EXPONENTIAL STABILITY OF SOLUTIONS TO NONLINEAR TIME-DELAY SYSTEMS OF NEUTRAL TYPE GENNADII V. DEMIDENKO, INESSA I. MATVEEVA

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1 Electronic Journal of Differential Equations, Vol (2016), No. 19, pp ISSN: URL: or ftp eje.math.txstate.eu EXPONENTIAL STABILITY OF SOLUTIONS TO NONLINEAR TIME-DELAY SYSTEMS OF NEUTRAL TYPE GENNADII V. DEMIDENKO, INESSA I. MATVEEVA Abstract. We consier a nonlinear time-elay system of neutral equations with constant coefficients in the linear terms `y(t) + Dy(t τ) = Ay(t) + By(t τ) + F (t, y(t), y(t τ)), t where F (t, u, v) q 1 u 1+ω 1 + q 2 v 1+ω 2, q 1, q 2, ω 1, ω 2 > 0. We obtain estimates characterizing the exponential ecay of solutions at infinity an estimates for attraction sets of the zero solution. 1. Introuction There is large number of works evote to the stuy of elay ifferential equations (see for instance the books 1, 3, 13, 15, 16, 17, 18, 20, 21, 22, 23, 29, 31] an the bibliography therein). The question of asymptotic stability of solutions is very important from the theoretical an practical viewpoints because elay ifferential equations arise in many applie problems when escribing the processes whose spees are efine by present an previous states (see for example 14, 24, 25] an the bibliography therein). This article presents a continuation of our works on stability of solutions to elay ifferential equations 4, 5, 6, 7, 8, 9, 10, 11, 12, 26, 27]. We consier the system of nonlinear elay ifferential equations ( ) y(t) + Dy(t τ) = Ay(t) + By(t τ) + F (t, y(t), y(t τ)), t > 0, (1.1) t where A, B, D are constant (n n) matrices, τ > 0 is the time elay, an F is a continuous vector function mapping 0, ) C n C n into C n. We assume that F (t, u, v) satisfies the Lipschitz conition with respect to u on every compact G 0, ) C n C n an the inequality F (t, u, v) q 1 u 1+ω1 + q 2 v 1+ω2, t 0, u, v C n, (1.2) 2010 Mathematics Subject Classification. 34K20. Key wors an phrases. Time-elay systems; neutral equation; exponential stability; attraction sets. c 2016 Texas State University. Submitte October 6, Publishe January 11,

2 2 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 for some constants q 1, q 2, ω 1, ω 2 > 0. Here an hereafter we use the following ot prouct an vector norm n x, z = x j z j, x = x, x. j=1 Our aim is to stuy the exponential stability of the zero solution; namely, to obtain estimates characterizing the ecay rate of solutions at infinity an estimates for attraction sets of the zero solution. To establish conitions of stability, researchers often use various Lyapunov or Lyapunov Krasovskii functionals. At present, there is large number of works in this irection; for example, see the bibliographies in the survey 2] an in the book 31] evote wholly to obtaining conitions of stability by the use of Lyapunov-Krasovskii functionals. However, not every Lyapunov- Krasovskii functional makes it possible to obtain estimates characterizing exponential ecay of solutions at infinity. 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 (for example, see 15, 20] an the bibliography therein). 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 of attraction sets for 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 y(t) = Ay(t) + By(t τ) + F (t, y(t), y(t τ)), (1.3) t using Lyapunov Krasovskii functionals associate with the exponentially stable linear system y(t) = Ay(t) + By(t τ), (1.4) t were obtaine in 4, 5, 6, 28]. To stuy asymptotic stability of solutions to (1.4) the authors in 4] propose to use the Lyapunov Krasovskii functional Hy(t), y(t) + where the matrices H an K(s) satisfy K(t s)y(s), y(s) s, (1.5) H = H > 0, K(s) = K (s) C 1 0, τ], K(s) > 0, K(s) < 0, (1.6) s for s 0, τ]. Here H > 0 means that H is positive efinite. Using (1.5), we obtaine estimates of exponential ecay of solutions to linear systems of the form (1.4). In 4, 5] the authors consiere nonlinear systems of elay ifferential equations of the form (1.3) with F (t, u, v) satisfying (1.2). 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 4, 5, 6, 11, 26, 27].

3 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 3 To stuy exponential stability of solutions to the system of linear ifferential equations of neutral type (y(t) + Dy(t τ)) = Ay(t) + By(t τ) (1.7) t the first author in 7] introuce the Lyapunov-Krasovskii functional V (ϕ) = H(ϕ(0) + Dϕ( τ)), (ϕ(0) + Dϕ( τ)) + 0 τ K( s)ϕ(s), ϕ(s) s, ϕ(s) C τ, 0], (1.8) where the matrices H an K(s) satisfy (1.6). In particular, the following result was obtaine. Theorem 1.1. Suppose that there exist matrices H an K(s) satisfying (1.6) an the matrix ( ) HA + A C = H + K(0) HB + A HD B H + D HA D HB + B (1.9) HD K(τ) is positive efinite. Then the zero solution to (1.7) is exponentially stable. Using the functional (1.8), the stuy of exponential stability of solutions to timeelay systems of the form (1.1) was conucte in 7, 8, 9, 10, 12, 30]. Note that in 7, 8, 30] the estimates of exponential ecay of solutions to (1.1) were obtaine in the case D < 1 (here an hereafter 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 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]. If the spectrum of the matrix D belongs to the unit isk, the authors in 12] investigate the time-elay system (1.1) with F (t, u, v) satisfying (1.2) with ω 1 = ω 2 = 0. In this article we stuy a more complicate case; namely, we consier the nonlinear time-elay system (1.1) if ω 1, ω 2 > 0. Supposing that the spectrum of D belongs to the unit isk, we establish estimates characterizing exponential ecay of solutions at infinity an estimates for attraction sets of the zero solution. The main results are formulate in Theorems an their proofs are given in the next section. It shoul be note that some sufficient conitions for exponential stability of the zero solution to (1.1) are establishe in 15, Theorem 7.5] in the case D < Main results Suppose that the conitions of Theorem 1.1 are satisfie. Using the matrices H an K(s), introuce the following notation ( ) S11 S S = 12 S12, (2.1) S 22 S 11 = HA A H K(0), S 22 = K(τ) D K(0)D, S 12 = HAD + K(0)D HB, R = S 11 S 12 S 1 22 S 12, P = S 22. (2.2)

4 4 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 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 (see the proof of Theorem 2.1 for etails). Obviously, R an P are positive efinite if an only if the matrix S is positive efinite. Denote by r min > 0 an p min > 0 the minimal eigenvalues of R an P, respectively. Let κ > 0 be the maximal number such that We consier the initial value problem for (1.1), K(s) + κk(s) 0, s 0, τ]. (2.3) s (y(t) + Dy(t τ)) = Ay(t) + By(t τ) + F (t, y(t), y(t τ)), t > 0, t y(t) = ϕ(t), t τ, 0], y(+0) = ϕ(0), (2.4) where ϕ(t) C 1 τ, 0] is a given vector function. Let y(t) be a noncontinuable solution to the initial value problem (2.4), efine for t 0, t ). Using the matrices H an K(s) inicate in Theorem 1.1, we consier the Lyapunov-Krasovskii functional (1.8). Introucing the conventional notation we have y t : θ y(t + θ), θ τ, 0], V (y t ) = H(y t (0) + Dy t ( τ)), (y t (0) + Dy t ( τ)) + = H(y(t) + Dy(t τ)), (y(t) + Dy(t τ)) + K(t s)y(s), y(s) s. 0 τ K( θ)y t (θ), y t (θ) θ (2.5) Theorem 2.1. Let the conitions of Theorem 1.1 be satisfie. Then t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2 κ for t 0, t ), where K(t s)y(s), y(s) s, (2.6) ε 0 = 2q 1 H (1 + ε 1 ) ω1, (2.7) h 1+ω1/2 min ( δ 1 (s) = S22 1 S ) δ 0 (s), δ 2 (s) = ε 2 δ 0 (s), s 0, (2.8) 4ε 2 ( 1 + ε1 ) ω1 ] δ 0 (s) = 2 H q 1 D 1+ω 1 s ω1 + q 2 s ω2, ε 1 ε 1 > 0, ε 2 > 0, z(t) = S 1 22 S 12(y(t) + Dy(t τ)) + y(t τ), (2.9) an h min > 0 is the minimal eigenvalue of the matrix H.

5 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 5 Proof. We use the proof scheme in 4]. Obviously, the time erivative of the functional V (y t ) is t V (y t) H(Ay(t) + By(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), (Ay(t) + By(t τ)) + HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) + K(0)y(t), y(t) K(τ)y(t τ), y(t τ) + K(t s)y(s), y(s) s. t Using the matrix C efine in (1.9), we obtain t V (y t) ( ) ( ) y(t) y(t) C, y(t τ) y(t τ) + HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) + K(t s)y(s), y(s) s. t We consier the first summan in the right-han sie of (2.10). Obviously, ( ) ( ) ( ) y(t) I D y(t) + Dy(t τ) =. y(t τ) 0 I y(t τ) Then ( ) ( y(t) y(t) C, y(t τ) y(t τ) where ) ( y(t) + Dy(t τ) S y(t τ) S = ( ) I 0 D C I ), ( ) I D. 0 I (2.10) ( ) y(t) + Dy(t τ), y(t τ) Taking into account (1.9), the matrix S has the form (2.1). By the conitions of Theorem 1.1, the matrix C is positive efinite. Clearly, S is positive efinite if an only if C is positive efinite. Using the representation ( I S12 S 1 ) ( S = 22 S11 S 12 S 1 ) ( ) 22 S 12 0 I 0 0 I 0 S 22 S22 1, S 12 I we have ( ) ( ) y(t) + Dy(t τ) y(t) + Dy(t τ) S, y(t τ) y(t τ) = R(y(t) + Dy(t τ)), (y(t) + Dy(t τ)) + P z(t), z(t), where R, P an z(t) are efine by (2.2) an (2.9), respectively. Obviously, the matrix S is positive efinite if an only if the matrices R an P are positive efinite. Consequently, we erive C ( y(t) y(t τ) ), ( y(t) y(t τ) ) rmin y(t) + Dy(t τ) 2 + p min z(t) 2, (2.11) where r min > 0 an p min > 0 are the minimal eigenvalues of R an P, respectively.

6 6 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 We consier the secon an the thir summans in the right-han sie of (2.10). In view of (1.2) we have HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) 2 H ( q 1 y(t) 1+ω1 + q 2 y(t τ) 1+ω2) y(t) + Dy(t τ) 2 H ( q 1 ( y(t) + Dy(t τ) + D y(t τ) ) 1+ω1 + q 2 y(t τ) 1+ω2) y(t) + Dy(t τ). It is not har to show that Hence, (a + b) 1+ω (1 + ε 1 ) ω a 1+ω + ( 1 + ε 1 ε 1 ) ωb 1+ω, a, b 0, ε 1 > 0. ( y(t) + Dy(t τ) + D y(t τ) ) 1+ω1 (1 + ε 1 ) ω1 y(t) + Dy(t τ) 1+ω1 + ( 1 + ε 1 ε 1 ) ω1 D 1+ω 1 y(t τ) 1+ω1. For example, choosing ε 1 = D, we have ( y(t) + Dy(t τ) + D y(t τ) ) 1+ω1 (1 + D ) ω1 y(t) + Dy(t τ) 1+ω1 + (1 + D ) ω1 D y(t τ) 1+ω1. Consequently, HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) 2q 1 H (1 + ε 1 ) ω1 y(t) + Dy(t τ) 2+ω1 ( 1 + ε1 ) ω1 ] + 2 H q 1 D 1+ω 1 y(t τ) ω1 + q 2 y(t τ) ω2 ε 1 y(t τ) y(t) + Dy(t τ). By the efinition of z(t), Hence, y(t τ) S 1 22 S 12 y(t) + Dy(t τ) + z(t). y(t τ) y(t) + Dy(t τ) S22 1 S 12 y(t) + Dy(t τ) 2 + z(t) y(t) + Dy(t τ) ( S22 1 S ) y(t) + Dy(t τ) 2 + ε 2 z(t) 2, ε 2 > 0. 4ε 2 Then we obtain HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) 2q 1 H (1 + ε 1 ) ω1 y(t) + Dy(t τ) 2+ω1 + δ 1 ( y(t τ) ) y(t) + Dy(t τ) 2 + δ 2 ( y(t τ) ) z(t) 2, where δ 1 (s), δ 2 (s) are efine by (2.8). (2.12)

7 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 7 By (2.11) an (2.12), we have ( ) ( ) y(t) y(t) C, y(t τ) y(t τ) + HF (t, y(t), y(t τ)), (y(t) + Dy(t τ)) + H(y(t) + Dy(t τ)), F (t, y(t), y(t τ)) 2q 1 H (1 + ε 1 ) ω1 y(t) + Dy(t τ) 2+ω1 (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2. It follows from (2.10) that By (2.3), we obtain t V (y t) 2q 1 H (1 + ε 1 ) ω1 y(t) + Dy(t τ) 2+ω1 (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2 + t K(t s)y(s), y(s) s. t V (y t) 2q 1 H (1 + ε 1 ) ω1 y(t) + Dy(t τ) 2+ω1 Using the matrix H, we have (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2 κ 1 H(y(t) + Dy(t τ)), (y(t) + Dy(t τ)) H y(t) + Dy(t τ) 2 1 h min H(y(t) + Dy(t τ)), (y(t) + Dy(t τ)), where h min > 0 is the minimal eigenvalue of H. Hence, K(t s)y(s), y(s) s. (2.13) t V (y t) 2q 1 H (1 + ε 1 ) ω1 1+ω1/2 H(y(t) + Dy(t τ)), (y(t) + Dy(t τ)) h 1+ω1/2 min (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2 κ By the efinition of V (y t ), we obtain K(t s)y(s), y(s) s. t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 ( y(t τ) )) y(t) + Dy(t τ) 2 (p min δ 2 ( y(t τ) )) z(t) 2 κ where ε 0 is efine by (2.7). The proof is complete. K(t s)y(s), y(s) s,

8 8 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 The parameters ε 1, ε 2 > 0 allow us to control ε 0, δ 1 (s), δ 2 (s) in (2.6). Let σ > 0 be a number such that { r min δ 0 (σ) < min ( S 1 22 S 12 + ), p } min 1. ε 4ε 2 2 Then r min δ 1 (σ) > 0 an p min δ 2 (σ) > 0. We introuce the notation γ = min{r min δ 1 (σ), κ H } > 0, β = γ 2 H. (2.14) 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. We istinguish three cases D l < e lβτ, D l = e lβτ, e lβτ < D l < 1. We escribe in every case an attraction set for the initial function ϕ(t) an show that the solution to the initial value problem (2.4) with this function is efine for t > 0. We establish estimates characterizing exponentially ecay of this solution at infinity. Theorem 2.2. Let the conitions of Theorem 1.1 be satisfie an D l < e lβτ. (2.15) Suppose that ϕ(t) E 1, where { ( E 1 = ϕ(s) C 1 γ ) 2/ω1, τ, 0] : Φ < σ, V (ϕ) < ε 0 H ( 1 D l e lβτ ) 1 } + max{ D,..., D l }Φ < σ, (2.16) = 1 ε 0 H V ω1/2 (ϕ) ] 1/ω 1 V (ϕ), Φ = max ϕ(s). (2.17) γ h min s τ,0] Then the solution to the initial value problem (2.4) is efine for t > 0 an y(t) ( 1 D l e lβτ ) 1 + max { D e βτ,..., D l e lβτ } ] Φ e βt, t > 0. The proof of the above theorem is base on the next two lemmas. Lemma 2.3. Let Then D l < e lβτ. D j e β(t jτ) + D k+1 Φ ( 1 D l e lβτ ) 1 + max { D e βτ,..., D l e lβτ } ] Φ e βt (2.18) (2.19) for t kτ, (k + 1)τ), k = 0, 1,..., where, β, an Φ are efine in (2.14), (2.17).

9 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 9 Proof. Obviously, D j e β(t jτ) + D k+1 Φ ] + D k+1 e (k+1)βτ Φ e βt, In view of the conition on D l, we obtain the estimate D j e β(t jτ) + D k+1 Φ t kτ, (k + 1)τ). + max { D e βτ,..., D l e lβτ } ] Φ e βt. We consier the series Dj e jβτ. Obviously, = 2 + j=l 3 + j=2l D l e lβτ + ( D l e lβτ ) = (1 + D l e lβτ + ( D l e lβτ ) ) By (2.15), we have ( 1 D l e lβτ ) 1. Using this inequality, we erive (2.19) from (2.20). The proof is complete. (2.20) Lemma 2.4. Let the conitions of Theorem 2.2 be satisfie. Then the solution to the initial value problem (2.4) is efine for t > 0; moreover, on each segment t kτ, (k + 1)τ), k = 0, 1,..., it satisfies the estimate y(t) D j e β(t jτ) + D k+1 Φ, (2.21) where, β, an Φ are efine in (2.14), (2.17). Proof. Let t 0, τ). If the initial function ϕ(t) belongs to the attraction set E 1 efine by (2.16), then max s τ,0] ϕ(s) < σ. Consequently, r min δ 1 ( y(t τ) ) = r min δ 1 ( ϕ(t τ) ) > r min δ 1 (σ) > 0,

10 10 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 p min δ 2 ( y(t τ) ) = p min δ 2 ( ϕ(t τ) ) > p min δ 2 (σ) > 0. By Theorem 2.1, t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 (σ)) y(t) + Dy(t τ) 2 κ K(t s)y(s), y(s) s for t 0, t 1 ), where t 1 = min{τ, t }. Using (2.13), we have t V (y t) ε 0 V 1+ω1/2 (y t ) r min δ 1 (σ) H(y(t) + Dy(t τ)), (y(t) + Dy(t τ)) H κ K(t s)y(s), y(s) s. Taking into account (2.5), we obtain t V (y t) ε 0 V 1+ω1/2 (y t ) γ H V (y t), where γ is efine in (2.14). If ϕ(t) E 1 then ε 0 H V ω1/2 (ϕ) < 1. γ By a Gronwall-like inequality (for example, see 19]), we have the estimate V (y t ) 1 ε 0 H V ω1/2 (ϕ) ] 2/ω 1V ( γt ) (ϕ) exp. γ H Using the efinition of the functional (2.5), we obtain V (y t ) y(t) + Dy(t τ). h min Consequently, Hence, y(t) + Dy(t τ) 1 ε 0 H γ y(t) y(t) + Dy(t τ) + Dy(t τ) ] V ω1/2 1/ω1 V (ϕ) ( (ϕ) exp h min e βt + D ϕ(t τ) e βt + D Φ, t 0, t 1 ), γt ). 2 H (2.22) where, β, an Φ are efine in (2.14) an (2.17). The function in the righthan sie of (2.22) is continuous an boune for t > 0. Then t 1 = τ an the noncontinuable solution y(t) to (2.4) is efine for t 0, τ]. Consequently, t > τ. It follows from (2.22) that y(t) satisfies (2.21) for k = 0. Obviously, if the initial function ϕ(t) belongs to the attraction set E 1 then Let t τ, 2τ). Consequently, y(t) < σ, t 0, τ]. r min δ 1 ( y(t τ) ) > r min δ 1 (σ) > 0,

11 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 11 By Theorem 2.1, p min δ 2 ( y(t τ) ) > p min δ 2 (σ) > 0. t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 (σ)) y(t) + Dy(t τ) 2 κ K(t s)y(s), y(s) s for t 0, t 2 ), where t 2 = min{2τ, t }. Repeating the same reasoning as above, we have y(t) + Dy(t τ) e βt, t 0, t 2 ), where an β are efine in (2.17) an (2.14), respectively. Hence, y(t) y(t) + Dy(t τ) + Dy(t τ) e βt + Dy(t τ) D 2 y(t 2τ) + D 2 y(t 2τ) e βt + D e β() + D 2 Φ, t τ, t 2 ). (2.23) The function in the right-han sie of (2.23) is continuous an boune for t > 0. Then t 2 = 2τ an the noncontinuable solution y(t) to (2.4) is efine for t 0, 2τ]. Consequently, t > 2τ. It follows from (2.23) that y(t) satisfies (2.21) for k = 1. By Lemma 2.3, e βt + D e β() + D 2 Φ ( 1 D l e lβτ ) 1 ] e βt + D 2 Φ, t τ, 2τ]. Consequently, if the initial function ϕ(t) belongs to the attraction set E 1, then y(t) < σ, t 0, 2τ]. Repeating the same reasoning, we obtain that the solution to (2.4) is efine for t > 0, an it satisfies (2.21) on each segment t kτ, (k + 1)τ), k N. By Lemma 2.3 an the conition D l < 1, we have y(t) ( 1 D l e lβτ ) 1 ] e βt + max{ D,..., D l }Φ, t > 0. Consequently, if the initial function ϕ(t) belongs to the attraction set E 1, then y(t) < σ, t > 0. The proof is complete. By Lemmas 2.3 an 2.4, we obtain that the solution to (2.4) satisfies (2.18). Therefore, Theorem 2.2 is prove. Theorem 2.5. Let the conitions of Theorem 1.1 be satisfie an D l = e lβτ. (2.24)

12 12 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 Suppose that ϕ(t) E 2, where { E 2 = ϕ(s) C 1 τ, 0] : Φ < σ, V (ϕ) < ( γ ) 2/ω1, ε 0 H } lτβ elτβ 1 + max{ D,..., D l }Φ < σ. (2.25) Then the solution to the initial value problem (2.4) is efine for t > 0 an the estimate hols y(t) ( 1 + t ) lτ { } ] + max 1, D e βτ,..., D e ()βτ Φ e βt, t > 0, where, β, an Φ are efine in (2.14), (2.17). The proof of the above theorem is base on the next two lemmas. (2.26) Lemma 2.6. Let Then D l = e lβτ. D j e β(t jτ) + D k+1 Φ ( 1 + t ) lτ { } ] + max 1, D e βτ,..., D e ()βτ Φ e βt (2.27) for t kτ, (k + 1)τ), k = 0, 1,.... where, β, an Φ are efine in (2.14), (2.17). Proof. Obviously, D j e β(t jτ) + D k+1 Φ ] + D k+1 e (k+1)βτ Φ e βt, t kτ, (k + 1)τ). In view of the conition (2.24) on D l, we obtain the estimate D j e β(t jτ) + D k+1 Φ { } ] + max 1, D e βτ,..., D e ()βτ Φ e βt. (2.28) If k l 1 then (2.27) follows from (2.28).

13 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 13 Let l k 2l 1; i.e., 1 Clearly, Then we have = + t lτ < 2. We consier the sum k Dj e jβτ. j=l k l + D l e lβτ k l = +. + t. lτ Using this inequality, (2.27) follows from (2.28). Let ml k (m + 1)l 1, m = 2, 3,... ; i.e., m t lτ sum k Dj e jβτ. It is not ifficult to see that 2 = j=l j=ml < m + 1. We consier the k ml + D l e lβτ + + D ml e mlβτ k ml (1 + m). Consequently, ( 1 + t lτ. ) In view of this estimate, (2.27) follows from (2.28). Owing to the arbitrariness of m, the proof is complete. Lemma 2.7. Let the conitions of Theorem 2.5 be satisfie. Then the solution to the initial value problem (2.4) is efine for t > 0; moreover, it satisfies (2.21) on each segment t kτ, (k + 1)τ), k = 0, 1,.... Proof. Recall that the noncontinuable solution to (2.4) is efine for t 0, t ). Let t 0, τ). Repeating the same reasoning as in the proof of Lemma 2.4, we obtain that y(t) satisfies (2.22) an is efine for t 0, τ]. Consequently, t > τ. It follows

14 14 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 from (2.22) that y(t) satisfies (2.21) for k = 0. Obviously, if the initial function ϕ(t) belongs to the attraction set E 2 efine by (2.25), then Let t τ, 2τ). Consequently, By Theorem 2.1, y(t) < σ, t 0, τ]. r min δ 1 ( y(t τ) ) > r min δ 1 (σ) > 0, p min δ 2 ( y(t τ) ) > p min δ 2 (σ) > 0. t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 (σ)) y(t) + Dy(t τ) 2 κ K(t s)y(s), y(s) s for t 0, t 2 ), where t 2 = min{2τ, t }. By a similar way as in the proof of Lemma 2.4, we have y(t) + Dy(t τ) e βt, t 0, t 2 ), where an β are efine in (2.17) an (2.14), respectively. Hence, y(t) y(t) + Dy(t τ) + Dy(t τ) e βt + Dy(t τ) D 2 y(t 2τ) + D 2 y(t 2τ) e βt + D e β() + D 2 Φ, t τ, t 2 ). (2.29) The function in the right-han sie of (2.29) is continuous an boune for t > 0. Then t 2 = 2τ an the noncontinuable solution y(t) to (2.4) is efine for t 0, 2τ]. Consequently, t > 2τ. It follows from (2.29) that y(t) satisfies (2.21) for k = 1. By Lemma 2.6, e βt + D e β() + D 2 Φ ( 1 + t ) ] e βt + D 2 Φ, lτ We consier the function f(t) = ( 1 + t ) e βt, t 0. lτ It is not ifficult to show that { 1 lτβ f(t) elτβ 1, lτβ 1, 1, lτβ 1. Obviously, 1 lτβ elτβ 1 1 for τ, β > 0. Then f(t) 1 lτβ elτβ 1, t 0, t τ, 2τ]. for any τ, β > 0. Taking into account the last inequality, from (2.30) we have e βt + D e β() + D 2 Φ lτβ elτβ 1 + D 2 Φ, t τ, 2τ]. (2.30)

15 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 15 Consequently, if the initial function ϕ(t) belongs to the attraction set E 2 then y(t) < σ, t 0, 2τ]. Repeating the same reasoning, we obtain that the solution to (2.4) is efine for t > 0, an it satisfies (2.21) on each segment t kτ, (k + 1)τ), k N. By Lemma 2.6 an the conition D l < 1, we have y(t) ( 1 + t lτ ] e βt + max{ D,..., D l }Φ ) lτβ elτβ 1 + max{ D,..., D l }Φ, t > 0. Consequently, if the initial function ϕ(t) belongs to the attraction set E 2 then The proof is complete. y(t) < σ, t > 0. By Lemmas 2.6 an 2.7, we obtain that the solution to (2.4) satisfies (2.26). Therefore, Theorem 2.5 is prove. Theorem 2.8. Let the conitions of Theorem 1.1 be satisfie an e lβτ < D l < 1. (2.31) Suppose that ϕ(t) E 3, where { E 3 = ϕ(s) C 1 τ, 0] : Φ < σ, V (ϕ) < ( γ ) 2/ω1, ε 0 H (1 ( D l e lβτ ) ) 1 1 } + max{ D,..., D l }Φ < σ. Then the solution to the initial value problem (2.4) is efine for t > 0 an y(t) (1 ( D l e lβτ ) ) D l 1 l 1 max { 1, D,..., D } ] ( t ) Φ exp lτ ln Dl, for t > 0, where, β, an Φ are efine in (2.14), (2.17). The proof of the above theorem is base on the next two lemmas. Lemma 2.9. Let Then D j e β(t jτ) + D k+1 Φ e lβτ < D l < 1. (1 ( D l e lβτ ) ) D l 1 l 1 max { 1, D,..., D } ] ( t ) Φ exp lτ ln Dl (2.32) (2.33) (2.34)

16 16 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 for t kτ, (k + 1)τ), k = 0, 1,.... where, β, an Φ are efine in (2.14), (2.17). Proof. First we consier the first summan in the left-han sie of (2.34). k l 1 we obviously have For. Let ml k (m + 1)l 1, m = 1, 2, 3,.... Clearly, 2 = j=l j=ml k ml + D l e lβτ + + D ml e mlβτ 1 + D l e lβτ + + D l m e Consequently, mlβτ ]. D l m e mlβτ 1 + ( D l e lβτ ) 1 ( + + D l e lβτ ) ] m D l m e mlβτ 1 + ( D l e lβτ ) 1 ( + + D l e lβτ ) ] m Since D l e lβτ > 1 owing to (2.31), we have D l m e mlβτ 1 ( D l e lβτ ) ] 1 1. Taking into account that mlτ t < (m + 1)lτ, we obtain D j e β(t jτ) D l m e β(t mlτ) 1 ( D l e lβτ ) ] 1 1 D l t lτ 1 ( D l e lβτ ) ] 1 1.

17 EJDE-2016/19 EXPONENTIAL STABILITY OF SOLUTIONS 17 As a result, we erive the estimate for the first summan in (2.34) for every k, D j e β(t jτ) 1 ( D l e lβτ ) ] 1 1( ) exp ( t lτ ln Dl ). (2.35) We now we consier the secon summan in the left-han sie of (2.34). Obviously, for 0 k l 2 we have D k+1 max { D,..., D }. Let ml 1 k (m + 1)l 2, m = 1, 2,.... Hence, D k+1 D l m D k+1 ml D l m max { 1, D,..., D }. Since D l < 1 an t < ((m + 1)l 1)τ, it follows that ( D l m D l t ()τ lτ = D l 1 t ) l 1 exp lτ ln Dl. Owing to arbitrariness of m, we infer that D k+1 D l 1 l 1 max { 1, D,..., D } exp ( t lτ ln Dl ) for every k. Taking into account the estimate (2.35), we erive (2.34). The proof is complete. Lemma Let the conitions of Theorem 2.8 be satisfie. Then the solution to the initial value problem (2.4) is efine for t > 0; moreover, it satisfies (2.21) on each segment t kτ, (k + 1)τ), k = 0, 1,.... Proof. Recall that the noncontinuable solution to (2.4) is efine for t 0, t ). Let t 0, τ). Repeating the same reasoning as in the proof of Lemma 2.4, we obtain that y(t) satisfies (2.22) an is efine for t 0, τ]. Consequently, t > τ. It follows from (2.22) that y(t) satisfies (2.21) for k = 0. Obviously, if the initial function ϕ(t) belongs to the attraction set E 3 efine by (2.32), then Let t τ, 2τ). Consequently, By Theorem 2.1, y(t) < σ, t 0, τ]. r min δ 1 ( y(t τ) ) > r min δ 1 (σ) > 0, p min δ 2 ( y(t τ) ) > p min δ 2 (σ) > 0. t V (y t) ε 0 V 1+ω1/2 (y t ) (r min δ 1 (σ)) y(t) + Dy(t τ) 2 κ K(t s)y(s), y(s) s for t 0, t 2 ), where t 2 = min{2τ, t }. Repeating the same reasoning as above, we have y(t) + Dy(t τ) e βt, t 0, t 2 ),

18 18 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 where an β are efine in (2.17) an (2.14), respectively. Hence, y(t) y(t) + Dy(t τ) + Dy(t τ) e βt + Dy(t τ) D 2 y(t 2τ) + D 2 y(t 2τ) e βt + D e β() + D 2 Φ, t τ, t 2 ). (2.36) The function in the right-han sie of (2.36) is continuous an boune for t > 0. Then t 2 = 2τ an the noncontinuable solution y(t) to (2.4) is efine for t 0, 2τ]. Consequently, t > 2τ. It follows from (2.36) that y(t) satisfies (2.21) for k = 1. By Lemma 2.9, e βt + D e β() + D 2 Φ 1 ( D l e lβτ ) ] 1 1( ) ( t ) exp lτ ln Dl + D 2 Φ, for t τ, 2τ]. Consequently, if the initial function ϕ(t) belongs to the attraction set E 3, then y(t) < σ, t 0, 2τ]. Repeating the same reasoning, we obtain that the solution to (2.4) is efine for t > 0, an it satisfies (2.21) on each segment t kτ, (k + 1)τ), k N. By Lemma 2.9 an the conition D l < 1, we have y(t) 1 ( D l e lβτ ) ] 1 1( ) exp ( t lτ ln Dl ) + max{ D,..., D l }Φ, t > 0. Consequently, if the initial function ϕ(t) belongs to the attraction set E 3 then y(t) < σ, t > 0. The proof is complete. By Lemmas 2.9 an 2.10, we obtain that the solution to (2.4) satisfies (2.33). Therefore, Theorem 2.8 is prove. Conclusion. In this article, we investigate the nonlinear time-elay system (1.1) of neutral type with constant coefficients in the linear terms. Supposing that the spectrum of D belongs to the unit isk, we inicate sufficient conitions uner which the zero solution to (1.1) is exponentially stable. Depening on the norms of the powers of D, we establishe the constructive estimates for solutions to (1.1) an attraction sets of the zero solution to (1.1) (see Theorems 2.2, 2.5, 2.8). All the values characterizing the exponential ecay rate of the solutions at infinity an the attraction sets are written out in explicit form. Acknowlegments. The authors are grateful to the anonymous referee for the helpful comments an suggestions. This research was supporte by the Russian Founation for Basic Research (project no ).

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20 20 G. V. DEMIDENKO, I. I. MATVEEVA EJDE-2016/19 25] N. MacDonal; Biological Delay Systems: Linear Stability Theory, Cambrige Stuies in Mathematical Biology, 8, Cambrige University Press, Cambrige, ] I. I. Matveeva, A. A. Shcheglova; Some estimates of the solutions to time-elay ifferential equations with parameters, J. Appl. Inust. Math., 5 (2011), ] I. I. Matveeva; Estimates of solutions to a class of systems of nonlinear elay ifferential equations, J. Appl. Inust. Math., 7 (2013), ] D. Melchor-Aguilar, S.-I. Niculescu; Estimates of the attraction region for a class of nonlinear time-elay systems, IMA J. Math. Control Inform., 24 (2007), ] 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, ] M. A. Skvortsova; Estimates for solutions an attraction omains of the zero solution to systems of quasi-linear equations of neutral type, Bulletin of South Ural State University. Series: Mathematical Moelling, Programming & Computer Software, 37 (2011), (Russian). 31] M. Wu, Y. He, J-H. She; Stability Analysis an Robust Control of Time-Delay Systems, Science Press Beijing, Beijing; Springer-Verlag, Berlin, 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