Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324 DOI 10.1186/s13662-016-1046-5

RESEARCH

Open Access

New delay-range-dependent exponential stability criteria for certain neutral differential equations with interval discrete and distributed time-varying delays Watcharin Chatbupapan and Kanit Mukdasai* * Correspondence: [email protected] Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Abstract In this research, we investigate the problem of delay-dependent exponential stability analysis for certain neutral differential equations with discrete and distributed time-varying delays. The time-varying delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for the time-varying delays are available. The restrictions on the derivative of interval time-varying delays are needed. Based on a class of novel augmented Lyapunov-Krasovskii functionals, a model transformation, the decomposition technique of constant coefficients, the Leibniz-Newton formula, and utilization of a zero equation, new delay-range-dependent exponential stability criteria are derived in terms of the linear matrix inequality (LMI) for the equations considered. Numerical examples suggest for the results given to illustrate the effectiveness and improvement over some existing methods.

1 Introduction The neutral differential equation is a retarded system that often appears in many scientific and engineering fields such as aircraft, chemical and process control systems, and biological systems [–]. The problem of various stability analyses for dynamical systems with state delays has been intensively studied in the past years by several researchers in mathematics [–]. However, delay-dependent stability criteria for neutral differential equations have been attracting the attention of several researchers. Delay-dependent stability criteria make use of information on the length of delays. A certain neutral differential equation (CNDE) with constant delays is of the form  d x(t) + px(t – τ ) = –ax(t) + b tanh x(t – σ ), dt

t ≥ ,

(.)

where a, b, τ , σ are positive real constants and |p| < . For each solution x(t) of (.), we assume the initial condition x (t) = φ(t),

t ∈ [–r, ],

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Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

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where φ ∈ C([–r, ]; R) denotes the space of all continuous vector functions mapping [–r, ] into R with r = max{τ , σ }. Over the past decades, the problem of asymptotic stability analysis for (.) has been discussed in [, , , , , ] by using several model transformation methods and the Lyapunov-Krasovskii functional approach, while the problem of exponential stability analysis has been studied with the use of the model transformation technique and the Lyapunov-Krasovskii functional approach in []. In [, , ], the authors studied the problem of exponential stability analysis for CNDE with time-varying delays of the form     d x(t) + px t – τ (t) = –ax(t) + b tanh x t – σ (t) , dt

t ≥ ,

(.)

where a, b are positive real constants and |p| < . τ (t) and σ (t) are neutral and discrete time-varying delays, respectively,  ≤ τ (t) ≤ τ ,

τ˙ (t) < τd ,

 ≤ σ (t) ≤ σ ,

σ˙ (t) < σd ,

where τ , σ , τd , and σd are given positive real constants. For each solution x(t) of (.), we assume the initial condition x (t) = φ(t),

t ∈ [–r, ],

where φ ∈ C([–r, ]; R). In [], the results are derived without the use of the model transformation method and the bounding technique, while the authors have used the model transformation method, radially unboundedness, and the Lyapunov-Krasovskii functional approach in []. Stability analysis of uncertain neutral stochastic systems with time-varying delays has received the attention of a lot of theoreticians and engineers in this field over the last few decades [–]. Moreover, the authors have studied the problem of stability for systems with discrete and distributed delays such as [], which presented some stability conditions for uncertain neutral systems with discrete and distributed delays. The robust stability of uncertain dynamical systems with discrete and distributed delays has been studied in [, , , ]. This research presents new criteria based on new methods with mixed model transformation techniques. We investigate the problem of exponential stability criteria for CNDE with discrete and distributed time-varying delays. The time-varying delays are assumed to belong to the given lower and upper bound delays and restrictions on the derivative of the time-varying delays are needed. Based on the combination of a mixed model transformation, decomposition technique of constant coefficients, utilization of a zero equation, and a new Lyapunov-Krasovskii functional, sufficient conditions for exponential stability are obtained and formulated in terms of LMIs for the systems. Finally, numerical examples suggest that the proposed criteria are effective and an improvement over previous ones.

2 Problem formulation and preliminaries Consider the CNDE with mixed interval time-varying delays of the form     d x(t) + px t – τ (t) = –ax(t) + b tanh x t – σ (t) + c dt



t

x(s) ds, t–ρ(t)

t ≥ ,

(.)

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where a, b, c are positive real constants and |p| < . τ (t), σ (t), and ρ(t) are neutral, discrete, and distributed interval time-varying delays, respectively,  ≤ τ ≤ τ (t) ≤ τ ,

τ˙ (t) ≤ τd < ∞,

 ≤ σ ≤ σ (t) ≤ σ ,

(.)

σ˙ (t) ≤ σd < ∞,

(.)

 ≤ ρ ≤ ρ(t) ≤ ρ ,

(.)

where τ , τ , σ , σ , ρ , ρ , τd , and σd are given positive real constants. For each solution x(t) of (.), we assume the initial condition t ∈ [–ω, ],

x (t) = φ(t),

where φ ∈ C([–ω, ]; R) and ω = max{τ , σ , ρ }. Definition . ([]) Equation (.) is exponentially stable, if there exist positive real constants α, β such that, for each φ(t) ∈ C([–ω, ], R), the solution x(t, φ) of the system satisfies   x(t, φ) ≤ βφe–αt ,

t ≥ .

Lemma . ([] (Jensen’s inequality)) For any symmetric positive definite matrix Q, positive real number h, and vector function x˙ (t) : [–h, ] → Rn the following integral is well defined: 



 x˙ (s + t)Q˙x(s + t) ds ≤ –



T

–h –h

T  x˙ (s + t) ds Q

–h



x˙ (s + t) ds .

–h

Lemma . For any constant symmetric positive definite matrix Q ∈ Rn×n , h(t) a discrete time-varying delay with (.), the vector function ω : [–h , ] → Rn such that the integrations concerned are well defined, we have 

–h

–[h – h ]

ωT (s)Qω(s) ds

–h



–h

≤–



ωT (s) dsQ

–h(t)



–h

–h(t)



–h(t)

–h(t)

ωT (s) dsQ

ω(s) ds – –h

ω(s) ds. –h

Proof It is easy to see that 

–h

ωT (s)Qω(s) ds

[h – h ] –h



–h

= [h – h ]



–h(t)

  ≥ h(t) – h =

 



–h –h(t)





–h(t)

ωT (s)Qω(s) ds + [h – h ]

–h

ωT (s)Qω(s) ds –h

  ωT (s)Qω(s) ds + h – h(t)



–h(t) –h

–h(t)

ωT (s)Qω(s) + ωT (ξ )Qω(ξ ) ds dξ

–h(t)

ωT (s)Qω(s) ds –h

Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

+ ≥

  + 

  



–h(t)  –h(t)

ωT (s)Qω(s) + ωT (ξ )Qω(ξ ) ds dξ –h –h



–h(t)

 

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–h

=

–h –h

T

ωT (s)Q/ Q/ ω(ξ ) ds dξ

–h(t)

–h(t)  –h(t)

T

ωT (s)Q/ Q/ ω(ξ ) ds dξ –h

–h



ωT (s) dsQ

–h(t)



–h



–h(t) –h

–h(t)

–h(t)

ωT (s) dsQ

ω(s) ds +

ω(s) ds. –h



This completes the proof. Remark . In Lemma ., we have modified the method of [].

3 Main results In this section, we investigate the exponential stability problem for equation (.) with interval time-varying delays satisfying (.)-(.). From the model transformation method, we have the Leibniz-Newton formula of the form    = x(t) – x t – τ (t) –



t

x˙ (s) ds,

(.)

t–τ (t)

   = x(t) – x t – γ τ (t) –



t

x˙ (s) ds,

(.)

t–γ τ (t)

where γ is a given positive real constant. We utilize the zero equations and obtain    = r x(t) – r x t – τ (t) – r



t

x˙ (s) ds,

(.)

t–τ (t)

   = r x(t) – r x t – γ τ (t) – r



t

x˙ (s) ds,

(.)

t–γ τ (t)

where r , r ∈ R will be chosen to guarantee the exponential stability of equation (.). By (.)-(.), equation (.) can be represented by the form 

 t  t     d x˙ (s) ds – p x˙ (s) ds p x(t) + p x t – τ (t) + x t – γ τ (t) + dt t–γ τ (t) t–τ (t)  t   = –(a – r – r )x(t) – (a + r )x t – τ (t) – (a + r ) x˙ (s) ds   – r x t – γ τ (t) – r

t–τ (t)



t

x˙ (s) ds

t–γ τ (t)

  + b tanh x t – σ (t) + c



t

(.)

x(s) ds. t–ρ(t)

For convenience, we define a new variable,     D(t) = p x(t) + p x t – τ (t) + x t – γ τ (t) +



t t–γ τ (t)

 x˙ (s) ds – p

t

t–τ (t)

x˙ (s) ds.

(.)

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Rewrite equation (.) in the following equation:   ˙ = –(a – r – r )x(t) – (a + r )x t – τ (t) – (a + r ) D(t)   – r x t – γ τ (t) – r



t

x˙ (s) ds

t–τ (t)



t

x˙ (s) ds

t–γ τ (t)

  + b tanh x t – σ (t) + c



t

x(s) ds.

(.)

t–ρ(t)

We introduce the following notations for later use:  = [ (i,j) ]× ,

(.)

where (i,j) = (j,i) , (,) = k α – q , (,) = q p – q + m + n + k r + k r – k a , (,) = q p – q – m – k a – k r , (,) = –q p – q – m – k a – k r , (,) = q – q – n – k r , (,) = q – q – n – k r , (,) = (,) = (,) = (,) = , (,) = k b, (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –m , (,) = (,) = , (,) = k c, (,) = q p + k + k + k + k + k τ + k γ  τ + k + k σ + m + n + w + w + w τ + w γ  τ + w (τ – τ ) + w γ  (τ – τ ) + w (σ – σ ) + k + ω(ρ – ρ ) , (,) = q p + q p + m + n – m , (,) = –q p + q p + m + n – m , (,) = q + q p + m + n – n , (,) = q + q p + m + n – n , (,) = (,) = (,) = (,) = , (,) = m + n , (,) = m + n ,

Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

(,) = –q a , (,) = q r + q r – q a , (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = m + n , (,) = (,) = (,) = , (,) = q p – k e–ατ + k td – m , (,) = –q p + q p – m – m , (,) = q + q p – m – n , (,) = q + q p – m – n , (,) = (,) = (,) = (,) = , (,) = –m , (,) = –m , (,) = –q a , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –m , (,) = (,) = (,) = , (,) = –q p – m , (,) = q – q p – m – n , (,) = q – q p – m – n , (,) = (,) = (,) = (,) = , (,) = –m , (,) = –m , (,) = –q a , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –m , (,) = (,) = (,) = , (,) = q – k e–αγ τ + k td – n , (,) = q + q – n – n , (,) = (,) = (,) = (,) = , (,) = –n , (,) = –n , (,) = , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = ,

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Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

(,) = –n , (,) = (,) = (,) = , (,) = q – n , (,) = (,) = (,) = (,) = , (,) = –n , (,) = –n , (,) = , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –n , (,) = (,) = (,) = , (,) = –(k + w )e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –k e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –(k + w )e–αγ τ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –k e–αγ τ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –k e–ασ + k sd – k , (,) = , (,) = q b, (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –k e–ασ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –q , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = ,

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Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

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(,) = (w – w )e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = (w – w )e–αγ τ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –w e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –w e–αγ τ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –w e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –w e–ατ , (,) = (,) = (,) = (,) = (,) = (,) = , (,) = –w e–αγ τ , (,) = (,) = (,) = (,) = (,) = , (,) = –w e–αγ τ , (,) = (,) = (,) = (,) = , (,) = –w e–ασ , (,) = (,) = (,) = , (,) = –w e–ασ , (,) = (,) = , (,) = –w e–ασ , (,) = , (,) = –ωe–αρ . The exponential stability for the CNDE with time-varying delays in equation (.) will be represented as follows. Theorem . For given positive real constants σ , σ , σd , τ , τ , τd , ρ , ρ and γ , equation (.) is exponentially stable with a decay rate α if there exist positive real constants ω, ki , wi where i = , , . . . , , and real constants r , r , mk , nk where k = , , . . . ,  such that the following symmetric linear matrix inequality holds:  < .

(.)

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Proof For ω, ki and wi are positive real constants where i = , , . . . , , we consider the Lyapunov-Krasovskii functional candidate for equation (.) of the form

V (t, xt ) =



Vi (t, xt ),

(.)

i=

where V (t, xt ) = k D (t),  t  eα(s–t) x (s) ds + k V (t, xt ) = k

eα(s–t) x (s) ds t–τ (t)

t–τ



t



t

t–γ τ



eα(s–t) x (s) ds 

t

t–γ τ (t) t

eα(s–t) x (s) ds + w

+ w t–τ





t–τ

α(s–t) 

t–τ

eα(s–t) x (s) ds,

t–γ τ





t–γ τ

x (s) ds + w

e 

eα(s–t) x (s) ds t–γ τ

+ w

t







t

eα(θ–t) x (θ ) dθ ds + k γ τ

V (t, xt ) = k τ –τ







eα(θ–t) x (θ ) dθ ds –γ τ

t+s

t+s

t

eα(θ–t) x (θ ) dθ ds

+ w τ –τ

t+s







t

eα(θ–t) x (θ ) dθ ds

+ w γ τ –γ τ

t+s



–τ



t

eα(θ–t) x (θ ) dθ ds

+ w (τ – τ ) –τ



t+s –γ τ



t

eα(θ–t) x (θ ) dθ ds,

+ w γ (τ – τ ) 

t

eα(s–t) x (s) ds + k

+ k

–γ τ

t+s

t

eα(s–t) tanh x(s) ds

V (t, xt ) = k t–σ (t)







t

eα(θ–t) tanh x(θ ) dθ ds

+ k σ  –σ





t+s



t

eα(θ–t) tanh x(θ ) dθ ds

+ w σ –σ

t+s



–σ



t

eα(θ–t) tanh x(θ ) dθ ds,

+ w (σ – σ ) 

–σ –ρ



t+s t

eα(θ–t) x (θ ) dθ ds.

V (t, xt ) = ω(ρ – ρ ) –ρ

t+s

Calculating the time derivatives of V (t, xt ) along the solution of equation (.) yields V˙ (t, xt ) =

 i=

V˙ i (t, xt ).

(.)

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The time derivatives of V (t, xt ) and V (t, xt ) are calculated as ˙ V˙  (t, xt ) = k D(t)D(t)

   = k D(t) –(a – r – r )x(t) – (a + r )x t – τ (t) – (a + r )   – r x t – γ τ (t) – r



  x˙ (s) ds + b tanh x t – σ (t) + c

t

t–γ τ (t)

    + q D(t) –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t) 



t

t

x˙ (s) ds – p

+ t–γ τ (t)



t

x˙ (s) ds

t–τ (t)



t

x(s) ds t–ρ(t)

 x˙ (s) ds

t–τ (t)

    + q x(t) –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t) 



t

t

x˙ (s) ds – p

+ t–γ τ (t)

 x˙ (s) ds

t–τ (t)

      + q x t – τ (t) –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t) 



t

x˙ (s) ds – p

+ t–γ τ (t)



t–τ (t)



t

x˙ (s) ds – p

+

x˙ (s) ds

    x˙ (s) ds –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t)

t

t



t–τ (t)

+ q 

t

t–γ τ (t)

 x˙ (s) ds

t–τ (t)

      + q x t – γ τ (t) –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t) 



t

t–γ τ (t)



t–τ (t)

t–γ τ (t) t



t

x˙ (s) ds – p

+

 x˙ (s) ds

    x˙ (s) ds –D(t) + p x(t) + p x t – τ (t) + x t – γ τ (t)

t

+ q 

t

x˙ (s) ds – p

+

t–γ τ (t)

 x˙ (s) ds

t–τ (t)

  ˙ ˙ – a x(t) – a x t – τ (t) + q D(t) –D(t) 

  x˙ (s) ds + b tanh x t – σ (t) + c

t

– a t–τ (t)





t

x(s) ds t–ρ(t)



+ αk D (t) – αV (t), V˙  (t, xt ) = (k + k + k + k + w + w )x (t) – (k + w )e–ατ x (t – τ )     – k  – τ˙ (t) e–ατ (t) x t – τ (t) – (k + w )e–αγ τ x (t – γ τ )     – k  – γ τ˙ (t) e–αγ τ (t) x t – γ τ (t) + (w – w )e–ατ x (t – τ )(w – w )e–αγ τ x (t – γ τ ) – αV (t)

(.)

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≤ (k + k + k + k + w + w )x (t) – (k + w )e–ατ x (t – τ )     – k e–ατ x t – τ (t) + k τd x t – τ (t)   – (k + w )e–αγ τ x (t – γ τ ) – k e–αγ τ x t – γ τ (t)   + k γ τd x t – γ τ (t) + (w – w )e–ατ x (t – τ )(w – w )e–αγ τ x (t – γ τ ) – αV (t).

(.)

Obviously, for any a scalar s ∈ [t –τ , t], we get e–ατ ≤ eα(s–t) ≤  and e–αγ τ ≤ eα(s–t) ≤ , for any a scalar s ∈ [t – γ τ , t]. Together with Lemma . and ., we obtain V˙  (t, xt ) = k τ









x (t) ds – k τ –τ

esα x (t + s) ds –τ









x (t) ds – k γ τ

+ k γ τ 

–γ τ





esα x (t + s) ds –γ τ





+ w τ

esα x (t + s) ds

x (t) ds – w τ –τ



–τ







x (t) ds – w γ τ

+ w γ τ

esα x (t + s) ds

–γ τ

–γ τ



–τ

+ w (τ – τ )



–τ

x (t) ds – w (τ – τ )

–τ

esα x (t + s) ds

–τ



–γ τ

+ w (γ τ – γ τ )



x (t) ds – w (γ τ – γ τ )

–γ τ

– αV (t)

esα x (t + s) ds

–γ τ



≤ k (τ ) x (t) – k e–ατ



t

x(s) ds t–τ





t

+ k (γ τ ) x (t) – k e–αγ τ

x(s) ds



t–γ τ



t

+ w (τ ) x (t) – w e–ατ

x(s) ds t–τ

  

–γ τ



t

–αγ τ

+ w (γ τ ) x (t) – w e

x(s) ds t–γ τ

+ w (τ – τ ) x (t) + w (γ τ – γ τ ) x (t)  t–τ   x(s) ds – w e–ατ – w e–ατ t–τ (t)

 – w e–γ ατ

x(s) ds

x(s) ds t–τ





t–γ τ



t–τ (t)

t–γ τ (t)



t–γ τ (t)

– w e–γ ατ

x(s) ds t–γ τ

– αV (t).

(.)

From Lemma ., ., and tanh x(t) ≤ x (t), we have     V˙  (t, xt ) = k tanh x(t) – k  – σ˙ (t) e–ασ (t) tanh x t – σ (t)     + k σ  tanh x(t) ds – k σ eαs tanh x(t + s) ds –σ

–σ

Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324









tanh x(t) ds – w σ

+ w σ –σ

Page 12 of 18

eαs tanh x(t + s) ds –σ



–σ

+ w (σ – σ )

tanh x(t) ds

–σ



–σ

– w (σ – σ )

eαs tanh x(t + s) ds

–σ

– αV (t)     ≤ k tanh x(t) – k e–ασ tanh x t – σ (t) + k sd tanh x t – σ (t)  t  tanh x(s) ds + k σ x (t) – k e–ασ t–σ

 + w σ x (t) – w e–ασ



t

tanh x(s) ds t–σ



+ w (σ – σ ) x (t) – w e–ατ



t–τ

tanh x(s) ds t–τ (t)





+ k x (t) – k tanh x(t)   t–τ (t) tanh x(s) ds – w e–ατ t–τ

– αV (t), V˙  (t, xt ) ≤ ω(ρ – ρ ) x (t) – ωe–αρ

(.) 



t

.

x(s) ds

(.)

t–ρ(t)

From the Leibniz-Newton formula, the following equations are true for any real constants mi , ni , i = , , . . . ,  with appropriate dimensions:

      m D(t) + m x(t) + m x t – τ (t) + m x t – γ τ (t) + m 

  x˙ (s) ds + m tanh x t – σ (t) + m

t

+ m 

t

tanh x(s) ds t–σ



   tanh x(s) ds × x(t) – x t – τ (t) –

t

t

 x˙ (s) ds = ,

      n D(t) + n x(t) + n x t – τ (t) + n x t – γ τ (t) + n 

  x˙ (s) ds + n tanh x t – σ (t) + n

t



t–γ τ (t) t

+ n

(.)

t–τ (t)

t–σ

+ n

x˙ (s) ds

t–τ (t)



t–γ τ (t)

+ m

t



x˙ (s) ds

t–τ (t) t

tanh x(s) ds t–σ



   tanh x(s) ds × x(t) – x t – γ τ (t) –

t–σ

t

t

 x˙ (s) ds = .

(.)

t–γ τ (t)

According to (.)-(.), it is straightforward to see that V˙ (t, xt ) + αV (t, xt ) ≤ ξ T (t)ξ (t),

(.)

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where

  ζ (t) = D(t), x(t), x t – τ (t) ,





t–τ (t)



t

t

x(s) ds, x(t – γ τ ), t–τ

 

t–γ τ

t–γ τ (t)





 x(s) ds, tanh x t – σ (t) , 

t

t–τ

t–γ τ



t

t–σ

 x˙ (s) ds ,

t

t–τ

t–τ (t)



˙ tanh x(s) ds, D(t), 

t–τ (t)

x(s) ds, t–τ

t–σ

tanh x(s) ds, t–σ

tanh x(s) ds,



x(s) ds,

t

x(s) ds, t–γ τ

x˙ (s) ds, x(t – τ ),

t–σ

x(s) ds,



t–γ τ (t)

t–σ (t)



t

x(s) ds,

x(s) ds,

t t–γ τ (t)

t–γ τ

x(t – τ ), x(t – γ τ ), 



  x˙ (s) ds, x t – γ τ (t) ,

t

T

tanh x(s) ds, t–σ (t)

t–ρ

and  is defined in (.). It is true that if condition (.) holds, then V˙ (t, xt ) + αV (t, xt ) ≤ ,

∀t ∈ R+ .

(.)

From (.), it is easy to see that

x(t, φ) ≤ βφe–αt ,

t ∈ R+ .

This means that equation (.) is exponentially stable. The proof of the theorem is complete.  We now present the delay-dependent criteria for exponential stability of equation (.) where τ = σ = c = . We introduce the following notations for later use: ˜ = [ ˜ (i,j) ]× , 

(.)

˜ (i,j) = (i,j) , except where ˜ (,) = q p + k + k + k + k + k τ + k γ  τ + k + k σ + m + n + k , ˜ (,) = –k e–ατ , ˜ (,) = –k e–αγ τ . Corollary . For given positive real constants σ , τ , σd , τd and γ , equation (.) where τ = σ = c =  is exponentially stable with a decay rate α if there exist positive real constants ki where i = , , . . . ,  and we have the real constants r , r , mk , nk where k = , , . . . ,  such that the following symmetric linear matrix inequality holds: ˜ < . 

(.)

Proof For ki , i = , , . . . ,  are positive real numbers, we consider the Lyapunov-Krasovskii functional candidate for equation (.) where τ = σ = c =  of the form V (t, xt ) =

 i=

Vi (t, xt ),

(.)

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where V (t, xt ) = k D (t),  t  eα(s–t) x (s) ds + k V (t, xt ) = k

eα(s–t) x (s) ds t–τ (t)

t–τ



t



t







eα(s–t) x (s) ds, t–γ τ (t)

t–γ τ

t

eα(θ–t) x (θ ) dθ ds

V (t, xt ) = k τ –τ



t+s





t

eα(θ–t) x (θ ) dθ ds,

+ k γ τ 

t

eα(s–t) x (s) ds + k

+ k

–γ τ

t+s

t

eα(s–t) tanh x(s) ds

V (t, xt ) = k t–σ (t)







t

eα(θ–t) tanh x(θ ) dθ ds.

+ k σ  –σ

t+s

According to Theorem ., we have the delay-dependent exponential stability criteria  (.) of equation (.) where τ = σ = c = .

4 Numerical examples In this section, four numerical examples are given to present the effectiveness of our main results by comparing the upper bounds of the delays σ and the parameter b as well as investigating the rate of convergence. Example . Consider the following equation with mixed interval time-varying delays:     d x(t) + .x t – τ (t) = –.x(t) + b tanh x t – σ (t) + . dt



t

(.)

x(s) ds. t–ρ(t)

Decompose the constants a and p as a = a + a and p = p + p , respectively, where a = ,

a = .,

p = .,

p = .. 



Solving the LMI (.) when b = ., α = ., τ (t) = . + sin (t) , σ (t) = . + cos (t) , and , we can obtain a set of parameters guaranteeing exponential stability as ρ(t) = . + | cos(t)|  follows: q = .,

q = –.,

q = –.,

q = –.,

q = ., q = .,

q = ., k = ., k = .,

k = .,

k = .,

k = .,

k = .,

k = .,

k = .,

k = .,

ω = .,

w = .,

w = .,

w = .,

w = .,

w = .,

w = .,

w = .,

w = .,

w = .,

w = ..

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Table 1 Upper bounds of b for Example 4.1 when γ = 0.05 τd = σd

α=0

α = 0.1

α = 0.2

α = 0.3

α = 0.4

0.1 0.2 0.3 0.4 0.5

1.4906 1.4910 1.4883 1.4861 1.4834

1.3865 1.3852 1.3835 1.3812 1.3774

1.2747 1.2740 1.2710 1.2661 1.2610

1.1601 1.1580 1.1512 1.1450 1.1305

1.0433 1.0380 1.0271 1.0126 0.9830

Table 2 Upper bounds of α for Example 4.1 when γ = 0.05 τd = σd

b = 0.4

b = 0.6

b = 0.8

b=1

b = 1.2

0.1 0.2 0.3 0.4 0.5

0.9105 0.8728 0.8095 0.7020 0.5549

0.7610 0.7382 0.7031 0.6414 0.5394

0.6015 0.5895 0.5730 0.5426 0.4897

0.4364 0.4302 0.4217 0.4094 0.3892

0.2660 0.2623 0.2591 0.2543 0.2477

Table 3 Upper bounds of b for Example 4.1 when γ = 0.05 c

α=0

α = 0.1

α = 0.2

α = 0.3

α = 0.4

0.5 1 2 3 4

1.4834 1.4830 1.4791 1.4780 1.4750

1.3774 1.3761 1.3727 1.3704 1.3681

1.2610 1.2610 1.2580 1.2534 1.2511

1.1305 1.1288 1.1249 1.1228 1.1270

0.9830 0.9806 0.9781 0.9720 0.9704

Table 4 Upper bounds of α for Example 4.1 when γ = 0.05 c

b = 0.4

b = 0.6

b = 0.8

b=1

b = 1.2

0.5 1 2 3 4

0.5549 0.5548 0.5546 0.5546 0.5546

0.5394 0.5391 0.5400 0.5400 0.5377

0.4897 0.4887 0.4880 0.4880 0.4843

0.3892 0.3877 0.3890 0.3850 0.3840

0.2477 0.2480 0.2460 0.2430 0.2401

Moreover, the upper bounds of the parameter b which guarantees the exponential and asymptotic stabilities are . and ., respectively. The maximum upper bounds b of this example can be found in Table  for different values of α, τd , σd . The maximum upper bounds as regards the rate of convergent α for this example can be found in Table  with different values of b, τd , σd . The maximum upper bounds b for exponential and asymptotic stabilities of Example . are listed in Table  for different values of α, c. The maximum upper bounds α for exponential and asymptotic stabilities of Example . are listed in Table  for different values of c, b. Example . Consider the following equation studied in [, ]:     d x(t) + .x t – τ (t) = –.x(t) + . tanh x t – σ (t) , dt 

(.)

when σ (t) = sin(t) and τd = .. Decompose the constants a and p thus: a = a + a and p = p + p , respectively, where a = .,

a = .,

p = .,

p = ..

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Table 5 The upper bound of time delay σ (t) for Example 4.2 when γ = 0.02 Methods

α = 0.0038

α = 0.02

α = 0.028

Chen and Meng (2011) [8] Keadnarmol and Rojsiraphisal (2014) [23] Corollary 3.2

Infeasible 7.5231 3.5 × 103

Infeasible 0.5234 564.9979

Infeasible 0.0321 448.9991

Table 6 Upper bounds of b for Example 4.3 when γ = 0.5 Methods σ2 = τ2 = 0.5

b A.S. (α = 0)

E.S. (α = 0.177)

Agarwal and Grace (2000) [4] El-Morshedy and Gopalsamy (2000) [15] Park and Kwon (2008) [32] Kwon and Park (2008) [21] Li (2009) [24] Deng et al. (2009) [14] Nam and Phat (2009) [29] Rojsiraphisal and Niamsup (2010) [34] Chen and Meng (2011) [8] Chen (2012) [7] Keadnarmol and Rojsiraphisal (2014) [23] Corollary 3.2

0.318 0.424 0.422 1.49 0.699 0.889 1.405 1.405 1.346 1.405 1.405 1.4051

0.722 0.478 1.092 1.1089 1.2114

By solving the linear matrix inequality (.), the maximum upper bounds σ for exponential stability of this example is listed fore comparison in Table , for different values of α. We can see that our results in Corollary . are much less conservative than those obtained in [, ]. Example . Consider the following equation, which is considered in [, , , , , , , , , , ]:  d x(t) + .x(t – .) = –.x(t) + b tanh x(t – .). dt

(.)

Decompose constants a and p as a = a + a and p = p + p , respectively, where a = .,

a = .,

p = .,

p = ..

Table  lists for comparison the upper bounds b for asymptotic stability (α = ) and exponential stability (α = .) of equation (.) by different methods. We can see from Table  that our result (Corollary .) is better than other existing work. Example . Consider the following equation in [, , , , , , , , ]:  d x(t) + .x(t – .) = –.x(t) + . tanh x(t – σ ). dt

(.)

Decompose the matrix a and p in the same way as the decomposition in Example .. Table  lists for comparison the upper bounds delay for asymptotic stability (α = ) and exponential stability (α = .) of (.) by different methods. It is clear that our results (Corollary .) are significantly better than some existing criteria.

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Table 7 Upper bounds of σ2 for Example 4.4 when γ = 0.05 Methods τ2 = 0.1

σ2 A.S. (α = 0)

E.S. (α = 0.0038)

Park (2004) [30] Park and Kwon (2008) [32] Kwon and Park (2008) [21] Li (2009) [24] Nam and Phat (2009) [29] Rojsiraphisal and Niamsup (2010) [34] Chen and Meng (2011) [8] Chen (2012) [7] Corollary 3.2

0.444 1.90 107 2.07 2.32 2.32 1021 1.34 × 1021 3.15 × 109

1.947 175.289 3,700

5 Conclusions In this paper, we proposed the delay-range-dependent exponential stability criteria for certain NDE with discrete and distributed interval time-varying delays. Then we presented the delay-dependent exponential stability criteria for certain NDE with time-varying delays. The method combining an augmented Lyapunov-Krasovskii functional, a mixed model transformation, the decomposition technique of constant coefficients, and utilization of zero equations has been adopted to study the paper. New stability criteria have been formulated in terms of LMI. Finally, four numerical examples are given to show that the proposed criteria are less conservative than some existing stability criteria. Competing interests The authors declare to have no competing interests. Authors’ contributions The paper was carried out in collaboration between both authors. KM proposed the problem and initiated techniques used in this paper. WC presented the new method and numerical reasoning tests for analysis. Both authors wrote the manuscript and approved the final manuscript. Acknowledgements This work was supported by DPST Research (Grant number 001/2557), National Research Council of Thailand and Khon Kaen University, Thailand (Grant number 590044). Received: 26 August 2016 Accepted: 29 November 2016 References 1. Fridman, E: Stability of linear descriptor systems with delays: a Lyapunov-based approach. J. Math. Anal. Appl. 273, 24-44 (2002) 2. Kwon, OM, Park, JH, Lee, SM: Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays. Appl. Math. Comput. 207, 202-212 (2009) 3. Liao, X, Liu, Y, Guo, S, Mai, H: Asymptotic stability of delayed neural networks: a descriptor system approach. Commun. Nonlinear Sci. Numer. Simul. 14, 3120-3133 (2009) 4. Agarwal, RP, Grace, SR: Asymptotic stability of certain neutral differential equations. Math. Comput. Model. 31, 9-15 (2000) 5. Antsaklis, PJ, Michel, AN: A Linear Systems Primer. Birkhäuser, Berlin (2007) 6. Boyd, S, Ghaou, LE, Feron, E, Balakrishnan, V: Linear Matrix Inequalities in Control Theory. Studies in Applied Mathematics. SIAM, Philadelphia (1994) 7. Chen, H: Some improved criteria on exponential stability of neutral differential equation. Adv. Differ. Equ. 2012, 170 (2012) 8. Chen, H, Meng, X: An improved exponential stability criterion for a class of neutral delayed differential equations. Appl. Math. Lett. 24, 1763-1767 (2011) 9. Chen, H, Zhang, Y, Hu, P: Novel delay-dependent robust stability criteria for neutral stochastic delayed neural networks. Neurocomputing 73(13-15), 2554-2561 (2010) 10. Chen, H, Wang, L: New result on exponential stability for neutral stochastic linear system with time-varying delay. Appl. Math. Comput. 239(15), 320-325 (2014) 11. Chen, H, Wang, L: Delay-dependent robust L2 -L∞ filter design for uncertain neutral stochastic systems with mixed delays. Digit. Signal Process. 30, 184-194 (2014) 12. Chen, H, Zhang, Y, Zhao, Y: Stability analysis for uncertain neutral systems with discrete and distributed delays. Appl. Math. Comput. 218(23), 11351-11361 (2012)

Chatbupapan and Mukdasai Advances in Difference Equations (2016) 2016:324

Page 18 of 18

13. Chen, H, Zhao, Y: Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay. Int. J. Syst. Sci. 46(14), 2584-2597 (2015) 14. Deng, S, Liao, X, Guo, S: Asymptotic stability analysis of certain neutral differential equations: a descriptor system approach. Math. Comput. Simul. 71, 4297-4308 (2009) 15. El-Morshedy, HA, Gopalsamy, K: Nonoscillation, oscillation and convergence of a class of neutral equations. Nonlinear Anal. 40, 173-183 (2000) 16. Gu, K, Kharitonov, VL, Chen, J: Stability of Time-Delay Systems. Birkhäuser, Berlin (2003) 17. Han, QL: A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 40(10), 1791-1796 (2004) 18. Hale, JK, Verduyn Lunel, SM: Introduction to Functional Differential Equations. Springer, New York (1993) 19. Jiang, X, Han, QL: Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica 42, 1059-1065 (2006) 20. Kwon, OM, Park, JH: Exponential stability of uncertain dynamic systems including state delay. Appl. Math. Lett. 19, 901-907 (2006) 21. Kwon, OM, Park, JH: On improved delay-dependent stability criterion of certain neutral differential equations. Appl. Math. Comput. 199, 385-391 (2008) 22. Fridman, E: New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett. 43, 309-319 (2001) 23. Keadnarmol, P, Rojsiraphisal, T: Globally exponential stability of a certain neutral differential equation with time-varying delays. Adv. Differ. Equ. 2014, 32 (2014) 24. Li, X: Global exponential stability for a class of neural networks. Appl. Math. Lett. 22, 1235-1239 (2009) 25. Li, X, Fu, X: Effect of leakage time-varying delay on stability of nonlinear differential systems. J. Franklin Inst. 350, 1335-1344 (2013) 26. Li, X, Rakkiyappan, R: Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays. Commun. Nonlinear Sci. Numer. Simul. 18(6), 1515-1523 (2013) 27. Li, X, Song, S: Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans. Neural Netw. 24, 868-877 (2013) 28. Li, X, Wu, J: Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63-69 (2016) 29. Nam, PT, Phat, VN: An improved stability criterion for a class of neutral differential equations. Appl. Math. Lett. 22, 31-35 (2009) 30. Park, JH: Delay-dependent criterion for guaranteed cost control of neutral delay systems. J. Optim. Theory Appl. 124, 491-502 (2005) 31. Park, JH: Delay-dependent criterion for asymptotic stability of a class of neutral equations. Appl. Math. Lett. 17, 1203-1206 (2004) 32. Park, JH, Kwon, OM: Stability analysis of certain nonlinear differential equation. Chaos Solitons Fractals 27, 450-453 (2008) 33. Pinjai, S, Mukdasai, K: New delay-dependent robust exponential stability criteria of LPD neutral systems with mixed time-varying delays and nonlinear perturbations. J. Appl. Math. 2013, Article ID 268905 (2013) 34. Rojsiraphisal, T, Niamsup, P: Exponential stability of certain neutral differential equations. Appl. Math. Lett. 17, 3875-3880 (2010) 35. Sun, YG, Wang, L: Note on asymptotic stability of a class of neutral differential equations. Appl. Math. Lett. 19, 949-953 (2006) 36. Xiong, L, Zhong, S, Tian, J: New robust stability condition for uncertain neutral systems with discrete and distributed delays. Chaos Solitons Fractals 42(2), 1073-1079 (2009) 37. Zhang, XM, Wu, M, She, JH, He, Y: Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41, 1405-1412 (2005)