Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients

Volume 2010, Article ID 956121,20pages doi:10.1155/2010/956121

Research Article

Exponential Stability and Estimation of Solutions

of Linear Differential Systems of Neutral Type with

Constant Coefficients

J. Ba ˇstinec,

_{J. Dibl´ık,}

_{D. Ya. Khusainov,}

_{and A. Ryvolov ´a}

Brno University of Technology, 61600 Brno, Czech Republic

2_{Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Veveˇr´ı331/95,}

Brno University of Technology, 60200 Brno, Czech Republic

3_{Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National}

University of Kyiv, Vladimirskaya Str., 64, 01033 Kyiv, Ukraine

Correspondence should be addressed to J. Dibl´ık,[email protected]

Received 6 July 2010; Accepted 12 October 2010

Academic Editor: Julio Rossi

Copyrightq2010 J. Baˇstinec et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the exponential-type stability of linear neutral delay differential systems with constant coefficients using Lyapunov-Krasovskii type functionals, more general than those reported in the literature. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. The approach developed is used to characterize the decay of solutionsby inequalities for the norm of an arbitrary solution and its derivativein the case of stability, as well as in a general case. Illustrative examples are shown and comparisons with known results are given.

1. Introduction

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay:

˙

xt Dxt˙ −τ Axt Bxt−τ, 1.1

wheret≥0 is an independent variable,τ >0 is a constant delay,A, B, andDaren×nconstant matrices, andx : −τ,∞ → Rn _{is a column vector-solution. The sign “·” denotes the}

solutionxxtof problem1.1,1.2on−τ,∞where

xt ϕt, xt ˙ ϕt,˙ t∈−τ,0 1.2

is defined in the classical sensewe refer, e.g., to1 as a function continuous on−τ,∞

continuously differentiable on−τ,∞except for pointsτp,p 0,1, . . ., and satisfying1.1 everywhere on0,∞except for pointsτp,p0,1, . . ..

The paper finds an estimate of the norm of the difference between a solutionxxt

of problem1.1,1.2and the steady statext≡0 at an arbitrary momentt≥0. LetFbe a rectangular matrix. We will use the matrix norm:

F:

λmax

FT_{F, 1.3}

where the symbolλmaxFTFdenotes the maximal eigenvalue of the corresponding square symmetric positive semidefinite matrix FT_{F. Similarly, λ}

minFTF denotes the minimal eigenvalue ofFT_{F. We will use the following vector norms:}

xt:

n

i1 x2

it,

xtτ: sup

−r≤s≤0

{xst},

xtτ,β: t

t−r

e−βt−s_xs2_d s,

1.4

whereβis a parameter.

The most frequently used method for investigating the stability of functional-differential systems is the method of Lyapunov-Krasovskii functionals2,3 . Usually, it uses positive definite functionals of a quadratic form generated from terms of1.1and the integral

over the interval of delay4 of a quadratic form. A possible form of such a functional is then

xt−Dxt−τ THxt−Dxt−τ

t

t−τ

xTsGxsds, 1.5

whereHandGare suitablen×npositive definite matrices.

Regarding the functionals of the form1.5, we should underline the following. Using a functional1.5, we can only obtain propositions concerning the stability. Statements such as that the expression

t

t−τ

xTsGxsds 1.6

Literature on the stability and estimation of solutions of neutral differential equations is enormous. Tracing previous investigations on this topic, we emphasize that a Lyapunov functionvx xT_{Hxhas been used to investigate the stability of systems1.1in5 see}

6 as well. The stability of linear neutral systems of type1.1, but with different delaysh1 andh2, is studied in1 where a functional

xtc1

t

t−h1

xsdsc2

t

t−h2

xs˙ ds 1.7

is used with suitable constants c1 and c2. In7, 8 , functionals depending on derivatives are also suggested for investigating the asymptotic stability of neutral nonlinear systems. The investigation of nonlinear neutral delayed systems with two time dependent bounded delays in9 to determine the global asymptotic and exponential stability uses, for example, functionals

xTtP xt

0

−h1

xTtsQxsds

0

−h2 ˙

xTtsxt˙ sds,

e2γt_xT_{tP xt}

₀

−h1

e2γts_xT_tsQxsds

₀

−h2

e2γts_x˙T_{tsxt˙ sds,}

1.8

wherePandQare positive matrices andγis a positive scalar.

Delay independent criteria of stability for some classes of delay neutral systems are developed in10 . The stability of systems1.1with time dependent delays is investigated in11 . For recent results on the stability of neutral equations, see9,12 and the references therein. The works in 12, 13 deal with delay independent criteria of the asymptotical stability of systems1.1.

In this paper, we will use Lyapunov-Krasovskii quadratic type functionals of the dependent coordinates and their derivatives

V0xt, t xTtHxt

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds 1.9

andVxt, t ept_V

0xt, t , that is,

Vxt, t ept

xTtHxt

t

t−τ

e−βt−sxTsG1xs x˙2sG2x˙2s

ds

, 1.10

where xis a solution of1.1,βand p are real parameters, then×x matricesH,G1, and G2 are positive definite, and t > 0. The form of functionals 1.9 and 1.10is suggested by the functionals1.7-1.8. Although many approaches in the literature are used to judge the stability, our approach, among others, in addition to determining whether the system

initial interval. If, in the literature, estimates of solutions are given, then, as a rule, estimates of derivatives are not investigated.

To the best of our knowledge, the general functionals1.9 and1.10 have not yet been applied as suggested to the study of stability and estimates of solutions of1.1.

2. Exponential Stability and Estimates of the Convergence of

Solutions to Stable Systems

First we give two definitions of stability to be used later on.

Definition 2.1. The zero solution of the system of equations of neutral type 1.1 is called exponentially stable in the metricC0 _{if there exist constantsN}

i > 0,i 1,2 andμ >0 such

that, for an arbitrary solutionxxtof1.1, the inequality

xt ≤N1x0_τN2x˙ 0_τ e−μt 2.1

holds fort >0.

Definition 2.2. The zero solution of the system of equations of neutral type 1.1 is called exponentially stable in the metricC1_{if it is stable in the metricC}0_{and, moreover, there exist} constantsRi > 0,i 1,2, andν > 0 such that, for an arbitrary solutionxxtof1.1, the

inequality

xt˙ ≤R1x0_τR2x˙ 0_τ e−νt 2.2

holds fort >0.

We will give estimates of solutions of the linear system1.1on the interval 0,∞

using the functional1.9. Then it is easy to see that an inequality

λminHxt2

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs

ds

≤V0xt, t

≤λmaxHxt2

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs

ds

2.3

holds on0,∞. We will use an auxiliary 3n×3n-dimensional matrix:

SSβ, G1, G2, H

:

⎛ ⎜ ⎜ ⎝

−ATH−HA−G1−ATG2A −HB−ATG2B −HD−ATG2D −BT_H−BT_G

2A e−βτG1−BTG2B −BTG2D −DT_H−DT_G

2A −DTG2B e−βτG2−DTG2D

⎞ ⎟ ⎟

⎠,

depending on the parameterβand the matricesG1,G2,H. Next we will use the numbers

ϕH: λmaxH

λminH, ϕ1G1, H:

λmaxG1

λminH, ϕ2G2, H:

λmaxG2

λminH. 2.5

The following lemma gives a representation of the linear neutral system1.1on an interval

m−1τ, mτ in terms of a delayed system derived by an iterative process. We will adopt the customary notationk_iksOi 0 wherekis an integer,sis a positive integer, andOdenotes the function considered independently of whether it is defined for the arguments indicated or not.

Lemma 2.3. Letmbe a positive integer and t ∈ m−1τ, mτ. Then a solutionx xtof the initial problem1.1,1.2is a solution of the delayed system

˙

xt Dmxt˙ −mτ Axt DAB

m−1

i1

Di−1xt−iτ Dm−1Bxt−mτ 2.6

fort∈m−1τ, mτwherext−mτ ϕt−mτandxt˙ −mτ ϕt˙ −mτ.

Proof. Form1 the statement is obvious. Ift∈τ,2τ, replacingtbyt−τ, system1.1will turn into

˙

xt−τ Dxt˙ −2τ Axt−τ Bxt−2τ. 2.7

Substituting2.7into1.1, we obtain the following system of equations:

˙

xt D2xt˙ −2τ Axt DABxt−τ DBxt−2τ, 2.8

wheret∈τ,2τ. Ift∈2τ,3τ, replacingtbyt−τin2.7, we get

˙

xt−2τ Dxt˙ −3τ Axt−2τ Bxt−3τ. 2.9

We do one more iteration substituting2.9into2.8, obtaining

˙

xt D3xt˙ −3τ Axt DABxt−τ

DDABxt−2τ D2Bxt−3τ

2.10

fort∈2τ,3τ. Repeating this procedurem−1-times, we get the equation

˙

xt Dmxt˙ −mτ Axt DAB

m−1

i1

Di−1xt−iτ Dm−1Bxt−mτ 2.11

Remark 2.4. The advantage of representing a solution of the initial problem 1.1,1.2as a solution of2.6is that, although2.6remains to be a neutral system, its right-hand side does not explicitly depend on the derivative ˙xtfort∈0, mτ depending only on the derivative of the initial function on the initial interval−τ,0.

Now we give a statement on the stability of the zero solution of system 1.1 and estimates of the convergence of the solution, which we will prove using Lyapunov-Krasovskii functional1.9.

Theorem 2.5. Let there exist a parameterβ > 0and positive definite matricesG1,G2,H such that matrixSis also positive definite. Then the zero solution of system1.1is exponentially stable in the metricC0_{. Moreover, for the solutionxxtof1.1,1.2the inequality}

xt ≤ϕHx0

τϕ1G1, Hx0τ

τϕ2G2, Hx˙ 0τ

e−γt/2 2.12

holds on0,∞whereγ≤γ0:minβ, λminS/λmaxH.

Proof. Lett >0. We will calculate the full derivative of the functional1.9along the solutions of system1.1. We obtain

d

dtV0xt, t Dxt˙ −τ Axt Bxt−τ

T_Hxt

xTtHDxt˙ −τ Axt Bxt−τ

xTtG1xt−e−βτxTt−τG1xt−τ

x˙TtG2xt˙ −e−βτx˙Tt−τG2xt˙ −τ

−β

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

2.13

For ˙xt, we substitute its value from1.1to obtain

d

dtV0xt, t Dxt˙ −τ Axt Bxt−τ

T_Hxt

xTtHDxt˙ −τ Axt Bxt−τ

xTtG1xt−e−βτxTt−τG1xt−τ

Dxt˙ −τ Axt Bxt−τ TG2Dxt˙ −τ Axt Bxt−τ

−e−βτx˙Tt−τG2xt˙ −τ

−β

_t

t−τ

e−βt−s_xT_sG

1xs x˙TsG2xs˙

ds.

Now it is easy to verify that the last expression can be rewritten as

d

dtV0xt, t −

xTt, xTt−τ,x˙Tt−τ

×

⎛ ⎜ ⎜ ⎝

−AT_H−HA−G

1−ATG2A −HB−ATG2B −HD−ATG2D −BT_H−BT_G

2A e−βτG1−BTG2B −BTG2D −DTH−DTG2A −DTG2B e−βτG2−DTG2D

⎞ ⎟ ⎟ ⎠

×

⎛ ⎜ ⎜ ⎝

xt xt−τ

˙

xt−τ

⎞ ⎟ ⎟

⎠−β

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds

2.15

or

d

dtV0xt, t −

xTt, xTt−τ,x˙Tt−τ×S×

⎛ ⎜ ⎜ ⎝

xt xt−τ

˙

xt−τ

⎞ ⎟ ⎟ ⎠

−β

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

2.16

Since the matrixSwas assumed to be positive definite, for the full derivative of Lyapunov-Krasovskii functional1.9, we obtain the following inequality:

d

dtV0xt, t ≤ −λminS

xt2_xt−τ2_{xt˙ −τ}2

−β

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

2.17

We will study the two possible casesdepending on the positive value ofβ: either

β > λminS

λmaxH

2.18

is valid or

β≤ λminS

λmaxH

2.19

1Let2.18be valid. From2.3follows that

−xt2_{≤ −} 1

λmaxHV0xt, t

1 λmaxH

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

2.20

We use this expression in2.17. SinceλminS>0, we obtainomitting termsxt−τ2and xt˙ −τ2

d

dtV0xt, t ≤λminS

×

− 1

λmaxHV0xt, t 1

λmaxH

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds

−β

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds

2.21

or

d

dtV0xt, t ≤ −

λminS

λmaxHV0xt, t

−

β− λminS λmaxH

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

2.22

Due to2.18we have

d

dtV0xt, t ≤ −

λminS

λmaxHV0xt, t . 2.23

Integrating this inequality over the interval0, t, we get

V0xt, t ≤V0x0,0 exp

−λminS λmaxH ·t

≤V0x0,0 e−γ0t. 2.24

2Let2.19be valid. From2.3we get

−

_t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds≤ −V0xt, t λmaxHxt2. 2.25

We substitute this expression into inequality2.17. SinceλminS > 0, we obtainomitting termsxt−τ2andxt˙ −τ2

d

dtV0xt, t ≤ −λminSxt

2_β−V

0xt, t λmaxHxt2

or

d

dtV0xt, t ≤ −βV0xt, t −

λminS−βλmaxH

xt2_{. 2.27}

Since2.19holds, we have

d

dtV0xt, t ≤ −βV0xt, t . 2.28

Integrating this inequality over the interval0, t, we get

V0xt, t ≤V0x0,0 e−βt≤V0x0,0 e−γ0t. 2.29

Combining inequalities2.24,2.29, we conclude that, in both cases2.18,2.19, we have

V0xt, t ≤V0x0,0 e−γ0t≤V0x0,0 e−γt 2.30

and, obviouslysee1.9,

V0x0,0 ≤λmaxHx02λmaxG1x02τ,βλmaxG2x˙ 02τ,β. 2.31

We use inequality2.30to obtain an estimate of the convergence of solutions of system

1.1. From2.3follows that

xt2_≤ 1 λminH

λmaxHx02λmaxG1x02τ,βλmaxG2x˙ 02τ,β

e−γt 2.32

orbecause√ab≤√a√bfor nonnegativeaandb

xt ≤ϕHx0

ϕ1G1, Hx0τ,β

ϕ2G2, Hx˙ 0τ,β

e−γt/2. 2.33

The last inequality implies

xt ≤ϕHx0

τϕ1G1, Hx0τ

τϕ2G2, Hx˙ 0τ

e−γt/2. 2.34

Theorem 2.6. Let the matrixDbe nonsingular andD < 1. Let the assumptions ofTheorem 2.5 withγ <2/τln1/Dandγ≤γ0be true. Then the zero solution of system1.1is exponentially stable in the metricC1_{. Moreover, for a solutionxxtof 1.1,1.2, the inequality}

xt˙ ≤

B DM

ϕH τϕ1G1,H

x0_τ

1M

τϕ2G2,H

xt˙ _τ

e−γτ/2

2.35

where

M ADABeγτ/2_{1− De}γτ/2−1 _2.36

holds on0,∞.

Proof. Lett >0. Then the exponential stability of the zero solution in the metricC0_{is proved} inTheorem 2.5. Now we will show that the zero solution is exponentially stable in the metric

C1_{as well. As follows fromLemma 2.3, for derivative ˙xt, the inequality}

xt˙ ≤ Dmx˙ 0_τDm−1Bx0_τAxt

DAB

m−1

i1

Di−1xt−iτ

2.37

holds ift ∈ m−1τ, mτ. We estimate xtandxt−iτusing2.12and inequality x0 ≤ x0_τ. We obtain

xt˙ ≤ Dmx˙ 0_τDm−1Bx0_τ

A

ϕH τϕ1G1, H

x0_τ

τϕ2G2, Hx˙ 0τ

e−γt/2

DABD−1

ϕH τϕ1G1, H

x0_τ

τϕ2G2, Hx˙ 0τ

×

_m−1

i1

Dieγiτ/2

e−γt/2.

2.38

Since

m−1

i1

Dieγiτ/2<

∞

i1

Dieγiτ/2 De

γτ/2

inequality2.38yields

xt˙ ≤ Dmx˙ 0_τDm−1Bx0_τ

ADABD−1 De

γτ/2 1− Deγτ/2

×ϕH τϕ1G1, H

x0_τ

τϕ2G2, Hx˙ 0τ

e−γt/2

Dmx˙ 0_τDm−1Bx0_τ

M

ϕH τϕ1G1, H

x0_τ

τϕ2G2, Hx˙ 0τ

e−γt/2.

2.40

Becauset∈m−1τ, mτ, we can estimate

Dm

1 D ₋m < 1 D ₋t/τ exp −t τ ln 1 D ,

Dm−1 1 DD

m_< 1

Dexp

−t τ ln 1 D . 2.41 Then

Dmx˙ 0_τDm−1Bx0_τ ≤

x˙ 0_τ B

Dx0τ

exp −t τln 1 D

. 2.42

Now we get from2.40

xt˙ ≤

x˙ 0_τ B

Dx0τ

exp −t τ ln 1 D M

ϕH τϕ1G1, H

x0_τ

τϕ2G2, Hx˙ 0τ

e−γt/2.

2.43 Since exp −t τ ln 1 D ≤exp −γt 2

, 2.44

the last inequality implies

xt˙ ≤

B DM

ϕH τϕ1G1, H

x0_τ

1M

τϕ2G2, H

x˙ 0_τ

e−γt/2.

The positive numbermcan be chosen arbitrarily large. Therefore, the last inequality holds for every t > 0. We have obtained inequality 2.35 so that the zero solution of 1.1 is exponentially stable in the metricC1_.

3. Estimates of Solutions in a General Case

Now we will estimate the norms of solutions of1.1and the norms of their derivatives in the case of the assumptions ofTheorem 2.5orTheorem 2.6being not necessarily satisfied. It means that the estimates derived will cover the case of instability as well. For obtaining such type of results we will use a functional of Lyapunov-Krasovskii in the form1.10. This functional includes an exponential factor, which makes it possible, in the case of instability, to get an estimate of the “divergence” of solutions. Functional1.10is a generalization of1.9 because the choicep0 givesVxt, t V0xt, t . For1.10the estimate

ept

λminHxt2

_t

t−τ

e−βt−s_xT_sG

1xs x˙2sG2x˙2s

ds

≤Vt, t

≤ept

λmaxHxt2

t

t−τ

e−βt−sxTsG1xs x˙2sG2x˙2s

ds

3.1

holds. We define an auxiliary 3n×3nmatrix

S∗S∗β, G1, G2, H, p

:

⎛ ⎜ ⎜ ⎝

−AT_H−HA−G

1−ATG2A−pH −HB−ATG2B −HD−ATG2D −BT_H−BT_G

2A e−βτG1−BTG2B −BTG2D −DT_H−DT_G

2A −DTG2B e−βτG2−DTG2D

⎞ ⎟ ⎟ ⎠

3.2

depending on the parametersp,βand the matricesG1,G2, andH. The parameterpplays a significant role for the positive definiteness of the matrixS∗. Particularly, a proper choice of

p 0 can cause the positivity ofS∗. In the following,ϕH,ϕ1G1, Handϕ2G2, H, have the same meaning as in Part 2. The proof of the following theorem is similar to the proofs of Theorems2.5and 2.6and its statement in the case ofp 0 exactly coincides with the statements of these theorems. Therefore, we will restrict its proof to the main points only.

Theorem 3.1. ALetpbe a fixed real number,βa positive constant, andG1,G2, andHpositive definite matrices such that the matrixS∗is also positive definite. Then a solutionxxtof problem

1.1,1.2satisfies on0,∞the inequality

xt ≤ϕHx0

τϕ1G1, Hx0τ

τϕ2G2, Hx˙ 0τ

e−γt/2, 3.3

BLet the matrixD be nonsingular andD < 1. Let all the assumptions of part (A) with

γ < 2/τln1/Dand γ ≤ γ∗ be true. Then the derivative of the solutionx xtof problem

1.1,1.2satisfies on0,∞the inequality

xt˙ ≤

B DM

ϕH τϕ1G1, H

x0_τ

1M

τϕ2G2, H

x˙ 0_τ

e−γt/2,

3.4

whereMis defined by2.36.

Proof. Lett >0. We compute the full derivative of the functional1.10along the solutions of

1.1. For ˙xt, we substitute its value from1.1. Finally we get

d

dtVxt, t −e

pt_xT_{t, x}T_t−τ,x˙T_t−τ×S∗

×

⎛ ⎜ ⎜ ⎝

xt xt−τ

˙

xt−τ

⎞ ⎟ ⎟

⎠−eptβ−p

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds. 3.5

Since the matrixS∗is positive definite, we have

d

dtVxt, t ≤ −λminS

∗_ept_xt2_xt−τ2_{xt˙ −τ}2

−eptβ−p

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds.

3.6

Now we will study the two possible cases: either

β−p > λminS∗

λmaxH

3.7

is valid or

β−p≤ λminS

∗

λmaxH

3.8

holds.

1Let3.7be valid. SinceλminS∗>0, from inequality3.1follows that

−eptxt2 ≤ − 1

λmaxHVxt, t

ept λmaxH

_t

t−τ

e−βt−s_xT_sG

1xs x˙TsG2xs˙

ds.

We use this inequality in3.6. We obtain

d

dtVxt, t ≤ −

λminS∗

λmaxHVxt, t −e

pt

β−p− λminS

∗

λmaxH

×

_t

t−τ

e−βt−s_xT_sG

1xs x˙TsG2xs˙

ds.

3.10

From inequality3.7we get

d

dtVxt, t ≤ −

λminS∗

λmaxHVxt, t . 3.11

Integrating this inequality over the interval0, t, we get

Vxt, t ≤Vx0,0 exp

−λminS∗ λmaxHt

≤Vx0,0 e−γ∗−pt_{. 3.12}

2Let3.8be valid. From inequality3.1we get

−ept

t

t−τ

e−βt−sxTsG1xs x˙TsG2xs˙

ds≤ −Vxt, t eptλmaxHxt2. 3.13

We use this inequality in3.6again. SinceλminS∗>0, we get

d

dtVxt, t ≤ −

β−pVxt, t −λminS∗−

β−pλmaxH

eptxt2. 3.14

Because the inequality3.8holds, we have

d

dtVxt, t ≤ −

β−pVxt, t . 3.15

Integrating this inequality over the interval0, t, we get

Vxt, t ≤Vx0,0 e−β−pt≤Vx0,0 e−γ∗−pt. 3.16

Combining inequalities3.12,3.16, we conclude that, in both cases3.7,3.8, we have

From this, it follows

ept

xT_tHxt

_t

t−τ

e−βt−s_xT_sG

1xs x˙TsG2xs˙

ds

≤

xT_0Hx0

₀

−τ

eβs_xT_sG

1xs x˙TsG2xs˙

ds

e−γ∗−pt_,

ept_λ

minHxt2

≤λmaxHx02λmaxG1x02β,τ λmaxG2x˙ 02β,τ

e−γ∗−pt.

3.18

From the last inequality we derive inequality3.3 in a way similar to that of the proof of

Theorem 2.5. The inequality to estimate the derivative3.4can be obtained in much the same way as in the proof ofTheorem 2.6.

Remark 3.2. As can easily be seen fromTheorem 3.1, partA, if

pλminS∗ λmaxH >

0, 3.19

we deal with an exponential stability in the metricC0_{. If, moreover, partBholds and3.19} is valid, then we deal with an exponential stability in the metricC1_.

4. Examples

In this part we consider two examples. Auxiliary numerical computations were performed by using MATLAB & SIMULINK R2009a.

Example 4.1. We will investigate system1.1wheren2,τ1,

D

0.5 0

0 0.5

, A

−1 0.1 0.1 −1

, B

0.1 0

0 0.1

, 4.1

that is, the system

˙

x1t 0.5 ˙x1t−1−x1t 0.1x2t 0.1x1t−1,

˙

x2t 0.5 ˙x2t−1 0.1x1t−x2t 0.1x2t−1,

4.2

with initial conditions1.2. Setβ0.1 and

G1

1 0

0 1

, G2

1 1

1 3

, H

2 0.1

0.1 5

For the eigenvalues of matricesG1,G2, andH, we getλminG1 λmaxG1 1,λminG2 . 0.5858, λmaxG2 . 3.4142, λminH . 1.9967, and λmaxH . 5.0033. The matrix S Sβ, G1, G2, Htakes the form

S .

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2.1500 −1.1100 −0.1100 0.0600 −0.5500 0.3000 −1.1100 6.1700 0.0800 −0.2100 0.4000 −1.0500 −0.1100 0.0800 0.8948 −0.0100 −0.0500 −0.0500 0.0600 −0.2100 −0.0100 0.8748 −0.0500 −0.1500 −0.5500 0.4000 −0.0500 −0.0500 0.6548 0.6548

0.3000 −1.0500 −0.0500 −0.1500 0.6548 1.9645

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

4.4

and λminS . 0.1445. Because all the eigenvalues are positive, matrix S is positive definite. Since all conditions ofTheorem 2.5are satisfied, the zero solution of system4.2 is asymptotically stable in the metricC0_{. Further we have}

ϕH. 5.0033

1.9967

.

2.5058, ϕ1G1, H. 1 1.9967

.

0.5008,

ϕ2G2, H. 3.4142 1.9967

.

1.7099, γ0min

0.1, 0.1445

5.0033

.

min0.1,0.0289 0.0289,

A1.1, B0.1, D0.5, DAB0.45, M2.0266.

4.5

Since γ0 < 2/τln1/D . 1.3863, all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of4.2,35is asymptotically stable in the metricC1_{. Finally,} from2.12and2.35follows that the inequalities

xt ≤√2.5058x0√0.5008x0₁√1.7099x˙ 0₁e−0.0289t/2 . ₁

.5830x00.7077x0₁1.3076x˙ 0₁ e−0.0289t/2,

xt˙ ≤0.22.0266√2.5058√0.5008x0₁

12.0266√1.7099x˙ 0₁e−0.0289t/2 . ₄

.8422x0₁3.6500x˙ 0₁ e−0.0289t/2

4.6

hold on0,∞.

Example 4.2. We will investigate system1.1wheren2,τ1,

D

0.1 0

0 0.1

, A

−3 −2

1 0

, B

0 0.6213

0.6213 0

that is, the system

˙

x1t 0.1 ˙x1t−1−3x1t−2x2t 0.6213x2t−1,

˙

x2t 0.1 ˙x2t−1 1x1t 0.6213x1t−1,

4.8

with initial conditions1.2. Setβ0.1 and

G1

0.5 0.1

0.1 0.1

, G2

0.1 0

0 0.1

, H

0.6 0.4

0.4 0.6

. 4.9

For the eigenvalues of matricesG1,G2, andH, we getλminG1. 0.0764,λmaxG1. 0.5236, λminG2 λmaxG2 0.1λminH 0.2, andλmaxH 1. The matrixS Sβ, G1, G2, H takes the form

S .

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1.3000 1.1000 −0.3106 −0.1864 −0.0300 −0.0500

1.1000 1.1000 −0.3728 −0.1243 −0.0200 −0.0600

−0.3106 −0.3728 0.4138 0.0905 0 −0.0062 −0.1864 −0.1243 0.0905 0.0519 −0.0062 0 −0.0300 −0.0200 0 −0.0062 0.0895 0 −0.0500 −0.0600 −0.0062 0 0 0.0895

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

4.10

and λminS . 0.00001559. Because all eigenvalues are positive, matrix S is positive definite. Since all conditions ofTheorem 2.5are satisfied, the zero solution of system4.8 is asymptotically stable in the metricC0_{. Further we have}

ϕH 1

0.2 5, ϕ1G1, H

. 0.5236

0.2

.

2.618, ϕ2G2, H 0.1 0.2 0.5,

γ0. min0.1, 0.00001559 0.00001559,

A. 3.7025, B. 0.6213, D0.1, DAB. 0.8028, M. 4.5945. 4.11

2.12and2.35follows that the inequalities

xt ≤√5x0√2.618x0₁√0.5 x˙ 0₁e−0.00001559t/2

. ₂

.2361x01.6180x0₁0.7071x˙ 0₁ e−0.00001559t/2,

xt˙ ≤6.2134.5945√5√2.618x0₁

14.5945√0.5x˙ 0₁e−0.00001559t/2

. _23.9206x0

14.2488x˙ 01 e−0.00001559t/2

4.12

hold on0,∞.

Remark 4.3. In12 an example can be found similar toExample 4.2with the same matrices

A,D, arbitrary constant positiveτ, and with a matrix

BBα

0 α

α 0

, 4.13

whereαis a real parameter. The stability is established for|α|<0.4. In recent paper13 , the stability of the same system is even established for|α|<0.533.

Comparing these particular results with Example 4.2, we see that, in addition to stability, our results imply the exponential stability in the metricC0_{as well as in the metricC}1_. Moreover, we are able to prove the exponential stabilityinC0_{as well as inC}1_{inExample 4.2} with the matrix B Bα for|α| ≤ 0.6213 and for an arbitrary constant delay τ. The latter

statement can be explained easily—for an arbitrary positiveτ, we setβ0.1/τ. Calculating the characteristic equation for the matrixSwhereBis changed byBαwe get

P6λ: 6

i0

piαλi0, 4.14

where

p6α −1,

p5α −0.2α2−3.1219,

p4α −0.01α4−1.3105α22.0830,

p3α −0.0998α40.5717α2−0.4943,

p2α −0.0366α4−0.096828α20.053858,

p1α −0.004204382α40.0073α2−0.0028,

p0α −0.00015392α4−0.00020116α20.000059723.

It is easy to verify that−1ipiα>0 fori0,1, . . . ,6 and|α| ≤0.6213, and for the equation

P₆∗λ P6−λ 6

i0

p∗_iαλi_{0, 4.16}

we havep∗_iα −1ipiα>0. Then, due to the symmetry of the real matrixS, we conclude

that, by Descartes’ rule of signs, all eigenvalues ofSi.e., all roots ofP6λ 0are positive. This means that the exponential stabilityin the metricC0 _{as well as in the metricC}1 _for |α| ≤0.6213 is proved. Finally, we note that the variation ofαwithin the interval indicated or the choiceβ 0.1/τ does not change the exponential stability having only influence on the form of the final inequalities forxtandxt˙ .

5. Conclusions

In this paper we derived statements on the exponential stability of system1.1as well as on estimates of the norms of its solutions and their derivatives in the case of exponential stability and in the case of exponential stability being not guaranteed. To obtain these results, special Lyapunov functionals in the form 1.9 and 1.10 were utilized as well as a method of constructing a reduced neutral system with the same solution on the intervals indicated as the initial neutral system 1.1. The flexibility and power of this method was demonstrated using examples and comparisons with other results in this field. Considering further possibilities along these lines, we conclude that, to generalize the results presented to systems with bounded variable delay τ τt, a generalization is needed of Lemma 2.3to the above reduced neutral system. This can cause substantial difficulties in obtaining results which are easily presentable. An alternative would be to generalize only the part of the results related to the exponential stability in the metric C0 _{and the} related estimates of the norms of solutions in the case of exponential stability and in the case of the exponential stability being not guaranteed omitting the case of exponential stability in the metric C1 _{and estimates of the norm of a derivative of solution. Such an} approach will probably permit a generalization to variable matricesA At,B Bt,

DDtand to a variable delayττtor to two different variable delays. Nevertheless, it seems that the results obtained will be very cumbersome and hardly applicable in practice.

Acknowledgments

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