Novel Criteria for Exponential Stability of Linear Neutral ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2015.2478125, IEEE Transactions on Automatic Control IEEE TRANSACTIONS ON AUTOMATIC CONTROL

1

Novel criteria for exponential stability of linear neutral time-varying differential systems Pham Huu Anh Ngoc and Trinh Hieu Abstract—Using a novel approach, we get explicit criteria for exponential stability of linear neutral time-varying differential systems. A brief discussion to the obtained results is given. To the best of our knowledge, the results of this paper are new. Index Terms—Time-varying system, neutral delay differential system, exponential stability

I. Introduction

N

EUTRAL delay differential equations have numerous applications in science and engineering. They are used as models of steam or water pipes, heat exchangers [12], lossless transmission lines, partial element equivalent circuits [2] and control of constrained manipulators with delay measurements in mechanical engineering [22]. Problems of stability of delay differential equations of neutral type have been investigated intensively during the past decades, see e.g. [1]-[2], [5]-[6], [8]-[26] and references therein. In general, stability analysis of delay differential equations of neutral type is hard. The traditional approaches to problems of stability of delay differential equations of neutral type are Lyapunov’s method and its variants, see e.g. [5], [8]-[9], [12]-[13], [17]. Most of existing results in the literature are given in terms of matrix inequalities and hard to use. As far as we know, there are not many explicit criteria for exponential stability of delay time-varying differential systems of neutral type, see e.g. [1], [6], [9]. In this paper, we investigate exponential stability of linear neutral time-varying differential systems of the form d (x(t) − D(t)x(t − r)) = A(t)x(t) + B(t)x(t − τ ) dt Z

0

+

C(t, θ)x(t + θ)dθ,

t ≥ σ,

(1)

−h

x(θ + σ) = ϕ(θ),

θ ∈ [−d, 0];

d := max{r, τ, h},

(2)

where A(·), B(·), D(·) : R → Rn×n and C(·, ·) : R × [−h, 0] → Rn×n , are given continuous functions and r, τ, h > 0, are time delays. Although there are many works dedicated to stability of differential systems of neutral type, to the best of our Pham Huu Anh Ngoc is with the Department of Mathematics, Vietnam National University-HCMC, International University, Thu Duc, Sai Gon, Viet Nam, Tel. (84)-(8)-7.242.181, Fax (84)-(8)7.242.1, Email: [email protected]. This author is supported by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2015-28-01/HD-KHCN. Trinh Hieu, School of Engineering, Deakin university, Geelong 3217, Australia. Email address: [email protected]

knowledge, there are no explicit spectral criteria for exponential stability of the linear neutral time-varying differential system (1). The main purpose of the present paper is to fill this gap. We present in this paper a novel approach to exponential stability of (1). Our approach is simple and based on spectral properties of Metzler matrices and a comparison principle. Consequently, we get explicit criteria for exponential stability of the linear neutral time-varying differential system (1). A brief discussion to the obtained results is given. Finally, it is to note that the method of comparison (applied to delay differential equations) has been used widely in the literature, see e.g. [4], [14], [18]-[21], [24], [25]. However, a result like Theorem III.3 of this technical paper cannot be found in the literature. II. Preliminaries Let N be the set of all natural numbers. For given m ∈ N, let m := {1, 2, ..., m}. For given integers l, q ≥ 1, Rl denotes the l-dimensional vector space over R and Rl×q stands for the set of all l × q-matrices with entries in R. For A = (aij ) ∈ Rl×q and B = (bij ) ∈ Rl×q , A ≥ B means that aij ≥ bij for i = 1, · · · , l, j = 1, · · · , q. In particular, if aij > bij for i = 1, · · · , l, j = 1, · · · , q, then we write A B instead of A ≥ B. Denote by Rl×q + the set of all nonnegative matrices. Similar notations are adopted for vectors. For x ∈ Rn and P ∈ Rl×q we define |x| = (|xi |) and |P | = (|pij |). Then one has |P Q| ≤ |P ||Q|, ∀P ∈ Rl×q , ∀Q ∈ Rq×r . Let In be the identity matrix in Rn×n . For any matrix M ∈ Rn×n the spectral abscissa (resp. the spectral radius) of M is defined by s(M ) := max{ 0, are time delays. Furthermore, we assume that there exists a constant matrix D0 ∈ Rn×n such that

Denote by C := C([−d, 0], Rn ), the Banach space of all continuous functions on [−d, 0] with values in R×n , endowed with the norm kϕk := maxθ∈[−d,0] kϕ(θ)k. Definition III.1. [9] Let σ ∈ R and ϕ ∈ C be given. A continuous function x(·) : [−d + σ, ∞) → Rn , is said to be a solution of (1) through (σ, ϕ) if the function x(t) − D(t)x(t − r) is continuously differentiable on (σ, ∞) with a right hand derivative at σ and (1) and (2) hold. It is well-known that for fixed σ ∈ R and given ϕ ∈ C, there exists a unique solution of (1) through (σ, ϕ), denoted by x(·; σ, ϕ), see e.g. [9, Theorem 1.1, page 256] Definition III.2. [9] The system (1) is said to be exponentially stable if there exist K, β > 0 such that kx(t; σ, ϕ)k ≤ Ke−β(t−σ) kϕk,

We are now in the position to prove the main result of this paper. For given A := (aij ) ∈ Rn×n , we associate the Metzler matrix M (A) := (ˆ aij ) where a ˆij = |aij | if i 6= j, for i, j ∈ n and a ˆii = aii , for i ∈ n. Theorem III.3. Suppose there exist A0 , B0 ∈ Rn×n and a continuous function C0 (·) : [−h, 0] → Rn×n such that + M (A(t)) ≤ A0 , ∀t ∈ R; |A(t)D(t)| + |B(t)| ≤ B0 , ∀t ∈ R; (4) |C(t, θ)| ≤ C0 (θ), ∀(t, θ) ∈ R × [−h, 0]. (5) Let D0 ∈ Rn×n satisfy (3). Then (1) is exponentially stable if one of the following conditions holds: (a) Z 0 ρ(D0 ) < 1; s A0 + B0 + C0 (θ)dθ (In − D0 )−1 < 0; −h

(b) there exist p, q ∈ Rn+ , p 0, q 0 such that Z 0 A0 p + B 0 + C0 (θ)dθ q 0, p + D0 q q. −h

(c) s(A0 ) < 0; ρ (−A0 )−1 B0 +

∀t ∈ R.

(3)

Z

0

C0 (θ)dθ + D0 < 1.

−h

R0 Proof: Let A := A0 , B := B0 + −h C0 (θ)dθ, C := In and D := D0 . It follows from Lemma II.2 that (a), (b) and (c) of Theorem III.3 are equivalent. Thus, it remains to show that (1) is exponentially stable provided (b) of Theorem III.3 holds. Suppose (b) of Theorem III.3 holds. Then there are p := (p1 , p2 , ..., pn )T ∈ Rn+ and q := (q1 , q2 , ..., qn )T ∈ Rn+ , p 0, q 0 such that Z 0 A0 p + (B0 + C0 (θ)dθ)q 0, p + D0 q q. −h

By continuity, βd

|D(t)| ≤ D0 ,

∀t ≥ σ, ∀ϕ ∈ C.

Z

0

C0 (θ)dθ)q −βp,

A0 p + e (B0 + −h

0018-9286 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

(6)

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2015.2478125, IEEE Transactions on Automatic Control P.H.A. Ngoc : EXPONENTIAL STABILITY OF NEUTRAL DELAY DIFFERENTIAL SYSTEMS n×n R+ , θ ∈ [−h, 0]. Taking (4)-(5) into account, we get

and p + eβd D0 q q,

(7)

n X d (0) (0) |yi (t)| ≤ aii |yi (t)| + aij |yj (t)| dt

for some sufficiently small β > 0. Let x(t) := x(t; σ, ϕ), t ∈ [−d + σ, ∞) be the unique solution of (1) through (σ, ϕ). Define y(t) := x(t) − D(t)x(t − r), t ∈ [σ, ∞). Then x(·) and y(·) satisfy the following system dy(t) = A(t)y(t) + A(t)D(t)x(t − r) + B(t)x(t − τ ) dt Z

C(t, θ)x(t + θ)dθ,

t ≥ σ,

j=1,j6=i n X

n X

|mij (t)||xj (t − r)| + |bij (t)||xj (t − τ )| j=1 j=1 n Z 0 X (0) + cij (θ)|xj (t + θ)|dθ j=1 −h

+

for almost any t ∈ [σ, +∞). It follows that for any t ∈ [σ, +∞)

0

+

(8)

−h

|yi (t + ξ)| − |yi (t)| ξ

D+ |yi (t)| := lim sup

x(t) = y(t) + D(t)x(t − r),

t ≥ σ.

|x(t)| ≤ u(t), t ∈ [σ, t1 ] and there is i0 ∈ n such that |y(t)| ≤ v(t),

ξ→0+

(9)

Choose K > 0 such that |ϕ(t)| Kq and |ϕ(0)| + D0 |ϕ(t)| Kp, for any t ∈ [−d, 0] and for any ϕ ∈ C, kϕk ≤ 1. Define u(t) := Ke−β(t−σ) q, t ∈ [σ − d, ∞) and v(t) := Ke−β(t−σ) p, t ∈ [σ, ∞). It follows that |x(t)| u(t), ∀t ∈ [σ − d, σ] and |y(σ)| v(σ). We claim that |x(t)| ≤ u(t), ∀t ≥ σ and |y(t)| ≤ v(t), ∀t ≥ σ. Assume on contrary that there exists t0 > σ such that either |x(t0 )| u(t0 ) or |y(t0 )| v(t0 ). Set t1 := inf{t ∈ (σ, ∞) : (|x(t)|, |y(t)|) (u(t), v(t))}. By continuity, t1 > σ and one of the following statements holds: (C1 )

3

∀t ∈ [σ, t1 );

1 = lim sup ξ + ξ→0 ≤

Zt+ξ

d |yi (ζ)|dζ dζ

t

(0) aii |yi (t)| +

n X

(0) aij |yj (t)| +

n X

|bij (t)||xj (t − τ )| +

n Z X

0

j=1 −h

j=1

|mij (t)||xj (t − r)|

j=1

j=1,j6=i

+

n X

(0)

cij (θ)|xj (t + θ)|dθ

where D+ denotes the Dini upper-right derivative, see [23], page 195. In particular, it follows from (i) that +

D |yi0 (t1 )| ≤

(0) ai0 i0 Ke−β(t1 −σ) pi0

n X

+

(0)

ai0 j Ke−β(t1 −σ) pj

j=1,j6=i0

|yi0 (t1 )| = vi0 (t1 );

|yi0 (t)| > vi0 (t), ∀t ∈ (t1 , t1 +), (10)

+

for some > 0. (C2 )

|y(t)| ≤ v(t), t ∈ [σ, t1 ] and there is j0 ∈ n such that |x(t)| ≤ u(t),

∀t ∈ [σ, t1 );

+

n X

|mij (t1 )|Ke−β(t1 −σ) eβr qj +

j=1 n Z 0 X j=1 −h

(0)

ci0 j (θ)Ke−β(t1 −σ) eβh qj dθ

≤ Ke |xj0 (t1 )| = uj0 (t1 ); |xj0 (t)| > uj0 (t), ∀t ∈ (t1 , t1 + ), (11)

X n

n X

(|mij (t1 )| + |bij (t1 )|)qj + eβd

−β(t1 −σ)

≤ Ke

j=1,j6=i n X

j=1

|bij (t)||xj (t − τ )| +

+

j=1

n Z X

0

|cij (t, θ)||xj (t + θ)|dθ,

j=1 −h

for almost any t ∈ [σ, +∞) and every i ∈ n. Let A0 := (0) (0) (0) n×n (aij ) ∈ Rn×n and C0 (s) := (cij (θ)) ∈ + , B0 := (bij ) ∈ R+

n Z X

0

j=1 −h

j=1

X n j=1

d |yi (t)| = sgn(yi (t))y˙ i (t) ≤ aii (t)|yi (t)|+ dt n n X X |aij (t)||yj (t)| + |mij (t)||xj (t − r)|

(0)

ai0 j pj +

j=1

eβd

Suppose (C1 ) holds. Let A(t) := (aij (t)), A(t)D(t) := (mij (t)), B(t) := (bij (t)), C(t, s) := (cij (t)), t ∈ R. It follows from (8) that

|bij (t1 )|Ke−β(t1 −σ) eβτ qj

j=1

−β(t1 −σ)

for some > 0.

n X

(0) ai0 j pj

+e

βd

n X

(0) bi0 j qj

j=1

+

(0)

ci0 j (θ)qj dθ n Z X

0

j=1 −h

(0) ci0 j (θ)qj dθ

(6)

< −βKe−β(t1 −σ) pi0 = D+ vi0 (t1 ).

However, this conflicts with (10). Assume that (C2 ) holds. It follows from (3) and (9) that |x(t1 )| ≤ |y(t1 )| + |D(t1 )||x(t1 − r)| ≤ Ke−β(t1 −σ) p +D0 Ke−β(t1 −σ) eβd q = Ke−β(t1 −σ) p + eβd D0 q (7)

Ke−β(t1 −σ) q = u(t1 ).


This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2015.2478125, IEEE Transactions on Automatic Control 4

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

This conflicts with the last inequality in (11). Thus, we have |x(t)| ≤ u(t), ∀t ≥ σ and |y(t)| ≤ v(t), ∀t ≥ σ. In particular, this yields |x(t; σ, ϕ)| ≤ Ke−β(t−σ) q, ∀t ≥ σ, ∀ϕ ∈ C, kϕk ≤ 1. Since (1) is linear, it follows that for any ϕ ∈ C, ϕ 6= 0, |x(t; σ,

ϕ 1 )| = |x(t; σ, ϕ)| ≤ Ke−β(t−σ) q, ∀t ≥ σ. kϕk kϕk

Therefore,

IV. Discussion and illustrative example As mentioned in Introduction, stability analysis of linear time-varying delay differential systems of neutral type is very hard. To the best of our knowledge, there are not many explicit criteria for exponential stability of the linear time-varying delay differential system of neutral type (1). In [1], the authors considered a scalar linear neutral differential equation of the form d (x(t) − d(t)x(t − r)) + a(t)x(t) + b(t)x(t − τ ) = 0, (16) dt where a(·), b(·), d(·) : R → R+ are continuous functions such that p1 ≤ a(t) ≤ p2 ; q1 ≤ b(t) ≤ q2 , 0 ≤ d(t) ≤ d1 < 1, ∀t ∈ R and |d0 (t)| ≤ d2 , ∀t ∈ R for some p1 , p2 , q1 , q2 , d1 , d2 ∈ R+ . Using a Lyapunov functional, it has been shown in [1, Theorem 2] that (16) is locally asymptotically stable if

|x(t; σ, ϕ)| ≤ Kkϕke−β(t1 −σ) q, ∀t ≥ σ, ∀ϕ ∈ C. Then, we have kx(t; σ, ϕ)k ≤ Kkqke−β(t1 −σ) kϕk, ∀t ≥ σ, ∀ϕ ∈ C. So (1) is exponentially stable. This completes the proof. The following results are immediate from Theorem III.3. Corollary III.4. Suppose A, B, D ∈ Rn×n are given and C(·) : [−h, 0] → Rn×n is a given continuous function. The linear neutral time-invariant differential system Z 0 d (x(t)−Dx(t−r)) = Ax(t)+Bx(t−τ )+ C(θ)x(t+θ)dθ, dt −h (12) is exponentially stable if one of the following conditions holds (a) ρ(|D|) < 1 and Z 0 s M (A) + |AD| + |B| + |C(θ)|dθ (In − |D|)−1 < 0. −h

(b) there exist p, q ∈ Rn+ , p 0, q 0 such that Z

|C(θ)|dθ q 0;

p+|D|q q.

s(M (A)) < 0 and

ρ (−M (A))

Note that we need not assume that |d0 (t)| ≤ d2 ; b(t) ≥ q1 , ∀t ∈ R. As far as we know, a result like Theorem III.3 cannot be found in the literature. A remarkable result on asymptotic stability of the linear neutral time-invariant differential systems can be found in [15]. It has been shown [15, Theorem 3.1] that (12) with C(·) ≡ 0 is asymptotically stable if kDk < 1

−h

−1

Some similar results can be found in [6], [9]. Clearly, −a(t) ≤ −p1 , ∀t ∈ R and |a(t)d(t)| + |b(t)| ≤ p2 d1 + q2 , ∀t ∈ R. By Theorem III.3, (16) is (globally) exponentially stable if −p1 + (p2 d1 + q2 )(1 − d1 )−1 < 0, or equivalently, p1 > q2 + d1 (p1 + p2 ).

and µ(A) +

kBk + kAkkDk < 0, 1 − kDk

(17)

0

M (A)p+ |AD|+|B|+ (c)

p1 > q2 + d1 (p2 + q2 ).

Z

0

|AD| + |B| +

|C(θ)|dθ + |D| < 1.

−h

Corollary III.5. The linear delay differential system of retarded type Z 0 d x(t) = A(t)x(t) + B(t)x(t − τ ) + C(t, θ)x(t + θ)dθ, dt −h (13) is exponentially stable if there exist A0 , B0 ∈ Rn×n and a continuous function C0 (·) : [−h, 0] → Rn×n such that + M (A(t)) ≤ A0 , ∀t ∈ R; |B(t)| ≤ B0 , ∀t ∈ R;

Z

−h

C0 (θ)dθ < 0.

and µ1 (M (A))+

It follows that µ1 M (A) + (|A||D| + |B|)(In − |D|)−1 < 0.

0

s A0 + B 0 +

k|B|k1 + k|A|k1 k|D|k1 < 0. 1 − k|D|k1 (18) The latter inequality in (18) implies µ1 (M (A)) + µ1 (|A||D| + |B|)(In − |D|)−1 < 0. k|D|k1 < 1

(14)

|C(t, θ)| ≤ C0 (θ), ∀(t, θ) ∈ R × [−h, 0] and

where µ(A) := lim→0+ kIn +hAk−1 is the matrix measure of A, see e.g. [7]. Some similar results can be found in [10]-[11], [26]. Suppose Rn is endowed with the norm k · k1 . Then the operator norm associated with the norm k · k1 on Pvector n Rn×n is given by kM k1 = maxj∈n i=1 |mij | and the corresponding matrix Pn measure of M is given by µ1 (M ) := maxj∈n [mjj + i=1,i6=j |mij |], for M := (mij ) ∈ Rn×n , see e.g. [7]. Thus, (17) can be rewritten as

(15)

Since µ1 (M ) ≥ s(M ) for any M ∈ Rn×n , we get s M (A) + (|A||D| + |B|)(In − |D|)−1 < 0.


This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2015.2478125, IEEE Transactions on Automatic Control P.H.A. Ngoc : EXPONENTIAL STABILITY OF NEUTRAL DELAY DIFFERENTIAL SYSTEMS

Thus, (17) is more conservative than Corollary III.4 (i) (with C(·) ≡ 0). To end this paper, we give an example to illustrate Theorem III.3. Consider a linear neutral system of the form d [x(t) − D(t)x(t − r)] = A(t)x(t) + B(t)x(t − τ ), (19) dt 0 α cos t 2 where x(t) ∈ R , D(t) := ; A(t) = −2I2 , t ∈ 0 0 R and 2 0 α sin t B(t) := , t ∈ R; 0 ≤ α < . sin t 0 3 It is clear that |D(t)| ≤ D :=

0 0

α 0

|A(t)D(t)| + |B(t)| ≤ B0 := Let A0 := −2I2 . Then −2 −1 A0 +B0 (I2 −D) = 0 = Since

−2 1

0 −2

∀t ∈ R; 0 1

3α 0

0 + 1

3α −2 + α

3α 0

, ∀t ∈ R.

1 0

α 1

.

3α 1 0, −2 + α 1 −2 3α the matrix is Hurwitz stable, by Theo1 −2 + α rem II.1 (ii). Thus, (19) is exponentially stable, by Theorem III.3. −2 1

References [1] R.P Agarwal, S.R Grace, Asymptotic stability of certain neutral differential equations, Mathematical and Computer Modelling 31 (2000), 9-15. [2] A. Bellen, N. Guglielmi and A.E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits and Systems: Fundamental theory and Applications, 46 (1999), 212-216. [3] A. Berman, R.J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Acad. Press, New York, 1979. [4] A. Goubet Bartholomeus, M. Dambrine and J. P. Richard, Stability of perturbed systems with time-varying delays, Systems & Control Letters 31 (1997), 155-163. [5] S. Cong, On exponential stability conditions of linear neutral stochastic differential systems with time-varying delay, International Journal of Robust and Nonlinear Control 23 (2013), 12651276. [6] M.A. Cruz, J.K. Hale, Stability of functional differential equations of neutral type, Journal of Differential Equations 7 (1970), 334-355. [7] C. Desoer, H. Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory 19 (1972), 480-486. [8] E. Fridman, New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters 43 (2001), 309-319. [9] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag Berlin, Heidelberg, New york, 1993.

5

[10] P. He, D.Q. Cao, Algebraic stability criteria of linear neutral systems with multiple time delays, Applied Mathematics and Computation 155 (2004), 643-653. [11] G.D Hui, G.D Hu, Simple criteria for stability of neutral systems with multiple delays, International Journal of Systems Science 28 (1997), 1325-1328. [12] V.B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Dordrecht, the Neitherlands: Kluwer 1996. [13] V. Kharitonov, S. Mondie, J. Collado, Exponential estimates for neutral time-delay systems: An LMI Approach, IEEE Transactions on Automatic Control 50 (2005), 666-670. [14] B. Lehman, K. Shujaee, Delay independent stability conditions and decay estimates for time-varying functional differential equations, Automatic Control, IEEE Transactions on 39(1994), 16731676. [15] L.M. Li, Stability of linear neutral delay-differential systems, Bulletin of the Australian Mathematical Society 38 (1988), 339344. [16] H. Li, S. Zhong, Hou-biao Li, Some new simple stability criteria of linear neutral systems with a single delay, Journal of Computational and Applied Mathematics 200 (2007), 441-447. [17] Yajuan Liu, S.M. Lee, O.M. Kwon, Ju H. Park, Delay-dependent exponential stability criteria for neutral systems with interval time-varying delays and nonlinear perturbations, Journal of the Franklin Institute 350 (2013), 3313-3327. [18] P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012), 1208-1213. [19] P.H.A. Ngoc, Stability of positive differential systems with delay, IEEE Transactions on Automatic Control 58 (2013), 203-209. [20] P. H. A. Ngoc, Novel criteria for exponential stability of functional differential equations, Proceedings of the American Mathematical Society 141 (2013), 3083-3091. [21] P.H.A. Ngoc, Novel criteria for exponential stability of nonlinear differential systems with delay. IEEE Trans. Automat. Control 60 (2015), 485-490. [22] S.I. Niculescu, B. Brogliato, Force measurement time-delays and contact instability phenomenon, European Journal of Control 5 (1999), 279-289. [23] P. Sahoo, T. Riedel, Mean Value Theorems and Functional Equations, World Scientific, 1998. [24] A.P. Tchangani, M. Dambrine, J.P. Richard, V. Kolmanovskii, Stability of Nonlinear Differential Equations with Distributed Delay, Nonlinear Analysis: Theory, Methods and Applications, Pergamon-Elsevier Sc. Publ., 1998, 34, 1081-1095. [25] A.P. Tchangani, M. Dambrine, J.P. Richard, Stability, attraction domains, and ultimate boundedness for nonlinear neutral systems. Mathematics and computers in simulation 45 (1998), 291-298. [26] K. Zhang, D.Q. Cao, Further results on asymptotic stability of linear neutral systems with multiple delays, Journal of the Franklin Institute 344 (2007), 858-866.
