Differ Equ Dyn Syst DOI 10.1007/s12591-014-0198-6 ORIGINAL RESEARCH

The L p -Version of the Generalized Bohl–Perron Principle for Neutral Type Functional Differential Equations Michael Gil’

© Foundation for Scientific Research and Technological Innovation 2014

Abstract We consider a vector linear neutral type homogeneous functional differential equation. It is proved that the considered equation is exponentially stable, provided the corresponding non-homogeneous equation with the zero initial function and an arbitrary free term from L p ([0, ∞), Cn ), has a solution belonging to L p ([0, ∞), Cn ). Keywords Functional differential equation · Neutral type equation · Linear equation · Exponential stability Mathematics Subject Classification

34K20

Introduction and Statement of the Main Result Recall that the Bohl–Perron principle means that the homogeneous ordinary differential equation (ODE) y˙ = A(t)y (t ≥ 0) with a variable n × n-matrix A(t), bounded on [0, ∞) is exponentially stable, provided the nonhomogeneous ODE x˙ = A(t)x + f (t) with the zero initial function has a bounded solution for any bounded vector valued function f , cf. [6]. In [17, Theorem 4.15] the Bohl–Perron principle was generalized to a class of retarded systems with finite delays; besides the asymptotic (not exponential) stability was proved. The result from [17] was afterwards considerably developed, cf. the books [2] and very interesting papers [4,5], in which the generalized Bohl–Perron principle was effectively used for the stability analysis of the first and second order scalar equations. In particular, in [4] the scalar non-autonomous linear functional differential equation x(t)+a(t)x(h(t)) ˙ = 0 is considered. The authors give sharp conditions for exponential stability, which are suitable in the case that the coefficient function a(t) is periodic, almost periodic or asymptotically almost periodic, as often encountered in applications. In the paper [5], the authors provide sufficient conditions

M. Gil’ (B) Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beersheba 84105, Israel e-mail: [email protected]

123

Differ Equ Dyn Syst

for the stability of rather general second-order delay differential equations. In the paper [14] the generalized Bohl–Perron principle has been extended to neutral type equations. In the paper [12] (see also [15]) a result similar to the Bohl–Perron principle has been derived in the terms of the norm of the space L p for a class of delay-difference equations. That result is called the L p -version of the generalized Bohl–Perron principle. In the present paper we generalize the main result from [12] to the neutral type functional differential equations. Although in the present paper we consider equations having invertible neutral parts and they can be theoretically reduced to differential-delay equations, the results from [12] can not be applied to the reduced equations, since the reduced equations do not belong, in general, to the class of equations considered in that paper. In the monograph [2] the generalized Bohl–Perron principle discussed in the sup-norm, only, and the L p -version is not considered. In the third section, we show that our results can be effectively used for the stability analysis. As it is well-known, the basic method for the stability analysis of vector timevariant equations is the direct Lyapunov method. By that method many very strong results are obtained. But finding Lyapunov’s type functionals for nonautonomous neutral type functional differential equations is usually difficult. At the same time, in the third section we suggest explicit stability conditions. Let Cn be an n-dimensional complex Euclidean space with a norm .n and the unit matrix I ; for an n × n-matrix A, An = supv∈Cn Avn /vn . C(ω) ≡ C(ω, Cn ) is the space of continuous functions defined on a (finite or infinite) real segment ω with values in Cn and the finite sup-norm  · C(ω) ; L p (ω) ≡ L p (ω, Cn ) ( p ≥ 1) is the space of functions u defined on a set ω ⊆ R with values in Cn and the finite norm ⎡ ⎤1/ p  p (1 ≤ p < ∞), u L p = u L p (ω) := ⎣ u(t)n dt ⎦ ω

and u L ∞ (ω) = ess supt∈ω u(t)n . In addition, C 1 (ω) ≡ C 1 (ω, Cn ) is the space of functions u ∈ C(ω, Cn ) with u˙ ∈ C(ω, Cn ) (u(t) ˙ = du/dt) and the finite norm uC 1 (ω) := uC(ω) + u ˙ C(ω) . Let η < ∞ be a positive constant, A˜ j (t), Ak (t) (t ≥ 0; j = 1, ..., m; ˜ k = 1, ..., m), and ˜ τ ) (t ≥ 0; τ ∈ [0, η]) be n × n variable matrices. In addition, Ak (t), A(t, τ ) A(t, τ ), A(t, ˜ τ ) have piece-wise continuous derivatives in t. are piece-wise continuous in t; A˜ j (t), A(t, Define operators E 0 , E 1 : L p (−η, ∞) → L p (0, ∞) by (E 0 f )(t) =

m 

η Ak (t) f (t − h k (t)) +

k=1

A(t, s) f (t − s)ds 0

and (E 1 f )(t) =

m˜ 

A˜ k (t) f (t − h˜ k ) +

k=1

η

˜ s) f (t − s)ds ( f ∈ L p (0, ∞); t ≥ 0), A(t,

0

where 0 < h˜ 1 < ... < h˜ m˜ ≤ η (m˜ < ∞) are constants, and h j (t) are real continuously differentiable functions, such that 0 ≤ h j (t) ≤ η

and

− ∞ ≤ h˙ j (t) < 1

Our main object in this paper is the equation

123

(t ≥ 0; j = 1, ..., m).

Differ Equ Dyn Syst

d [y(t) − (E 1 y)(t)] = (E 0 y)(t) dt with the initial condition y(t) = φ(t)

(t > 0)

(−η ≤ t ≤ 0)

(1.1)

(1.2)

for a given φ ∈ C 1 (−η, 0). We consider also the non-homogeneous equation d [x(t) − (E 1 x)(t)] = (E 0 x)(t) + f (t) (t > 0) dt with a given vector function f ∈ L p (0, ∞) and the initial condition x(t) ≡ 0

(−η ≤ t ≤ 0).

(1.3)

(1.4)

˜ Everywhere below it is assumed that A j (t)n ( j = 1, ..., m) and  A˜ k (t)n (k = 1, ..., m) are bounded on [0, ∞), and η 0

sup( A˜ t (t, s)n + A(t, s)n )ds < ∞. t≥0

A solution of problem (1.1) and (1.2) is a continuous function, satisfying the problem t y(t) − (E 1 y)(t) = φ(0) − (E 1 φ)(0) +

(E 0 y)(t1 )dt1 (t ≥ 0),

(1.5a)

0

y(t) = φ(t) (−η ≤ t ≤ 0).

(1.5b)

Here (E 1 φ)(0) =

m˜ 

A˜ k (0)φ(−h˜ k (0)) +

k=1

η

˜ s)φ(−s)ds. A(0,

0

A solution of problem (1.3) and (1.4) is defined as a continuous function x(t), which satisfies the equation t x(t) − (E 1 x)(t) =

t (E 0 x)(t1 )dt1 +

0

f (t1 )dt1 (t ≥ 0)

(1.6)

0

and condition (1.4). The existence of and uniqueness of solutions is assumed. Various existence and uniqueness results under the condition η m˜  ˜ s)n ds < 1 V1 := sup  A˜ k (t)n + sup  A(t, (1.7) k=1 t≥0

0

t≥0

as well as stability definitions can be found for instance in [18,20]. Now we are in a position to formulate the main result of the section. Theorem 1.1 Let condition (1.7) hold, and for a p ∈ [1, ∞) and any f ∈ L p (0, ∞) problem (1.3) and (1.4) have a unique solution x(t) ∈ L p (0, ∞). Then Eq. (1.1) is exponentially stable. This theorem is proven in the next section.

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Differ Equ Dyn Syst

Proof of Theorem 1.1 We need the following Lemma 2.1 Let h(t) be a differentiable function with the properties 0 < h(t) ≤ η and −∞ < h˙ j (t) < 1 (t ≥ 0). Then for any u ∈ L p (−η, ∞) one has u(t − h(t)) L p (0,T ) ≤

 p

χ(h)u L p (−h(0),T ) (T > 0),

where χ(h) =

1 . ˙ inf t≥0 (1 − h(t))

Proof Obviously, T u(t

p − h(t))n dt

T =

0

0

T u(t

≤ χ(h)

˙ dt ˙ 1 − h(t)

p 1 − h(t)

u(t − h(t))n

p ˙ − h(t))n (1 − h(t))dt

T ≤ χ(h)

p

u(s)n ds.

−h(0)

0



As claimed. Rewrite (1.1) as y˙ (t) − (E 1 y˙ )(t) = (E 1 y)(t) + (E 0 y)(t) (t ≥ 0),

(2.1)

where (E 1 y)(t) =

m˜ 

A˜ k (t)y(t − h˜ k ) +

k=1

η

A˜ t (t, s)y(t − s)ds.

0

Lemma 2.2 For any T > 0 and a p ≥ 1 one has E 1 u L p (0,T ) ≤ V1 u L p (−η,T ) and there is a constant V0 , such that (E 1 + E 0 )u L p (0,T ) ≤ V0 u L p (−η,T )

(u ∈ L p (−η, T )).

Proof Let u ∈ L ∞ (−η, T ). We have (E 1 u)(t)n ≤

m˜ 

 A˜ k (t)u(t − h˜ k )n +

k=1

η 0

˜ s)u(t − s)n ds  A(t,

⎞ ⎛ η m˜  ˜ s)n ds ⎠ ≤ V1 u L ∞ (−η,T ) (0 ≤ t ≤ T ).  A˜ k (t)n +  A(t, ≤ u L ∞ (−η,T ) ⎝ k=1

123

0

Differ Equ Dyn Syst

So for L ∞ the first inequality is proved. Now let u ∈ L 1 (−η, T ). Then E 1 u L 1 (0,T ) ≤

m˜ 

 A˜ k (t)u(t − h˜ k ) L 1 (0,T ) +

k=1

⎛

≤ u L 1 (−η,T ) ⎝

m˜  k=1

η 0

sup  A˜ k (t)n +

η

t

˜ s)u(t − s) L 1 (0,T ) ds  A(t, ⎞

˜ s)n ds ⎠ = V1 u L 1 (−η,T ) . sup  A(t, t

0

Hence for all p ≥ 1 the first inequality is due to the Riesz–Thorin theorem, cf. [10, section VI.10.11]. Similarly by Lemma 2.1 the second inequality can be proved.  Lemma 2.3 If for any f ∈ L p (0, ∞) (1 ≤ p < ∞) a solution of problem (1.3) and (1.4) and its derivative are in L p (0, ∞), and V0 < ∞, V1 < ∞, then any solution of problem (1.1) and (1.2) and its derivative are also in L p (−η, ∞). Proof Let y(t) be a solution of problem (1.1) and (1.2). For a positive constant ν > 0 put

−νt e φ(0) if t ≥ 0, ζ (t) = φ(t) if −η ≤ t < 0 and x0 (t) = y(t) − ζ (t). We can write dζ (t)/dt = −νe−νt φ(0) (t ≥ 0) and d [x0 (t) − (E 1 x0 )(t)] = (E 0 x0 )(t) + ψ(t) (t > 0), dt where ψ(t) = −ζ˙ (t) +

d(E 1 ζ )(t) + (E 0 ζ )(t) = −ζ˙ (t) + (E 1 ζ˙ )(t) + (E 1 ζ )(t) + (E 0 ζ )(t). dt

Besides, (1.4) holds with x(t) = x0 (t) and f (t) = ψ(t). Since ζ ∈ L p (−η, ∞) and ζ˙ ∈ L p (−η, ∞), by the previous lemma we have ψ ∈ L p (−η, ∞). Due to the hypothesis of this lemma, x0 , x˙0 ∈ L p (0, ∞). Thus y = x0 + ζ, y˙ ∈ L p (−η, ∞). As claimed.  Lemma 2.4 Let condition (1.7) hold. Then for any solution x(t) of problem (1.3) and (1.4) and all T > 0, one has x ˙ L p (0,T ) ≤ (1 − V1 )−1 (V0 x L p (0,T ) +  f  L p (0,T ) ). Proof By Lemma 2.2 from (1.3) we have ˙ L p (0,T ) + V0 x L p (0,T ) +  f  L p (0,T ) . x ˙ L p (0,T ) ≤ V1 x Hence the condition V1 < 1 implies the required result.



Lemma 2.5 Any function ξ ∈ L p (0, ∞) (1 ≤ p < ∞) with ξ˙ ∈ L p (0, ∞) is bounded on [0, ∞). Moreover, ξ C(0,∞) ≤ pξ  L p (0,∞) ξ˙  L p (0,∞) p

p−1

if 1 < p < ∞,

and ξ C(0,∞) ≤ ξ˙  L 1 (0,∞) .

Proof For simplicity, in this proof put ξ(t)n = |ξ(t)|. First consider the case p = 1. Since d|ξ(t)| |ξ(t + h)| − |ξ(t)| |ξ(t + h) − ξ(t)| = lim ≤ lim = |ξ˙ (t)|, h→0 h→0 dt h h

123

Differ Equ Dyn Syst

we obtain ∞ |ξ(t)| = − t

d|ξ(t1 )| dt1 ≤ dt1

∞

|ξ˙ (t1 )|dt1 ≤ ξ˙  L 1 (0,∞) (t ≥ 0).

t

Assume that 1 < p < ∞. Then by the Gólder inequality ∞ |ξ(t)| = − p

t

∞ ≤p

d|ξ(t1 )| p dt1 = − p dt1

∞ |ξ(t1 )| p−1 t

⎡

|ξ(t1 )| p−1 |ξ˙ (t1 )|dt1 ≤ p ⎣

t

∞

d|ξ(t1 )| dt1 dt1

⎤1/q ⎡ ∞ ⎤1/ p  |ξ(t1 )|q( p−1) dt1 ⎦ ⎣ |ξ˙ (t1 )| p dt1 ⎦ ,

t

t

where q = p/( p − 1). Since q( p − 1) = p, we get the required inequality.



Proof of Theorem 1.1 Substituting y(t) = y (t)e−t

(2.2)

with an  > 0 into (2.1), we obtain the equation

y˙ − y − E ,1 y˙ +  E ,1 y = (E ,1 + E ,0 )y ,

(2.3)

where m˜ 

(E ,1 f )(t) =

e

h˜ k 

A˜ k (t) f (t − h˜ k ) +

k=1

(E ,0 f )(t) =

m 

η

˜ s) f (t − s)ds, es A(t,

0

Ak (t)e

h k (t)

η f (t − h k (t)) +

k=1

A(t, s)es f (t − s)ds 0

and

(E ,1

f )(t) =

m˜  k=1

e

h˜ k 

A˜ k (t) f (t

− h˜ k ) +

η

es A˜ t (t, s) f (t − s)ds.

0

Rewrite (2.3) as d [y − E ,1 y ] = Z  y + E ,0 y , dt

(2.4)

where Z  :=  I −  E ,1 . Furthermore, introduce in L p (0, ∞) the operator Gˆ : f → x where x(t) is the solution of problem (1.3) and (1.4). That is, Gˆ solves problem (1.3) and (1.4). By the hypothesis of the theorem, we have x = Gˆ f ∈ L p (0, ∞) for any f ∈ L p (0, ∞). So Gˆ is defined on the whole space L p (0, ∞). It is closed, since we assume the uniqueness of solutions to problem (1.3) and (1.4). Therefore Gˆ : L p (0, ∞) → L p (0, ∞) is bounded ˆ L p (0,∞) is finite. according to the Closed Graph Theorem [10, p. 57]. So the norm G Consider now the equation d [x − E ,1 x ] = Z  x + E ,0 x + f dt

123

(2.5)

Differ Equ Dyn Syst

with the zero initial function. Subtract (1.3) from (2.5), with w(t) = x (t) − x(t), where x and x are solutions of problems (1.3), (1.4) and (2.5), (1.4), respectively. Then d [w − E 1 w] = E 0 w + F , dt

(2.6)

where F = Z  x + (E ,0 − E 0 )x +

d (E ,1 − E 1 )x . dt

(2.7)

It is simple to check that Z  → 0, E ,1 → E 1 and E ,1 → E 1 in the operator norm of L p (0, ∞) as  → 0. For the brevity in this proof put . L p (0,T ) = |.|T for a finite T > 0. We have d (E ,1 − E 1 )x = (E ,1 − E 1 )x˙ + (E ,1 − E 1 )x . dt So

  d   (E ,1 − E 1 )x  ≤ a1 ()|x˙ |T + a2 ()|x |T ,  dt 

(2.8)

T

where a j () → 0 as  → 0 for an index j. But according to Lemma 2.2, for a sufficiently small , we have |E ,1 x |T ≤ eη V1 |x |T with eη V1 < 1.

(2.9)

Due to Lemma 2.3, from (2.5), the inequality |x˙ |T ≤ (1 − eη V1 )−1 (|E ,1 + Z  |T |x |T + | f |T )

+ Z | is bounded uniformly with → E 1 , the norm |E ,1 follows. Since Z  → 0, E ,1  T respect to  and T > 0. So for a sufficiently small 0 > 0, there is a constant c1 , such that

|x˙ |T ≤ c1 (|x |T + | f |T ) ( < 0 ; T > 0). Now (2.8) implies    d  (E ,1 − E 1 )x  ≤ a1 ()c1 (|x |T + | f |T ) + a2 ()|x |T ≤ a3 ()(|x |T + | f |T ).   dt T (2.10) Furthermore, |(E ,0 − E 0 )x |T ≤ a4 ()|x |T . Thus from (2.10) we deduce that |F |T ≤ a5 ()(|x |T + | f |T ) uniformly in T > 0. By (2.6) x − x = Gˆ F . So ˆ L p (0,∞) a5 ()(|x |T + | f |T ). |x − x|T ≤ G ˆ L p (0,∞) a5 () < 1. Thus For a sufficiently small , we have q() := G ˆ L p (0,∞) a5 ()| f |T ). |x |T ≤ (1 − q())−1 (|x|T + G

123

Differ Equ Dyn Syst

By the hypothesis of this theorem, x ∈ L p (0, ∞). This gives us the inequality ˆ L p (0,∞)  f  L p (0,∞) ). |x |T ≤ (1 − q())−1 (x L p (0,∞) + a5 ()G So, letting T → ∞, we get x ∈ L p (0, ∞). Due to Lemma 2.4 x˙ ∈ L p (0, ∞). Hence, by Lemma 2.3, a solution y of (2.3) and its derivative are in L p (0, ∞). Making use of Lemma 2.5 we can assert that a solution of (2.3) is bounded. Now (2.2) proves the exponential stability. As claimed. 

Time-Variant Linear Systems “Close” to of Autonomous Ones Let us consider the equation x(t) ˙ − A˜

η

η x(t ˙ − s)d μ(s) ˜ +C

0

x(t − s)dμ(s) = E 0 x(t) + f (t) (t ≥ 0),

(3.1)

0

where E 0 is defined as above, A˜ and C are constant n × n-matrices and μ, μ˜ are scalar nondecreasing functions with finite number of jumps. Without loss of generality suppose that var(μ) = var(μ) ˜ = 1. Here var(μ) is the variation of μ. ˜ n < 1,  A

(3.2)

Inequality (3.2) provides condition (1.7). Certainly (3.1) can be written in the form (1.3). Put K (z) = z I − z A˜

η e

−zs

η d μ(s) ˜ +C

0

e−zs dμ(s).

0

Assuming that all the characteristic values of K (z), that is the zeros of det K (z), are in the open left half plane C− . Then G 0 (t) =

1 2π

∞

eiωt K −1 (iω)dω

−∞

is the Green function of the autonomous equation z˙ (t) − A˜

η

η z˙ (t − s)d μ(s) ˜ +C

0

z(t − s)dμ(s) = 0 (t ≥ 0).

(3.3)

0

Besides, due to Lemma 2.1 η

q := E 0  L 2 (−η,∞)→L 2 (0,∞)

 m  supt≥0 Ak (t)n  + sup A(t, s)n ds. (3.4) ≤ ˙ t≥0 k=1 inf t≥0 1 − h k (t) 0

Recall that we assume that −∞ < h˙ j (t) < 1.

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Differ Equ Dyn Syst

Furthermore, use the operator Gˆ defined on L 2 (0, ∞) by Gˆ f (t) =

t G 0 (t − t1 ) f (t1 )dt1 ( f ∈ L 2 (0, ∞)). 0

ˆ 0 x + f ). Assume that Then due to (3.1) we get x = G(E ˆ L 2 (0,∞) < 1. q G

(3.5)

ˆ L 2 (0,∞) )−1 . Now Theorem 1.1 implies Then x L 2 (0,∞) ≤ Gˆ f  L 2 (0,∞) (1 − qG Lemma 3.1 Let conditions (3.2) and (3.5) hold. Then the equation y˙ (t) − A˜

η

η y˙ (t − s)d μ(s) ˜ +C

0

y(t − s)dμ(s) = E 0 y(t) (t ≥ 0),

(3.6)

0

is exponentially stable. Let fˆ(z) be the Laplace transform of f . Then by the Parseval equality 1 K −1 (iω) fˆ(iω)2L 2 (−∞,∞) 2π 1 ≤ sup K −1 (iω)2n  fˆ(iω)2L 2 (−∞,∞) 2π ω∈R

Gˆ f 2L 2 (0,∞) =

= sup K −1 (iω)2n  f 2L 2 (0,∞) . ω∈R

˜ n )−1 . Due to Lemma 2 from [13] supω∈R K −1 (iω)n = θ (K ), Denote v0 = 2Cn (1− A where θ (K ) := sup−v0 ≤ω≤v0 K −1 (iω)n . Thus condition (3.5) can be written as qθ (K ) < 1. To estimate θ (K ) introduce the notations. For an n × n-matrix A, λk (A) (k = 1, ..., n) denote its eigenvalues enumerated in an arbitrary order with their multiplicities, A∗ is the adjoint one , and A−1 is the inverse one. N2 (A) is the Hilbert–Schmidt (Frobenius) norm: N22 (A) = Trace A A∗ , and A I = (A − A∗ )/2i. Introduce the quantity g(A) = (N22 (A) −

n 

|λk (A)|2 )1/2 .

k=1

It is not hard to check that g 2 (A) ≤ N 2 (A) − |Trace A2 |. In Section 2.2 of [11] it is proved that g 2 (A) ≤ 2N22 (A I )

and

g(eiτ A + z I ) = g(A)

(3.7)

for all τ ∈ R and z ∈ C. If A1 and A2 are commuting matrices, then g(A1 + A2 ) ≤ g(A1 ) + g(A2 ).

(3.8)

From Corollary 2.1.2 [11], it follows that for any invertible n × n-matrix A, the inequality A−1 n ≤

n−1  k=0

g k (A) √ k!ρ k+1 (A)

(3.9)

123

Differ Equ Dyn Syst

is true, where ρ(A) is the smallest modulus of the eigenvalues of A: ρ(A) = mink=1,...,n |λk (A)|. Put B(z) = A˜

η

η z exp(−zs)d μ(s) ˜ −C

0

exp(−zs)dμ(s). 0

So K (z) = z I − B(z). By (3.7) g(B(z)) = g(K (z)). Thanks to (3.9), for any regular value z of K (.), the inequality [K (z)]−1 n ≤ (K (z))

(z ∈ C)

(3.10)

is valid, where (K (z)) =

n−1  k=0

g k (B(z)) √ k!ρ k+1 (K (z))

and ρ(K (z)) is the smallest absolute value of the eigenvalues of K (z) for a fixed z: ρ(K (z)) = min |λk (K (z))|. k=1,...,n

Thus ˆ L 2 (0,∞) ≤ θ (K ) ≤ 0 (K ), where 0 (K ) := G

sup

−v0 ≤ω≤v0

(K (iω)).

Hence, due to Lemma 3.1 we arrive at the following result. Theorem 3.2 Let all the zeros of K be in C− and the conditions (3.2) and q0 (K ) < 1 hold. Then Eq. (3.6) is exponentially stable. Denote g(B) ˆ :=

sup

ω∈[−v0 ,v0 ]

g(B(iω))

and

ρ(K ˆ ) :=

inf

ω∈[−v0 ,v0 ]

ρ(K (iω)).

Then we have ˆ ), where (K ˆ ) := 0 (K ) ≤ (K

n−1  k=0

gˆ k (B) . √ k!ρˆ k+1 (K )

Now Theorem 3.2 implies ˆ )<1 Corollary 3.3 Let all the zeros of det K (z) be in C− and the conditions (3.2), and q (K hold. Then (3.6) is exponentially stable. Thanks to the definition of g(A), for all ω ∈ R one can write g(B(iω)) ≤ N2 (B(iω)) ≤ ˜ + N2 (C). If |ω|N2 ( A) ˜ K (z) = z I − z Ae

−z h˜

η +C

e−zs dμ(s),

(3.11)

0

then by (3.7) ˜

g(B(iω)) = g(ieiωh B(iω)) ⎡ ⎛ η ⎞⎤  η 1 ⎣ ˜ ˜ |ω|N2 ( A˜ − A˜ ∗ ) + N2 ⎝ e−iω(s−h) dμ(s)C + eiω(s−h) dμ(s)C ∗ ⎠⎦ . ≤ √ 2 0

123

0

Differ Equ Dyn Syst

So in the case (3.11) we get g(K ˆ )≤

v0 √ N ( A˜ 2 2

− A˜ ∗ ) +

√

2N2 (C). ˜ In the rest of this section we suppose that A and C commute. So the eigenvalues of K (z) for a fixed z can be written as η λ j (K (z)) = z − z

e

−zs

˜ + d μ(s)λ ˜ j ( A)

0

η

e−zs dμ(s)λ j (C),

0

˜ + g(C). So and, in addition, according to (3.8), g(B(iω)) ≤ |ω|g( A) ˜ + g(C) (ω ∈ [−v0 , v0 ]). ˜ := v0 g( A) g(B(iω)) ≤ g(C, A) ˜ (k = 1, ..., n) are positive and put Furthermore, suppose λk (C) and λk ( A) vk =

2λk (C) . ˜ 1 − λk ( A)

If η ηvk < π/2

and

dk (μ, μ) ˜ := λk (C)

˜ cos (τ vk )dμ − vk λk ( A)

0

η sin (τ vk )d μ˜ > 0 0

(3.12) (k = 1, ..., n), then by Corollary 3 from [13] all the characteristic values of K are in C− and inf |λ j (K (iω))| ≥ d˜com := min dk (μ, μ) ˜ ( j = 1, ... , n).

ω∈R

(3.13)

k

So ˆ ) ≤ com (K ) := (K

n−1 k  ˜ g (C, A) √ k+1 . k!d˜com k=0

Now Corollary 3.3 implies Corollary 3.4 Let A˜ and C be commuting matrices with positive eigenvalues. Let the conditions (3.2) and (3.12) and qcom (K ) < 1 be fulfilled. Then (3.6) is exponentially stable.

Example Consider the system ˜ + y˙ j (t) − a˜ y˙ j (t − h)

2  k=1

c jk yk (t) =

2 

a jk (t)yk (t − h(t)) ( j = 1, 2; t ≥ 0), (4.1)

k=1

where 0 < a˜ < 1, c jk are real constants, A(t) = (a jk (t))2j,k=1 is piece-wise continuous, and ˙ < 1. 0 ≤ h(t) ≤ h˜ and h(t) ˜

˜

˜ −z h I − C and So K (z) = z(1 − ae ˜ −z h )I + C with C = (c jk )2j,k=1 , B(z) = z ae by (3.7), g(B(z)) = g(C) ≤ gC = |c12 − c21 |. The eigenvalues of K (z) for a fixed z are ˜ k (C) λ j (K (z)) = z −z ae ˜ −z h +λ j (C). Suppose λk (C) (k = 1, 2) are positive and put vk = 2λ1− a˜ . If ˜ k < π/2 hv

and

˜ k) > 0 dk = λk (C) − vk a˜ sin (hv

(k = 1, 2),

(4.2)

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Differ Equ Dyn Syst

then by (3.13), the characteristic values of K are in C− , and

  1 gC ˆ ) ≤ 1 := inf |λk (K (iω))| ≥ dˆ := min dk and (K 1+ . ω∈R k=1,2 dˆ dˆ

In the considered example E 0 y(t) = A(t)y(t − h(t))

and

q = E 0  L 2 (−h,∞)→L 2 (0,∞) ≤ ˜

supt A(t)n  . ˙ inf t 1 − h(t)

Thanks to Corollary 3.3 we can assert that the zero solution to system (4.1) is exponentially stable provided the conditions (4.2) and q1 < 1 hold. Comments The strong inequality in (1.7) cannot be replaced by the condition V1 ≤ 1. Indeed, consider the equation   d 10 [y(t) − Ay(t)] = −y(t), where A = 0a dt with 0 < a < 1, then V1 = 1 and for the constant initial function φ(t) = column (1, 0) (t ≤ 0) that equation do not have a solution. The study of the neutral functional differential equations is essentially based on the questions of the action of difference operators of the type (E f )(t) = A(t) f (τ (t)), τ (t) ≤ t (t ≥ 0) with in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions. The problem of action of the pointed operators is important in the context of this paper. In order to achieve the action of the considered operator in the space of essentially bounded or summable functions one has assume that mes{t : τ (t) = c} = 0 for every constant c. This is necessary condition. cf. the interesting papers [8] and [9]. Sufficient conditions are written in these two papers via the Radon derivatives. In this connection, see also the book [1], where the similar problems are discussed. In our paper the condition mes{t : τ (t) = c} = 0 is fulfilled, since τ (t) = t − h(t) and −∞ < h˙ j (t) < 1. Note that the idea to rewrite the neutral differential equation of the type (1.1) in the form x˙ = (I − E 1 )−1 E 0 x is not new. The similar idea was used in the paper [7] to study boundary value problems and oscillation properties of scalar equations. Then in [16], this idea has been used to prove the non-oscillations of a first order neutral equation. In [16] also various tests of the exponential stability are presented. The material of the paper [3] is close to the part of the present paper about perturbations of autonomous systems. In that paper the exponential stability of systems of neutral delay differential equations is studied. For more details see the deep book by Kurbatov [21].

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Differ Equ Dyn Syst 2. Azbelev, N.V., Simonov, P.M.: Stability of Differential Equations with Aftereffects. Stability Control Theory Methods Applications, vol. 20. Taylor & Francis, London (2003) 3. Bainov, D., Domoshnitsky, A.: Non-negativity of the Cauchy matrix and exponential stability of a neutral type system of functional–differential equations. Extracta Mathematicae 8(1), 75–82 (1993) 4. Berezansky, L., Braverman, E.: On exponential stability of a linear delay differential equation with an oscillating coefficient. Appl. Math. Lett. 22(12), 1833–1837 (2009) 5. Berezansky, L., Braverman, E., Domoshnitsky, A.: Stability of the second order delay differential equations with a damping term. Differ. Equ. Dyn. Syst. 16(3), 185–205 (2008) 6. Daleckii, Yu L., Krein, M.G.: Stability of solutions of differential equations in Banach space. Am. Math. Soc., Providence, RI (1971) 7. Domoshnitsky, A.: Extension of Sturm’s theorem to equations with time-lag. Differentsial’nye Uravnenija 19, 1475–1482 (1983). in Russian 8. Drakhlin, M.E.: Operator of the internal superposition in the space of summable functions. Izv. VUZov, Math. 5, 18–23 (1986). in Russian 9. Drakhlin, M.E., Plyshevskaya, T.K.: To the theory of functional differential equations. Differentsialnye Uravnenia 14, 1347–1361 (1978). in Russian 10. Dunford, N., Schwartz, J.T.: Linear Operators, Part I. Interscience Publishers Inc., New York (1966) 11. Gil’, M.I.: Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, vol. 1830. Springer, Berlin (2003) 12. Gil’, M.I.: The L p - version of the generalized Bohl–Perron principle for vector equations with delay. Int. J. Dyn. Syst. Differ. Equ. 3(4), 448–458 (2011) 13. Gil’, M.I.: Exponential stability of nonlinear neutral type systems. Arch. Control Sci. 22(2), 125–143 (2012) 14. Gil’, M.I.: The generalized Bohl–Perron principle for the neutral type vector functional differential equations. Math. Control Signals Syst. (MCSS) 25(1), 133–145 (2013) 15. Gil’, M.I.: Stability of Vector Differential Delay Equations. Birkhäuser, Basel (2013) 16. Gusarenko, S.A., Domoshnitskii, A.I.: Asymptotic and oscillation properties of first order linear scalar functional–differential equations. Differentsial’nye Uravnenija 25(12), 2090–2103 (1989). In Russian 17. Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York (1966) 18. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) 19. Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1999) 20. Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, London (1986) 21. Kurbatov, V.: Functional Differential Operators and Equations. Kluwer Academic Publishers, Dordrecht (1999)

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