∀x ∈ Rn , ∀λ > 0.

Note that the linear mapping f (x) = Ax is homogeneous of degree zero with respect to the
standard dilation map.
The following result, which is a direct consequence of Theorem 2, establishes a necessary
and sufficient condition for global asymptotic stability of homogeneous monotone systems with
time-varying delays. By global asymptotic stability, we mean that the origin is asymptotically
stable for all nonnegative initial conditions.
Corollary 1: For the time-delay system (24), suppose Assumption 2 holds. Suppose also that
f (x) := g(x, x) is homogeneous of degree p ∈ R+ with respect to the dilation map δλr . Then,
the following statements are equivalent:
1) There exists a vector w > 0 such that f (w) < 0.
2) The homogeneous monotone system (24) has a globally asymptotically stable equilibrium

point at the origin for all ϕ ∈ C [−τmax , 0], Rn+ and all time-delays satisfying Assumption 1.

Proof: The implication 2) ⇒ 1) follows directly from Theorem 2 and Remark 2. We will
show that 1) implies 2).

1) ⇒ 2) : Let ρi (s) = sri /rmax wi , where
rmax = max ri .
i∈In

For any positive constant s̄ > 0, it is clear that ρi is of class K on [0, s̄], and ρ−1
is continuously
i
differentiable and satisfy (18). As f is homogeneous of degree p ∈ R+ with respect to the

dilation map δλr , it follows that



f ◦ ρ(s) = g ρ(s), ρ(s) = g δsr1/rmax (w), δsr1/rmax (w)

= sp/rmax δsr1/rmax g(w, w)

= sp/rmax δsr1/rmax f (w) .

(32)

Since f (w) < 0, the right-hand side of equality (32) is negative definite for all s ∈ [0, s̄].
Therefore, according to Theorem 2 and stability test T2, the time-delay system (24) has an
asymptotically stable equilibrium point at the origin for any 0 ≤ ϕ(t) ≤ ρ(s̄), t ∈ [−τmax , 0].
To prove global asymptotic stability, suppose we are given a nonnegative initial condition

ϕ ∈ C [−τmax , 0], Rn+ . Since wi > 0 for each i ∈ In and ϕ is continuous (hence, bounded) on
[−τmax , 0], there exists a sufficiently large s̄ > 0 such that ϕ(t) ≤ ρ(s̄) for all t ∈ [−τmax , 0].

It now follows immediately from the argument in the previous paragraph that the origin is
asymptotically stable with respect to any nonnegative initial condition ϕ.
Remark 5: Delay-independent stability of homogeneous monotone systems with time-varying
delays satisfying Assumption 1 was previously considered in [40] by using a max-separable
Lyapunov function with components
Vi (xi ) =

ρ−1
i (xi )

=



xi
wi

rmax /ri

,

i ∈ In .

Note, however, that the proof of Theorem 2 differs significantly from the analysis in [40]. The
main reason for this is that the homogeneity assumption, which plays a key role in the stability
proof in [40], is not satisfied for general monotone systems.

⊳

B. Sub-homogeneous monotone systems
Another important class of monotone nonlinear systems are those with sub-homogeneous
vector fields:

Definition 4: A vector field f : Rn+ → Rn is said to be sub-homogeneous of degree p ∈ R if
f (λx) ≤ λp f (x),

∀x ∈ Rn+ , ∀λ ≥ 1.

Theorem 2 allows us to show that global asymptotic stability of the origin for monotone
systems whose vector fields are sub-homogeneous is insensitive to bounded and unbounded
time-varying delays.
Corollary 2: Consider the time-delay system (24) under Assumption 2. Suppose also that
f (x) := g(x, x) is sub-homogeneous of degree p ∈ R+ . If the origin for the delay-free monotone
system (29) is globally asymptotically stable, then the sub-homogeneous monotone system (24)
has a globally asymptotically stable equilibrium at the origin for all time-varying delays satisfying Assumption 1.
Proof: The origin for the sub-homogeneous monotone system (29) is globally asymptotically
stable. Thus, for any constant α > 0, there exists a vector w > 0 such that α1n ≤ w and the
solution x(t, w) to the delay-free system (29) converges to the origin [37, Theorem 4.1]. It
follows from Theorem 2 and stability test T3 that the time-delay system (24) is asymptotically
stable with respect to initial conditions satisfying 0 ≤ ϕ(t) ≤ α1n , t ∈ [−τmax , 0].

To complete the proof, let ϕ ∈ C [−τmax , 0], Rn+ be an arbitrary initial condition. As α > 0

and ϕ is continuous (hence, bounded) on [−τmax , 0], we can find α > 0 such that ϕ(t) ≤ α1n for
t ∈ [−τmax , 0]. This together with the above observations implies that the origin is asymptotically
stable for all nonnegative initial conditions.
Remark 6: In [41], it was shown that global asymptotic stability of the origin for subhomogeneous monotone systems is independent of bounded time-varying delays. In this work,
we establish insensitivity of sub-homogeneous monotone systems to the general class of possibly
unbounded delays described by Assumption 1, which includes bounded delays as a special case.
⊳

C. Sub-homogeneous (non-monotone) positive systems
Finally, motivated by results in [54], we consider the time-delay system


ẋ(t) = g(x(t), x(t − τ (t))) = h x(t) + d x(t − τ (t)) .
We assume that h and d satisfy Assumption 3.

(33)

Assumption 3: The following properties hold:
1) For each i ∈ In , hi (x) ≥ 0 for x ∈ Rn+ with xi = 0;
2) For all x ∈ Rn+ , d(x) ≥ 0;

3) Both h and d are sub-homogeneous of degree p ∈ R+ ;

4) For any x ∈ Rn+ \ {0}, there is i ∈ In such that



sup di (z ′ ) | 0 ≤ z ′ ≤ x < − sup hi (z) | 0 ≤ z ≤ x, zi = xi .
Note that under Assumption 3, the time-delay system (33) is not necessarily monotone.
However, Assumptions 3.1 and 3.2 ensure the positivity of (33) [19, Theorem 5.2.1].
In [54], it was shown that if Assumption 3 holds, then the positive nonlinear system (33) with
constant delays (τ (t) = τmax , t ∈ R+ ) has a globally asymptotically stable equilibrium at the
origin for all τmax ∈ R+ . Theorem 2 helps us to extend the result in [54] to time-varying delays
satisfying Assumption 1.
Corollary 3: For the time-delay system (33), suppose Assumption 3 holds. Then, the origin
is globally asymptotically stable for all time-delays satisfying Assumption 1.
Proof: For any x, y ∈ Rn+ and each i ∈ In , define

ḡi (x, y) = sup hi (z) + di (z ′ ) | 0 ≤ z ≤ x, zi = xi , 0 ≤ z ′ ≤ y .
It is straightforward to show that ḡ(x, y) satisfies Assumption 2. Thus, the time-delay system


ẋ t = ḡ x(t), x(t − τ (t)) ,

(34)

is monotone. Under Assumption 3, the sub-homogeneous monotone system (34) without delays
(τ (t) = 0) has a globally asymptotically stable equilibrium at the origin [54, Theorem III.2].
Therefore, according to Corollary 2, the origin for the time-delay system (34) is also globally
asymptotically stable for any time-delays satisfying Assumption 1.
As g(x, y) ≤ ḡ(x, y) for any x, y ∈ Rn+ , it follows from [19, Theorem 5.1.1] that for any
initial condition ϕ,
x(t, ϕ, g) ≤ x(t, ϕ, ḡ),

t ∈ R+ ,

(35)

where x(t, ϕ, g) and x(t, ϕ, ḡ) are solutions to (33) and (34), respectively, for a common initial condition ϕ. Since x = 0 is a globally asymptotically stable equilibrium point for (34),

x(t, ϕ, ḡ) → 0 as t → ∞. Moreover, as (33) is a positive system, x(t, ϕ, g) ≥ 0 for t ∈ R+ . We
can conclude from (35) and the above observations that for any nonnegative initial condition ϕ,
x(t, ϕ, g) converges to the origin. Hence, for the time-delay system (33), the origin is globally
asymptotically stable.
VI. C ONCLUSIONS
In this paper, we have presented a number of results that extend fundamental stability properties
of monotone linear systems to monotone nonlinear systems. Specifically, we have shown that
for such nonlinear systems, equivalence holds between asymptotic stability of the origin, the
existence of a max-separable Lyapunov function on a compact set, and D-stability. In addition,
we have demonstrated that if the origin for a delay-free monotone system is asymptotically stable,
then the corresponding system with bounded and unbounded time-varying delays also has an
asymptotically stable equilibrium point at the origin. We have derived a set of necessary and
sufficient conditions for establishing delay-independent stability of monotone nonlinear systems,
which allow us to extend several earlier works in the literature. We have illustrated the main
results with several examples.
A PPENDIX
Before proving the main results of the paper, namely, Theorems 1 and 2, we first state a key
lemma which shows that all components of a max-separable Lyapunov function are necessarily
monotonically increasing.
Lemma 1: Consider the monotone system (1) and the max-separable function V : Rn+ → R+
as in (7) with Vi : R+ → R+ differentiable for all i ∈ In . Suppose that there exist functions
ν1 , ν2 of class K such that
ν1 (xi ) ≤ Vi (xi ) ≤ ν2 (xi ),

(36)

for all xi ∈ R+ and all i ∈ In . Suppose also that there exists a positive definite function µ such
that
D + V (x) ≤ −µ(V (x)),

(37)

for all x ∈ Rn+ . Then, the functions Vi satisfy, for all xi > 0,
∂Vi
(xi ) > 0.
∂xi

(38)

Proof: For some j ∈ In , consider the state x = ej xj , where ej is the j th column of the

identity matrix I ∈ Rn×n , and xj ∈ R+ . From (36), Vj (xj ) > 0 for any xj > 0 and Vi (xi ) = 0

for i 6= j. Thus, the set J in (10) satisfies J (ej xj ) = {j} for all xj > 0. Evaluating (37) through
Proposition 2 leads to
D + V (ej xj ) =

∂Vj
(xj )fj (ej xj ) ≤ −µ(V (ej xj )) < 0
∂xj

(39)

for xj > 0. The strict inequality in (39) implies that ∂Vj /∂xj is nonzero and that its sign is
constant for all xj > 0. As a negative sign would yield Vj (xj ) < 0 (and, hence violate (36)) and
j is chosen arbitrarily, (38) holds.
A. Proof of Theorem 1
The theorem will be proven by first showing the equivalence 1) ⇔ 2) ⇔ 3). Next, this
equivalence will be exploited to subsequently show 3) ⇒ 4) and 4) ⇒ 1). Consequently, we
have
1) ⇔ 2) ⇔ 3) ⇒ 4) ⇒ 1),
which proves the desired result.
First, we note that the implication 2) ⇒ 1) follows directly from Lyapunov stability theory
(e.g., [46]) and we proceed to prove that 1) implies 2).
1) ⇒ 2): By asymptotic stability of the origin as in Definition 2, the region of attraction of
x = 0 defined as

A := x0 ∈ Rn+ | limt→∞ x(t, x0 ) = 0
is nonempty. In fact, we can find some δ > 0 such that all states satisfying 0 ≤ x < δ1 are in
A. Then, as the system (1) is monotone, the reasoning from [11, Theorem 3.12] (see also [51,
Theorem 2.5] for a more explicit statement) can be followed to show the existence of a vector
v such that 0 < v < δ1 and f (v) < 0.

Let ω(t) = x(t, v), t ∈ R+ be the solution to (1) starting from such a v. By the local Lipschitz
continuity of f , ω is continuously differentiable. As v ∈ A, wi (t) → 0 when t → ∞ for each
i ∈ In . Moreover, note that v is an element of the set
Ω = {x ∈ Rn+ | f (x) < 0}.
According to [19, Proposition 3.2.1], Ω is forward invariant so ω(t) ∈ Ω for all t ∈ R+ . Thus,
the components ωi (t) are strictly decreasing in t, i.e., ω̇i (t) < 0 for t ∈ R+ . This further implies
that, for a given state component xi ∈ (0, vi ], there exists a unique t ∈ R+ such that xi = ωi (t).
Let

Ti (xi ) = t ∈ R+ | xi = ωi (t) .

From the definition, it is clear that Ti (xi ) = ωi−1(xi ). Since ω̇i (t) < 0 for t ∈ R+ , the inverse of
ωi , i.e., the function Ti , is continuously differentiable and strictly decreasing for all xi ∈ (0, vi ].
We define, as in [43], [44], the component functions
Vi (xi ) = e−Ti (xi ) ,

i ∈ In .

(40)

Note that Vi (0) = 0. Moreover, Vi are continuously differentiable and strictly increasing for all
xi ∈ (0, vi ] as a result of the properties of the function Ti . Therefore, the component functions
Vi in (40) satisfy (16) for some functions ν1 , ν2 of class K. Moreover, from [44, Theorem 3.2],
the upper-right Dini derivative of the max-separable Lyapunov function (7) with components
(40) is given by
D + V (x) ≤ −V (x),
for all x ∈ X with X = {x ∈ Rn+ | 0 ≤ x ≤ v}. This shows that (17) holds, and hence the


proof is complete.

We now proceed to show equivalence between 2) and 3). We begin with the implication
2) ⇒ 3).
2) ⇒ 3): According to Lemma 1, the functions Vi are monotonically increasing. Thus, their
inverses
ρi (s) = Vi−1 (s)

(41)

can be defined for all s ∈ [0, Vi (vi )]. Define
s̄ := min Vi (vi ).
i∈In

Since v > 0, it follows from (16) that s̄ > 0. Moreover, due to continuous differentiability of Vi
and Lemma 1, ρi is of class K and satisfies (18).
In the remainder of the proof, it will be shown that the function ρ with components ρi defined
in (41) satisfies (19) for all s ∈ [0, s̄]. Thereto, consider a state xs , parameterized by s according

to the definition xs := ρ(s). Since s̄ ≤ Vi (vi ) for all i ∈ In , it holds that xs ∈ X for all s ∈ [0, s̄].
Evaluating (17) for such an xs yields




∂Vj s 
xj fj xs ≤ −µ V (xs ) .
D + V xs = maxs
j∈J (x ) ∂xj

(42)

The definition xsi = ρi (s), i ∈ In , implies, through (41), that Vi (xsi ) = s for all s ∈ [0, s̄].

Consequently, V (xs ) = s and the set J (xs ) in (42) satisfies J (xs ) = In , such that (42) implies



∂Vi
ρi (s) fi ρ(s) ≤ −µ s
∂xi

(43)

for s ∈ [0, s̄] and i ∈ In . Since Vi is strictly increasing and µ is positive definite, fi (ρi (s)) ≤ 0.
Define functions ri : [0, s̄] → R as


∂Vi
(zi ) ρi (s) ≤ zi ≤ ρi (s̄) .
ri (s) := sup
∂xi


By continuous differentiability of Vi and the result of Lemma 1, it follows that ri exists and
satisfies ri (s) > 0 for all s ∈ [0, s̄]. Moreover, it is easily seen that
ri (s) ≥

∂Vi
(ρi (s)).
∂xi

This together with (43) implies that
ri (s)fi (ρ(s)) ≤

∂Vi
(ρi (s))fi (ρ(s)) ≤ −µ(s)
∂xi

for all s ∈ [0, s̄]. Here, the inequality follows from the observation that fi (ρi (s)) ≤ 0. Then,
strict positivity of ri implies that
fi (ρ(s)) ≤ −

µ(s)
,
ri (s)

for all s ∈ [0, s̄] and any i ∈ In . Since ri is strictly positive and µ is positive definite, the
function αi (s) = µ(s)/ri (s) is positive definite and (19) holds.
We continue with the reverse implication 3) ⇒ 2).



3) ⇒ 2) : Define v := ρ(s̄). Since ρi , i ∈ In , are of class K and s̄ > 0, we have v > 0. Let
Vi be such that
Vi (xi ) = ρ−1
i (xi )

(44)

for xi ∈ [0, vi ]. Note that the inverse of ρi exists on this compact set as ρi is of class K and,
hence, is strictly increasing. Because of the same reason, it is clear that Vi as in (44) satisfies
(16) for some functions ν1 , ν2 of class K.
The remainder of the proof will show that the max-separable Lyapunov function (7) with
components (44) satisfies (17). By Proposition 2, the upper-right Dini derivative of V along
solutions of (1) is given by
∂Vj
(xj )fj (x).
j∈J (x) ∂xj

D + V (x) = max

(45)

Define X := {x ∈ Rn+ | 0 ≤ x ≤ v}, choose any x ∈ X and consider any j ∈ J (x), where
J (x) is defined in (10). Then, Vj (xj ) = V (x), such that the use of equality (44) leads to
xj = ρj (Vj (xj )) = ρj (V (x)).

(46)

Note that when i be different from j, a similar argument establishes that Vi (xi ) ≤ V (x) and,
thus, xi ≤ ρi (V (x)). Combining this with (46) gives
x ≤ ρ(V (x)).

(47)

Since f satisfies Kamke condition (4), it follows from of (46) and (47) that fj (x) ≤ fj (ρ(V (x))).
Moreover, ∂Vj /∂xj > 0 for all xj ∈ (0, vj ] due to (44) and the fact that ρi satisfies (18). The
above observations together with (45) imply that

∂Vj
(xj )fj ρ(V (x)) .
j∈J (x) ∂xj

D + V (x) ≤ max

(48)

Next, define functions r̄i : [0, s̄] → R+ as


∂Vi
(zi ) ρi (s) ≤ zi ≤ vi .
r̄i (s) := inf
∂xi

(49)

Note that r̄i is strictly positive for s ∈ (0, s̄]. Moreover, for any j ∈ J (x), the equality (46) is
recalled, from which it follows that


∂Vj
r̄j (V (x)) = r̄j (Vj (xj )) = inf
(zj ) xj ≤ zj ≤ vi
∂xj
∂Vj
(xj ).
≤
∂xj


(50)

Then, returning to the Dini derivative of V in (48), the use of (19), and subsequently, the
application of (50), leads to
D + V (x) ≤ max −
j∈J (x)

∂Vj
(xj )αj (V (x)),
∂xj

≤ max −r̄j (V (x))αj (V (x)),
j∈J (x)

(51)

for all x ∈ X . Recall that the functions αi are positive definite, whereas the functions r̄i in (49)
are strictly positive for s ∈ (0, s̄]. As a result, the functions r̄i (s)αi (s) are positive definite and
there exists a positive definite function µ such that µ(s) ≤ r̄i (s)αi (s) for all s ∈ [0, s̄] and all
i ∈ In . Applying this result to (51) yields
D + V (x) ≤ −µ(V (x)),
for all x ∈ X , proving (17) and finalizing the proof.



We now show that 3) implies 4) by exploiting the equivalence 1) ⇔ 2) ⇔ 3).
3) ⇒ 4): First, we show that the system (21) is monotone. Thereto, recall that monotonicity
of (1) implies
x′ ≤ x, x′i = xi ⇒ fi (x′ ) ≤ fi (x)


⇒ ψi x′i , fi (x′ ) ≤ ψi xi , fi (x) ,

(52)

where the latter implication follows from (20). Then, (52) represents the Kamke condition for
the vector field ψ(x, f ), such that monotonicity of (21) follows from Proposition 1.

Next, note that ϕ 0, f (0) = 0 implying that the origin is an equilibrium point of (21). In

order to prove asymptotic stability of the origin, we recall that f satisfies, by assumption, (19)
for some function ρ. From this, we have


ψ ρ(s), f (ρ(s) ≤ ψ ρ(s), −α(s) ,

(53)

where the inequality is maintained due to the fact that ψi is monotonically increasing in the
second argument for all i ∈ In . The functions αi and ρ are positive definite, hence −α(s) < 0
and ρ(s) > 0 for all s ∈ (0, s̄]. Then, from (20), we have


ψ ρ(s), −α(s) < ψ ρ(s), 0 = 0,

for s ∈ (0, s̄], where the equality follows from ϕ(x, 0) = 0 for any x ∈ Rn+ . Hence, ψ ρ(s), −α(s)



is negative definite and (53) is again of the form (19). From the implication 3) ⇒ 2) ⇒ 1), we
conclude the origin is asymptotically stable.



Finally, we prove that 4) implies 1).
4) ⇒ 1): Assume that the system (21) is asymptotically stable for any Lipschitz continuous
function ψ satisfying statement 4). Particularly, let ψ(x, y) = y. Then, the monotone system (1)


is asymptotically stable.
B. Proof of Theorem 2

1) ⇒ 2): Assume that x = 0 for the time-delay system (24) is asymptotically stable for
all delays satisfying Assumption 1. Particularly, let τ (t) = 0. Then, the non-delayed monotone
system (29) has an asymptotically stable equilibrium point at the origin.
2) ⇒ 1): For investigating asymptotic stability of the time-delay monotone system (24), we
employ Lyapunov-Razumikhin approach which allows us to impose minimal restrictions on timevarying delays [52]. In particular, we make use of the max-separable Lyapunov function that
guarantees asymptotic stability of the origin without time-delays as a Lyapunov-Razumikhin
function.
Let ρ be a function such that the delay-free system (29) satisfies (19), i.e.,


g ρ(s), ρ(s) ≤ −α s ,

(54)

holds for s ∈ [0, s̄]. Define v := ρ(s̄) and let X be the compact set (15). First, we show that for
any ϕ(t) ∈ X , t ∈ [−τmax , 0], the solution x(t, ϕ) satisfies x(t, ϕ) ∈ X for all t ∈ R+ .
Clearly, x(0, ϕ) = ϕ(0) ∈ X . In order to establish a contradiction, suppose that the statement
x(t, ϕ) ∈ X , t ∈ R+ , is not true. Then, there is i ∈ In and a time t̂ ∈ R+ such that x(t, ϕ) ∈ X
for all t ∈ [0, t̂], xi (t̂, ϕ) = vi , and

D + xi (t̂, ϕ) ≥ 0.

(55)

As x(t̂, ϕ) ≤ v, it follows from Assumption 2.1 that


gi x(t̂, ϕ), y ≤ gi v, y ,

(56)

for y ∈ Rn+ . As t̂−τ (t̂) ∈ [−τmax , t̂] and ϕ(t) ∈ X for t ∈ [−τmax , 0], we have x(t̂−τ (t̂), ϕ) ∈ X
irrespectively of whether t̂ − τ (t̂) is nonnegative or not. Thus, from Assumption 2.2,


gi y, x(t̂ − τ (t̂), ϕ) ≤ gi y, v ,

(57)

for any y ∈ Rn+ . Using (56) and (57), the Dini-derivative of xi (t, ϕ) along the trajectories of (24)
at t = t̂ is given by
D + xi (t̂, ϕ) = gi x(t̂, ϕ), x(t̂ − τ (t̂), ϕ)

≤ gi (v, v) = gi ρ(s̄), ρ(s̄)



≤ −αi (s̄) < 0,

which contradicts (55). Therefore, x(t, ϕ) ∈ X for t ∈ R+ .
We now prove the asymptotic stability of the origin. According to the proof of Theorem 1, if
the delay-free system (29) satisfies (54), then it admits a max-separable Lyapunov function (7)
with components Vi (xi ) = ρ−1
i (xi ) defined on X , see (44), such that


∂Vj
xj gj x, x ≤ −µ(V (x)),
j∈J (x) ∂xj

D + V (x) = max

(58)

holds for all x ∈ X . For any ϕ(t) ∈ X , from (44), we have


xj t, ϕ = ρj V (x(t, ϕ)) ,

j ∈ J (x(t, ϕ)),

(59)

with J as in (10). Combining (59) with the observation that


xi t, ϕ ≤ ρi V (x(t, ϕ)) ,
for any i ∈ In , implying that

x(t, ϕ) ≤ ρ V (x(t, ϕ)) =: x̄(t)

(60)

for ϕ(t) ∈ X . At this point we recall that x(t, ϕ) ∈ X for all t ∈ R+ . Thus, V (x(t, ϕ)) ≤ V (v).

This in turn implies that ρ(V (x(t, ϕ)) ≤ v and, hence, x̄(t) ∈ X for any t ∈ R+ . Next, it
follows from Assumption 2.1 that



gj x(t, ϕ), y ≤ gj x̄(t), y ,

j ∈ J (x(t, ϕ)),

for any ϕ(t) ∈ X and any y ∈ Rn+ . This condition will be exploited later in the proof.

(61)

In the remainder of the proof, the max-separable Lyapunov function V of the delay-free system
(29) will be used as a candidate Lyapunov-Razumikhin function for the time-delay system (24).
To establish a Razumikhin-type argument [52], it is assumed that


V x(s, ϕ) < q V (x(t, ϕ)) ,

(62)

for all s ∈ [t − τ (t), t], where q : R+ → R+ is a continuous non-decreasing function satisfying
q(r) > r for all r > 0. We will specify q later. By the definition of V , it follows that the
assumption (62) implies that
Vi (xi (t − τ (t), ϕ)) ≤ q(V (x(t, ϕ))),
for all i ∈ In . As a result,


x t − τ (t), ϕ ≤ ρ q(V (x(t, ϕ))) =: x̃(t),

(63)

such that the application of Assumption 2.2 yields


gi y, x(t − τ, ϕ) ≤ gi y, x̃(t) ,

(64)

for all i ∈ In and any y ∈ R+
n . Returning to the candidate Lyapunov-Razumikin function V , its
upper-right Dini derivative along trajectories x(·, ϕ) reads


∂Vj
xj (t, ϕ) gj x(t, ϕ), x(t − τ (t), ϕ)
j∈J (x(t,ϕ)) ∂xj


∂Vj
≤ max
xj (t, ϕ) gj x̄(t), x(t − τ (t), ϕ)
j∈J (x(t,ϕ)) ∂xj


∂Vj
= max
x̄j (t) gj x̄(t), x(t − τ (t), ϕ) ,
j∈J (x(t,ϕ)) ∂xj


D + V x(t, ϕ) =

max

(65)

where (61) was used to obtain the inequality (exploiting Proposition 1 as before) and (59) to get
the second equality. Next, recall that the assumption (62) implies (64), such that (65) is bounded
as


∂Vj
x̄j (t) gj x̄(t), x̃(t)
j∈J (x(t,ϕ)) ∂xj


∂Vj
≤ max
x̄j (t) gj x̄(t), x̃(t) .
j∈J (x̄(t)) ∂xj


D + V x(t, ϕ) ≤

max

(66)

Here, the second inequality follows from the observation that J (x̄(t)) = In for any t, such that
J (x(t, ϕ)) ⊆ J (x̄(t)).

We recall that the value of x̃(t) in (66) is dependent on the choice of the function q, see (63).
At this point, we assume that q can be chosen such that

 µ(V (x̄(t)))
,
0 ≤ gi x̄(t), x̃(t) − gi x̄(t), x̄(t) ≤
2kD

(67)

holds for all x̄(t) ∈ X and for any i ∈ In . Here, k ≥ 1 will be chosen later and D > 0 is such
that
∂Vi
(xi ) ≤ D,
∂xi

∀x ∈ X , ∀i ∈ In .

(68)

Note that such D exists by continuous differentiability of V and the fact that X is a compact
set. Also, we stress that the first inequality in (67) follows from the observation that x̄(t) ≤ x̃(t)
(compare (60) with (63) and recall that q(r) > r for all r > 0 and the functions ρi are of class
K).
Now, under the assumption (67), it follows from (66) that



 µ V (x̄(t))
 ∂Vj

∂Vj
+
x̄j (t) gj x̄(t), x̄(t) +
x̄j (t)
D V x(t, ϕ) ≤ max
j∈J (x̄(t)) ∂xj
∂xj
2kD



1
µ V (x̄(t)) .
≤− 1−
2k

(69)

Here, (58) as well as the bound (68) are used and it is recalled that V (x̄(t)) = V (x(t, ϕ)) by the
choice of x̄(t) as (60). As (69) holds for any trajectory that satisfies (62) and k ≥ 1, asymptotic
stability of the origin follows from the Razumikhin stability theorem, see [52, Theorem 7],
provided that the assumption (67) holds.
In the final part of the proof, we will construct a function q that satisfies the assumption (67).
To this end, define the compact set
X̃ := {x ∈ Rn+ | 0 ≤ x ≤ 2v}.
Clearly, X ⊂ X̃ . Since g in (24) is locally Lipschitz, there is a constant LX˜ > 0 such that
kg(x, y ′) − g(x, y)k∞ ≤ LX˜ ky ′ − yk∞

(70)

for all x, y, y ′ ∈ X˜ and with kxk∞ = maxi |xi |. Since x̄(t) ∈ X , we have x̄(t) ∈ X̃ . If x̃(t) ∈ X˜ ,
it follows from (70) that


gj x̄(t), x̃(t) − gj x̄(t), x̄(t) ≤ LX̃ max {x̃i (t) − x̄i (t)},
i∈In




= LX˜ max ρi q(V (x̄(t))) − ρi V (x̄(t)) ,
i∈In

(71)
(72)

where the property x̄(t) ≤ x̃(t) and the first inequality in (67) are used to obtain inequality (71).
Equality (72) follows from the definitions (60) and (63). The desired condition (67) holds if

 µ̃(V (x̄(t)))
ρi q(V (x̄(t))) − ρi V (x̄(t))) ≤
2kDLX̃
µ(V (x̄(t)))
≤
2kDLX̃

(73)

for all i ∈ In . Here, µ̃ : [0, s] → R+ is a function of class K that lower bounds the positive
definite function µ. Such a function µ̃ exists by the fact that µ is positive definite on the compact
set [0, s] [55]. At this point, we note that, even though x̄(t) ∈ X̃ , this does not necessarily hold
for x̃(t). However, the condition (73) implies that
x̃(t) ≤ x̄(t) +

µ(V (x̄(t)))
µ(V (x̄(t)))
1n ≤ v +
1n ,
2kDLX̃
2kDLX̃

such that choosing k sufficiently large guarantees x̃(t) ∈ X̃ . For this choice, the Lipschitz
condition (70) indeed holds.
After fixing k ≥ 1 as above and denoting s = V (x̄), the function q : [0, s̄] → R+ defined as


µ̃(s)
−1
.
q(s) := min ρi ρi (s) +
i∈In
2kDLX̃
This function satisfies (73) and, hence, (67). In addition, since the functions ρi and µ̃ are of
class K, q is nondecreasing. Also, it can be observed that q(s) > s for all s > 0 as required.
Therefore, all conditions on q are satisfied. This finalizes the proof of the implication 2) ⇒ 1).
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