Improved results on passivity analysis of neutral-type neural networks with mixed time-varying delays

Improved results on passivity analysis of neutral-type neural networks with mixed time-varying delays

THONGCHAI BOTMART Khon Kaen University Department of Mathematics Khon Kaen 40000, THAILAND

thongbo@kku.ac.th

NARONGSAK YOTHA^∗

Rajamangala University of Technology Isan Department of Applied Mathematics and Statistics

Nakhon Ratchasima 30000, THAILAND narongsak.yo@rmuti.ac.th

∗Corresponding author KANIT MUKDASAI

Khon Kaen University Department of Mathematics Khon Kaen 40000, THAILAND

kanit@kku.ac.th

WAJAREE WEERA University of Pha Yao Department of Mathematics Pha Yao 56000, THAILAND

wajaree@up.ac.th

Abstract: This paper is considered with problem of the passivity analysis for neutral-type neural networks with mixed time-varying delays. By constructing an augmented Lyapunov-Krasovskii functionals and using the double integral inequality with approach to estimate the derivative of the Lyapunov-Krasovskii functionals, sufficient conditions are established to ensure the passivity of the considered neutral-type neural networks, in which some useful information on the neuron activation function ignored in the existing literature is taken in to account. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Key–Words: Passivity, Neutral-type neural networks, Time delay, Integral inequality

1 Introduction

In the past few decades, delayed neural networks (NNs) have been an important issue due to their appli- cations in many areas such as signal processing, pat- tern recognition, associative memories, fixed-point, computations, parallel computation, control theory and optimization solvers [1, 2, 3]. The state esti- mation problems for NNs with discrete interval and distributed time-varying delays have been extensively studied in [4, 5, 6]. On the other hand, it is com- mon that the time delay of system state, have been many phenomenon such as automatic control, chem- ical reactors, distributed networks, heat exchanges, etc. The systems containing the information of past state derivatives are called neutral-type neural net- works (NTNNs). The existing work on the state es- timator of NTNNs with mixed delays are only [7, 8]

at present. In [9], the authors considered the prob- lem of global passivity analysis of interval neural net- works with discrete and distributed delays of neutral type. Consequently, the passivity analysis of NTNNs has also been received considerable attention and lots of works were reported in recent years.

The problem of passivity performance analysis has also been extensively applied in many areas such

as signal processing, sliding mode control, and net- worked control [10, 11, 12]. The main idea of the passivity theory is that the passive properties of a system can keep the system internally stable. In [13, 14, 15, 16, 17], authors investigated the passiv- ity of neural networks with time-varying delay, and gave some criteria for checking the passivity of neu- ral networks with time-varying delay. Passivity anal- ysis for neural networks of neutral type with Marko- vian jumping parameters and time delay in the leakage term has been presented in [18] . Robust exponential passive filtering for uncertain neutral-type neural net- works with time-varying mixed delays via wirtinger- based integral inequality has been presented in [19] is Wirtinger-based integral inequality [20]. Recently, a new double integral inequality for time-delay system was proposed in [21, 22], which is less conservative.

These motivate our research.

Motivated by above discussing, this paper inves- tigates the passivity analysis for NTNNs with dis- crete and continuous distributed time-varying delays.

Based on the constructed Lyapunov-Krasovskii func- tional, free-weighting matrix approach, and double in- tegral inequality for estimating the derivative of the LyapunovKrasovskii functional, the delay-dependent

passivity conditions are derived in terms of LMIs, which can be easily calculated by MATLAB LMIs control toolbox. Numerical example is provided to demonstrate the feasibility and effectiveness of the proposed criteria.

Notation Rⁿ is the n-dimensional Euclidean space; R^m×n denotes the set of m × n real ma- trices; I_n represents the n-dimensional identity ma- trix; λ(A) denotes the set of all eigenvalues of A;

λ_max(A) = max{Reλ; λ ∈ λ(A)}; C([0, t], Rⁿ) de- notes the set of all Rⁿ-valued continuous functions on [0, t]; L₂([0, t], R^m) denotes the set of all the R^m- valued square integrable functions on [0, t]; The nota- tion X ≥ 0 (respectively, X > 0 ) means that X is positive semidefinite (respectively, positive definite);

diag(· · ·) denotes a b lock diagonal matrix; Matrix di- mensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

2 Problem formulation and prelimi- naries

Consider the following NTNNs with time-varying dis- crete and distributed delays described by:

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x(t) = −Ax(t) + W g(x(t))˙ +W₁g(x(t − τ (t))) +W₂^R_t−k(t)^t g(x(s))ds +W₃x(t − h(t)) + u(t),˙ y(t) = g(x(t)),

x(t) = φ(t), t ∈ [−τmax, 0], τ_max= max{τ2, k₂, h₂},

(1)

where x(t) = [x1(t), x₂(t), ..., x_n(t)] ∈ Rⁿ is the state of the neural, A = diag(a₁, a₂, . . . , a_n) >

0 represents the self-feedback term, W, W₁, W₂ and W3 represents the connection weight matrices, g(·) = (g1(·), g2(·), . . . , gn(·))^T represents the acti- vation functions, u(t) and y(t) represents the input and output vectors, respectively; φ(t) is an initial con- dition. The variables τ (t), k(t) and h(t) represents the interval time-varying and time-varying delays with satisfy the following conditions:

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0 < τ₁≤ τ (t) ≤ τ2, τ (t) ≤ τ˙ 3, 0 ≤ h(t) ≤ h2, h(t) ≤ h˙ 3, 0 ≤ k(t) ≤ k2, ∀t ≥ 0,

(2)

where known scalars τ₁, τ₂, τ₃, h₂, h₃ and k₂.

The neural activation functions g_k(·), k = 1, 2, . . . , n satisfy gk(0) = 0 and for s₁, s₂ ∈ R, s16= s2,

l⁻_k ≤ g_k(s₁) − gk(s₂)

s₁− s2 ≤ l⁺k, (3) where l⁻_k, l⁺_k are known real scalars.

Definition 1 [13]: The system (1) is said to be pas- sive if there exists a scalar γ such that for all t_f ≥ 0,

2 Z tf

0 y^T(s)u(s) ds ≥ −γ Z tf

0 u^T(s)u(s) ds, and for all solutions of (1) with x(0) = 0.

Lemma 2 [21]: For a positive definite matrix S >

0, and any continuously differentiable function x : [a, b] → Rⁿ, The following inequality holds:

Z b a

x˙^T(s)S ˙x(s) ds ≥ 1

b − aΠ^T₁SΠ₁+ 3 b − a

×Π^T2SΠ₂+ 5

b − aΠ^T₃SΠ₃, where

Π₁= x(b) − x(a),

Π₂= x(b) + x(a) − _b−a^{2 R}_a^bx(s)ds, Π₃= x(b) − x(a) + _b−a^{6 R}_a^bx(s)ds

−_(b−a)¹²2

Rb a

Rb

θ x(s)dsdθ.

Lemma 3 [22]: For a positive definite matrix S >

0, and any continuously differentiable function x : [a, b] → Rⁿ, the following inequality holds:

Z b a

Z b θ

x˙^T(s)S ˙x(s) dsdθ ≥ 2Π^T₄SΠ₄+ 4Π^T₅SΠ₅ +6Π^T₆SΠ₆,

where

Π₄= x(b) − _b−a^{1 R}_a^bx(s)ds,

Π₅= x(b) + _b−a^{2 R}_a^bx(s)ds − _(b−a)⁶ 2

Rb a

Rb

θ x(s)dsdθ, Π₆= x(b) − _b−a^{3 R}_a^bx(s)ds + _(b−a)²⁴2

Rb a

Rb

θ x(s)dsdθ

−_(b−a)^{60 3 R}a^b

Rb θ

Rb

s x(λ)dλdsdθ.

3 Main results

For presentation convenience, in the following, we denote

L⁺= diag(l⁺₁, l⁺₂, · · · , l⁺_n), L⁻= diag(l⁻₁, l⁻₂, · · · , l⁻n),

ζ(t) = [ x^T(t), x^T(t − τ (t)), x^T(t − τ1), x^T(t − τ2), g^T(x(t)), g^T(x(t − τ (t))), g^T(x(t − τ1)), g^T(x(t − τ2)), ^R_t−τ^t ₁x^T(s)ds, Rt

t−τ2x^T(s)ds, ^R_t−τ^t ₁^R_θ^tx^T(s)dsdθ, Rt

t−τ2

Rt

θx^T(s)dsdθ, ^R_t−τ^t ₁^R_θ^tR_s^tx^T(λ)dλdsdθ, Rt

t−τ₂

Rt θ

Rt

sx^T(λ)dλdsdθ, ^R_t−k(t)^t g^T(x(s))ds, x˙^T(t − h(t)), u^T(t) ]^T,

and ei ∈ R^n×17nis defined as

e_i= [0_n×(i−1)n, I_n, 0_n×(17−i)n] for i = 1, 2, . . . , 17.

Theorem 4 For given scalars τ₁, τ₂, τ₃, h₂, h₃and k₂ the system (1) with (3) is passive for any delays sat- isfying (2), if there exist real positive matrices P ∈ R^7n×7n, Q, S ∈ R^n×n, (i = 1, 2, 3), real positive diagonal matrices U₁, U₂, T_s, T_ab(s = 1, 2, 3, 4; a = 1, 2, 3; b = 2, 3, 4; a < b), with appropriate dimen- sions, and a scalar γ > 0 such that the following LMIs holds:

Ω = Ω₁+ Ω₂+ Ω₃+ Ω₄+ Ω₅+ Ω₆+ Ω₇ < 0, (4) where

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Ω₁= Π^T₁P Π₂+ Π^T₂P Π₁+ (Π₃+ Π₄)^T +Π₃+ Π₄,

Ω₂= 3e^T₁Q₁e₁− e^T3Q₁e₃− e^T4Q₁e₄ +2e^T₅Q₂e₅− e^T7Q₂e₇− e^T8Q₂e₈ +C₀^TQ₃C0− (1 − τ3)e^T₆Q₁e₆

−(1 − h3)e^T₁₆Q₃e₁₆, Ω₃= e^T₅k²₂S₁e₅− e^T15S₁e₁₅,

Ω₄= C₀^T(τ₁²S₂+ τ₂²S₂)C0− Π^T5S₂Π₅

−3Π^T6S₂Π₆− 5Π^T7S₂Π₇− Π^T8S₂Π₈

−3Π^T9S₂Π₉− 5Π^T10S₂Π₁₀, Ω₅= C₀^T(0.5τ₁²S₃+ 0.5τ₂²S₃)C0

−2Π^T11S₃Π₁₁− 4Π^T12S₃Π₁₂

−6Π^T13S₃Π₁₃− 2Π^T14S₃Π₁₄

−4Π^T15S₃Π₁₅− 6Π^T16S₃Π₁₆, Ω₆=^P4_s=1(Π^T₁₇T_sΠ₁₈+ Π^T₁₈T_sΠ₁₇)

+^P3_a=1^P4_b=2,b>aΠ^T₁₉T_abΠ₂₀ +^P3_a=1^P4_b=2,b>aΠ^T₂₀TabΠ₁₉, Ω₇= −γe^T₁₇e₁₇− e^T5e₁₇− e^T17e₅,

(5)

with

Π₁ = [e^T₁, e^T₉, e^T₁₀, e^T₁₁, e^T₁₂, e^T₁₃, e^T₁₄]^T,

Π₂ = [C^T₀, e^T₁ − e^T3, e^T₁ − e^T4, τ₁e^T₁ − e^T9, τ₂e^T₁ − e^T10, 0.5τ₁²e^T₁ − e^T11, 0.5τ₂²e^T₁ − e^T12]^T,

Π₃ = e^T₅(U₁− U2)C0, Π₄ = e^T₁(L⁺U₂− L⁻U₁)C0, Π₅ = e₁− e3,

Π₆ = e₁+ e₃−_τ²₁e₉, Π₇ = e₁− e3+ ⁶

τ₁e₉−¹²_τ2 1

e₁₁, Π₈ = e₁− e4,

Π₉ = e₁+ e₄−_τ²₂e₁₀, Π₁₀ = e₁− e4+ ⁶

τ2e₁₀− ¹²_τ2 2

e₁₂, Π₁₁ = e₁−_τ¹₁e₉,

Π₁₂ = e₁+ ²

τ1e₉−_τ⁶2 1e₁₁, Π₁₃ = e₁+ e₂−_τ³₁e₉+ ²⁴

τ₁²e₁₁−⁶⁰_τ3 1e₁₃, Π₁₄ = e₁−_τ¹₂e₁₀,

Π₁₅ = e₁+ ²

τ2e₁₀−_τ⁶2 2

e₁₂,

Π₁₆= e₁+ e₂−_τ³₂e₁₀+²⁴

τ₂²e₁₂− ⁶⁰_τ³

2

e₁₄, Π₁₇= e_s+4− L⁻e_s,

Π₁₈= L⁺e_s− es+4,

Π₁₉= (e_a+4− eb+4) − L⁺(ea− eb), Π₂₀= L⁺(e_a− eb) − (ea+4− eb+4),

C0 = Ae₁+ W e₅+ W₁e₆+ W₂e₁₅+ W₃e₁₆+ e₁₇. Proof : Consider a Lyapunov-Krasovskii functionai candidate:

V (x(t)) =

5

X

i=1

Vi(x(t)), (6) where

V₁(x(t)) = η^T(t)P η(t) +2

n

X

k=1

ρ_k Z _x(t)

0

hg_k(s) − l_k⁻sⁱds

+2

n

X

k=1

σ_k Z x(t)

0

hl_k⁺s − gk(s)ⁱds,

V₂(x(t)) =

2

X

i=1

Z t t−τi

hx^T(s)Q₁x(s)

+g^T(x(s))Q₂g^T(x(s))ⁱds +

Z _t

t−τ(t)

x^T(s)Q₁x(s)ds +

Z t t−h(t)

x˙^T(s)Q₃x(s)ds,˙ V₃(x(t)) = k₂

Z _t

t−k2

Z _t

θ

g^T(x(s))S₁g(x(s))dsdθ, V₄(x(t)) =

2

X

i=1

τ_i Z t

t−τi

Z t θ

x˙^T(s)S₂x(s)dsdθ,˙

V₅(x(t)) =

2

X

i=1

Z t t−τi

Z t θ

Z t s

x˙^T(λ)S₃x(λ)dλdsdθ,˙

where η(t) =^hx^T(t),^R_t−τ^t

1x^T(s)ds,^R_t−τ^t

2x^T(s)ds, R_t

t−τ1

R_t

θx^T(s)dsdθ,^R_t−τ^t ₂^R_θ^tx^T(s)dsdθ, Rt

t−τ1

Rt θ

Rt

sx^T(λ)dλdsdθ,^R_t−τ^t ₂^R_θ^tR_s^tx^T(λ)dλdsdθ^iT, U₁ = diag{ρ1, ρ₂, · · · , ρn} ≥ 0, and

U₂ = diag{σ1, σ₂, · · · , σn} ≥ 0 are to be determined, The time derivative of V (x(t)) can be computed as follows:

V˙₁(x(t)) = 2 ˙η^T(t)P η(t) +2

n

X

k=1

nρ_kx(t)[g˙ _k(x(t)) − l_k⁻x(t)]

+σ_kx(t)[l˙ _k⁺x(t) − gk(x(t))]^o,

≤ ζ^T(t) Ω₁ζ(t),

V˙₂(x(t)) ≤ 3x^T(t)Q₁x(t) + ˙x^T(t)Q₃x(t)˙

−x^T(t − τ1)Q₁x(t − τ1)

−x^T(t − τ2)Q₁x(t − τ2) +2g^T(x(t))Q₂g(x(t))

−g^T(x(t − τ1))Q₂g(x(t − τ1))

−g^T(x(t − τ2))Q₂g(x(t − τ2))

−(1 − τ3)x^T(t − τ (t))Q1x(t − τ (t))

−(1 − h3) ˙x^T(t − h(t))Q3x(t − h(t)),˙

= ζ^T(t) Ω₂ζ(t), V˙₃(x(t)) = k₂²g^T(x(t))S₁g(x(t))

−k2

Z t t−k₂

g^T(x(s))S₁g(x(s))ds,

≤ k₂²g^T(x(t))S₁g(x(t))

−k2

Z t t−k(t)

g^T(x(s))S₁g(x(s))ds,

≤ k₂²g^T(x(t))S₁g(x(t))

− Z t

t−k(t)

g^T(x(s))ds S₁

× Z t

t−k(t)

g(x(s))ds,

= ζ^T(t) Ω₃ζ(t),

V˙₄(x(t)) = x˙^T(t)(τ₁²S₂+ τ₂²S₂) ˙x(t)

−τ1

Z t t−τ₁

x˙^T(s)S₂x(s)ds˙

−τ2

Z t t−τ₂

x˙^T(s)S₂x(s)ds,˙

≤ ζ^T(t) Ω₄ζ(t),

V˙₅(x(t)) = 0.5 ˙x^T(t)(τ₁²S₃+ τ₂²S₃) ˙x(t)

− Z t

t−τ1

Z t θ

x˙^T(s)S₃x(s)dsdθ˙

− Z t

t−τ2

Z t θ

x˙^T(s)S₃x(s)dsdθ,˙

≤ ζ^T(t) Ω₅ζ(t),

where Ωi, (i = 1, 2, 3, 4, 5) is defined in (5).

From (3), the nonlinear function gk(x_k) satisfies l⁻_k ≤ g_k(x_k)

x_k ≤ l⁺_k, k = 1, 2, · · · , n, xk6= 0.

Thus, for any t_k> 0, (k = 1, 2, · · · , n), we have 2t_k[g_k^T(x(θ)) − l⁻_kx(θ)] [l⁺_kx(θ) − gk(x(θ))] ≥ 0, which

2[g^T(x(θ)) − x^T(θ)L⁻]^TT [L⁺x(θ) − g(x(θ))] ≥ 0,

where T = diag{t1, t₂, · · · , tn}.

Let θ be t, t − τ (t), t − τ1 and t − τ2, and replace T with Ts(s = 1, 2, 3, 4), then, we have (s = 1, 2, 3, 4)

2ζ^T(t)Π^T₁₇TsΠ₁₈ζ(t) ≥ 0, (7) Another observation from (3), we have

l⁻_k ≤ g_k(x(θ₁)) − gk(x(θ₂))

x(θ₁) − x(θ2) ≤ lk⁺, k = 1, 2, · · · , n.

Thus, for any t_k> 0, (k = 1, 2, · · · , n), and Λ = gk(x(θ₁)) − gk(x(θ₂)), we have

2t_k[Λ − l⁻_k(x(θ₁) − x(θ2))]

×[l⁺_k(x(θ₁) − x(θ2)) − Λ] ≥ 0, which

2[Λ − L⁻(x(θ₁) − x(θ2))]^T

×T [L⁺(x(θ₁) − x(θ2)) − Λ] ≥ 0, where Λ = col{Λ1, Λ₂, · · · , Λn}.

Let θ₁and θ₂take values in t, t−τ (t), t−τ1and t−τ2, and replace T with Tab(a = 1, 2, 3; b = 2, 3, 4; b >

a), then, we have

2ζ^T(t)Π^T₁₉T_abΠ₂₀ζ(t) ≥ 0, (8) where a = 1, 2, 3, b = 2, 3, 4, b > a.

From (7) and (8), it can be shown that

ζ^T(t) Ω₆ζ(t) ≥ 0, (9) where Ω₆is defined in (5).

Therefore, we conclude that

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V (x(t)) − γu˙ ^T(t)u(t) − 2y^T(t)u(t)

≤^P5i=1Vi(x(t)) + Ω₆− γu^T(t)u(t)

−2y^T(t)u(t),

= ζ^T(t) Ω ζ(t),

(10)

where Ω is defined in (4). If we have Ω < 0, then V (x(t)) − γu˙ ^T(t)u(t) − 2y^T(t)u(t) ≤ 0, (11) for any ζ(t) 6= 0. Since V (x(0)) = 0 under zero ini- tial condition, let x(t) = 0 for t ∈ [τmax, 0], after integrating (11) with respect to t over the time period from 0 to tf, we get

2 Z tf

0 y^T(s)u(s)ds

≥ V (x(tf)) − V (x(0)) − γ Z tf

0

u^T(s)u(s)ds,

≥ −γ Z tf

0 u^T(s)u(s)ds.

Thus, the NTNNs (1) with (3) is passive in the sense of Definition 1. This completes the proof.

4 Numerical example

In this section, we present example to illustrate the effectiveness and the reduced conservatism of our re- sult. Consider the NTNNs (1) with the following pa- rameters:

A =

"

8.4 0

0 9

# , W =

"

−0.21 −0.19

−0.24 0.1

# ,

W₁ =

"

−0.09 −0.2 0.2 0.1

# ,

W₂ =

"

−0.52 0

0.2 −0.09

# ,

W₃ =

"

−0.5 0 0 −0.5

# .

The activation functions are assumed to be g_i(x_i(t)) = 0.5(|xi + 1| − |xi − 1|), i = 1, 2.

It is easy to see:

L⁻=

"

0 0 0 0

# , L⁺=

"

1 0 0 1

# .

For 0.2 ≤ τ (t) ≤ 3, h(t) ≤ 0.5, k(t) ≤ 2, h3 = 0.5, τ₃ = 0.5 and γ = 0.34 by using MATLAB LMIs control toolbox and by solving the LMIs in Theorem 4, in our paper we obtain the feasible solutions:

P =

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1.0087 0.11967 −0.0038 −0.0013 0.1196 0.7258 −0.0067 −0.0088

−0.0038 −0.0067 0.0128 0.0084

−0.0013 −0.0088 0.0084 0.0429 0.0000 0.0000 −0.0016 −0.0017 0.0000 0.0000 −0.0015 −0.0056

−0.0286 −0.0289 0.0098 0.0892

−0.0035 −0.0472 −0.0263 0.0502 0.0000 0.0000 −0.0022 −0.0013 0.0000 0.0000 −0.0011 −0.0050

−0.0060 0.0007 0.0174 0.0289 0.0030 0.0050 −0.0033 0.0096 0.0000 0.0000 −0.0001 0.0000 0.0000 0.0000 0.0000 −0.0002 0.0000 0.0000 −0.0286 −0.0035 0.0000 0.0000 0.0000 −0.0289 −0.0472 0.0000

−0.0016 −0.0015 0.0098 −0.0263 −0.0022

−0.0017 −0.0056 0.0892 0.0502 −0.0013 0.0012 0.0011 0.0007 0.0001 0.0002 0.0011 0.0040 0.0000 0.0007 0.0003 0.0007 0.0000 1.4958 0.7472 −0.0009 0.0001 0.0007 0.7472 3.0171 −0.0004 0.0002 0.0003 −0.0009 −0.0004 0.0006 0.0003 0.0010 −0.0003 −0.0020 0.0004 0.0005 0.0001 −0.0140 −0.0364 0.0001 0.0002 0.0006 −0.1875 −0.3777 0.0002 0.0000 0.0000 −0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 −0.0001 0.0000

0.0000 −0.0060 0.0030 0.0000 0.0000 0.0000 0.0007 0.0050 0.0000 0.0000

−0.0011 0.0174 −0.0033 −0.0001 0.0000

−0.0050 0.0289 0.0096 0.0000 −0.0002 0.0003 0.0005 0.0002 0.0000 0.0000 0.0010 0.0001 0.0006 0.0000 0.0000

−0.0003 −0.0140 −0.1875 −0.0001 0.0000

−0.0020 −0.0364 −0.3777 0.0000 −0.0001 0.0004 0.0001 0.0002 0.0000 0.0000 0.0014 0.0002 0.0004 0.0000 0.0000 0.0002 5.1380 0.2050 0.0000 0.0000 0.0004 0.2050 5.7050 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000



































 ,

Q₁ =

"

0.2383 0.3599 0.3599 0.7875

# ,

Q₂ =

"

0.0019 0.0023 0.0023 0.0093

# ,

Q₃ =

"

0.3563 0.0107 0.0107 0.2513

# ,

S₁ =

"

2.2291 0.0099 0.0099 0.2674

# ,

S₂ = 10⁻⁵×

"

2.6041 1.1983 1.1983 5.4488

# ,

S₃ = 10⁻⁸×

"

5.4115 −0.0914

−0.0914 5.2782

# ,

U₁ =

"

2.0621 0 0 1.5664

# ,

U₂ =

"

3.9452 0 0 3.2423

# ,

T₁ =

"

23.2649 0 0 16.9973

# ,

T₂ =

"

0.0792 0 0 0.1857

# ,

T₃ =

"

0.0562 0 0 0.1528

# ,

T₄ =

"

0.0025 0 0 0.0141

# ,

T₅ =

"

1.2748 0 0 1.3177

# ,

T₆ =

"

1.2244 0 0 1.3813

# ,

T₇ =

"

1.2681 0 0 1.2956

# ,

T₈ =

"

1.3421 0 01.2576

# ,

T₉ =

"

1.2748 0 0 1.3177

# ,

T₁₀ =

"

1.0461 0 0 1.0442

# .

5 Conclusion

In this paper, the passivity analysis for NTNNs with discrete and distributed time-varying delays has been studied. By employing the LKFs method, double inte- gral inequality was developed to guarantee the passiv- ity performance of NTNNs. A new passivity analysis criterion has been given in terms of LMIs, which is dependent on time-varying delays. Finally, a numer- ical example has been presented which illustrate the effectiveness and usefulness of the proposed method.

Acknowledgements: The first author was supported by Faculty of Science, Khon Kaen University. The second author was supported by the National Re- search Council of Thailand 2018 and Faculty of Sci- ence and Liberal Arts, Rajamangala University of Technology Isan (RMUTI), Thailand. Also, study is part of the project “Robust passivity analysis for neutral-type neural networks with mixed time-varying delays and applications”.

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