ESTIMATES FOR SOLUTIONS TO A CLASS OF NONLINEAR TIME-DELAY SYSTEMS OF NEUTRAL TYPE

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1 Electronic Journal of Differential Equations Vol ) No. 34 pp ISSN: URL: or ftp eje.math.txstate.eu ESTIMATES FOR SOLUTIONS TO A CLASS OF NONLINEAR TIME-DELAY SYSTEMS OF NEUTRAL TYPE GENNADII V. DEMIDENKO INESSA I. MATVEEVA Abstract. We consier nonlinear time-elay systems of neutral type with constant coefficients in the linear terms `yt) + Dyt τ) = Ayt) + Byt τ) + F t yt) yt τ)). t We obtain estimates characterizing the exponential ecay of solutions at infinity an epenening on the norms of the powers of D. 1. Introuction There is large number of works evote to the stuy of elay ifferential equations see for instance [ ]. The question of asymptotic stability is very important from the theoretical an practical viewpoints because elay ifferential equations arise in many applie problems when escribing the processes whose rates of change are efine by present an previous states; see [ ] an the bibliography therein. This article presents a continuation of our work on stability of solutions to elay ifferential equations [ ]. We consier the system of nonlinear elay ifferential equations yt) + Dyt τ) = Ayt) + Byt τ) + F t yt) yt τ)) t > 0 1.1) t where A B D are constant n n) matrices τ > 0 is the time elay an F t u v) is a real-value vector function satisfying the Lipschitz conition with respect to u an the inequality F t u v) q 1 u + q 2 v 1.2) for some constants q 1 q 2 0. When D 0 this system is calle one of neutral type [12]. Our aim is to obtain new estimates on the exponential ecay of solutions to 1.1) without fining roots of characteristic quasipolynomials efine by the linear part of 1.1) when F t u v) 0). In recent years the stuy in this irection has evelope rapily. For constant coefficients there are a lot of works for linear elay ifferential equations incluing equations of neutral type. It shoul be note 2000 Mathematics Subject Classification. 34K20. Key wors an phrases. Time-elay systems; neutral type; exponential stability; Lyapunov-Krasovskii functional. c 2015 Texas State University - San Marcos. Submitte November Publishe February

2 2 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 that various Lyapunov-Krasovskii functionals are use for obtaining exponential estimates see the bibliography in [16]). The case of nonlinear equations is of special interest an is more complicate in comparison with the case of linear equations. Along with estimates of exponential ecay of solutions a very important question is eriving estimates for attraction sets of nonlinear equations. The natural problem is to obtain such estimates by means of the Lyapunov-Krasovskii functionals use for exponential stability analysis of equations efine by the linear part. To the best of our knowlege the first constructive estimates of attraction sets for the system yt) = Ayt) + Byt τ) + F t yt) yt τ)) 1.3) t using Lyapunov-Krasovskii functionals associate with the exponentially stable linear system yt) = Ayt) + Byt τ) 1.4) t were obtaine in [ ]. To stuy the asymptotic stability of solutions to 1.4) the authors in [4] propose to use the Lyapunov-Krasovskii functional Hyt) yt) + t t τ where the real matrices H an Ks) satisfy Kt s)ys) ys) s 1.5) H = H > 0 Ks) = K s) C 1 [0 τ] Ks) > 0 s Ks) < 0 s [0 τ] 1.6) where H > 0 means that H is postive efinite. The usage of 1.5) allowe us to obtain estimates for the exponential ecay of solutions to the linear system 1.4). The authors in [4 5] consiere 1.3) with F t u v) q 1 u 1+ω1 + q 2 v 1+ω2 q 1 0 q 2 0 ω 1 0 ω 2 0. Using the functional in 1.5) conitions of asymptotic stability of the zero solution were obtaine estimates characterizing the ecay rate at infinity were establishe an estimates of attraction sets of the zero solution were erive. Using a generalization of the functional in 1.5) analogous results were obtaine for linear an nonlinear systems of elay ifferential equations with perioic coefficients in the linear terms see [ ]. To stuy the exponential stability of solutions to the systems of linear ifferential equations yt) + Dyt τ)) = Ayt) + Byt τ) 1.7) t the first author in [7] introuce the Lyapunov-Krasovskii functional V ϕ) = H ϕ0) + Dϕ τ) ) ϕ0) + Dϕ τ)) + 0 τ K s)ϕs) ϕs) s ϕs) C[ τ 0] 1.8) where the matrices H an Ks) satisfy 1.6). In particular the following result was obtaine.

3 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 3 Theorem 1.1. Suppose that there exist matrices H an Ks) satisfying 1.6) an that the matrix HA + A C = H + K0) HB + A HD B H + D HA D HB + B 1.9) HD Kτ) is positive efinite. Then the zero solution to 1.7) is exponentially stable. Using the functional in 1.8) the stuy of exponential stability of solutions to 1.1) was conucte in [ ]. There conitions for exponential stability of the zero solution estimates for the exponential ecay of solutions at infinity an estimates of attraction sets of the zero solution were obtaine. Note that in [7 8] the estimates of exponential ecay of solutions to 1.1) were obtaine when D < 1 here an thereafter we use the spectral norm of matrices). In [9] for the linear case F t u v) 0) analogous estimates were establishe when the spectrum of the matrix D belongs to the unit isk {λ C : λ < 1}. However in the case of D < 1 the estimates are weaker in comparison with the estimates obtaine in [7]. More precise exponential estimates for the linear systems were obtaine in [10 11]. Moreover in [11] the authors establishe estimates of exponential ecay of solutions of the linear time-elay systems of neutral type with perioic coefficients. In this article we consier the nonlinear time-elay system 1.1) when the spectrum of the matrix D belongs to the unit isk. Our aim is to obtain estimates characterizing exponential ecay of solutions at infinity epenening on the norms D j. 2. Estimates of solutions Consier the initial value problem for 1.1) yt) + Dyt τ)) = Ayt) + Byt τ) + F t yt) yt τ)) t > 0 t yt) = ϕt) t [ τ 0] y0+) = ϕ0) 2.1) where ϕt) C 1 [ τ 0] is a given vector function. Suppose that the conitions of Theorem 1.1 are satisfie. Using the matrices H an Ks) we introuce HA A S = H K0) HAD + K0)D HB D A H + D K0) B H Kτ) D 2.2) K0)D ) q = q 1 + q 21 + q 1 D + q 2 ) 2 H 2.3) [ R = HA A H K0) qi HAD + K0)D HB) Kτ) 1HAD 2.4) D K0)D qi] + K0)D HB) where I is the unit matrix. It is not har to verify that the matrix C in 1.9) is positive efinite if an only if the matrix S is positive efinite. Note that R is positive efinite if the matrix S qi is positive efinite.

4 4 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 Theorem 2.1. Let the conitions of Theorem 1.1 be satisfie. Suppose that the parameters q 1 q 2 are such that the matrix S qi is positive efinite. Let k > 0 be the maximal number such that Ks) + kks) 0 s [0 τ]. 2.5) s Let r min > 0 be the minimal eigenvalue of the matrix R. Then each solution to 2.1) satisfies yt) + Dyt τ) V ϕ) exp γt ) t > 0 2.6) h min 2 H where V ϕ) is efine by 1.8) h min > 0 is the minimal eigenvalue of the matrix H an γ = min{r min k H } > ) Proof. We follow the strategy in [4]. Let yt) be a solution to 2.1). Using the matrices H an Ks) inicate in Theorem 1.1 we consier the Lyapunov-Krasovskii functional efine in 1.8). Introucing the conventional notation we have y t : θ yt + θ) θ [ τ 0] V y t ) = Hy t 0) + Dy t τ)) y t 0) + Dy t τ)) + = Hyt) + Dyt τ)) yt) + Dyt τ)) + 0 τ t t τ K θ)y t θ) y t θ) θ The time erivative of this functional is t V y t) HAyt) + Byt τ)) yt) + Dyt τ)) + Hyt) + Dyt τ)) Ayt) + Byt τ)) + HF t yt) yt τ)) yt) + Dyt τ)) + Hyt) + Dyt τ)) F t yt) yt τ)) + K0)yt) yt) Kτ)yt τ) yt τ) + t t τ t Kt s)ys) ys) s. Using the matrix C efine in 1.9) we obtain yt) yt) t V y t) C yt τ) yt τ) + HF t yt) yt τ)) yt) + Dyt τ)) + Hyt) + Dyt τ)) F t yt) yt τ)) + t t τ t Kt s)ys) ys) s. Consier the first summan in the right-han sie of 2.8). Since yt) I D yt) + Dyt τ) = yt τ) 0 I yt τ) Kt s)ys) ys) s. 2.8)

5 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 5 it follows that yt) C yt τ) where yt) S yt τ) S = I 0 D C I yt) + Dyt τ) yt τ) I D S11 S = 12 0 I S12 S 22 yt) + Dyt τ) yt τ) which is efine in 2.2). Now we consier the secon an the thir summans in the right-han sie of 2.8). In view of 1.2) we have HF t yt) yt τ)) yt) + Dyt τ)) + Hyt) + Dyt τ)) F t yt) yt τ)) 2 H q 1 yt) + q 2 yt τ) ) yt) + Dyt τ) 2q 1 H yt) + Dyt τ) 2 + 2q 1 D + q 2 ) H yt τ) yt) + Dyt τ) q yt) + Dyt τ) 2 + yt τ) 2) where q is given in 2.3). Hence yt) yt) C + HF t yt) yt τ)) yt) + Dyt τ)) yt τ) yt τ) + Hyt) + Dyt τ)) F t yt) yt τ)) yt) + Dyt τ) S qi) yt τ) yt) + Dyt τ) yt τ) 2.9) By the conitions of Theorem 2.1 the matrix S qi is positive efinite. Using the representation ). I S12 S S qi = 22 qi) 1 S11 qi S 12 S 22 qi) 1 S I 0 S 22 qi ) I 0 S 22 qi) 1 S12 I we have S qi) yt) + Dyt τ) yt τ) yt) + Dyt τ) yt τ) [S 11 qi S 12 S 22 qi) 1 S 12]yt) + Dyt τ)) yt) + Dyt τ)). Since the matrix S qi is positive efinite the matrix R = S 11 qi S 12 S 22 qi) 1 S12 is positive efinite. Taking into account 2.2) the matrix R has the form 2.4). Consequently from 2.9) we obtain yt) yt) C yt τ) yt τ) + HF t yt) yt τ)) yt) + Dyt τ)) + Hyt) + Dyt τ)) F t yt) yt τ)) Ryt) + Dyt τ)) yt) + Dyt τ)) r min yt) + Dyt τ) )

6 6 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 where r min > 0 is the minimal eigenvalue of R. Using the matrix H we have yt) + Dyt τ) 2 1 Hyt) + Dyt τ)) yt) + Dyt τ)). H By 2.10) from 2.8) we erive Using 2.5) we have t V y t) r min Hyt) + Dyt τ)) yt) + Dyt τ)) H t + t Kt s)ys) ys) s. t τ t V y t) r min Hyt) + Dyt τ)) yt) + Dyt τ)) H k t t τ Kt s)ys) ys) s. Taking into account the efinition of the functional 1.8) we obtain t V y t) γ H V y t) where γ = min{r min k H } > 0. From this ifferential inequality we obtain the estimate V y t ) V ϕ) exp γt ). H Clearly yt) + Dyt τ) 2 1 h min Hyt) + Dyt τ)) yt) + Dyt τ)) where h min is the minimal eigenvalue of H. Then using the efinition of the functional in 1.8) we have V y t ) V ϕ) yt) + Dyt τ) exp γt ). h min h min 2 H The proof is complete. In the next theorem base on 2.6) we prove estimates for the solution to 2.1). These estismates will be use for proving our main results. We introuce the following values: V ϕ) α = β = γ h min 2 H Φ = max ϕs). 2.11) s [ τ0] Theorem 2.2. Let the conitions of Theorem 2.1 be satisfie. Then on each segment t [kτ k + 1)τ) k = the solution y to 2.1) satisfies yt) α D j e βt jτ) + D k+1 Φ 2.12) where α β an Φ are efine in 2.11).

7 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 7 Proof. Obviously taking into account 2.11) by 2.6) for t [0 τ) we have the inequality yt) αe βt + Dyt τ) αe βt + D Φ which gives us 2.12) for k = 0. Let t [kτ k + 1)τ) k = It is not har to show the sequence of the inequalities yt) αe βt + Dyt τ) αe βt + Dyt τ) + D 2 yt 2τ) + D 2 yt 2τ) + D 3 yt 3τ) D k yt kτ) + D k+1 yt k + 1)τ) + D k+1 yt k + 1)τ) αe βt + D yt τ) + Dyt 2τ) + D 2 yt 2τ) + Dyt 3τ) + + D k yt kτ) + Dyt k + 1)τ) + D k+1 yt k + 1)τ). By 2.6) we erive the estimate yt) αe βt + α D e βt τ) + α D 2 e βt 2τ) α D k e βt kτ) + D k+1 Φ which implies 2.12). The proof is complete. Next we obtain estimates for solutions to 2.1) on the whole half-line {t > 0}. Analogy as in [7] we istinguish three cases allowing us to obtain more precise estimates. Since the spectrum of the matrix D belongs to the unit isk {λ C : λ < 1} it follows that D j 0 as j. Let l > 0 be the minimal integer such that D l < 1. In Theorems below we establish estimates if respectively where β = Theorem 2.3. Assume that D l < e lβτ D l = e lβτ e lβτ < D l < 1 γ 2 H with γ efine in 2.7). D l < e lβτ. 2.13) Then the solution to the initial value problem 2.1) satisfies yt) [α 1 D l e lβτ ) 1 l 1 ] D j e jβτ + max{ D e βτ... D l e lβτ }Φ e βt for t > 0 where α β an Φ are efine in 2.11). 2.14) Proof. Using 2.12) on each segment t [kτ k + 1)τ) k = one can write the inequality yt) [ α ] D j e jβτ + D k+1 e k+1)βτ Φ e βt. In view of the conition on D l we obtain the estimate on the whole half-line {t > 0} [ yt) α D j e jβτ + max { D e βτ... D l e lβτ } ] Φ e βt. 2.15)

8 8 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 Consier the series Dj e jβτ. Obviously D j e jβτ = 2l 1 D j e jβτ + j=l 3l 1 D j e jβτ + j=2l D j e jβτ +... D j e jβτ + D l e lβτ D j e jβτ + D l e lβτ ) 2 D j e jβτ +... = 1 + D l e lβτ + D l e lβτ ) ) l 1 D j e jβτ. Since D l e lβτ < 1 by 2.13) we have D j e jβτ 1 D l e lβτ ) 1 l 1 D j e jβτ. Using this inequality from 2.15) we erive the require estimate 2.14). Theorem 2.4. Assume that D l = e lβτ. 2.16) Then the solution to the initial value problem 2.1) satisfies yt) [ α 1 + t lτ D j e jβτ + max{1 D e βτ... ) l 1 ] D l 1 e l 1)βτ }Φ e βt t > 0 where α β an Φ are efine in 2.11). 2.17) Proof. By Theorem 2.2 the solution to 2.1) satisfies 2.12) on each segment t [kτ k + 1)τ) k = Consequently yt) [ α ] D j e jβτ + D k+1 e k+1)βτ Φ e βt. Taking into account conition 2.16) on D l we obtain [ ] yt) α D j e jβτ + max{1 D e βτ... D l 1 e l 1)βτ }Φ e βt. 2.18) If k l 1 then 2.17) follows from 2.18) for t [0 lτ). Let l k 2l 1; i.e. 1 t lτ < 2. Consier the sum k Dj e jβτ. Clearly D j e jβτ = D j e jβτ + D j e jβτ j=l k l D j e jβτ + D l e lβτ D j e jβτ

9 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 9 Then we have k l = D j e jβτ + D j e jβτ. D j e jβτ D j e jβτ + t D j e jβτ. lτ By this inequality 2.17) follows from 2.18) for t [lτ 2lτ). Let ml k m + 1)l 1 m = ; i.e. m t lτ < m + 1. Consier the sum k Dj e jβτ. It is not ifficult to see that D j e jβτ 2l 1 = D j e jβτ + D j e jβτ + + j=l D j e jβτ j=ml k ml D j e jβτ + D l e lβτ D j e jβτ + + D ml e mlβτ D j e jβτ k ml D j e jβτ + D j e jβτ + + D j e jβτ 1 + m) D j e jβτ. Consequently D j e jβτ 1 + t lτ D j e jβτ. ) l 1 In view of this estimate 2.17) follows from 2.18) for t [mlτ m + 1)lτ). Owing to arbitrariness of m 2.17) is vali for all t > 0. Theorem 2.5. Assume that e lβτ < D l < ) Then the solution to the initial value problem 2.1) satisfies yt) [α D l e lβτ D l e lβτ 1) 1 D j e jβτ ] + D l 1 l 1 max{1 D... D l 1 }Φ exp t lτ ln Dl ) for t > 0 where α β an Φ are efine in 2.11). 2.20) Proof. In view of Theorem 2.2 a solution to 2.1) satisfies 2.12) on each segment t [kτ k + 1)τ) k =

10 10 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 At first we consier the first summan in the right-han sie of 2.12). k l 1 we obviously have D j e jβτ D j e jβτ. Let ml k m + 1)l 1 m = Clearly D j e jβτ 2l 1 = D j e jβτ + D j e jβτ + + j=l D j e jβτ j=ml k ml D j e jβτ + D l e lβτ D j e jβτ + + D ml e mlβτ D j e jβτ [ 1 + D l e lβτ + + D l m e Consequently D j e jβτ mlβτ ] l 1 D j e jβτ. D l m e mlβτ [1 + D l e lβτ ) D l e lβτ ) m ] D j e jβτ D l m e mlβτ [1 + D l e lβτ ) D l e lβτ ) m +... ] D j e jβτ. Since D l e lβτ > 1 owing to 2.19) D j e jβτ D l m e mlβτ [ 1 D l e lβτ ) 1] 1 l 1 D j e jβτ. Taking into account that mlτ t < m + 1)lτ we obtain D j e βt jτ) D l m e βt mlτ)[ 1 D l e lβτ ) l 1 1 ] 1 D j e jβτ D l t lτ [ 1 D l e lβτ ) l 1 1 ] 1 D j e jβτ. As result we erive the estimate for the first summan in 2.12) for every k α D j e βt jτ) α [ 1 D l e lβτ ) 1 ] l 1 1 ) D j e jβτ exp t lτ ln Dl ). For 2.21)

11 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 11 We now consier the secon summan in the right-han sie of 2.12). Obviously for 0 k l 2 we have D k+1 max{ D... D l 1 }. Let ml 1 k m + 1)l 2 m = Hence D k+1 D l m D k+1 ml D l m max { 1 D... D l 1 }. Since D l < 1 an t < m + 1)l 1)τ D l m D l t l 1)τ lτ Owing to arbitrariness of m we infer that = D l 1 l 1 exp t ) lτ ln Dl. D k+1 D l 1 l 1 max { 1 D... D l 1 } exp t lτ ln Dl ) for every k. Taking into account the estimate 2.21) for the first summan in the right-han sie of 2.12) we erive 2.20). We remark that the results obtaine above give us the assertions on robust stability for 1.7). Inee consier uncertain systems of the form yt) + Dyt τ)) = Ayt) + Byt τ) + At)yt) + Bt)yt τ) 2.22) t where At) an Bt) are unknown n n) matrices such that At) q 1 Bt) q 2. Obviously in this case the vector function F t u v) = At)u + Bt)v satisfies 1.2). Then Theorem 2.1 gives us the conitions of robust exponential stability for 1.7). From Theorems we have the estimates of exponential ecay of solutions to 2.22). Consier the system 1.1) where D = A = Illustrative examples 3 2 B = a a 0 a is a real parameter F t u v) is a real-value vector function satisfying the Lipschitz conition with respect to u an the inequality 1.2). First we consier the linear case F t u v) 0); i.e. q 1 = q 2 = 0. In [27] in the case of arbitrary positive τ stability was shown for a < 0.4. The same system was stuie in [21] where stability was establishe for a < In [3] exponential stability was shown for a Moreover in the case of a = an τ = 1 the following estimate for solutions was obtaine yt) c 1 y0) + c 2 sup ys) + c 3 sup )e s ys) t/2 1 s 0 1 s 0 with c j > 0. In the same case using our results we establish the following inequality yt) max 1 s 0 ys) e t/2 > )

12 12 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 Inee we choose the matrices H an Ks) as follows H = Ks) = e ks K k = K 0 = Obviously these matrices satisfy 1.6) an 2.5). Since the matrix C = is positive efinite then by Theorem 1.1 the zero solution to the system is exponentially stable. To establish 3.1) we nee to calculate for q = 0 the matrix R its minimal eigenvalue r min H an β = 1 rmin 2 min{ H k}. In our case R = r min = H = β = min{ } =. 2 2 Since D < e βτ by Theorem 2.3 we have 3.1). It shoul be note that using the same matrices H an K 0 it is not har to establish exponential stability in the case of arbitrary positive τ for 0.9 a It is enough to take for example k = 0.015/τ. Changing slightly H an K 0 the bounaries for a may be enlarge. We now consier the case of F t u v) 0. Let a = τ = 1 q 1 = 0.01 q 2 = As mentione above in [8] the authors establishe estimates of exponential ecay for solutions of systems of the form 1.1) in the case of D < 1. Using [8 Theorem 2] one can write own the inequality where β 1 = 1 2 min { yt) 1 max t 1 s 0 ys) e β1 1 > 0 3.2) c min 1 + D 2 ) H q1 + D q D 2 )q q2 2 )) 1 + D 2 ) } k c min is the minimal eigenvalue of C efine by 1.9). Choosing the same matrices H K 0 an k = 0.1 we have C = c min = β 1 = min{ } =. 2 2 At the same time by Theorem 2.3 we have the estimate yt) 2 max 1 s 0 ys) e β t 2 > 0 3.3) where β = /2. Inee in our case R = r min =

13 EJDE-2015/34 ESTIMATES FOR SOLUTIONS 13 Consequently β = min{ r min H k} = min{ } =. 2 Obviously 3.3) is more strong than 3.2) because β characterizing the exponential ecay rate of the solutions to 1.1) at infinity is larger than β 1. All the numerical computations were performe by using Scilab Acknowlegments. The authors are grateful to the anonymous referee for the helpful comments an suggestions. The authors were supporte by the Russian Founation for Basic Research project no ) an the Siberian Branch of the Russian Acaemy of Sciences interisciplinary project no. 80). References [1] R. P. Agarwal L. Berezansky E. Braverman A. Domoshnitsky; Nonoscillation Theory of Functional Differential Equations with Applications Springer New York [2] N. V. Azbelev V. P. Maksimov L. F. Rakhmatullina; Introuction to the Theory of Functional Differential Equations: Methos an Applications Contemporary Mathematics an Its Applications 3 Hinawi Publishing Corporation Cairo [3] J. Baštinec J. Diblík D. Ya. Khusainov A. Ryvolová; Exponential stability an estimation of solutions of linear ifferential systems of neutral type with constant coefficients Boun. Value Probl Art. ID pp. [4] G. V. Demienko I. I. Matveeva; Asymptotic properties of solutions to elay ifferential equations Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Inform ) Russian). [5] G. V. Demienko I. I. Matveeva; Matrix process moelling: Asymptotic properties of solutions of ifferential-ifference equations with perioic coefficients in linear terms Proceeings of the Fifth International Conference on Bioinformatics of Genome Regulation an Structure Novosibirsk Russia July ). Novosibirsk: Institute of Cytology an Genetics ) [6] G. V. Demienko I. I. Matveeva; Stability of solutions to elay ifferential equations with perioic coefficients of linear terms Siberian Math. J ) [7] G. V. Demienko; Stability of solutions to linear ifferential equations of neutral type J. Anal. Appl ) [8] G. V. Demienko T. V. Kotova M. A. Skvortsova; Stability of solutions to ifferential equations of neutral type Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Inform ) Russian); English transl. in: J. Math. Sci ) [9] G. V. Demienko E. S. Voop yanov M. A. Skvortsova; Estimates of solutions to the linear ifferential equations of neutral type with several elays of the argument J. Appl. Inust. Math ) [10] G. V. Demienko I. I. Matveeva; On exponential stability of solutions to one class of systems of ifferential equations of neutral type J. Appl. Inust. Math ) [11] G. V. Demienko I. I. Matveeva; On estimates of solutions to systems of ifferential equations of neutral type with perioic coefficients Siberian Math. J ) [12] L. E. El sgol ts S. B. Norkin; Introuction to the Theory an Application of Differential Equations with Deviating Arguments Acaemic Press New York Lonon [13] T. Erneux; Applie Delay Differential Equations Surveys an Tutorials in the Applie Mathematical Sciences 3 Springer New York [14] K. Gu V. L. Kharitonov J. Chen; Stability of Time-Delay Systems Control Engineering Boston Birkhäuser [15] J. K. Hale; Theory of Functional Differential Equations Springer-Verlag New York Heielberg Berlin [16] V. L. Kharitonov; Time-Delay Systems. Lyapunov Functionals an Matrices Control Engineering Birkhauser Springer New York [17] D. Ya. Khusainov A. V. Shatyrko; The Metho of Lyapunov Functions in the Investigation of the Stability of Functional Differential Systems Kiev University Press Kiev 1997 Russian).

14 14 G. V. DEMIDENKO I. I. MATVEEVA EJDE-2015/34 [18] V. B. Kolmanovskii A. D. Myshkis; Introuction to the Theory an Applications of Functional Differential Equations Mathematics an its Applications 463 Kluwer Acaemic Publishers Dorrecht [19] D. G. Korenevskii; Stability of Dynamical Systems uner Ranom Perturbations of Parameters. Algebraic Criteria Naukova Dumka Kiev [20] Y. Kuang; Delay Differential Equations with Applications in Population Dynamics Mathematics in Science an Engineering 191 Acaemic Press Boston [21] X.-X. Liu B. Xu; A further note on stability criterion of linear neutral elay-ifferential systems J. Franklin Inst ) [22] N. MacDonal; Biological Delay Systems: Linear Stability Theory Cambrige Stuies in Mathematical Biology 8 Cambrige University Press Cambrige [23] I. I. Matveeva A. A. Shcheglova; Some estimates of the solutions to time-elay ifferential equations with parameters J. Appl. Inust. Math ) [24] I. I. Matveeva; Estimates of solutions to a class of systems of nonlinear elay ifferential equations J. Appl. Inust. Math ) [25] D. Melchor-Aguilar S.-I. Niculescu; Estimates of the attraction region for a class of nonlinear time-elay systems IMA J. Math. Control Inform ) [26] W. Michiels S.-I. Niculescu; Stability an Stabilization of Time-Delay Systems. An Eigenvalue-Base Approach Avances in Design an Control 12 Philaelphia Society for Inustrial an Applie Mathematics [27] Ju.-H. Park S. Won; A note on stability of neutral elay-ifferential systems J. Franklin Inst ) Gennaii V. Demienko Laboratory of Differential an Difference Equations Sobolev Institute of Mathematics 4 Aca. Koptyug avenue Novosibirsk Russia. Department of Mechanics an Mathematics Novosibirsk State University 2 Pirogov street Novosibirsk Russia aress: emienk@math.nsc.ru Inessa I. Matveeva Laboratory of Differential an Difference Equations Sobolev Institute of Mathematics 4 Aca. Koptyug avenue Novosibirsk Russia. Department of Mechanics an Mathematics Novosibirsk State University 2 Pirogov street Novosibirsk Russia aress: matveeva@math.nsc.ru