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A Robust Stability Condition for Subhomogeneous Cooperative Time-delay Systems*

Conference Paper

A Robust Stability Condition for Subhomogeneous Cooperative Time-delay Systems*

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.
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 xRnand i= 1, . . . , n,xidenotes
the ith coordinate of x. We define Rn
+:= {xRn:xi
0,1in}.Rn
+is called the positive orthant of Rn.
The interior of Rn
+is defined as int (Rn
+) := {xRn:
xi>0,1in}. 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: xyif
xiyi;x > y if xyand x6=y; and xyif xi> yi
for 1in.
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 tfRwith tf0. For xC([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 be 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
xC([τ, tf),Rn),x(t)and x(t)satisfies (1) for all
t[0, tf)[7].
For a given φC([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 x0y0, 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 aband 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 be a subset of
C([τ, 0],Rn)and let x(t, φ)represent the trajectory of the
system (1) with respect to initial condition φat time
t. Then the system (1) is said to be monotone if for every
φ, ψ , satisfying
φ(θ)ψ(θ),for all τθ0
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 τθ0and 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 be a subset of C([τ , 0],Rn). The
system (1) is monotone in , if the quasimonotone condition
is satisfied for every φ, ψ .
¿From now on, for any pRnwe define ˆp
C([τ, 0],Rn)to be:
ˆp(θ)p, θ[τ, 0] (3)
Let be an open subset of C([τ, 0],Rn). The equilibria
of (1) in are those φsuch 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={ˆpΩ : pRnand hp)=0}(5)
If ˆpEwith p= 0, then we say the system (1) has an
equilibrium at the origin. If pint (Rn
+), then we say the
system (1) has an equilibrium in the interior of the positive
orthant. If pbd (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) = hy)(7)
with yRnand ˆyas defined in (3).
Lemma 2.3: Let h: Ω 7→ Rn, where is 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 be an open subset of
C([τ, 0],Rn). As already discussed, the set of equilibria
of the system (1) in is:
E={ˆpΩ : pRnand hp) = 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 , where is an open subset of C([τ, 0],Rn).
If vRnis such that ˆvand h(ˆv)0(hv)0), then
x(t, ˆv), solution of (1) through ˆv, is non-increasing (non-
decreasing) for all t0.
Note that since H(v) = hv), as defined in (7), then H(v)
0implies hv)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 , where is an open subset of C([τ, 0],Rn)
and let Hbe defined as (7). Assume that there exists a vector
v0with H(v)0(H(v)0). Then the trajectory x(t, ˆv)
of the system (1) is non-increasing (non-decreasing) for all
t0.
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 of
C([τ, 0],Rn). Let x(t, ˆx0), the trajectory of the system (1),
be bounded for all t0and for all ˆx0. If x(t, ˆx0)
ˆx0(or x(t, ˆx0)ˆx0) for tbelonging to some non-empty
subinterval of (0,), then x(t, ˆx0)ˆpEas 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 vRn
+,λR
with λ1. 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. Since α(0,1), we have λ > 1.
Therefore:
H(λz)λH (z)zRn, 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 be 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 φω,λRwith λ1. 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(tτi)) τi0(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
1imτi
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 τi0, 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) = hv), we can conclude that
hv)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 φC([τ, 0] that satisfies φˆv.
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
x0Rn
+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 x0Rn
+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 τi0, 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 φC([τ, 0],Rn
+), we can find a v0
such that φˆvand hv)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 x0Rn
+, 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
pint (Rn
+), then the system (9) has an asymptotically
stable equilibrium at ˆp(as defined in (3)) for all τi0, 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) = hv), we can conclude
that hv)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 px0v,
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 wx0p,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 pint (Rn
+)then for every
x0pthere exists vx0such that H(v)0and for
every x0pthere 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 λ1,
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 x0λv 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 pint (Rn
+)
then the system (9) has a globally asymptotically stable
equilibrium at ˆpfor all τi0, 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 φC([τ, 0],Rn
+), we can find a vpand
wpsuch that ˆwφˆvwith hv)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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... Such a system can include terms such as x τ a+x τ for a > 0 and τ > 0, which arise frequently in models of biochemical reaction networks [25]. However, for subhomogeneous cooperative systems, research has focused mainly on stability [24], [26], [27], such as the robust stability of delay [28]. Subhomogeneous cooperative systems that are disturbed by exogenous input have not been studied in the framework of ISS theory. ...
... It can be easily checked that f is C 1 on R n + , cooperative and subhomogeneous of degree 2 3 . Hence, (28) with w ≡ 0 is positive and monotone. Additionally, f (x) = 0 has one equilibrium at (2.0235, 2.2624) T . ...
... Consider the systeṁ x = f (x) + g(x)w w.(28) ...
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Preface. 1. Models. 2. General Theory. 3. Stability of Retarded Differential Equations. 4. Stability of Neutral Type Functional Differential Equations. 5. Stability of Stochastic Functional Differential Equations. 6. Problems of Control for Deterministic FEDs. 7. Optimal Control of Stochastic Delay Systems. 8. State Estimates of Stochastic Systems with Delay. Bibliography. Index.
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