New Exponential Stability Analysis for Certain Neutral Integro-Differential Equations with Non-Differentiable Interval Time-Varying Delays

New Exponential Stability Analysis for Certain

Neutral Integro-Differential Equations with

Non-Differentiable Interval Time-Varying Delays

Watcharin Chartbupapan, Kanit Mukdasai

Abstract—The problems of delay-range-dependent exponen-tial stability analysis for certain neutral differenexponen-tial equation with discrete and distributed time-varying delays are studied. The time-varying delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for time-varying delays are available. Based on new class of augmented Lyapunov-Krasovskii functional, descriptor model transformation, decomposition technique of coefficient constant, Leibniz-Newton formula, utilization of zero equation, improved integral inequalities and Peng-Park’s integral inequal-ity, new delay-range-dependent exponential stability criteria is derived in terms of linear matrix inequalities (LMIs) for the equation. Numerical example suggests that the results given to illustrate the effectiveness.

Index Terms—exponential stability analysis, neutral differ-ential equation, linear matrix inequality (LMIs), discrete and distributed time-varying delays

I. INTRODUCTION

I

N the last few decades, the problem of various stability analysis for uncertain neutral systems with state delays has been intensively studied by several researchers [1]-[19]. Since neutral delayed systems have already been applied in many fields, such as population ecology, distributed net-works containing lossless transmission lines, propagation and diffusion models, and partial element equivalent circuits in very large scale integration systems [10]. Furthermore, the delay-dependent stability analysis of certain neutral differ-ential equations (CNDE) has been received considerable attention in recent years. Since delay-dependent stability criteria makes use of information on the length of delays. Usually, the range of delay considered in most of the existing references is from zero to an upper bound. However, the delay may vary in a range for which the lower bound is not restricted to being zero. Many researchers investigate the problem of delay-range-dependent stability criteria for differ-ential systems with interval time-varying delays. The lower and upper bounds for the time-varying delays are available and the delay functions are necessary to be differentiable. Manuscript received December 6, 2017; revised February 2, 2018. This work was supported by the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), Khon Kaen University (Grant number MRG6080042), Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2017 and National Research Council of Thailand and Khon Kaen University, Thailand (Grant number 600054).

Watcharin Chartbupapan is with the Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. E-mail : [email protected].

Kanit Mukdasai is with the Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. E-mail : [email protected].

Correspondence should be addressed to Kanit Mukdasai, [email protected].

Delay-dependent asymptotic stability criteria for CNDE with constant delays has been discussed in [4], [9], [11], [12], [15], [17] by using several model transformation method and Lyapunov-Krasovskii functional approach, while the problem of exponential stability analysis has been studied with model transformation method and Lyapunov-Krasovskii functional approach in [15]. In [2], [3], [7], the authors studied the problem of exponential stability analysis for CNDE with time-varying delays by several methods. In [2], the results are derived without the use of the model transformation method and the bounding technique, while authors have been studied by using model transformation method, radially unbounded-ness and Lyapunov-Krasovskii functional approach in [7]. Moreover, the authors have studied the problem of stability for systems with discrete and distributed delays such as [18] which presented some stability conditions for uncertain neu-tral systems with discrete and distributed delays. The robust stability of uncertain linear neutral systems with discrete and distributed delays has been studied in [6]. However, neutral and discrete time-varying delays are required to be differentiable and information on bounded derivative of time-varying delays from existing methods.

Motivated by above discussions, we aim to establish the new delay-range-dependent exponential stability criteria for certain neutral differential equation with discrete and dis-tributed interval time-varying delays. The restrictions on the derivatives of the discrete and distributed time-varying delays for the equation are removed. Based on the combination of mixed model transformation, mixed integral inequalities, decomposition technique of coefficient constant, utilization of zero equation and new Lyapunov-Krasovskii functional, sufficient condition for exponential stability is obtained and formulated in terms of LMIs for the equation. Finally, numerical example suggests that the proposed criteria is effective.

Notation:

The following notations will be accounted in this paper: let Rn and Rn×m denotes n-dimensional Euclidean space with vector norm||.||and set ofn×mmatrices, respectively.

A matrix P is symmetric positive definite, write P >0, if

II. PRELIMINARIES

Consider the certain neutral integro-differential equation with interval time-varying delays of the form

d

dt[x(t) +px(t−τ(t))] = −ax(t) +btanhx(t−σ(t))

+c

∫ t

t₋ρ(t) x(s)ds,

t≥0, (1)

wherea, bare positive real constants and c, pare real con-stants with|p|<1. τ(t),σ(t)andρ(t)are neutral, discrete and distributed interval time-varying delays, respectively,

0≤τ1≤τ(t)≤τ2, τ˙(t)≤τd (2)

0≤σ1≤σ(t)≤σ2, (3)

0≤ρ1≤ρ(t)≤ρ2, (4)

where τ1, τ2, τd,σ1,σ2, ρ1 and ρ2 are given positive real

constants. For each solutionx(t)of (1), we assume the initial condition

x0(t) =ϕ(t), t∈[−ω,0],

where ϕ ∈ C([−ω,0];R) and ω = max{τ2, σ2, ρ2}. Let x(t, ϕ) denote the state trajectory of (1) and x(t,0) is the corresponding trajectory with zero initial condition. We consider the Leibniz-Newton formula of two forms

0 =x(t)−x(t−τ(t))−∫_tt_−τ(t)x˙(s)ds, (5)

0 =x(t)−x(t−σ(t))−∫_tt_−σ(t)x˙(s)ds. (6) By utilizing the following zero equations, we obtain

0 =ϵ1x(t)−ϵ1x(t−τ(t))−ϵ1

∫t

t−τ(t)x˙(s)ds, (7) 0 =ϵ2x(t)−ϵ2x(t−σ(t))−ϵ2

∫t

t₋σ(t)x˙(s)ds, (8)

where ϵ1, ϵ2 ∈ R. By descriptor model transformation and

(5)-(8), (1) can be represented by the form

˙

x(t) = y(t) +ϵ1x(t)−ϵ1x(t−τ(t))−ϵ1

∫ t

t₋τ(t) y(s)ds

+ϵ2x(t)−ϵ2x(t−σ(t))−ϵ2

∫ t

t−σ(t)

y(s)ds, (9)

y(t) = −ax(t) +btanhx(t−σ(t)) +c

∫ t

t−ρ(t) x(s)ds

−py(t−τ(t)). (10) For a = a1 +a2 +a3, we rewrite the (9)-(10) in the

following equations.

˙

x(t) = y(t) +ϵ1x(t)−ϵ1x(t−τ(t))−ϵ1

∫ t

t₋τ(t) y(s)ds

+ϵ2x(t)−ϵ2x(t−σ(t))−ϵ2

∫ t

t₋σ(t)

y(s)ds,(11)

y(t) = −a1x(t)−a2x(t−τ(t))−a2

∫ t

t−τ(t) y(s)ds

−a3x(t−σ(t))−a3

∫ t

t−σ(t) y(s)ds

+btanhx(t−σ(t)) +c

∫ t

t−ρ(t) x(s)ds

−py(t−τ(t)). (12)

Definition 1. [8] The equation (1) isexponentially stable, if there exist positive real constants α,β such that for each

ϕ(t) ∈ C([−ω,0], R), the solution x(t, ϕ) of the system satisfies

∥x(t, ϕ)∥ ≤β∥ϕ∥e−αt, t≥0.

Lemma 2. [16] For any constant symmetric positive definite matrixQ∈Rn×n_,0≤h

1≤h(t)≤h2 and vector function ω: [−h2,0]→Rn such that the integrations concerned are

well defined, then

−[h2−h1]

∫ ₋h1

−h2

ωT(s)Qω(s)ds

≤ − ∫ ₋h1

−h(t)

ωT(s)dsQ

∫ ₋h1

−h(t) ω(s)ds

− ∫ ₋h(t)

−h2

ωT(s)dsQ

∫ ₋h(t)

−h2

ω(s)ds.

Lemma 3. [16] For any constant matrices Q1, Q2, Q3 ∈ Rn×n, Q1≥0, Q3>0,

[

Q1 Q2

∗ Q3

]

≥ 0, 0 ≤h1 ≤ h(t) ≤h2 and vector function ˙

x: [−h2,0]→Rnsuch that the following integration is well

defined, then

−[h2−h1]

∫ t₋h1

t−h2

[

x(s) ˙

x(s)

]T[

Q1 Q2

∗ Q3

] [

x(s) ˙

x(s)

]

ds

≤       

x(t−h1) x(t−h(t))

x(t−h2)

∫t₋h1

t−h(t)x(s)ds

∫t₋h(t) t−h2 x(s)ds

      

T

     

−Q3 Q3 0 −QT2 0

∗ −Q3−Q3 Q3 QT2 −QT2

∗ ∗ −Q3 0 QT2

∗ ∗ ∗ −Q1 0

∗ ∗ ∗ ∗ −Q1

     

      

x(t−h1) x(t−h(t))

x(t−h2)

∫t₋h1

t−h(t)x(s)ds

∫t₋h(t) t−h2 x(s)ds

      

.

X, Mi∈Rn×n, i= 1,2, . . . ,5 and0≤h1≤h(t)≤h2,

− ∫ t−h1

t₋h2

˙

xT(s)Xx˙(s)ds

≤ 

xx((tt−−hh(1t)))

x(t−h2)

 

T



M1+M

T

1 −M1T +M2 0

∗ M1+M1T −M2−M2T −M1T+M2

∗ ∗ −M2−M2T

  

xx((tt−−hh(1t)))

x(t−h2)

 

+[h2−h1]



xx((tt−−hh(1t)))

x(t−h2)

 

T

M_∗3 M3M+4M5 M04

∗ ∗ M5

  

xx((tt−−hh(1t)))

x(t−h2)

 ,

where



X_∗ MM13 MM24

∗ ∗ M5

 _≥0.

Lemma 5. (Peng-Park’s integral inequality.) [14], [13] Any matrix

[

Z S

∗ Z

]

≥ 0, positive scalars τ and τ(t)

satisfying 0 < τ(t)< τ, vector function x˙ : [−τ,0]→Rn

such that the concerned integrations are well defined,then

−τ

∫ t

t₋τ

˙

xT(s)Zx˙(s)ds≤ωT(t)Ωω(t),

where ω(t) = [xT(t), xT(t−τ(t)), xT(t−τ)]T and Ω =



−_∗Z ₋2ZZ+−SS+ST _ZS_−S

∗ ∗ −Z

 .

III. MAIN RESULTS

In this section, we give our main results. We introduce the following notations for later use:

∑

=[Ω(i,j)

]

26×26, (13)

where Ω(i,j)= Ω(j,i),

Ω(1,1) = 2αk1+n9+k7+n10+k8+ 2z1+ 2z4 +2k1ϵ1+ 2k1ϵ2−2q2a1+k2τ22+k3σ22 +n1τ22−2n1τ2τ1+n1τ12+n2σ22−2n2σ2σ1 +n2σ21−k4e(−2ατ2)−k5e(−2ασ2)+τ12f1 +τ2

2r1−r3e(−2ατ2)+f4τ22−2τ1τ2+τ12f4 +σ2

1f7+σ22r4+σ22f10−2σ1σ2f10+σ12f10

−r6e(−2ασ2)+n7σ12+k6σ22+n8σ22

−2n8σ1σ2+n8σ21+n11ρ22−2n11ρ1ρ2 +n11ρ21,

Ω(1,2) = q1ϵ1+q1ϵ2+τ12f2+τ22r2+τ22f5−2τ1τ2f5 +τ2

1f5+σ12f8+σ22r5+σ22f11−2σ1σ2f11 +σ2

1f11+k1−q2−q3a1,

Ω(1,3) = −z1+z2−k1ϵ1−q2a2−q4a1+k4e(−2ατ2)

−s1+r3e(−2ατ2),

Ω(1,4) = −z1+z3−k1ϵ1−q2a2−q5a1,

Ω(1,5) = −z4+z5−k1ϵ2−q2a3−q6a1+k5e(−2ασ2)

−s2+r6e(−2ασ2),

Ω(1,6) = −z4+z6−k1ϵ2−q2a3−q7a1, Ω(1,7) = q2b−q8a1,

Ω(1,8) = q2c−q10a1, Ω(1,9) = −q2p−q9a1, Ω(1,10) = −r2e(−2ατ2), Ω(1,12) = −r5e(−2ασ2), Ω(1,17) = s1,

Ω(1,19) = s2,

Ω(2,2) = k9+n3τ12+k4τ22+n4σ21+k5σ22+n5τ22

−2n5τ1τ2+n5τ12+n6σ22−2n6σ1σ2 +n6σ21+τ12f3+τ22r3+τ22f6−2τ1τ2f6 +τ2

1f6+σ12f9+σ22f12−2σ1σ2f12+σ21f12

−2q3+σ22r6, Ω(2,3) = −q1ϵ1−q3a2−q4, Ω(2,4) = −q1ϵ1−q3a2−q5, Ω(2,5) = −q1ϵ2−q3a3−q6, Ω(2,6) = −q1ϵ2−q3a3−q7, Ω(2,7) = q3b−q8,

Ω(2,8) = q3c−q10, Ω(2,9) = −q3p−q9,

Ω(3,3) = −2z2−2q4a2−2k4e(−2ατ2)+ 2s1+ 2m1

−2m2+τ2m3−τ1m3+τ2m5−τ1m5

−2r3e(−2ατ2)−2f6e(−2ατ2), Ω(3,4) = −z2−z3−q4a2−q5a2, Ω(3,5) = −q4a3−q6a2,

Ω(3,6) = −q4a3−q7a2, Ω(3,7) = q4b−q8a2, Ω(3,8) = q4c−q10a2, Ω(3,9) = −q4p−q9a2, Ω(3,10) = r2e(−2ατ2),

Ω(3,11) = −r2e(−2ατ2)−f5e(−2ατ2), Ω(3,14) = f5e(−2ατ2),

Ω(3,16) = −m1+m2+τ2m4−τ1m4+f6e(−2ατ2), Ω(3,17) = k4e(−2ατ2)−s1−m1+m2+τ2m4−τ1m4

+r3e(−2ατ2)+f6e(−2ατ2), Ω(4,4) = −2z3−2q5a2,

Ω(4,5) = −q5a3−q6a2, Ω(4,6) = −q5a3−q7a2, Ω(4,7) = q5b−q8a2, Ω(4,8) = q5c−q10a2, Ω(4,9) = −q5p−q9a2,

Ω(5,5) = w−2z5−2q6a3−2k5e(−2ασ2)+ 2s2 +2m6−2m7+σ2m8−σ1m8+σ2m10

Ω(5,6) = −z5−z6−q6a3−q7a3, Ω(5,7) = q6b−q8a3,

Ω(5,8) = q6c−q10a3, Ω(5,9) = −q6p−q9a3, Ω(5,12) = r5e(−2ασ2),

Ω(5,13) = −r5e(−2ασ2)−f11e(−2ασ2), Ω(5,15) = f11e(−2ασ2),

Ω(5,18) = −m6+m7+σ2m9−σ1m9 +f12e(−2ασ2),

Ω(5,19) = k5e(−2ασ2)−s2−m6+m7+σ2m9

−σ1m9+r6e(−2ασ2)+f12e(−2ασ2), Ω(6,6) = −2z6−2q7a3,

Ω(6,7) = q7b−q8a3, Ω(6,8) = q7c−q10a3, Ω(6,9) = −q7p−q9a3, Ω(7,7) = −w+ 2q8b, Ω(7,8) = q8c+q10b, Ω(7,9) = −q8p+q9b,

Ω(8,8) = 2q10c−n11e(−2αρ2), Ω(8,9) = −q10p+q9c,

Ω(9,9) = −k9e(−2ατ2)+k9τd−2q9p, Ω(10,10) = −k2e(−2ατ2)−r1e(−2ατ2),

Ω(11,11) = −k2e(−2ατ2)−n1e(−2ατ2)−r1e(−2ατ2)

−f4e(−2ατ2),

Ω(11,17) = r2e(−2ατ2)+f5e(−2ατ2), Ω(12,12) = −k3e(−2ασ2)−r4e(−2ασ2),

Ω(13,13) = −k3e(−2ασ2)−n2e(−2ασ2)−r4e(−2ασ2)

−f10e(−2ασ2),

Ω(13,19) = r5e(−2ασ2)+f11e(−2ασ2), Ω(14,14) = −n1e(−2ατ2)−f4e(−2ατ2), Ω(14,16) = −f5e(−2ατ2),

Ω(15,15) = −n2e(−2ασ2)−f10e(−2ασ2), Ω(15,18) = −f11e(−2ασ2),

Ω(16,16) = −n9e(−2ατ1)+ 2m1+τ2m3−τ1m3

−f6e(−2ατ2),

Ω(17,17) = −k7e(−2ατ2)−k4e(−2ατ2)−2m2+τ2m5

−τ1m5−r3e(−2ατ2)−f6e(−2ατ2), Ω(18,18) = −n10e(−2ασ1)+ 2m6+σ2m8−σ1m8

−f12e(−2ασ2),

Ω(19,19) = −k8e(−2ασ2)−k5e(−2ασ2)−2m7+σ2m10

−σ1m10−r6e(−2ασ2)−f12e(−2ασ2), Ω(20,20) = −n3e(−2ατ1)−f3e(−2ατ1),

Ω(20,22) = −f2e(−2ατ1),

Ω(21,21) = −n4e(−2ασ1)−f9e(−2ασ1), Ω(21,23) = −f8e(−2ασ1),

Ω(22,22) = −f1e(−2ατ1), Ω(23,23) = −f7e(−2ασ1), Ω(24,24) = −n7e(−2ασ1), Ω(25,25) = −k6e(−2ασ2), Ω(26,26) = −n8e(−2ασ2),

other terms are 0.

Theorem 6. For given positive real constants

α, σ1, σ2, τ1, τ2, τd, ρ1 and ρ2, the certain neutral

integro-differential equation with interval time-varying delays (1) is exponentially stable if there exist positive real constants r1, r3, r4, r6, f4, f6, f10, f12, w, ki, njwhere i = 1,2, . . . ,9, j = 1,2, . . . ,11, and real constants

s1, s2, r2, r5, f1, f2, f3, f5, f7,f8, f9, f11, qk, mk where k = 1,2, . . . ,10 such that the following symmetric linear

matrix inequality holds:

∑

<0, (14) [

k4e−2ατ2 s1

∗ k4e−2ατ2

]

≥0, (15) [

k5e−2ασ2 s2

∗ k5e−2ασ2

]

≥0, (16) 

n5(τ2−τ1)e

−2ατ2 _m 1 m2

∗ m3 m4

∗ ∗ m5



_≥0, (17) 

n6(σ2−σ1)e

−2ασ2 _m 6 m7

∗ m8 m9

∗ ∗ m10



_≥0, (18) [

r1 r2

∗ r3

]

≥0, (19) [

r4 r5

∗ r6

]

≥0, (20) [

f4 f5

∗ f6

]

≥0, (21) [

f10 f11

∗ f12

]

≥0. (22)

Proof.Forr1, r3, r4, r6, f4, f6, f10, f12, w, ki,nj are

pos-itive real constants where i = 1,2, . . . ,9, j = 1,2, . . . ,11

ands1, s2, r2, r5, f1, f2, f3, f5, f7, f8, f9, f11, qk, mk are real

constants where k = 1,2, . . . ,10, consider the Lyapunov-Krasovskii functional candidate for (11)-(12) of the form

V(t) = 7

∑

i=1

Vi(t), (23)

where

V1(t) = k1x2(t),

V2(t) = k2τ2

∫ 0

−τ2

∫ t

t+s

e2α(θ−t)x2(θ)dθds

+k3σ2

∫ 0

−σ2

∫ t

t+s

e2α(θ−t)x2(θ)dθds

+n1(τ2−τ1)

∫ ₋τ1

−τ2

∫ t

t+s

e2α(θ−t)x2(θ)dθds

+n2(σ2−σ1)

∫ ₋σ1

−σ2

∫ t

t+s

e2α(θ−t)x2(θ)dθds,

V3(t) = n3τ1

∫ 0

−τ1

∫ t

t+s

e2α(θ−t)y2(θ)dθds

+k4τ2

∫ 0

−τ2

∫ t

t+s

e2α(θ−t)y2(θ)dθds

+n4σ1

∫ 0

−σ1

∫ t

t+s

e2α(θ−t)y2(θ)dθds

+k5σ2

∫ 0

−σ2

∫ t

t+s

e2α(θ−t)y2(θ)dθds

+n5(τ2−τ1)

∫ −τ1

−τ2

∫ t

t+s

e2α(θ−t)y2(θ)dθds

+n6(σ2−σ1)

∫ ₋σ1

−σ2

∫ t

t+s

V4(t) = τ1

∫ 0

−τ1

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[ f1 f2

∗ f3

] [

x(θ)

y(θ)

]

dθds

+τ2

∫ 0

−τ2

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[ r1 r2

∗ r3

] [

x(θ)

y(θ)

]

dθds

+(τ2−τ1)

∫ ₋τ1

−τ2

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[ f4 f5

∗ f6

] [

x(θ)

y(θ)

]

dθds

+σ1

∫ 0

−σ1

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[ f7 f8

∗ f9

] [

x(θ)

y(θ)

]

dθds

+σ2

∫ 0

−σ2

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[ r4 r5

∗ r6

] [

x(θ)

y(θ)

]

dθds

+(σ2−σ1)

∫ ₋σ1

−σ2

∫ t

t+s

e2α(θ−t)

[

x(θ)

y(θ)

]T[

f10 f11

∗ f12

] [

x(θ)

y(θ)

]

dθds,

V5(t) = n7σ1

∫ 0

−σ1

∫ t

t+s

e2α(θ−t)tanh2x(θ)dθds

+k6σ2

∫ 0

−σ2

∫ t

t+s

e2α(θ−t)tanh2x(θ)dθds

+n8(σ2−σ1)

∫ ₋σ1

−σ2

∫ t

t+s

e2α(θ−t)

tanh2x(θ)dθds,

V6(t) = n9

∫ t

t−τ1

e2α(s−t)x2(s)ds

+k7

∫ t

t₋τ2

e2α(s−t)x2(s)ds

n10

∫ t

t−σ1

e2α(s−t)x2(s)ds

+k8

∫ t

t−σ2

e2α(s−t)x2(s)ds

+k9

∫ t

t₋τ(t)

e2α(s−t)y2(s)ds,

V7(t) = n11(ρ2−ρ1)

∫ ₋ρ1

−ρ2

∫ t

t+s

e2α(θ−t)x2(θ)dθds.

The time-derivative of the Lyapunov-Krasovskii functional along the trajectory of (11)-(12) is given by

˙

V(t) = 7

∑

i=1 ˙

Vi(t). (24)

From the utilization of zero equations, the following equations are true for any real constantszm,m= 1,2, ...,6,

2

[

z1x(t) +z2x(t−τ(t)) +z3

∫ t

t−τ(t) y(s)ds

]

×[x(t)−x(t−τ(t)))−

∫ t

t₋τ(t) y(s)ds

]

= 0,(25)

2

[

z4x(t) +z5x(t−σ(t)) +z6

∫ t

t₋σ(t) y(s)ds

]

×[x(t)−x(t−σ(t)))−

∫ t

t−σ(t) y(s)ds

]

= 0.(26) We obtain, for any positive real constantsw,

0≤wx2_{(t−σ(t))−wtanh}2_{x(t−σ(t)). (27)}

According to (24)-(27), it is straightforward to see that

˙

V(t) + 2αV(t)≤ξT(t)∑ξ(t), (28) where

ξ= [x(t), y(t), x(t−τ(t)),∫_tt_−τ(t)y(s)ds, x(t−σ(t)),

∫t

t−σ(t)y(s)ds,tanhx(t−σ(t)),

∫t

t−ρ(t)x(s)ds, y(t−τ(t)),

∫t

t−τ(t)x(s)ds,

∫t₋τ(t) t−τ2 x(s)ds,

∫t

t−σ(t)x(s)ds,

∫t₋σ(t) t−σ2 x(s)ds,

∫t₋τ1

t−τ(t)x(s)ds,

∫t₋σ1

t−σ(t)x(s)ds, x(t−τ1), x(t−τ2), x(t−σ1), x(t−σ2),

∫t

t−τ1y(s)ds,

∫t

t−σ1y(s)ds,

∫t

t₋τ1x(s)ds,

∫t

t₋σ1x(s)ds,

∫t

t₋σ1tanhx(s)ds,

∫t

t−σ2tanhx(s)ds,

∫t₋σ1

t−σ2 tanhx(s)ds]

and ∑ is defined in (13). It is true that if conditions (14)-(22) hold, then

˙

V(t) + 2αV(t)≤0, t≥0. (29) From (29), it is easy to see that

∥x(t, ϕ)∥ ≤β∥ϕ∥e−αt, t∈R+.

This means that equation (1) is exponentially stable. This completes the proof of this theorem.

IV. NUMERICALEXAMPLES

In this section, numerical example is given to present the effectiveness of our main results by comparing the upper bounds of the delays and the parameter b as well as investigating the rate of convergence.

Example 7. Consider the following equation with mixed interval time-varying delays:

d

dt[x(t) + 0.1x(t−τ(t))] = −1.5x(t)

+btanhx(t−σ(t)) +0.5

∫ t

t−ρ(t)

x(s)ds.(30) Decompose constants a as following a = a1+a2+a3,

respectively, where

a1= 0.5, a2= 0.5, a3= 0.5.

we can obtain a set of parameters guaranteeing exponential stability as follows:

k1= 2.7032, k2= 0.2344, k3= 0.2169, k4= 0.4090, k5= 0.4276, k6= 0.3210, k7= 0.1988, k8= 0.2886, k9= 0.5843, n1= 0.2663, n2= 0.2725, n3= 0.4393, n4= 0.4393, n5= 0.4855, n6= 0.4993, n7= 0.4513, n8= 0.4058, n9= 0.2142, n10= 0.2466, n11= 1.7784, f1= 0.4641, f2= 0.0067, f3= 0.0266, f4= 0.3016, f5= 0.0391, f6= 0.3447, f7= 0.4656, f8= 0.0273, f9= 0.0335, f10= 0.3390, f11= 0.0435, f12= 0.3765, r1= 0.3116, r2= 0.0568, r3= 0.2819, r4= 0.2992, r5= 0.0715, r6= 0.3063, m1= 0.0159, m2=−0.0018, m3= 0.2805, m4=−0.1840, m5= 0.2586, m6=−0.0044, m7= 0.0259, m8= 0.2482, m9=−0.1856, m10= 0.2196.

Moreover, the upper bounds of the parameter b which guarantees the exponential and asymptotic stabilities are 1.5220 and 2.4310, respectively. The upper boundsb of this example can be found in Table I for different values of

α, τd. The maximum upper bounds b for exponential and

asymptotic stabilities of Example 7 is listed in Table II for different values ofα, c.

TABLE I

UPPER BOUNDS OFbFOREXAMPLE7.

τd α= 0 α= 0.1 α= 0.2 α= 0.3 α= 0.4

0.1 2.5325 2.3401 2.1627 2.0011 1.8451 0.2 2.5102 2.3210 2.1402 1.9730 1.8049 0.3 2.4807 2.2920 2.1060 1.9354 1.7611 0.4 2.4640 2.2529 2.0690 1.8760 1.6860 0.5 2.4310 2.2112 2.0002 1.7749 1.5220

TABLE II

UPPER BOUNDS OFbFOREXAMPLE7.

c α= 0 α= 0.1 α= 0.2 α= 0.3 α= 0.4

0.5 2.4310 2.2112 2.0002 1.7749 1.5220 1 2.4243 2.2310 2.0110 1.7719 1.5170 2 2.4256 2.2210 2.0004 1.7786 1.5151 3 2.4410 2.2180 2.0001 1.7738 1.5166 4 2.4297 2.2191 2.0031 1.7831 1.5159

V. CONCLUSION

In this paper, we proposed the delay-range-dependent exponential stability criteria for certain neutral differen-tial equation with discrete and distributed interval time-varying delays by using new class of augmented Lyapunov-Krasovskii functional, descriptor model transformation, de-composition technique of coefficient constant, Leibniz-Newton formula, utilization of zero equation, improved inte-gral inequalities and Peng-Park’s inteinte-gral inequality. Finally, numerical example is given to show that the proposed criteria is effective.

ACKNOWLEDGMENT

The author would like to thank the editors and Reviewers for their valuable suggestion and useful comments for this

paper. Also, The author would like to thank the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), Khon Kaen University (Grant number MRG6080042), Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2017 and National Research Council of Thailand and Khon Kaen University, Thailand (Grant number 600054).

REFERENCES

[1] R. P. Agarwal and S. R. Grace, “Asymptotic stability of certain neutral differential equations,”Mathematical and Computer Modelling,Vol. 31, pp. 9-15, 2000.

[2] H. Chen, “Some improved criteria on exponential stability of neutral differential equation,”Advances in Difference Equations, Vol. 170, 9 pages, 2012.

[3] H. Chen and X. Meng, “An improved exponential stability criterion for a class of neutral delayed differential equations,”Applied Mathematics Letters,Vol. 24, pp. 1763-1767, 2011.

[4] S. Deng, X. Liao and S. Guo, “Asymptotic stability analysis of certain neutral differential equations: a descriptor system approach,” Mathematics and Computers in Simulation, Vol. 71, pp. 4297-4308, 2009.

[5] Y. Du, S. Zhong, J. Xu and N. Zhou,“Delay-dependent exponential pas-sivity of uncertain cellular neural networks with discrete and distributed time-varying delays,” ISA Transactions,Vol. 56, pp. 1-7, 2015. [6] Q. L. Han,“A descriptor system approach to robust stability of uncertain

neutral systems with discrete and distributed delays,” Automatica,Vol. 40, no. 10, pp. 1791-1796, 2004.

[7] P. Keadnarmol and T. Rojsiraphisal,“Globally exponential stability of a certain neutral differential equation with time-varying delays,” Advances in Difference Equations,Vol. 32, 10 pages, 2014.

[8] O. M. Kwon and J. H. Park, “Exponential stability of uncertain dynamic systems including state delay,” Applied Mathematics Letters,Vol.19, pp. 901-907, 2006.

[9] O. M. Kwon and J. H. Park, “On improved delay-dependent stability criterion of certain neutral differential equations,” Applied Mathematics and Computation,Vol.199, pp. 385-391, 2008.

[10] Z. W. Liu and H. G. Zhang, “Delay-dependent stability for systems with fast-varying neutral-type delays via a PTVD compensation,”Acta Automatica Sinica,Vol. 36, no. 1, pp. 147-152, 2010.

[11] P. T. Nam and V. N. Phat, “An improved stability criterion for a class of neutral differential equations,”Applied Mathematics Letters,Vol. 22, pp. 31-35, 2009.

[12] J. H. Park,“Delay-dependent criterion for guaranteed cost control of neutral delay systems,” Journal of Optimization Theory and Applica-tions,Vol. 124, pp. 491-502, 2005.

[13] P. G. Park, J. W. Ko and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,”Automatica,Vol. 47, pp. 235-238, 2011.

[14] C. Peng and M. R. Fei,“An improved result on the stability of uncertain T-S fuzzy systems with interval time-varying delay,” Fuzzy Sets and Systems,Vol. 212, pp. 97-109, 2013.

[15] T. Rojsiraphisal and P. Niamsup, “Exponential stability of certainneu-tral differential equations,” Applied Mathematics Letters,Vol. 17, pp. 3875-3880, 2010.

[16] P. Singkibud, P. Niamsup and K. Mukdasai, “Improved Results on Delay-range-dependent Robust Stability Criteria of Uncertain Neutral Systems with Mixed Interval Time-varying Delays,” IAENG Interna-tional Journal of Applied Mathematics, Vol. 47, no.2, pp. 209-222, 2017.

[17] Y. G. Sun and L. Wang, “Note on asymptotic stability of a class of neutral differential equations,” Applied Mathematics Letters,Vol. 19, pp. 949-953, 2006.

[18] L. Xiong, S. Zhong, and J. Tian, “New robust stability condition for uncertain neutral systems with discrete and distributeddelays,” Chaos Solitons Fractals,Vol. 42, no. 2, pp. 1073-1079, 2009.