DSpace at VNU: Exponential Stability of Functional Differential Systems

DSpace at VNU: Exponential Stability of Functional Differential Systems tài liệu, giáo án, bài giảng , luận văn, luận án...

DOI 10.1007/s10013-016-0193-z

Exponential Stability of Functional Differential Systems

Pham Huu Anh Ngoc 1 · Cao Thanh Tinh 2

Received: 13 April 2015 / Accepted: 22 September 2015

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Abstract We present a novel approach to exponential stability of functional differential

systems Our approach is relied upon the theory of positive linear functional differential sys-tems and a comparison principle Consequently, we get some comparison tests and explicit criteria for exponential stability of functional differential systems Two examples are given

to illustrate the obtained results

Keywords Functional differential systems· Exponential stability

Mathematics Subject Classification (2010) Primary 34K20· Secondary 34 K15

1 Introduction

Functional differential systems have numerous applications in science and engineering They are used as models for a variety of phenomena in the life sciences, physics and technology, chemistry, and economics, see, e.g., [10,15,26]

Problems of stability of functional differential systems have been studied intensively during the past decades, see, e.g., [1,5 20,26–29] and the references therein Recently, problems of exponential stability of functional differential systems have attracted much attention from researchers, see, e.g., [1,12,16,18–23,27–29]

 Pham Huu Anh Ngoc

phangoc@hcmiu.edu.vn

Cao Thanh Tinh

tinhct@uit.edu.vn

1 Department of Mathematics, International University, Vietnam National University-HCMC,

Thu Duc District, Ho Chi Minh City, Vietnam

2 Department of Mathematics, University of Information Technology, Vietnam National

University-HCMC, Thu Duc District, Ho Chi Minh City, Vietnam

The traditional approaches to analyze the stability of delay differential systems are Lyapunov’s method and its variants (Razumikhin-type theorems, Lyapunov–Krasovskii functional techniques), see, e.g., [6 13,27–29] That is why most existing stability criteria

in the literature for delay differential systems (even for linear delay time-invariant systems) are given in terms of matrix inequalities or differential inequalities To the best of our knowl-edge, there are not many explicit criteria for exponential stability of time-varying (linear and nonlinear) delay differential systems Furthermore, in general, it is difficult to construct Lyapunov functions for delay differential systems

In this paper, we present a novel approach to the exponential stability of functional differ-ential systems Our approach is based on the theory of positive linear functional differdiffer-ential systems (see [17]) and the comparison principle Consequently, we get some comparison tests for exponential stability of functional differential systems which are analogs of com-parison test for convergence of infinite series In particular, we obtain some new explicit criteria for exponential stability of functional differential systems Roughly speaking, the main result of this paper (Theorem 6) says that “If a nonlinear (time-varying) functional

dif-ferential system is bounded above by a positive linear time-invariant difdif-ferential system and

the linear system is exponentially stable then the nonlinear system is exponentially stable too” This is a nice surprise because it is very similar to the well-known Weierstrass M-test

in the theory of infinite series of functions, see, e.g., [25] To the best of our knowledge, Theorem 6 of this paper is original Furthermore, Theorem 6 has potential applications, for example, it can be used to study the exponential stability of equilibria of various classes of neural networks such as Cohen–Grossberg neural networks, Hopfield-type neural networks and cellular neural networks, etc

We proceed as follows In the next section, we give notation and preliminary results which will be used in what follows In Section3, we present some comparison tests for exponential stability of functional differential systems Consequently, we derive explicit cri-teria for exponential stability of functional differential systems A discussion to the obtained results and two illustrative examples are given

2 Preliminaries

Let R be the set of all real numbers and let C be the set of all complex numbers For

a complex number z, denote by z the real part of z For a natural number n, let n := {1, 2, , n} Let R l ×q be the set of all l × q-matrices with real entries In what follows,

inequalities between real matrices or vectors will be understood componentwise, i.e., for

two real l × q-matrices A = (a ij ) and B = (b ij ) , the inequality A ≥ B means a ij ≥ b ijfor

i = 1, , l, j = 1, , q The set of all nonnegative l × q-matrices is denoted by R l ×q

+ .

For x ∈ Rn and P ∈ Rl ×q, we define|x| = (|x i |) and |P | = (|p ij |) A norm  ·  on R n

is said to be monotonic if x ≤ y whenever x, y ∈ R n , |x| ≤ |y| Every p-norm on R n ( x p = (|x1|p + |x2|p + · · · + |x n|p ) p1,1≤ p < ∞ and x∞ = maxi =1,2, ,n |x i |) are

monotonic

A matrix A ∈ Rn ×n is called a Metzler matrix if all the off-diagonal entries of A are nonnegative With a given matrix A = (a ij ) ∈ Rn ×n, one associates the Metzler matrix

M(A) := (ˆa ij )∈ Rn ×n ,where ˆa ij := |a ij |, i = j, i, j ∈ n; ˆa ii := a ii , i ∈ n For a square matrix A∈ Rn ×n , the spectral abscissa of A is denoted by

μ(A) = max{λ : det(λI n − A) = 0}.

We now summarize some properties of Metzler matrices which will be used in what follows.

Theorem 1 [24] Suppose M ∈ Rn ×n is a Metzler matrix Then

(i) (Perron–Frobenius) μ(M) is an eigenvalue of M and there exists a nonnegative

eigenvector x = 0 such that Mx = μ(M)x.

(ii) Given α ∈ R, there exists a nonzero vector x ≥ 0 such that Mx ≥ αx if and only if

(iii) (tIn − M)−1exists and is nonnegative if and only if t > μ(M).

(iv) Given B∈ Rn ×n

+ , C∈ Cn ×n Then

The following is immediate from Theorem 1 and is used in what follows

Theorem 2 Let M ∈ Rn ×n be a Metzler matrix Then the following statements are

equivalent

(i) μ(M) < 0;

(iii) M is invertible and M−1≤ 0;

(iv) For given b∈ Rn , b 0, there exists x ∈ R n

+such that Mx + b = 0;

(v) For any x∈ Rn

+\ {0}, the row vector x T M has at least one negative entry.

LetRnbe endowed with a vector norm ·  andC := C([−h, 0], R n )be a Banach space

of all continuous functions on [−h, 0] with values in R n normed by the maximum norm

φ = max θ ∈[−h,0] φ(θ) Denote

C+:= {ϕ ∈ C : ϕ(θ) ∈ R n

+ ∀θ ∈ [−h, 0]},

the positive convex cone ofC For ϕ ∈ C, let |ϕ| ∈ C+be defined by|ϕ|(s) := |ϕ(s)|, s ∈ [−h, 0] An operator L : C → R n is called positive if Lϕ∈ Rn

+for any ϕ∈C+

Let N BV ( [−h, 0], R n ×n ) denote the Banach space of all matrix functions η( ·) of

bounded variation on[−h, 0], continuous from the left on [−h, 0], satisfying η(−h) = 0

and endowed with the normη = Var(η; −h, 0) Finally, η(·) is said to be increasing if

η(θ2) ≥ η(θ1)for−h ≤ θ1≤ θ2≤ 0

3 Exponential Stability of Functional Differential Systems

Consider a linear functional differential system of the form

where for each t ≥ 0, x t ∈C defined by xt (θ ) = x(t + θ), θ ∈ [−h, 0] and A ∈ R n ×nand

L:C → R nis a bounded linear operator given by

−h d [η(θ)]ϕ(θ), ϕ ∈ C,

with η( ·) ∈ NBV ([−h, 0], R n ×n ).

For given ϕ∈C, (1) always has a unique solution x( ·, ϕ) such that

see, e.g., [11] Then (1) is said to be exponentially stable, if there are constants M≥ 1 and

α >0 such that

x(t, ϕ) ≤ Me −αt ϕ ∀t ≥ 0, ∀ϕ ∈ C.

It is well-known that (1) is exponentially stable if and only if

sup



z : det



zI n − A −

 0

−h e

zθ d [η(θ)]



= 0



< 0,

see, e.g., [11]

Clearly, it is not easy to verify this condition So it is very interesting to find a subclass

of systems, for which stability criteria are explicit and easy to verify in practice

Definition 1 [17] The system (1) is said to be positive if for any initial function ϕ ∈C+,

the corresponding solution x( ·, ϕ) of (1)–(2) satisfies x(t, ϕ)∈ Rn

+for every t≥ 0

Theorem 3 [17]

(a) The following statements are equivalent: (i) (1) is positive; (ii) A∈ Rn ×n is a Metzler

matrix and L is positive; (iii) A∈ Rn ×n is a Metzler matrix and η( ·) is increasing.

(b) Let (1) be positive Then (1) is exponentially stable if and only if μ(A + η(0)) < 0.

Definition 2 Let A0∈ Rn ×n and η

0( ·) ∈ NBV ([−h, 0], R n ×n )be given The system

where L0ϕ:=0

−h d [η0(θ ) ]ϕ(θ), ϕ ∈ C, is said to be bounded above by (1) if

For example, it is easy to check that the linear time delay differential system

˙x(t) = A0x(t)+

m



i=1

A i x(t − h i )+

 0

−h C(s)x(t + s)ds

is bounded above by

˙x(t) = B0x(t)+

m



i=1

B i x(t − h i )+

 0

−h D(s)x(t + s)ds

if and only if

Ai ≤ B i , i ∈ {0, 1, , m} and C(s) ≤ D(s) ∀s ∈ [−h, 0].

We are now in the position to state the first result of this paper

Theorem 4 (Comparison stability test for positive functional differential systems) Suppose

(1) and (3) are positive and (3) is bounded above by (1) Then

(i) If (1) is exponentially stable then (3) is exponentially stable.

(ii) If (3) is not exponentially stable then (1) is not exponentially stable.

Proof Since (1) and (3) are positive systems, it remains to show that μ(A0 + η0( 0)) ≤

μ(A +η(0)), by Theorem 3 Fix x ∈ R n

+and let ϕ0(θ ) = x ∀θ ∈ [−h, 0] Clearly, ϕ0∈C+ Since (3) is bounded above by (1), (4) implies L0ϕ0≤ Lϕ0 This gives (η(0) −η0( 0))x≥ 0

Since this holds for any x∈ Rn

+, η(0) ≥ η0( 0) Thus, A + η(0) = A0+ (A − A0) + η(0) ≥

A0+η0( 0) By Theorem 1(iv), μ(A0+η0( 0)) ≤ μ(A+η(0)) This completes the proof.

Remark 1 (i) In the theory of infinite series, the comparison test for convergence of infinite

series with nonnegative terms (see, e.g., [25]) asserts that if 0 ≤ a n ≤ b nfor sufficiently

large n∈ N, then

n=1anconverges if so does ∞

n=1bn;

n=1b ndiverges if so does ∞

n=1a n Clearly, Theorem 4 is an analog of the comparison test for convergence of infinite series for positive linear functional differential systems

(ii) The assumption of positivity of systems imposed in Theorem 4 cannot be removed

To see it, we consider the following systems:

where b > 0 Clearly, (5) is positive and exponentially stable, by Theorem 3 Furthermore, (6) is bounded above by (5) However, (6) may not be exponentially stable In fact, the characteristic equation of (6) is given by z + 2 + be −z = 0, which is equivalent to (z + 2)e z + b = 0 This equation has a root z0withz0 ≥ 0 if b > π + 2, see [11, Theorem A.5,

p 416] Thus, (6) is not exponentially stable if b > π+ 2 Note that (6) is not a positive equation, by Theorem 3

As shown in Remark 1, (3) may not be exponentially stable although it is bounded above

by a positive and exponentially stable system Actually, (3) is exponentially stable under a slightly stronger condition

Theorem 5 (Comparison stability test for functional differential systems) Let A0 ∈ Rn ×n

and η0( ·) ∈ NBV ([−h, 0], R n ×n ) be given Suppose L:C → R n , Lϕ:=0

−h d [η(θ)]ϕ(θ),

with η( ·) ∈ NBV ([−h, 0], R n ×n ), is positive If |L0ϕ | ≤ L|ϕ| for any ϕ ∈ C and

˙x(t) = M(A0)x(t) + Lx t , t≥ 0

is exponentially stable then (3) is exponentially stable In other words, (3) is exponentially

stable if μ(M(A0) + η(0)) < 0.

Proof Theorem 5 is just a particular case of Theorem 6 given below So, we omit the proof.

Remark 2 It is well known that if |a n | ≤ b n for sufficiently large n ∈ N and ∞n=1bn converges then ∞

n=1an(absolutely) converges (see, e.g., [25]) Theorem 5 gives an analog

of the comparison test for convergence of infinite series, for exponential stability of linear functional differential systems

To end this paper, we state and prove an extension of Theorem 5 to nonlinear functional differential systems

Consider a nonlinear time-varying functional differential equation of the form

(i) For any t ∈ R+, x t ( ·) ∈ C is defined by xt (θ ) := x(t + θ), θ ∈ [−h, 0] for given

h >0;

(ii) f ( ·, ·) : R+× Rn→ Rnis a given continuous function and is locally Lipschitz in the

second argument, uniformly in t on compact intervals ofR+and f (t, 0)= 0 for all

t∈ R+;

(iii) g( ·; ·) : R+×C → R n is a given continuous function such that g(t ; 0) = 0 ∀t ∈ R+

and g(t ; u) is (locally) Lipschitz continuous with respect to u on each compact subset

ofR+×C.

It is well-known that for fixed σ ∈ R+and given ϕ ∈C, there exists a unique local solution

of (7), denoted by x( ·, σ, ϕ) satisfying the initial value condition

see, e.g., [11] This solution is defined and continuous on[σ − h, γ ) for some γ > σ and

satisfies (7) for every t ∈ [σ, γ ) see, e.g., [11, p 44] Furthermore, if the interval[σ − h, γ )

is the maximum interval of existence of the solution x( ·, σ, ϕ) then x(·, σ, ϕ) is said to be

noncontinuable The existence of a noncontinuable solution follows from Zorn’s lemma and the maximum interval of existence must be open

Definition 3 The zero solution of (7) is said to be (globally) exponentially stable if there

exist positive numbers K, β such that for each σ ∈ R+ and each ϕ ∈ C, the solution x( ·, σ, ϕ) of (7)–(8) exists on[σ − h, +∞) and furthermore satisfies

x(t, σ, ϕ) ≤ Ke −β(t−σ) ϕ ∀t ≥ σ.

Theorem 6 Let for each t ∈ R+, f (t, ·) be continuously differentiable on R n Suppose there exist a matrix A := (a ij )∈ Rn ×n and a positive linear bounded operator L:C → R n defined by Lϕ=0

−h d [η(θ)]ϕ(θ), with η(·) ∈ NBV ([−h, 0], R n ×n ) such that

∂fi

for any t∈ R+and for any x ∈ Rn and

The zero solution of (7) is exponentially stable provided (1) is exponentially stable In

other words, the zero solution of (7) is exponentially stable if μ(A + η(0)) < 0.

Remark 3 Theorem 6 is an analog of the Weierstrass M-test for infinite series of functions

(see, e.g., [25, Theorem 7.10, p 134]) Roughly speaking, if the nonlinear functional dif-ferential system (7) is “bounded above” by the positive linear functional differential system (1) then the zero solution of (7) is exponentially stable provided so is (1)

Proof of Theorem 6 Since A∈ Rn ×n is a Metzler matrix and L is a positive linear bounded

operator, the system (1) is positive Therefore, η(0) ≥ η(−h) = 0 and thus A + η(0) is also

a Metzler matrix It follows from μ(A + η(0)) < 0 that

for some p := (α1, α2, , α n ) T ∈ Rn , α i >0∀i ∈ n, by Theorem 2 For given ϕ ∈ C,

let x(t) := x(t, σ, ϕ), t ∈ [σ − h, γ ) be a noncontinuable solution of (7)–(8) and let

y(t) := y(t, |ϕ|), t ∈ [0, ∞) be the solution of (1) such that y(s) = |ϕ|(s), s ∈ [−h, 0] We

show that|x(t + σ )| ≤ y(t) ∀t ∈ [−h, γ − σ ).

Fix ζ > 0 and define u(t) := y(t) + ζp ∀t ∈ [−h, ∞) Clearly, |x(t + σ )| = |ϕ(t)| =

y(t)

Assume on the contrary that there exists t∗∈ [0, γ − σ ) such that |x(t∗+ σ )|  u(t∗) Set

tb := inf{t ∈ [0, γ − σ ) : |x(t + σ )|  u(t)} By continuity, t b > 0 and there is i0∈ n such

that

|x(t + σ )| ≤ u(t) ∀t ∈ [0, t b );

|x i0(tb + σ )| = u i0(tb);

|x i0(τk + σ )| > u i0(τk), τk ∈ (t b , tb+1

for some τ k ∈ (t b, tb + 1

k ), k ∈ N By the mean value theorem [3], we have for each

t ∈ [σ, γ ) and for each i ∈ n

˙x i (t) = (f i (t, x(t)) − f i (t, 0)) + g i (t, xt)=

n



j=1

 1 0

dfi dxj (t, sx(t))ds



xj(t) + g i (t, xt).

Thus,

d

dt |x i (t) | = sgn(x i (t)) ˙x i (t) ≤

 1 0

df i

dx i (t, sx(t))ds



|x i (t)| +

n



j =1,j=i

 1 0

df i

dx j (t, sx(t))



|x j (t) | + |g i (t, x t )|

for almost any t ∈ [σ, γ ) Taking (9) and (10) into account, we obtain

d

dt |x i (t) | ≤ a ii |x i (t)| +

n



j =1,j=i

aij |x j (t)| +

n



j=1

 0

−h d [η ij(θ ) ]|x j(t + θ)|

for almost any t ∈ [σ, γ ) It follows that for any t ∈ [σ, γ )

D+|x i (t)| := lim sup

→0 +

|x i (t + )| − |x i (t)|

→0 +

1



 t +

t

d

ds |x i (s) |ds

≤ a ii |x i (t)| +

n



j =1,j=i

a ij |x j (t)| +

n



j=1

 0

−h d [η ij (θ ) ]|x j (t + θ)|, where D+ denotes the Dini upper-right derivative In particular, it follows from (11)–(12) that

D+|x i0(tb + σ )| ≤

n



j=1

ai0j |x j (tb + σ )| +

n



j=1

 0

−h d [η i0j(θ ) ]|x j (tb + σ + θ)|

( 12)

≤

n



j=1

ai0j (yj (tb ) + ζα j)+

n



j=1

 0

−h d [η i0j (θ ) ](y j (tb + θ) + ζα j)

≤ D+y i0(t b ) + ζ

⎛

⎝n

j=1

a i0j α j+

n



j=1

(η i0j ( 0))α j

⎞

⎠( 11)

< D+y

i0(t b ).

However, this conflicts with (12) Therefore,

|x(t + σ )| ≤ y(t) + ζp ∀t ∈ [0, γ − σ ).

Without loss of generality, supposeRnis endowed with a monotonic norm This implies

x(t + σ ) = |x(t + σ )| ≤ u(t) = y(t) + ζp ≤ y(t) + ζp ∀t ∈ [0, γ − σ ).

(13)

Letting ζ tend to zero in (13), we obtain

x(t + σ ) = |x(t + σ )| ≤ y(t) ∀t ∈ [0, γ − σ ). (14) Since (1) is exponentially stable, there are K ≥ 1 and α > 0 such that

Then (14) and (15) imply

Finally, we show that γ = ∞ and so the zero of (7) is exponentially stable Seeing a

contradiction, we assume that γ <∞ Then it follows from (16) that x( ·, σ, ϕ) is bounded

on[σ, γ ) Furthermore, this together with (7), (9), and (10) imply that ˙x(·) is bounded on [σ, γ ) Thus, x(·) is uniformly continuous on [σ, γ ) Therefore, lim t →γ−x(t)exists and

x( ·) can be extended to a continuous function on [σ, γ ] Moreover, the closure of {x t : t ∈ [σ, γ )} is a compact set in C by the Arzela–Ascoli theorem [3] Note that{(t, x t ) : t ∈ [σ, γ )} ⊂ [σ, γ ]× the closure of {x t : t ∈ [σ, γ )} Thus, the closure of {(t, x t ) : t ∈ [σ, γ )}

is a compact set inR ×C Since (γ, xγ )belongs to this compact set, one can find a solution

of (7) through this point to the right of γ This contradicts the noncontinuability hypothesis

on x( ·) Thus, γ must be equal to ∞ This completes the proof.

4 Discussion and Illustrative Examples

Consider the linear delay differential equation

where a > 0, h > 0, and b( ·) is a bounded continuous function on R+ By applying a Razumikhin-type theorem to (17), it has been shown in [13, Example 5.1, p 74] that (17) is

exponentially stable if b:= supt∈R +|b(t)| < a.

Note that (17) is bounded above by the positive linear delay differential equation

Because of−a +b < 0, (18) is exponentially stable, by Theorem 3 Therefore, the above assertion is immediate from Theorem 5 More generally, the linear differential equation with delays of the form

˙x(t) = a(t)x(t) + b(t)x(t − h1) + c(t)x(t − h2), t∈ R+, h1, h2>0

is exponentially stable, by Theorem 5 provided a( ·) ∈ C(R+, R) satisfies a(t) < −a, t ∈

R+for some a > 0 and b( ·), c(·) ∈ C(R+, R) are bounded such that

sup

t∈R +

|b(t)| + sup

t∈R +

|c(t)| < a.

Next, using a Lyapunov–Krasovskii functional, it has been proven in [10, p 154] that the zero solution of the nonlinear delay differential system

˙x(t) = Ax(t) + f (x(t − h))

is asymptotically stable provided A ∈ Rn ×n is a Metzler matrix and f ( ·) : R n → Rnis

locally Lipschitz such that f (0) = 0, f (x) ≥ 0 ∀x ∈ R n and f (x) ≤ γ x ∀x ∈ R n

+ and

μ(A + γ I n) < 0 for some γ > 0 (compare with Theorem 6).

Furthermore, consider the nonlinear delay differential system

˙x(t) = Ax(t) + F (t; x(t − h1), , x(h − h m )) , t∈ R+, (19)

where h1, , hm > 0, A∈ Rn ×n and F ( ·) is continuous in all its arguments It has been

shown in [12, Theorem 3.1] that (19) is exponentially stable if

F (t; u1, , um) ≤

m



i=1

βi u i  ∀t ∈ R+, ∀u1, , um ∈ Rn , (20) and

γ (A)+

m



i=1

β i < 0,

where γ (A):= limε→0 +εA+In−1

ε is the matrix measure of A.

The spirit of this result is very close to our ideas Note that (20) means that (19) is bounded above by the scalar positive linear delay differential equation

˙y(t) = γ (A)y(t) +

m



i=1

Clearly, (21) is positive and exponentially stable, by Theorem 3 This ensures that (19)

is exponentially stable If, instead of (20), one assumes that

|F (t; u1, , u m )| ≤

m



i=1

β i |u i | ∀t ∈ R+, ∀u1, , u m∈ Rn , (22) then [12, Theorem 3.1] follows directly from Theorem 6

To end this section, we illustrate Theorem 6 by two examples to which Theorem 3.1 of [12] cannot be applied

Example 1 Consider the scalar differential equation with delay

˙x(t) = −2x(t) + sin(t + x(t)) +

 0

where r( ·, ·, ·) : R+× [−h, 0] × R → R is a given continuous function and r(t, s, u) is Lipschitz continuous with respect to u on each compact subset ofR+× [−h, 0] × R and

r(t, s, 0) = 0 ∀(t, s) ∈ R+× [−h, 0].

Assume that there exists a continuous function m( ·) : [−h, 0] → R+such that

|r(t, s, u)| ≤ m(s)|u| ∀t ∈ R+, ∀s ∈ [−h, 0], ∀u ∈ R.

Let

f (t, x) := −2x + sin(t + x), t ∈ R+, x ∈ R,

and

g(t ; ϕ) := 0

−h r(t, s, ϕ(s))ds, t ∈ R+, s ∈ [−h, 0], ϕ ∈ C([−h, 0], R).

∂f

∂x (t, x) = −2 + cos(t + x) ≤ −1, t ∈ R+, x ∈ R,

and

|g(t; ϕ)| ≤ 0

−h m(s) |ϕ(s)|ds, ϕ ∈ C([−h, 0], R).

Thus, the zero solution of (23) is exponentially stable if0

−h m(s)ds <1, by Theorem 6.

The next example gives an application of Theorem 6

Example 2 [2] Consider a bidirectional associative memory (BAM) model described by

˙u i(t) = −d i ui (t)+

n



j=1

where d i > 0, I i ∈ R, i ∈ n and E := (e ij )∈ Rn ×n and for each j ∈ n, r j : R+ → R is

bounded and globally Lipschitz with constant L j(i.e.,|r j(uj ) − r j(vj) | ≤ L j |u j − v j| for

all u j , v j)

Let D := diag(− d1

L1,−d2

L 2 , ,−d n

L n ) ∈ Rn ×n We show that if μ(D + |E|) < 0 then

(24) has a unique equilibrium point u∗which is globally exponentially stable.

Consider the continuous function F : Rn→ Rndefined by

u := (u1, u2, , u n ) T → F (u) := (ξ1, ξ2, , ξ n ) T , ξ i:= 1

d i

⎛

⎝n

j=1

e ij r j (u j ) + I i

⎞

⎠ , i ∈ n.

Since for each j ∈ n, r j : R+ → R is bounded, there exist M i > 0, i ∈ n such that for any u := (u1, u2, , un) T ∈ Rn

1

d i

⎛

⎝n

j=1

eij rj (uj) + I i

⎞

By the Brouwer’s fixed-point theorem, there exists u∗∈ Rn such that F (u∗) = u∗ Thus,

u∗is a equilibrium point of (24).

Since D + |E| is a Metzler matrix and μ(D + |E|) < 0, it follows from Theorem 2 that

−di

Li ζi+

n



j=1

|e ij |ζ j <0 ∀i ∈ n, for some (ζ1, ζ2, , ζ n ) T ∈ Rn , ζ i > 0, i ∈ n Hence,

− d i ζ i+

n



j=1

By the coordinate translation z(t) = u(t) − u∗, (24) can be written as

˙z i (t) = −d i z i (t)+

n



j=1

where s j (x) := r j (x +u∗

j ) −r j (u∗

j ) , x ∈ R, j ∈ n Furthermore, u∗is globally exponentially stable for (24) if and only if the trivial solution of (26) is globally exponentially stable