21

Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

  • Upload
    others

  • View
    5

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

� �������� � �� � � ����� � �� ����� �� ������������! #"%$'&($*)+�! -,/.10 ��2( �$*34��2()65�798�:�;<8�=?>@&A34��2()65

BDC�EF>G8�$*8G2()6�IHJ$'K<2()+3L��2M$'KN=O>P&Q2R)S34$T:9$UH6�WVKXVYKX2M$U34K

Z\[^](_*`ba�`dcfehg*]Rci`kjml!n*lXo pD`dqX_'q<cr`dsig�th`bn'ci`bsigUuvqwn

x#y{z!|~}S���~�m�w���

��q<ci`bsig�jmq\](�*uJ_'q<c��GqXsrg*n'l�a�]Mj�g'qwa�]Mj�`^���q<��lAcmj��!�X���!� ���Uci`d[G�(�A�!�

Page 2: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,
Page 3: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

Exponential estimates for time delay systems

V. L. KharitonovControl Automatico

CINVESTAV-IPN A.P. 14-74007000 Mexico D.F., Mexico

E-mail: [email protected]

D. HinrichsenInstitut fur Dynamische Systeme

Universitat BremenD-28334 Bremen, Germany

E-mail: [email protected]

April 23, 2003

Abstract

In this paper we demonstrate how Lyapunov-Krasovskii functionals can be used to

obtain exponential bounds for the solutions of time-invariant linear delay systems.

Keywords: time delay system, exponential estimate, Lyapunov-Krasovskii func-

tional.

1 Introduction

The objective of this note is to describe a systematic procedure of constructing quadraticLyapunov functionals for (exponentially) stable linear delay systems in order to obtainexponential estimates for their solutions.The procedure we propose is a counterpart to the well known method of deriving exponentialestimates for stable systems x = Ax by means of quadratic Lyapunov functions V (x) =〈x, Ux〉. Here U � 0 is the solution of a Lyapunov equation A∗U +UA = −W where W � 0is any chosen positive definite matrix. If 2ωU � W for some ω > 0 then

‖eAt‖ ≤ κ(U)1/2e−ωt, t ≥ 0 (1)

where ‖ · ‖ denotes the spectral norm and κ(U) = ‖U‖ ‖U−1‖ is the condition number of U ,see [6]. Note that this estimate guarantees not only a uniform decay rate ω for all solutionsof x = Ax but also a bound on the transients of the system.It is surprising that a similar constructive method does not exist for delay systems. It istrue, there exists an operator theoretic version of Lyapunov’s equation in the abstract semi-group theory of infinite dimensional time-invariant linear systems, see [2], but this does notprovide us with a constructive procedure. For constructive purposes more concrete Lya-punov functions must be considered. Since the fifties different types of Lyapunov functionshave been proposed for the stability analysis of delay systems , see the pioneering works ofRazumikhin [11] and Krasovskii [10]. Whereas Razumikhin [11] used Lyapunov type func-tions V (x(t)) depending on the current value x(t) of the solution, Krasovskii [10] proposedto use functionals V (xt) depending on the whole solution segment xt, i.e. the true state of

1

Page 4: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

the delay system. These functionals, which are defined on the space of (continuous) initialfunctions, are called Lyapunov-Krasovskii functionals. For a brief discussion of these twoapproaches and some historic comments, see [5, §5.5].The majority of results concerning Lyapunov’s direct method for delay systems provides suf-ficient criteria for stability and asymptotic stability. These results assume that Lyapunovfunctionals with certain properties are given, and so the exponential estimates derived bymeans of these functionals are not obtained in a constructive manner. For a constructiveprocedure converse results are needed which show how Lyapunov functionals of a specifictype can be constructed for a given class of stable delay equations. Such converse resultsare available for certain delay systems, see e.g. Halanay [4], but they do not abound as thesufficient stability criteria based on given Lyapunov functionals.Quadratic Lyapunov functionals have been proposed for time-invariant linear delay equa-tions by Repin [12], Datko [3], Infante and Castelan [8], Huang [7], Kharitonov and Zhabko[9]. However, with exception of the latter reference, these Lyapunov functionals cannot beused for deriving exponential estimates, if no additional a priori information is available.Infante and Castelan propose an interesting method of constructing exponential estimateswhose decay rate comes arbitrarily close to the spectral abscissa of the delay system. Buttheir method presupposes that a concrete exponential estimate is already available. Theconstants of the exponential estimate which they derive from a quadratic Lyapunov func-tional depend explicitely on the constants of the presupposed a priori estimate. (compare(3.6), (3.19) and (2.6) in [8]).In order to overcome this dependence on an a priori estimate we use a modified Lyapunov-Krasovskii functional introduced in [9]. This functional is constructed in a similar wayto that of Infante and Castelan but contains an additional integral term. As in [8] theconstruction is based on a solution of a matrix differential-difference equation on a finitetime interval satisfying additional symmetry and boundary conditions. The matrix bound-ary value problem plays, roughly speaking, a similar role for linear delay equations asLyapunov’s equation in the delay free case. Our differential-difference equation is morecomplicated than that in [8] since Infante and Castelan only considered the one delaycase. Since these matrix equations have only recently been discovered there is as yet nosystematic solution theory available. This will be a subject for future research. However,first algorithms have been developed for the solution of the corresponding matrix boundaryvalue problem.The note is organized as follows. After some preliminaries on delay systems in the nextsection we describe the construction of the Lyapunov-Krasovskii functional according to [9]in section 3. Section 4 contains the main results of this note. Finally, section 5 presentsan example illustrating the basic steps leading to an exponential estimate for a given ex-ponentially stable system with two (commensurate) delays.

2 Preliminaries

In this paper we consider time delay systems of the following form

dx(t)

dt= A0x(t) +

m∑

k=1

Akx(t − hk). (2)

2

Page 5: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

where A0, A1, ..., Am ∈ Rn×n are given matrices and 0 < h1 < ... < hm = h are positive

delays. For any continuous initial function ϕ : [−h, 0] → Rn there exists the unique solution,

x(t, ϕ), of (2) satisfying the initial condition

x(θ, ϕ) = ϕ(θ), θ ∈ [−h, 0].

If t ≥ 0 we denote by xt(ϕ) the trajectory segment

xt(ϕ) : θ 7→ x(t + θ, ϕ), θ ∈ [−h, 0].

Throughout this note we will use the Euclidean norm for vectors and the induced matrixnorm for matrices. The space of continuous initial functions C([−h, 0], Rn) is provided withthe supremum norm ‖ϕ‖

∞= maxθ∈[−h,0] ‖ϕ(θ)‖.

Definition 1 The system (2) is said to be exponentially stable if there exist σ > 0 andγ ≥ 1 such that for every solution x(t, ϕ), ϕ ∈ C([−h, 0], Rn) the following exponentialestimate holds

‖x(t, ϕ)‖ ≤ γe−σt ‖ϕ‖∞

, t ≥ 0. (3)

For simplicity we will call an exponentially stable system just stable. The matrix-valuedfunction K : [−h,∞] → R

n×n which solves the matrix differential equation

d

dtK(t) = A0K(t) +

m∑

j=1

AjK(t − hj), t ≥ 0,

with initial condition

K(t) = 0 for − h ≤ t < 0, K(0) = In,

is called the fundamental matrix of the system (2),here In is the identity matrix. It isknown that K(t) also satisfies the differential equation [1]

d

dtK(t) = K(t)A0 +

m∑

j=1

K(t − hj)Aj, t ≥ 0. (4)

The following result is known as the Cauchy formula for the solutions of system (2), see [1].

x(t, ϕ) = K(t)ϕ(0) +

m∑

j=1

∫ 0

−hj

K(t − hj − θ)Ajϕ(θ)dθ, t ≥ 0. (5)

Every column of K(t) is a solution of system (2), so if the system is exponentially stable,then the matrix satisfies an inequality of the form (3). As a consequence, the integral

U(τ ) =

∫∞

0

K>(t)WK(t + τ )dt (6)

is well defined for all τ ≥ −h and any matrix W ∈ Rn×n.

3

Page 6: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

3 Lyapunov-Krasovskii functionals

We will now construct quadratic Lyapunov functions for the system (2) in a similar wayas for linear differential equations without delays. In this latter case, given any stablesystem x = Ax, a quadratic Lyapunov function is determined in the following way. Foran arbitrarily chosen quadratic function w(x) = x>Wx with positive definite W � 0 oneconstructs a quadratic function v(x) = x>Ux, U � 0 such that

dv(x(t))/dt = −w(x(t)), t ∈ R (7)

for every solution, x(t), of x = Ax. Function v(x) = x>Ux satisfies (7) if and only if U isthe (uniquely determined, positive definite) solution of the Lyapunov equation

A∗U + UA = −W. (8)

Analogously we choose for the delay system (2) positive definite n× n matrices W0, W1,...,W2m and consider the following functional on C([−h, 0], Rn)

w(ϕ) = ϕ>(0)W0ϕ(0) +

m∑

k=1

ϕ>(−hk)Wkϕ(−hk) +

m∑

k=1

∫ 0

−hk

ϕ>(θ)Wm+kϕ(θ)dθ, (9)

where ϕ ∈ C([−h, 0], Rn) is arbitrary. If the system (2) is exponentially stable, then thereexists a unique quadratic functional v : C([−h, 0], Rn) → R such that t 7→ v(xt(ϕ)) isdifferentiable on R+ and

dv(xt(ϕ))

dt= −w(xt(ϕ)), t ≥ 0 (10)

for all solutions x(t, ϕ) of (2), ϕ ∈ C([−h, 0], Rn) [9]. Functional v(·) is called the Lyapunov-Krasovskii functional associated with (9). It has been shown in [9] that the functional isgiven by

v(ϕ) =ϕ>(0)U(0)ϕ(0) +

m∑

k=1

2ϕ>(0)

∫ 0

−hk

U(−hk − θ)Akϕ(θ)dθ+

+

m∑

k=1

m∑

j=1

∫ 0

−hk

ϕ>(θ2)A>

k

[∫ 0

−hj

U(θ2 − θ1 + hk − hj)Ajϕ(θ1)dθ1

]dθ2+

+m∑

k=1

∫ 0

−hk

ϕ>(θ) [Wk + (hk + θ)Wm+k]ϕ(θ)dθ

(11)

where

U(τ ) =

∫∞

0

K>(t)

[W0 +

m∑

k=1

(Wk + hkWm+k)

]K(t + τ)dt, τ ≥ −h. (12)

Note that by exponential stability of (2) the matrix U(τ ) is well defined for all τ ≥ −h andis of the form (6) with

W = W0 +

m∑

k=1

(Wk + hkWm+k) . (13)

We call U(τ ) the Lyapunov matrix function for the system (2) associated with the functionalw(·) (9). Note that the first (1+m+m2) terms in (11) are completely determined by U and

4

Page 7: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

hence only depend upon the weighted sum W of the positive definite matrices Wk. However,the last m terms in (11) depends on the individual Wk’s. We will see later that all theseterms are needed in order to derive exponential estimates for (2) by means of the abovequadratic functionals. For the case of a single delay in (2) a similar Lyapunov-Krasovskiifunctional has been considered by Infante and Castelan in [8]. However, in their paperthe matrix Wk + (hk + θ)Wm+k in each one of the last m terms of (11) is replaced by aconstant positive definite matrix. This is due to the fact that Infante and Castelan did notinclude the integral term in their definition of the functional w(·), see [8, (3.4)]. However,we will see in Remark 5 that this integral term is an essential ingredient for deriving anexponential estimate of the form (3) for an exponentially stable delay system without anyfurther a priori knowledge.

Remark 1 If (2) is without delays (hk = 0, k = 1, . . . , m) then the interval [−h, 0] isreduced to {0}, C([−h, 0], Rn) to C({0}, Rn) ∼= R

n and we have K(t) = eAt. Identifyingϕ ∈ C({0}, Rn) with x = ϕ(0), the quadratic functional w(·) (9) is given by w(x) = x>Wxwhere W is defined by (13), and v(·) is given by v(x) = x>U(0)x, x ∈ R

n. Note that inthis case U(0) is by definition equal to

U =

∫∞

0

eA>tWeAtdt (14)

so that U = U(0) satisfies the Lyapunov equation (8). So the above construction of theLyapunov-Krasovskii functional v(·) from w(·) generalizes the construction procedure viathe Lyapunov equation (8) in the delay free case. 2

Remark 2 In the delay free case the success of quadratic Lyapunov functions rely on thefact that for a given w(x) = x>Wx the corresponding v(x) = x>Ux is not obtained viathe integral expression (14) but can be computed from the linear Lyapunov equation (8).Similarly, the above construction of the Lyapunov-Krasovskii functional (11) would notbe practical if it required the evaluation of the integral (12) (and so, in particular, theknowledge of the fundamental matrix K(t) on R+). But it is not difficult to show (see [9])that U(τ ), τ ∈ [−h, h] solves the following matrix delay differential equation

d

dτU(τ ) = U(τ )A0 +

m∑

k=1

U(τ − hk)Ak, τ ∈ [0, h] (15)

and additionally the following conditions

• the symmetry condition

U(−τ ) = U>(τ), τ ∈ [−h, h], (16)

• the Lyapunov type linear matrix equation

U(0)A0 + A>

0 U(0) +

m∑

k=1

U>(hk)Ak + A>

k U(hk) + W = 0. (17)

A systematic study of the equations (15), (16), (17) has not yet been accomplished. Weconjecture that U(τ )) is the unique solution of this set of equations, but a proof of thisconjecture is an open problem. 2

5

Page 8: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

4 Main results

In this section we show how one can use functionals (9) and (11) in order to obtain expo-nential estimates for time delay systems.We first specify two conditions under which a pair of (not necessarily quadratic) functionalsv, w : C([−h, 0], Rn) → R satisfying (10) yields an exponential estimate of the form (3).

Proposition 3 Suppose that α1, α2, β1, β2 are positive constants and v, w : C([−h, 0], Rn) →R are continuous functionals such that t 7→ v(xt(ϕ)) is differentiable on R+. If the followingconditions are satisfied for all ϕ ∈ C([−h, 0], Rn)

1. α1 ‖ϕ(0)‖2 ≤ v(ϕ) ≤ α2w(ϕ),

2. β1 ‖ϕ(0)‖2 ≤ w(ϕ) ≤ β2 ‖ϕ‖2∞

,

3. ddt

v(xt(ϕ)) ≤ −w(xt(ϕ)) for all t ≥ 0,

then for all ϕ ∈ C([−h, 0], Rn)

‖x(t, ϕ)‖ ≤

√α2β2

α1

e−

1

2α2t‖ϕ‖

∞, t ≥ 0. (18)

Proof : Given any ϕ ∈ C([−h, 0], Rn), conditions 1. and 3. imply that

d

dtv(xt(ϕ)) ≤ −

1

α2

v(xt(ϕ)), t ≥ 0.

Integrating this inequality from 0 to t and applying Gronwall’s Lemma we get

v(xt(ϕ)) ≤ v(ϕ)e−

1

α2t, t ≥ 0.

Then conditions 1. and 2. yield

α1 ‖x(t, ϕ)‖2 ≤ v(xt(ϕ)) ≤ v(ϕ)e−

1

α2t≤ α2β2 ‖ϕ‖

2∞

e−

1

α2t, t ≥ 0.

Comparing the left and the right hand sides, the exponential estimate (18) follows. �

We will now show that conversely, if the system (2) is exponentially stable, then positiveconstants α1, α2, β1, β2 can be determined such that the quadratic functionals v(·), w(·) con-structed in the previous section satisfy the assumptions of Proposition 3. As a consequencewe obtain an exponential estimate of the form (3) for every set of positive definite n × nmatrices W0, . . . , W2m.

Theorem 4 If system (2) is exponentially stable and W0, W1, . . . , W2m are positive definitereal n × n matrices then there exist positive constants α1, α2, β1, β2 such that the quadraticLyapunov-Krasovskii functionals w(·) and v(·) defined by (9) and (11), respectively, satisfythe assumptions of Proposition 3.

6

Page 9: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

Proof : We have seen in Section 3 that the functional v(·) is well defined by (11) and(12) since (2) is exponentially stable by assumption. Moreover it follows easily from thedefinitions (9) and (11) that the two functionals v, w : C([−h, 0], Rn) → R are continuous.Let ϕ ∈ C([−h, 0], Rn). We know from Section 3 that w(·) and v(·) are related by (10),i.e. t 7→ v(xt(ϕ)) is differentiable on R+ and w(·), v(·) satisfy the third condition of Propo-sition 3 with equality. It remains to show that there exist positive constants α1, α2, β1, β2

such that conditions 1. and 2. of Proposition 3 are satisfied. Let

λmin = mink=0,...,2m

λmin(Wk), λmax = maxk=0,...,2m

λmax(Wk),

where λmin(Wk) and λmax(Wk) denote the smallest and the largest eigenvalue of the positivedefinite matrix Wk, respectively. Then

λmin

[‖ϕ(0)‖2 +

m∑

k=1

‖ϕ(−hk)‖2 +

m∑

k=1

∫ 0

−hk

‖ϕ(θ)‖2 dθ

]≤

≤ w(ϕ) ≤ λmax

[‖ϕ(0)‖2 +

m∑

k=1

‖ϕ(−hk)‖2 +

m∑

k=1

∫ 0

−hk

‖ϕ(θ)‖2 dθ

].

From these inequalities we conclude that

λmin ‖ϕ(0)‖2 ≤ w(ϕ) ≤ λmax

(1 + m +

m∑

k=1

hk

)‖ϕ‖2

∞,

i.e. the functional w(·) satisfies the second condition of proposition 3 with

β1 = λmin, β2 = (1 + m +

m∑

k=1

hk)λmax. (19)

In order to check the first condition, let

µ = maxτ∈[0,h]

{‖U(τ )‖} , a = maxk=1,...,m

‖Ak‖ .

Then one can easily verify the following inequalities

ϕ(0)>U(0)ϕ(0) ≤ µ ‖ϕ(0)‖2 ,

2ϕ(0)>∫ 0

−hk

U(−hk − θ)Akϕ(θ)dθ ≤ µahk ‖ϕ(0)‖2 + µa

∫ 0

−hk

‖ϕ(θ)‖2 dθ,∫ 0

−hk

ϕ(θ)> [Wk + (hk + θ)Wm+k]ϕ(θ)dθ ≤ (1 + hk)λmax

∫ 0

−hk

‖ϕ(θ)‖2 dθ,

for k = 1, ..., m.In order to find an upper estimate of the double integrals in (11) we make use of the factthat by the Cauchy-Schwartz inequality in L2(−hi, 0; Rn) we have

(∫ 0

−hi

‖ϕ(θ)‖ dθ

)2

≤ hi

∫ 0

−hi

‖ϕ(θ)‖2 dθ, i = 1, . . . , m.

So

7

Page 10: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

∫ 0

−hk

ϕ(θ2)>A>

k

[∫ 0

−hj

U(θ1 − θ2 + hk − hj)Ajϕ(θ1)dθ1

]dθ2 ≤

≤ µa2

(∫ 0

−hk

‖ϕ(θ2)‖ dθ2

)(∫ 0

−hj

‖ϕ(θ1)‖ dθ1

)≤

≤1

2µa2

(∫ 0

−hk

‖ϕ(θ2)‖ dθ2

)2

+

(∫ 0

−hj

‖ϕ(θ1)‖ dθ1

)2 ≤

≤1

2µa2hk

∫ 0

−hk

‖ϕ(θ2)‖2 dθ2 +

1

2µa2hj

∫ 0

−hj

‖ϕ(θ1)‖2 dθ1,

for all k, j = 1, ..., m. As a consequence we obtain the following upper bound for v(ϕ):

v(ϕ) ≤ µ(1 + a

m∑

k=1

hk

)‖ϕ(0)‖2 +

m∑

k=1

(µa + (1 + hk)λmax +

m + 1

2µhka

2

)∫ 0

−hk

‖ϕ(θ)‖2 dθ.

(20)If we select α2 > 0 such that the following two conditions hold

• α2λmin ≥ µ (1 + a∑m

k=1 hk),

• α2λmin ≥ µa + (1 + h)λmax + m+12

µha2,

thenv(ϕ) ≤ α2w(ϕ).

To obtain the required quadratic lower bound for v(ϕ), we consider the modified functional

v(ϕ) = v(ϕ) − α ‖ϕ(0)‖2 , ϕ ∈ C([−h, 0], Rn)

where α > 0. Then ddt

v(xt(ϕ)) = −w(xt(ϕ)) where, for any ϕ ∈ C([−h, 0], Rn),

w(ϕ) = −w(ϕ) − αϕ(0)>

[A0ϕ(0) +

m∑

k=1

Akϕ(−hk)

]− α

[A0ϕ(0) +

m∑

k=1

Akϕ(−hk)

]>ϕ(0).

Omitting the last term in the definition of w(·) (see (9)) we obtain

w(ϕ) ≥(ϕ(0)>, ϕ(−h1)

>, ..., ϕ(−hm)>)W (α)

ϕ(0)ϕ(−h1)

...ϕ(−hm)

, ϕ ∈ C([−h, 0], Rn)

where

W (α) =

W0 0 · · · 00 W1 · · · 0...

.... . .

...0 0 · · · Wm

+ α

A0 + A>0 A1 · · · Am

A>1 0 · · · 0...

.... . . 0

A>m 0 · · · 0

.

8

Page 11: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

Using Schur complements we see that the matrix W (α) is positive definite if and only if

W (α) = W0 + α(A0 + A>

0 ) − α2m∑

k=1

AkW−1k A>

k � 0.

Since W0 � 0 there exists α1 > 0 such that W (α1) � 0. Integrating ddt

v(xt(ϕ)) = −w(xt(ϕ))from 0 to ∞ we get for α = α1,

v(ϕ) − α1‖ϕ(0)‖2 = v(ϕ) =

∫∞

0

w(xt(ϕ))dt ≥ 0.

Altogether we have found α1, α2 > 0 such that the first condition in Proposition 3 is satisfiedand this concludes the proof. �

Remark 5 The above proof shows that every quadratic functional w(·) of the form (9) withpositive definite matrices W0, . . . , W2m satisfies the second condition of Proposition 3 withthe constants β1, β2 > 0 given by (19). The first (m + 1) terms in (9) were needed to provethat the corresponding functional v(·) (11) satisfies the first inequality in the first conditionof Proposition 3. The last m terms in (9), along with the first term, were used in the proof toderive the second inequality. So, w(·) defined by (9) can be viewed as a quadratic functionalwith a minimum number of quadratic terms to yield a Lyapunov-Krasovskii functional v(·)satisfying the first condition in Proposition 3. As mentioned above the integral terms in (9)are missing in the definition of w(·) in [8]. This is compensated by an additional exponentialfactor eδt, δ > 0 in the definition of v(·), see [8, (3.1)], where −δ is supposed to be strictlygreater than the exponential growth rate of the delay system. Therefore the constructionof Infante and Castelan requires some a priori knowledge about the spectral abscissa of thesystem. 2

Remark 6 Clearly the exponential estimate obtained by Theorem 4 depends on the choiceof the matrices Wk � 0, k = 0, 1, ..., 2m. These matrices may serve as free parameters inan optimization of the estimate. In this note we do not try to obtain tight estimates, weonly wish to demonstrate that the above Lyapunov-Krasovskii approach yields a systematicprocedure for determining exponential estimates for an exponentially stable delay system(2) without any additional a priori information. 2

5 Illustration

In this section we illustrate the results of the previous section by an example.

Example 7 Consider the system

x(t) =

(−1 00 −2

)x(t) +

(0 0.7

0.7 0

)x(t − 1) +

(−0.49 0

0 −0.49

)x(t − 2). (21)

The characteristic quasipolynomial of the system is

f(s) = s2 + 3s + 2 + 0.98e−2s + 0.98se−2s + [0.49]2 e−4s.

9

Page 12: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

All the roots of this quasipolynomial lie in the open left half complex plane. The rootsclosest to the imaginary axis are

s1,2 ' −0.363 ± j1.388

so that the spectral abscissa of the system is −0.363. Since delay systems satisfy thespectrum determined growth assumption [2], there exists for every ε > 0 a constant γε ≥ 1such that the solutions of (21) satisfy the inequality

‖x(t, ϕ)‖ ≤ γε ‖ϕ‖∞ e−(0.363−ε)t, t ≥ 0.

We will now apply Theorem 4 in order to obtain an exponential estimate for the system (21)without making use of any knowledge about its spectral abscissa. Let us choose Wk = In,k = 0, 1, ..., 4 so that the functional w(·) (9) is given by

w(ϕ) = ‖ϕ(0)‖2 + ‖ϕ(−1)‖2 + ‖ϕ(−2)‖2 +

∫ 0

−1

‖ϕ(θ)‖2 dθ +

∫ 0

−2

‖ϕ(θ)‖2 dθ.

for ϕ ∈ C([−2, 0], R2). Obviously

‖ϕ(0)‖2 ≤ w(ϕ) ≤ 6 ‖ϕ‖2∞

,

so we may choose β1 = 1 and β2 = 6.

The matrix W (α) used in the proof of Theorem 4 is given by

W (α) =

(1 00 1

)− α

(2 00 4

)− α2

(0.7301 0

0 0.7301

).

W (α) is positive definite for α = 0.23, so we may choose α1 = 0.23.The corresponding functional v(xt) is of the form

v(xt) =x>(t)U(0)x(t) + 1.4x>(t)

∫ 0

−1

U(−1 − θ)

(0 11 0

)x(t + θ)dθ−

− 0.98x>(t)

∫ 0

−2

U(−2 − θ)x(t + θ)dθ+

+ 0.49

∫ 0

−1

x>(t + θ2)

[∫ 0

−1

(0 11 0

)U(θ1 − θ2)

(0 11 0

)x(t + θ1)dθ1

]dθ2−

− 0.686

∫ 0

−1

x>(t + θ2)

[∫ 0

−2

(0 11 0

)U(θ1 − θ2 − 1)x(t + θ1)dθ1

]dθ2+

+ 0.2401

∫ 0

−2

x>(t + θ2)

[∫ 0

−2

U(θ1 − θ2)x(t + θ1)dθ1

]dθ2+

+

∫ 0

−1

(2 + θ) ‖x(t + θ)‖2 dθ +

∫ 0

−2

(3 + θ) ‖x(t + θ)‖2 dθ.

Here the Lyapunov matrix function

U(τ ) = 6

∫∞

0

K>(t)K(t + τ)dτ

10

Page 13: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

Figure 1: Matrix U(τ ), piecewise linear approximation

satisfies the equation

d

dτU(τ ) = −U(τ )

(1 00 2

)+ U(τ − 1)

(0 0.7

0.7 0

)− U(τ − 2)

(0.49 00 0.49

). (22)

In Fig. 1 we plot the four components of a piecewise linear approximation of the Lyapunovmatrix function. From the plot we get

µ = 3.44 and a = 0.7.

The upper bound (20) of v(·) has the following form

v(xt)≤ µ(1+3a)‖x(t)‖2+ (µa+2+3

2µa2)

∫ 0

−1

‖x(t+θ)‖2 dθ+ (µa+3+3µa2)

∫ 0

−2

‖x(t + θ)‖2dθ.

The value α2 should satisfy the conditions

α2 ≥ µ(1 + 3a) = 10.65, α2 ≥ µa + 3 + 3µa2 = 10.46

and so we may choose α2 = 10.65.As a result we obtain an exponential estimate (18) for the solutions of the system (21) withthe following constants

γ =

√α2β2

α1≈ 16.69, σ =

1

2α2≈ 0.047.

In order to verify how well the piecewise linear approximation represents the Lyapunovmatrix valued function we compute the solution of equation (22) with the initial condi-tion generated by the linear piecewise approximation as initial function on [−h, 0], seeRemark (16). The four components of the corresponding solution are plotted on Fig. 2.A comparison Fig. 1 shows a good fit between the solution U(τ ) and the piecewise linearapproximation on [0, h]. 2

11

Page 14: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

Figure 2: Matrix U(τ ), numerical solution

References

[1] R. Bellman and K. L. Cooke. Differential-Difference Equations. Number 6 in Mathe-matics in Science and Engineering. Academic Press, New York, 1963.

[2] R. F. Curtain and H. J. Zwart. An Introduction to Infinite-Dimensional Linear SystemsTheory. Number 21 in Texts in Applied Mathematics. Springer-Verlag, New York, 1995.

[3] R. Datko. Uniform asymptotic stability of evolutionary processes in a Banach space.SIAM J. Math. Anal., 3:428–445, 1972.

[4] A. Halanay. Differential Equations: Stability, Oscillations, Time Lags. Academic Press,New York, 1966.

[5] J. K. Hale. Theory of Functional Differential Equations. Number 3 in Applied Mathe-matical Sciences. Springer-Verlag, New York, 2nd edition, 1977.

[6] D. Hinrichsen, E. Plischke, and A. J. Pritchard. Liapunov and Riccati equations forpractical stability. In Proc. 6th European Control Conf. 2001, pages 2883–2888, 2001.Paper no. 8485 (CDROM).

[7] W. Z. Huang. Generalization of Liapunov’s theorem in a linear delay system. J. Math.Anal. Appl., 142(1):83–94, 1989.

[8] E. F. Infante and W. B. Castelan. A Liapunov functional for a matrix difference-differential equation. J. Diff. Eqns., 29:439–451, 1978.

[9] V. L. Kharitonov and A. P. Zhabko. LLyapunov-Krasovskii approach to the robuststability analysis of time-delay systems. Automatica, 39:15–20, 2003.

12

Page 15: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

[10] N. N. Krasovskiı. On the application of the second method of Lyapunov for equationswith time delays. Prikl. Mat. Meh., 20:315–327, 1956.

[11] B. S. Razumikhin. On stability of systems with a delay. Prikl. Mat. Meh., 20:500–512,1956.

[12] I. M. Repin. Quadratic Liapunov functionals for systems with delay. J. Appl. Math.Mech., 29:669–672, 1966. Translation of Prikl. Mat. Meh., 29:564–566 (1965).

13

Page 16: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,
Page 17: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

����������� ��������������������������� � ������ � �"! #%$&$ ' (*)�+�, -/.0212346565*798;:<:>=<=<=�?A@�BC5*49?ED6FHG>I>JLKNM>@OM>FP?RQLML:CSNMC5NMC@:TJ M<K�G*UV4N5NM�?E4N5W@X

Y Z�[]\;^_\_`a[]bdc e \_fhgic [%jlkmeL^onW\�Y

prq<s&t�uVvTw xyvVz {}|�~����2�&�rs�u��E���O����V���>�*�/���y�h�R�h�9�}���6�*�h���<�������V���<�/�V� ���< *�h��¡

¢�£¥¤;¦¨§N©«ª­¬hª¯®¯©«°²±�³o´_®¶µ/·o®A³"¸�¹6°%¦�§L©º³"¬�®¯ª¯¬�»�µ/£L¬_¼�½%´�¾l³�®a¿C½�¸ Àh½/·¨®a¿W³�¹6°%¦�§L©º³"¬�®¯ª¯¬�Áª�Ã>³o£6Ä%µ/©ÆÅ6³ Ç&·È½>Éo©º³o¦ �ÊTË<Ì �y���V�V�C�

�V���>�TÍC���������V���<�/�V� ���< <�h��¡¹<©Æª¯¸/ªa£>ÃÏÎЪa£L¸T½/Ñi¹L¬o¿*³o¦Ò³_´OÀh½/·}Ó�ªÔ´h¬h·"³h®A³ »³"µ/´_®¯ÕÖ¹L×hÅ<µ/·"³_´ ¢Â§T§L·R½�ØVªa¦¥µV®¯ª­½/£ÙÉo°ÛÚ<·;ª�Ã�½V£L½/¦Ò³h®¯·_ª¯¬�Ǩ½/©Æ°%Õ£L½/¦�ª­µ/©Ü´ � Ê�Ë<Ì �2���V�V�C�

�V���>�TÝC���y�h�R�h�9�}���6�*�h���6ÞÙ�/����� � Ìß �VàR�"� ÌÔÌ�á �*â��*���Ôã Ë �²äN�Lå Ë �º���"�V�6�%æÖç��"��èé ¡Ç�µ/·Rµ/©a©�³h© Ǩµ/·_®¯ª­µ/©H¹6®¶µ�Éoªa©Æª�¼hªê£>Ã¥¢9© Ã>½/·;ª¯®a¿�¦m´�Àh½/·9»µ/·¯ÃC³P»2ªa£N³"µV·ìëí½/£6®¯·R½/©y¹<°%´_®A³h¦î´ � ÊTË6Ì �2���V�V�C�

�V���>�/ï*���y�h�R�h�9�}���6�*�h��¡ëí½/¦�§LÅ*®¶µV®¯ª­½/£Lµ/©¾l³�®a¿C½�¸%´}Ào½/·9»2ªa£N³"µ/·oð<ñÂÅ<µT¸/·RµV®¯ª­¬óòH§N®¯ªa¦�ª�¼�µV®¯ª­½/£ �Lô Ë*õTË à"�����V�V�C�

�V���>�TöC���y�h�R�h�9�}���6�*�h���<÷¨� ̺ø6ù � úV�h��àh�6â}�*���ºã Ë ��äN�Lå Ë �Ô���"�V�<�%æÖç9�"��èé �6û��"� õVá ��� á å Ë �º�>�"�V�<�%æÖç�����èé ¡¹L½/©ÆÄ�ªa£>Ãr¢�© ÃC³�Éh·Èµ/ª­¬�±�ª­¬�¬;µV®¯ª¨Á�×�Å*µT®¯ª¯½/£<´ó½V£üÇ�µV·Èµ/©a©º³o©îëí½/¦�§LÅ*®A³h·�´þýH´;ªa£>ÃÏÿm³oÑ�®¶½V£�� ´¥¾i³h®a¿C½�¸lÑíª¯®a¿Á�Ø>µT¬�®í»2ªa£N³�¹N³"µV·È¬h¿ �Lô Ë*õTË àR�î���V�V�C�

�V���>���C���H�/��à�û����Ë �*�V�<�<�/�6�º�V��� ����� ù ¡ò£Үa¿W³¨·RµV®A³¨½RÀ¨¬�½/£<ÄV³o·­Ã>³h£L¬;³�½ÈÀªê£�£<ª¯®A³Â¿C½V·;ªº¼�½/£ó¸/ªê´o¬;½/ÅW£6®A³"¸m½;§N®¯ªa¦¥µ/©<Ä/µ/©«Å6³�À;ÅW£L¬�®¯ª¯½V£*´ ��� á� �� î� �h����V�V�C�

�V���>���>���y�h�R�h�9�}���6�*�h����� áTÌ �V�h�9Þó� ù �� ¥�V�<�O�<� á � õVõTË*á��9Ë ¡¢ ÿ�½T®A³î½V£Ï®a¿W³Pÿ²ÅW¦Ò³o·_ª¯¬;µ/©O¹L½/©ÆÅ*®¯ª¯½/£l½RÀÒëí½/¦�§L©�³�Ø��îµ/¦�ªê© ®¶½/£<ª­µ/£lµ/£L¸Û¹��T³oÑíÕ���µV¦mªa© ®¶½/£<ª­µ/£�Áª�Ã>³o£6ÕÄ%µ/©ÆÅ6³PÇ·R½�Éo©º³o¦m´ ��� á�� �� �� �h�����V�V�C�

�V���>�T�C��â��N�h� ù �/�� Ð����V�<à�� ù �6� Ë �"� ù �/�� Ð���áTù �O¡ÿPÅW¦Ò³o·_ª­¬�µ/©C´;ªa¦�ÅW© µV®¯ª­½/£Ð½ÈÀ�µ9´_ªê©Æª¯¬;½/£! ½�µT®¯ªê£>ÃP¼�½/£N³�Ñíªa®a¿�µíÀ_·R³;³�¬;µ;§Lªa©ê©�µ/·_°P´;ÅW·aÀoµ�¬;³ ��"��$#h�� î� �h�����V�V�C�

�V���>�*�/���������V���<�/�V� ���< <�h��¡Ú¿W³PÇ�µV·Èµ/¦Ò³�®A³o·_ª�¼]³"¸�%'& ¢�© Ã�½/·_ªa®a¿>¦ Àh½/·�¹<°/¦�§L©º³"¬�®¯ª¯¬)(+*�Å*®­®A³h·, °.-�¾dµV®¯·_ª­¬;³_´ �6�*�h�<� Ë �/�����V�V�C�

�V���>�TÍC���������V���<�/�V� ���< <�h��¡Á·_·È½V·�¢�£Nµ/©«°%´_ªÔ´ì½RÀÒ®a¿W³Û´_°%¦�§L©º³"¬�®¯ª¯¬ì»�µ/£N¬;¼�½/´�¾l³�®a¿C½�¸mÀh½/·Û®a¿*³Û´;°/¦�§L©º³"¬�®¯ª­¬ÛÁ&ª ÃC³o£<Ä/µ/©ÆÅ<³�Ç&·È½>Éo©º³o¦ �Þ/��/�0#Ò���V�V�C�

�V���>�TÝC��â��N�h� ù �/�� Ð����V�<à�� ù �6ô Ì21 ���� �ä3� ù Ò�º *�h¡¹<ªa¦mÅW©�µV®¯ª­½/£l½ÈÀî¸�³h£L¸/·_ªa®¯ª­¬�¬h·_°%´_®¶µ/© ÃV·R½/Ñ�®a¿�Ñíª¯®a¿ó®a¿W³o·_¦¥µ/©2¬;½/£<ÄV³"¬�®¯ª­½/£ �NÞ/��/��#Û���V�V�C�

�V���>�/ï*��â��N�h� ù �/�� Ð����V�<à�� ù ¡4 ªê£<ª¯®A³Ð³o©º³o¦Ò³h£6®�¸VªÔ´h¬h·"³h®¯ªº¼�µV®¯ª­½/£ ½ÈÀÛ®a¿W³�ÿ�µVÄ�ª¶³o·_ÕE¹6®¶½5�V³_´ì³�×hÅ<µV®¯ª­½/£*´¥Ñíª¯®a¿iµîÀ;·"³�³�¬;µ;§Lªa©ê©�µ/·_°Ù´_ÅW·êÀhµT¬;³ �Þ/��/�0#Ò���V�V�C�

�V���>�TöC���y�h�R�h�9�}���6�*�h��¡¾dµV®a¿*³o¦¥µV®¯ª,�񻐣 ¸�³h·!*9³o·_Å_À;´¶§N·Èµ�ØTªÔ´ � Ê�Ë<Ì �2���V�V�C�

�V���>���C��ô��< *�"�$67"m� ���L�2�V�8�o�V�N�<�/�6�º�V��÷²��� ���"� ù ¡±&½�ÉhÅ>´o®¯£N³_´;´m½ÈÀ²£N½/£<©Æªê£N³�µ/·²´_°%´_®A³o¦m´mµV£L¸ì®a¿W³oªa·î¸T½/¦¥µ/ªa£*´�½ÈÀîµV®­®¯·RµT¬�®¯ª­½/£ � ô Ë*õTË àR�²���V�V�C�

Page 18: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

�V���>���>���y�h�R�h�9�}���6�*�h���<â��<���ºã Ë ��äN�Lå Ë �º�>�"�V�<�%æÖç�����èé �6û��"� õVá ��� á å Ë �º�>�"�V�<�%æÖç��"��èé ¡*�µ/© µV£L¬;³�¸iÚ<·;ÅW£L¬;µV®¯ª¯½V£r¾d½�¸�³o©�±�³"¸VÅ*¬�®¯ª¯½V£ ½ÈÀP»�µ/·­Ã>³hÕE¹L¬;µ/©º³îÓ�³o£<´h³m¹<°%´o®A³o¦m´�½/£rÇ�µ/·Rµ/©a©�³h©Âëí½/¦�§LÅ*®¯Õ³o·;´ �NäW� ø �R�� �� �h�����V�V�C�

�V���>�T�C��÷ á �<� Ì  ÙäC��á � �h��¡ëí½/©a© ½�¬;µV®¯ª­½/£Ò¦Ò³h®a¿W½�¸%´LÀh½/·�´o½/©ÆÄ�ªa£>Ã9©Æªê£ ³"µ/·Â¸/ª � ³o·"³o£6®¯ª­µ/©ÆÕȵ/© Ã>³]Éo·Rµ/ª­¬�É"½VÅC£N¸Tµ/·_°PÄ%µ/©ÆÅ6³�§L·È½>Éo©º³o¦m´ ��äC� ø �R�� �æ� �h�²���V�V�C�

�V���>�T�C��� Ë àR�húW�º�Ðô����h��úN¡¾d½�¸>³o©a©«ªa£>ÃìÑíª¯®a¿ ò·o®a¿C½/£N½/·;¦¥µ/© *�µ%´_ªÔ´ 4 ÅC£N¬�®¯ª­½/£*´ �OäC� ø �R�� î� �h�²���V�V�C�

�V���L�h�W���������V���<�/�V� ���< <�h����"m�NäC�R� � ���ÏÞ�� ���V�húV����� Ì�áTË�1 �h�¨ÞÙ� �"�V�húN¡��µV¦mªa© ®¶½/£dµ/£L¸��Wµ�¬�½�Éhª}¬�½/¦Ò³}À_ÅW©ê©�¬hªa·R¬h©º³����*µT¬�½�Éhª}µ/© Ã�½/·_ª¯®a¿�¦m´�Ào½V·P´_®¯·_Å<¬�®¯ÅW·R³�¸ ��µ/¦�ªa©Æ®¶½V£<ª¯µV£i³oª�Ã>³o£6Õ§L·R½�Éo©º³o¦m´ �Oç��C� á �N�h�î���V�V�C�

�V���L�V�/���y�h�R�h�9�}���6�*�h�����9�º� �o���>�R���¨�h�;��è�V�< *�$#V�<ô���� á �6� á �2�VàR� á ��¡ò£Ϯa¿*³��î©�³hªê£<¦¥µ/£Ù¤_®A³h·ÈµT®¯ª¯½/£�Àh½/·�ÿî½/£*´o®¶µ�Éoªa©Æª�¼�µ�Éo©º³m¹<°%´_®A³h¦ �Nç��C� á �N�h�î���V�V�C�

�V���L��ÍC���y�h�R�h�9�}���6�*�h���<�������V���<�/�V� ���< *�h��¡¢ �P°�Éh·;ª­¸P¾l³h®a¿W½�¸ÂÀh½/·¨®a¿W³�ÿ²ÅW¦Ò³o·_ª¯¬;µ/©<¹L½/©ÆÅ*®¯ª¯½V£�½ÈÀ Ó�ªÔ´h¬h·"³h®A³hÕhÚHªa¦Ò³&¢�© ÃC³�Éh·Èµ/ª­¬�±¨ª¯¬;¬�µV®¯ª Á�×hÅ<µV®¯ª­½/£*´ �� á�� �� �� �h�m���V�V�C�

�V���L��ÝC���y�h�R�h�9�}���6�*�h���<â��<���ºã Ë ��äN�Lå Ë �º�>�"�V�<�%æÖç�����èé �6û��"� õVá ��� á å Ë �º�>�"�V�<�%æÖç��"��èé ¡ÿPÅW¦Ò³o·_ª­¬�µ/©*¹L½V©«Å*®¯ª­½/£ó½ÈÀ&¹L¬o¿>ÅC·&¹L®¶µ�Éo©º³Â»2ªa£N³"µ/·Â¾dµV®¯·_ª ØPÁ¨×hÅ<µV®¯ª­½/£*´¨½/£¥¾rÅW©Æ®¯ª­¬�½V¦¨§NÅW®A³h·�´ � � á� �� î� �h����V�V�C�

�V���L�oï*��â��N�h� ù �/�� Ð����V�<à�� ù �²�/� áTÌ Þ���� Ë<Ì �W¡¢¨¸�µ;§N®¯ªaÄ�ª¯®¯°Ûªa£��%Ó ¤;¦¥µ�Ã>³PÇ&·È½�¬;³_´;´;ªa£>à ��"��$#h�� �� �h�î���V�V�C�

�V�]�>�*�/���y�h�R�h�9�}���6�*�h����� áTÌ �V�h�9Þó� ù �� ¥�V�<�O�<� á � õVõTË*á��9Ë ¡Ç¨³h·_®¯ÅW·�É"µV®¯ª­½/£�¢9£Lµ/©Æ°]´_ªÔ´yÀo½V·�®a¿W³�Áª�Ã>³h£<Ä%µ/©ÆÅ6³&Ç·R½�Éo©º³o¦�½RÀ9µ 4 ½/·_¦�µV©<Ç&·È½�¸/Å<¬�®�½ÈÀ�¾dµT®¯·;ª­¬;³o´ � Ê �V� Ë �/�Í/�V�V�W�

�V�]�>�TÍC�� � ø �º� õ � Ë �V� õ ¡4 ªê£<ª¯®A³PÁ©º³o¦Ò³h£6®y¾i³h®a¿C½�¸�Àh½/·�¾ ª ØC³"¸�Ç·R½�Éh©�³h¦î´îÑíªa®a¿ÛǨ³h£Lµ/© ®¯° � Ê �V� Ë �/��Í/�V�V�W�

�V�]�>�TÝC�mû9�Ô�V� 1 �;�V� �o��à�� á ÞÙ�/�"�"�º�<� � á ¡±�³"¬�ÅW·�´_ªêÄV³î¦¥³o´�¿Ð·"³+�£N³h¦Ò³o£6® ªa£��]Ó �6�<�h�<� Ë �/��Í/�V�V�W�

�V�]�>�/ï*��â��N�h� ù �/�� Ð����V�<à�� ù � ßÂù �;�ºàR� áVø6ù â õ � �h��à��Lç Ì � � �h��Þ����º� �"�V�V����� � áTÌ �h�"�Ûä3� Ë ��� Ë ¡Ú6µV°%©�½/·_Õoëí½VÅ<³�®­®A³m¹<°%´_®A³h¦�Ñíª¯®a¿ì¢9´_°/¦m¦Ò³�®¯·;ª­¬ *�½/ÅW£L¸Tµ/·_°rëí½V£L¸/ª¯®¯ª¯½/£<´ � �*�h�<� Ë �/��Í/�V�V�W�

�V�]�>�TöC���y�h�R�h�9�}���6�*�h��¡¹<°/¦�§L©�³�¬�®¯ª­¬�*�µ/©�µ/£L¬hªa£>ÃнÈÀ ��µ/¦�ªa©Æ®¶½V£<ª¯µV£Ù¾dµV®¯·_ª¯¬_³_´ �6�*�h�<� Ë �/��Í/�V�V�W�

�V�]�>���C���<�/�L� áìß � ¥� ÌºÌ �­� ��/��à9û����Ë �*�V�*�<�/�L�º�V��� ���"� ù ¡¢ ·R³AÃVÅW©�µ/·;ªº¼�µT®¯ª¯½/£ ½ÈÀ���Å6É"½VÄ�� ´m³"×hÅ<µV®¯ª­½/£�Àh½/·²·R½�ÉoÅC´_®&¸T½V¦�µVªê£*´�½ÈÀîµV®­®¯·RµT¬�®¯ª­½/£ �OÞ ��/��#îÍ/�V�V�W�

�V�]�>���>�îÞÙ�8� ù �/� Ì � áTÌ�� �6â�� �h� ù �/�� �� ��V�6à�� ù �NÞ��8� ù �/� Ì ���áTù ó� " á Ò�º�<�8��"9� � �ºàh¡¾d½�¸>³o©a©«ª¶³o·_ÅW£>Ãи�³o·9¢²É.���ÅC¿�©ÆÅW£>ÃìÄ%½V£Ï¹L®¶µ%¿�©ºÉo·Rµ/¦�¦¥³h£ �OÞ/��/��#mÍ/�V�V�W�

�V�]�>�T�C�mäC�R� øLù �V��"9� ù<Ì �V�V�6�y�h�R�h�9Þ��V�/�*��û��h�� ��H��à�� ù �V�V¡¤;£6®A³o·A§W½/©�µV®¯ªa£>ÃйN¬�µ/©Æªê£Cà 4 ÅW£L¬�®¯ª­½/£*´îÑíªa®a¿ÛÓ�Å<µ/©Ü´ �6ô ø ��� Ì Í/�V�V�W�

�V�]�>�T�C� ÊTá � ù ����� � ù �"���6àh�<�<�/�6�º�V��� ����� ù ¡¢ ÃV©�½�É"µV©«ªº¼�µV®¯ª­½/£�§N·È½�¬_³"¸/ÅW·"³&Àh½/·²©�½�¬;µ/©ê©Æ°Û´_®¶µ�Éhªê©Æª�¼hªê£CÃЬ�½/£6®¯·R½/©a©�³h·�´ ��Þ��V�Í/�V�V�W�

Page 19: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

�V�]�L�h�W���y�h�R�h�PÞ��V�/�*�Oû��h�� ��H��à�� ù �V�V��� �h���*�h� � � ÌºÌ ¥�V�<���*û �Ë ���R�h� � áTÌÔÌ ¥�V�<�O¡Ó�³h®A³"¬�®¯ª¯½V£ µ/£L¸ ëy©�µ%´�´_ª ¬�µV®¯ª­½/£ ½RÀ�¾lµV®A³o·_ª¯µV©í¢9®­®¯·_ª¶ÉoÅ*®A³_´ÒðÙ¢�Ç·RµT¬�®¯ª­¬�µ/©�¢Â§T§L©Æª¯¬;µV®¯ª­½/£ ½ÈÀ Î�µ/ÄV³o©º³h®¢�£Lµ/©Æ°%´;ªê´ �LÞÙ�V�Í/�V�V�W�

�V�]�L�V�/�mäC�R� 1 �V��� á à�� ù �h���h�<ô Ì21 �"�� �ä3� ù Ò�º *�h� Ë �<��� �h�"�}û��*äW�º�h�N�h���h�>â�� �h� ù �/�� ¥����V�<à�� ù � Ì � Ë àRæ�� �h���*�h�� ����#V�Nû��h� ù �/�� �"!#�� Ë �N� � ù*á ¥�Và ²�V�ºàR�h��¡¹<ªa¦mÅW©�µV®¯ª­½/£l½ÈÀ�¤;£L¸VÅ>´o®¯·;ª­µ/©Âëy·;°%´_®¶µV© � ·R½/Ñ�®a¿rÉo°ì®a¿W³��6³h·_®¯ª­¬�µ/© *�·_ª­¸�ÃV¦¥µ/£Ù¾i³h®a¿C½�¸ �NÞÙ�V�Í/�V�V�W�

�V�]�L��ÍC� � áTÌ �V�h�!��� ù Ò�V�<�O�Lû��h��  ���à�� ù �V�V¡Î�µ/ÄV³o©º³h® *�µ/´h³"¸�¾i³h®a¿C½�¸/´�Ào½V·9¤;¦�§L·R½/Ä/³�¸þÎЪa£L¸mÇ&·È½ �©�³h·�¹<ª�ÃV£Nµ/©NÇ·R½�¬_³_´�´_ªê£Cà �NÞÙ�V�Í/�V�V�W�

�V�]�L��ÝC�mäC�R� øLù �V��"9� ù<Ì �V�V�6�y�h�R�h�9Þ��V�Và"à�¡¢ ÿ�½T®A³�½/£�¤;£6®A³o·A§W½/©�µV®¯ªa£>ÃйL¬;µ/©Æªê£>à 4 ÅW£L¬�®¯ª­½/£*´ �6ô Ë<õTË àR��Í/�V�V�W�

�V�]�L�oï*��÷ á �<��ú�÷¨� Ì � Ë �<÷ áTÌ81íßÂÌ � �"�W  á ú Ì �V�6�*�Wè��  è�h�;�8��� á>á � û9�����ºà ù �Â� Ì ¡¢¨¬;¬hÅW·RµV®A³9¢9®­®A³o£<Å<µV®¯ª­½/£üëí½/·;·"³"¬�®¯ª¯½V£Ïªa£�¹WÇ�Á²ë�Ú ¤;¦¥µ�ÃTªê£CÃÒÅC´_ªê£>ÃÙòH§N®¯ªa¦�ª�¼�µV®¯ª¯½V£r½ÈÀ *¨ªa©«ªa£N³"µV· 4 ÅW£L¬hÕ®¯ª¯½V£*´�µ/£N¸Ù¢P´�´;ÅW¦�ªa£>Ãlµ/£ ýH£��%£L½/Ñí£ ¹/§WµV®¯ª­µ/©a©«°/Õ��<µ/·_°%ªa£>à ¢9®­®A³h£<Å<µV®¯ª¯½V£ Ó�ªÔ´o®¯·;ª¶ÉoÅ*®¯ª­½/£ � äC� ø �R�� �� �h�Í/�V�V�W�

�V�]�L��öC���y�h�R�h� Ë �*�V� Ì �*÷ á �<� Ì  �äC��á � �h��¡¹<°/¦�¦¥³�®¯·;ª­¬l¬;½/©a© ½�¬;µV®¯ª­½/£ ¦Ò³h®a¿W½�¸%´�Ào½V·�©Æªê£N³�µ/· ¸/ª � ³o·"³o£6®¯ª­µ/©ÆÕȵ/© Ã>³]Éo·Rµ/ª­¬ É�½/ÅW£L¸Tµ/·_°þÄ%µ/©ÆÅ6³Ò§L·R½�Éo©º³o¦m´ �äC� ø �R�� î� �h��Í/�V�V�W�

�V�]�L���C���<�/�L�º�V��� ���"� ù ¡Ú¿W³9Ã>³h£N³o·Rµ/©Æª�¼]³"¸Û´A§*³"¬�®¯·Èµ/©y·RµT¸/ªaÅ>´�µ/£L¸ó³"Ø�®¯·R³h¦¥µ/©£L½V·;¦m´ �Oç��C� á �N�h��Í/�V�V�W�

�V�]�L��>���*�;�V�*��äC�R��� õ �h���6ô ù ¥�V Ð÷¨�$#������ õTù à ù æ ��� Ì � ù �­� Ê ���6��ú)���º�h�6à�� ù �<÷ á �<�>úì÷�� Ì � Ë ¡¢ ÅW£<ª ³"¸�µ;§T§L·R½�µT¬h¿Ï®¶½ì®a¿W³�µ;§�§L·È½�ØVªa¦�µT®A³�´h½/©«Å*®¯ª­½/£l½ÈÀ�Ç}Ó9Á � � á� �� î�N�h�9Í/�V�V�W�

�V�]�L���C���y�h�R�h�9�}���6�*�h���<â��<���ºã Ë ��äN�Lå Ë �º�>�"�V�<�%æÖç�����èé �6û��"� õVá ��� á å Ë �º�>�"�V�<�%æÖç��"��èé ¡Ç�µ/·Rµ/©a©�³h©yµ/© Ã�½V·;ª¯®a¿�¦m´}Àh½/·²¦¥½�¸�³h©·"³"¸/Å<¬�®¯ª¯½/£l½RÀî¸/ªê´o¬h·"³h®A³�ðT®¯ªa¦¥³m´;°%´_®A³h¦î´ � "��$#h�� î�N�h�9Í/�V�V�W�

�V�]�L���C��÷ á �<��ú�÷¨� Ì � Ë ¡¢�´_®A³;³E§*³o´_®P¸�³_´h¬;³h£6®�µV© Ã>½/·;ª¯®a¿�¦ Àh½/·Û®a¿W³�ÃT© ½>É"µ/©}¦mªa£<ªa¦�ª�¼�µV®¯ª¯½V£ ½ÈÀÏÚHª,��¿C½/£N½/Ä]ð]Ç¿�ªa©ê©Æª §L´�À_ÅC£N¬�®¯ª­½/£Lµ/© �"��$#h�� �� �h��Í/�V�V�W�

�*�;�>�*�/��â����h�����>� ��h� ù*á  <à&�º� ù ú ø �h�"� ù �h�0 ¥�º�m�R�"���/�� ����>� øLÌ �V�<�<�º� õ ¡Ú6½V·�´o®A³o£ � �½%¿�©º³o·��2Ç�³h®A³o·9¾dµTµ%´;´�2Ç�³h®A³h·óÎÐÅC´_®�y¾dµV·_®¯ªa£ ¹N³�³]É"µ%´;´ � Ê �V� Ë �/��Í/�V�*�/�

�*�;�>�TÍC���2�/��� ÌºÌ � Ì ô Ì�õVá �;��� ù ¥à 1aá � �yå ç ø �"� ¥� Ì�ß á ���R� áTÌ á 1 "9�ºà��o���h�R�oæ ���8 ��P�y�h��� á  <�8� ��Ô�*���/�¨äCúCà"�R�� ¥àh¡Ç¨³�®A³o· *9³h£<£N³o·���±&µ/© §6¿�*¨°�³o·;´��±&µÈÀhµ�³h©�¾lµ/°T½���Á£<·_ª¯×�Å6³m¹ ��ñÂÅCªa£6®¶µ/£Nµ/Õ_ò·o®��� ���ª¯¬_³o£6®A³��î³h·;£��µV£L¸�³�¼ ��*�h�<� Ë �/��Í/�V�*�/�

�*�;�>�TÝC���y�h�R�h�9�}���6�*�h���<â��<���ºã Ë ��äN�Lå Ë �º�>�"�V�<�%æÖç�����èé �6û��"� õVá ��� á å Ë �º�>�"�V�<�%æÖç��"��èé ¡Á�� ¬�ª­³h£6®yÿ²ÅW¦¥³h·;ª­¬�µV©N¢�© Ã�½/·_ªa®a¿>¦î´}Ào½/· *�µV© µ/£N¬;³"¸Ð¹6®¶½�¬o¿Wµ%´_®¯ª­¬�Ú<·;ÅW£L¬;µV®¯ª¯½V£ �NÞ ��/��#îÍ/�V�*�/�

�*�;�>�/ï*���y�h�R�h�9�}���6�*�h���6ÞÙ�/����� � Ìß �VàR�"� ÌÔÌ�á �*â��*���Ôã Ë �²äN�Lå Ë �º���"�V�6�%æÖç��"��èé ¡Ç�µ/·o®¯ª­µ/©¹6®¶µ>Éoªa©«ªº¼�µV®¯ª­½/£l½ÈÀ9»�µ/·­Ã>³hÕE¹L¬;µ/©º³²Ó�ªÔ´h¬h·"³h®A³oÕ�ÚHªa¦¥³9»íªê£N³�µ/·ìëí½V£6®¯·È½V©¹<°%´_®A³h¦m´ �NÞ ��/��#îÍ/�V�*�/�

�*�;�>�TöC�mäC�R� øLù �V��"9� ù<Ì �V�V¡*9³_´o½VÄî±�³¶ÃTÅC©�µ/·_ªa®¯°�Àh½/·�Á�¸�Ã>³�¹<ªa£>ÃVÅW©�µ/·;ª¯®¯ª¶³_´�ªa£ÐǨ½/©Æ°]¿*³"¸/·Rµ/©Óm½/¦¥µ/ªê£<´ �LÞÙ�V�HÍ/�V�*�/�

�*�;�>���C���<�/�L�º�V��� ���"� ù ¡¢ ©Æªa£N³"µ/·_ªº¼�µV®¯ª­½/£ §L·_ªê£L¬�ª §N©�³ÒÀh½/· ·È½>ÉoÅC´_®¯£N³o´�´ Ñíª¯®a¿ ·"³_´A§*³"¬�®Ò®¶½ ®¯ªê¦Ò³oÕÖÄ%µV·;°/ªa£>Ãr§<³o·o®¯ÅC·�É"µT®¯ª¯½/£<´ �îÞÙ�V�Í/�V�*�/�

Page 20: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

�*�;�>���>�mäC�R� øLù �V��"9� ù<Ì �V�V� � áTÌ81êõ �V� õ "�� ù ����O�²�/�;àR�R�������"�6�V�O¡¢¨¸�µ;§N®¯ªaÄ/³ Î�µ/ÄV³o©º³h®&¾l³�®a¿C½�¸%´¨Àh½/·¥¹NµT¸T¸W©º³�Ç�½/ªa£6®&Ç·R½�Éo©º³o¦m´¥Õ¥òH§N®¯ªa¦¥µ/©�ëí½V£<Ä/³h·¯ÃC³o£L¬_³�±&µV®A³_´ � Ê�Ë<Ì �Í/�V�*�/�

�*�;�>�T�C��÷ á �<��ú�÷¨� Ì � Ë ¡¾d½/·R½�¼�½/Ä�� ´�Ó�ªê´o¬h·"³E§*µ/£L¬h°�Ç&·;ªa£L¬hªÜ§L©º³�Àh½/·rÚHª,��¿C½V£L½/Ä�·"³¶ÃTÅW© µ/·_ªº¼�µV®¯ª­½/£ ½RÀÒ£L½/£6©«ªa£N³"µV·�½;§<³o·RµV®¶½/·;´ � Ê�Ë<Ì �Í/�V�*�/�

�*�;�>�T�C�îÞÙ�8� ù �/� Ì � áTÌ�� ¡Áªa£�À �ÅC¿�·_ÅW£>Ã�¸>³_´9Ó�·_Å*¬$��´�À �ÅW·�¸/ª¶³mªa£*´_®¶µT®¯ª¯½/£��µ/·R³h£i¹L®¶½ �V³o´_ð � ©º³oª­¬o¿�ÅW£>ÃC³o£d¦�ª¯®­®A³o©«´�¸�³h·9¾i³h®a¿C½�¸>³�Ä/½/£��µ;§L©�µ/£ � ÊTË<Ì �OÍ/�V�*�/�

�*�;�L�h�W�mäC�R� øLù �V��"9� ù<Ì �V�V�6�y�h�R�h�9Þ��V�/�*��û��h�� ��H��à�� ù �V�V¡±�³"¬;½/£*´o®¯·;Å<¬�®¯ª­½/£i½RÀ�±�³+ �³"¬�®¯ªêÄ]ªa®¯°�Ó�³_´;ª¯®¯ª¶³_´îÉo° Î�µ/Ä/³h©�³�®9Ú<·Rµ/£*´êÀh½/·_¦m´ � ô Ë*õTË àR�¨Í/�V�*�/�

�*�;�L�V�/�mäC�R� øLù �V��"9� ù<Ì �V�V¡*9³_´o½VÄî±�³¶ÃTÅC©�µ/·_ªa®¯°�Àh½/·�®a¿W³Pÿm³oÅW¦¥µ/£<£�Ç·R½�Éh©�³h¦ �Lô Ë*õTË àR��Í/�V�*�/�

�*�;�L��ÍC��� �h���<�/�;  ���V�Và"  á �*�N�mÞÙ�/��� á ç ù<Ì � �h� õ �h���PÞÙ�/�"�"�º��÷ Ë ø 1 �²ô Ì21 �"��  ä3� ù Ò�º *�h�� Ë �6��� �h�"��ûm�äW���h� �h�"�h¡� ð��yð]¾rÅW© ®¯ªê·"³_´h½/©«Å*®¯ª­½/£ �ªÔ´_Å<µ/©«ªº¼�µT®¯ª¯½/£l½RÀ�¢¨¸�µ;§N®¯ªaÄ/³ 4 ªê£6ªa®A³PÁ&©�³h¦¥³h£6®}¹<ªê¦�ÅW©�µV®¯ª¯½V£*´ �Oç��>� á � �h��Í/�V�*�/�

�*�;�L��ÝC�mäC�R� øLù �V��"9� ù<Ì �V�V�NûP�/�<����� Ì ��äC�R���Ô  Ì �6û��h�� ��H��à�� ù �V�V¡ëí½�½/·�Éoª¯®y¹/§WµT¬_³_´�µ/£N¸ *�µ/£NµT¬o¿ 4 ·Èµ/¦Ò³o´9½/£��î½/¦¥½"Ã>³h£N³"½/ÅC´¨¹V§WµT¬;³o´¨Ñíª¯®a¿�¢Â§T§L©Æª­¬�µV®¯ª­½/£*´9®¶½9¢9£Lµ/©Æ°�¼hªê£CÃ4 ÅW£L¬�®¯ª¯½/£<´î½/£r¹/§6¿W³h·R³o´ � ô Ë*õTË àR�¨Í/�V�*�/�

�TÍ��>�*�/�îÞÙ�8� ù �/� Ì � áTÌ�� �LÞÙ�8� ù �/� Ì � �áTù С��ÅW·}¾d½�¸>³o©a©«ª¶³o·_ÅW£>Ã�¸�³o·mÚ¿*³o·_¦�½]³o©�µ%´_®¶½/նǩ�µ%´o®¯ª�¼hªa® �µV®�¦mª¯®OÇ ¿Cµ%´�³o£<ÅW¦�Ñ�µ/£L¸*©«ÅW£>ÃC³o£ÙÉ�³hªO¹6® �µ%¿�©º³o£ì´h½/Ñíª­³¸�³o·�ýH¦�Ñ�µ/£L¸*©«ÅW£>ÃV´¶§L©�µ%´_®¯ªº¼oª¯® �µV® �O�*�h�<� Ë �/��Í/�V�TÍC�

�TÍ��>�TÍC�mäC�R� øLù �V��"9� ù<Ì �V�V�6�y�h�R�h�9Þ��V�/�*¡¢�£üòÂÅW®¯©Æªa£N³�½ÈÀ9¢�¸Tµ;§ ®¯ªêÄV³rÎ�µ/ÄV³o©º³h® � µ/©º³o·��]ªa£Ù¾l³h®a¿W½�¸%´�Àh½/·�ÚHª,��¿W½/£L½/Äm±¨³¶ÃVÅW©�µ/·_ª�¼�µV®¯ª¯½V£ ½ÈÀP¤;£<Ä/³h·�´�³Ç�µ/·Rµ�É�½/©«ª­¬�Ç&·È½�Éh©�³h¦m´ �Lô ø ��� Ì Í/�V�TÍC�

�TÍ��>�TÝC��ô Ì21 �"�� �ä3� ù ¥�º *�h¡¢ ¾rÅW© ®¯ªêÕ¶¾l³o´�¿ 4 ªa£<ª¯®A³9Á©º³o¦Ò³o£6®í¾l³�®a¿C½�¸�Àh½/·�Ç¿Cµ%´h³ 4 ª¶³o©�¸Û¹<ªê¦�ÅW©�µV®¯ª¯½V£*´ � ô ø ��� Ì Í/�V�TÍC�

�TÍ��>�/ï*�mäC�h� õ �hú)�î��"9� � ù � á� àR�C�¶�NÞÙ�8� ù �/� Ì � �áTù С¢ ÿ�½T®A³�½/£ 4 ªa£<ª¯®A³�Ú¿W³o·_¦¥½;§L©�µ%´_®¯ª­¬hª¯®¯°ÐÑíªa®a¿ÛÇ¿Cµ%´h³�ëH¿Wµ/£>Ã>³o´ � ÊTË<Ì �OÍ/�V�TÍC�

�TÍ��>�TöC�îÞÙ�8� ù �/� Ì � áTÌ�� �LÞÙ�8� ù �/� Ì � �áTù Ð¡Ç ¿Cµ/´h³o£6ÅC¦�Ñ�µ/£N¸W©«ÅW£>ÃC³o£¥ÅW£L¸Ûýy¦�Ñ�µ/£L¸W©ÆÅW£>Ã/´A§L© µ/´_®¯ªº¼oª¯® �µV®yÉ;³oª<¹6® �µ/¿�©�³h£¥ªa¦ ��½/£C¼]³E§N®�¸>³o·9Ú¿*³o·_¦�½]³o©�µ%´_®¶½/ÕÇ©�µ%´_®¯ªº¼oª¯® �µV® � ÊTË<Ì �OÍ/�V�TÍC�

�TÍ��>���C�mû��h�; ����à0� ù �V�V¡ëí½/£*´o®¯·;Å<¬�®¯ª­½/£ ½ÈÀ � ³h£N³o·Rµ/©Æª�¼]³"¸ ýH£L¬_³o·o®¶µ/ªa£6®¯°�Ç·_ªê£L¬�ª §N©�³o´ìµ/£L¸ Î�µ/Ä/³h©�³�®a´Òªa£i¢�£<ªê´o½V®¯·R½;§Lª­¬ó¹N½�É"½/©º³oĹ/§WµT¬_³_´ �Lô Ë*õTË àR�¨Í/�V�TÍC�

�TÍ��>���>��÷ á �<��ú�÷¨� Ì � Ë ¡ÚO¤ � ±�¢ ð�µ/£�ªa®A³h·ÈµT®¯ªêÄV³Pµ/© Ã�½/·_ªa®a¿>¦ Àh½/·�·"³¶ÃTÅC©�µ/·_ª�¼hªê£CÃ�£L½/£<©Æªa£N³"µ/·�ªê©a© ð_§W½%´�³"¸�§L·R½�Éh©�³h¦î´ �Cô Ë*õTË à"��Í/�V�TÍC�

�TÍ��>�T�C�îÞÙ�8� ù �/� Ì � Ë �%�Và�� ù �$6¨���"à0� ù �<�y�h�R�h�PÞ��V�/�*� Þ��8� ù �/� Ì �}�º  � á �"�N¡ÚHª �]¿C½/£L½VÄ¥·R³AÃVÅW©�µ/·;ªº¼�µT®¯ª¯½/£ÒÀo½/·9Á©º³"¬�®¯·_ª­¬�µ/©O¤;¦¨§<³"¸Tµ/£N¬;³�ÚL½/¦¥½"ÃT·Èµ;§L¿�°�½/£rÅW£NÉ"½/ÅW£L¸>³"¸Ð¸T½/¦¥µ/ªa£*´ �Oç��>æ� á � �h��Í/�V�TÍC�

Page 21: Exponential estimates for time delay systems...obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate,

�TÍ��>�T�C� � áTÌ �V�h�!"9�8�"�V���O�N�y�h�R�h��ÞÙ�V�/�*���È� õVá Þ�����#V� Ê ���6��ú)���º�h�6à�� ù �<÷ á �<�>úì÷�� Ì � Ë ¡¤;£<Ä/³h·�´�³ÙýH£6ÑíÅ*¬h¿>®¯ª­¸�³o£6®¯ª �VµV®¯ª¯½V£lµ/£ 4 ©ÆÅTÃT®¯·_ª­³]ÉoÑ}³h·0�T³o£r¦�ªa®�ñÂÅ6³h®a´o¬h¿ �½/©�¸��µ/¦�§]À�³o·_£ ��ç��>� á � �h��Í/�V�TÍC�

�TÍ��L�h�W� � á ��àR�R��� ��áTù<Ì �h���*�y�h�R�h�PÞ��V�/�*� Ê �V��²� Ì  *���O¡ÚHªê¦Ò³hÕA´�³o·_ª­³o´2Àh½/·"³"¬;µ%´_®¯ªa£>ÃPÀh½/·�®¶½T®¶µ/©Ä/½/©«ÅW¦Ò³�¸TµV®¶µÐµV£L¸Ð¬o¿CµV·¯ÃC³�É"µT¬$�Û¸TµV®¶µ ��� á�� �� �� �h�PÍ/�V�TÍC�

�TÍ��L�V�/��ô�� õ � Ì ���/��� Ë �<àR�oæÖû9�h��àR�"�*�h��¡¢ ¹*¿C½/·o®�¤;£6®¯·R½�¸/Å<¬�®¯ª­½/£l®¶½�¤_®A³o·RµV®¯ªaÄ/³²¾i³h®a¿C½�¸%´�Àh½/·�»�µV·¯ÃC³²»2ªa£N³"µV·²¹<°%´_®A³h¦î´ � � á� �� î� �h��Í/�V�TÍC�

�TÍ��L��ÍC���y�h�R�h� Ë �*�V� Ì �3� áTÌ �V�h�9Þó� ù �0 ¥�V�<�O�<÷ á �<� Ì  �äC��á � �h��¡¹<°/¦�¦¥³�®¯·;ª­¬Ðëí½/©a© ½�¬�µV®¯ª­½/£�Àh½/·ÐýH£<´_®¯·_Å*¬�®¯ÅW·R³�¸�ÿî½/£<©Æªa£N³"µ/·9Ó²ª � ³h·R³h£6®¯ª¯µV©«Õ¶¢�© Ã>³�Éh·ÈµVª¯¬PÁ�×�Å<µV®¯ª¯½V£*´�½ÈÀ�¢�·_ÕÉoª¯®¯·ÈµV·;°¥¤;£L¸�³�Ø ��� á� �� î� �h�9Í/�V�TÍC�

�TÍ��L��ÝC�îÞÙ�8� ù �/� Ì � áTÌ�� ¡±¨ªê£>ÃTÄ%½/·_©º³_´;ÅW£>à ��Ó�ªÔ´o®¶½/·_®¯ª­½/£ÙÁ&£>ÃVªa£N³�³h·;ªa£>Ã����½/£6®¯ªa£<ÅWÅW¦î´_¦Ò³"¬o¿Wµ/£<ªê´o¬o¿*³²¾d½�¸>³o©a©«ª¶³o·_ÅW£>Ã�¸�³_´�¾dµT®A³o·_ª¯µ/©ÆÄ/³h·�¿Wµ/©Æ®A³h£*´�Ä/½/£Ù¹L®¶µ%¿�©�ÅW£6®A³h· *�³h· �Å<¬ �]´;ª­¬o¿�Õ®¯ª ÃTÅW£>ÃÛÄ%½/£�Ç ¿Wµ%´h³h£<ÅW¦mÑ�µV£L¸W©ÆÅC£CÃ>³o£ � "��$#h�� �� �h�PÍ/�V�TÍC�

�TÍ��L�oï*�îÞÙ�8� ù �/� Ì ���áTù Ð�6ÞÙ�/�"�"�º�ó� Ë �*�V� Ì �<ô Ì21 �"�� �ä3� ù ¥�º <�h�*ÞÙ�8� ù �/� Ì � áTÌ�� ¡ÁÂÄ/µ/©«Å<µV®¯ª­½/£ ½ÈÀ�Ä%µ/·_ª¯½VÅ>´�§L¿Cµ%´h³hÕÖ®¯·Rµ/£*´_ªa®¯ª­½/£ ¦�½�¸�³o©«´îÀh½/·������*ëy·��Ào½/·rµ;§T§L©Æª­¬�µV®¯ª­½/£�ªa£�¬�½/¦�¦Ò³o·R¬hª­µ/©4 Á�¾ §L·R½"ÃV·Rµ/¦m´ � "��$#h�� �� �h�9Í/�V�TÍC�

�TÝ��>�*�/�îÞÙ�8� ù �/� Ì � áTÌ�� �LÞÙ�8� ù �/� Ì � �áTù Ð�NäC�h� õTË ��� "�� � ù � á� àR�W�­¡�<½/©ÆÅW¦¥³h£Lµ/£6®A³hªê©º³²Ä/³o·;´;ÅC´¨¾dµ%´;´h³h£Lµ/£6®A³hªê©º³PÕ&¸�³h·�Ó�ªa© µT®¶½/¦Ò³h®A³o·_Ä/³h·�´_Å<¬o¿�µVÅ>´²¸�³h·�¹6ª¯¬h¿>®�¸�³h· ��½V£6®¯ªê£6ÅCÕÅW¦î´_¦Ò³"¬o¿Wµ/£<ª,� � Ê �V� Ë �/��Í/�V�TÝC�

�TÝ��>�TÍC��"9�V�<��� Ì P��à"à Ì �h���<÷¨�8�h�/��  á �î��� á � ù �h�R� á � ô Ì21 �"�� �ä3� ù ¥�º <�h¡¢i§W½/´_®A³o·_ª­½/·;ª³h·;·R½/·�¬�½/£L®¯·È½/©oÀh½/·Â®a¿W³�¢�©a©º³o£<Õhëíµ%¿�£îÇ&·È½�Éh©�³h¦ ��¬hªa·È¬�ÅW¦mÄV³o£6®¯ªa£>à � ·R½/£<Ñ�µV©ê© � ´&ªa£N³"×�Å<µ/©«ª¯®¯° �Þ/��/�0#îÍ/�V�TÝC�

�TÝ��>�TÝC�îÞÙ�8� ù �/� Ì ���áTù Ð� Ê �á � õ P� áVø6ø �*ô� *�;�º�V�óÞ Ë ���R���V�O¡ò£ µóÇ·"³"¸/ª­¬�®¯ª­½/£ ¾l½�¸�³o©LÀo½V·Ïëí½/£L¬�·R³�®A³dëíµ/·�É"½V£LµV®¯ª­½/£ É�µ%´h³�¸d½/£i¾d½VÄ�ªa£>Ãó¤;£6®A³o·aÀhµT¬;³o´ÒÕ�¤;£6®A³h·êÀhµT¬;³¬�½/£N¬;³o£L®¯·ÈµV®A³�¸Ò±�³�µT¬�®¯ª­½/£*´ �Nô ø ��� Ì Í/�V�TÝC�

�TÝ��>�/ï*�îÞÙ�8� ù �/� Ì ���áTù Ð� Ê �á � õ P� áVø6ø �*ô� *�;�º�V�óÞ Ë ���R���V�O¡¢ ÚHÑ�½/Õ­±¨³"µT¬�®¯ª¯½/£6Õ �H½V£N³_´ì¾l½/Ä�ªa£>ÃVÕ¶¤;£6®A³h·êÀhµT¬;³�¾l½�¸�³o©OÀh½/·ÛÇ&·R³�¸/ª¯¬�®¯ªê£CÃ� ����� ����� ëíµ/·hÉ�½/£LµV®¯ª­½/£ ªa£ëí½/£L¬�·R³�®A³ �6ô ø ��� Ì Í/�V�TÝC�

�TÝ��>�TöC� � Ì �V <�8 Ò��� ��� ù �/�;��� á � á� ��"9���� *�h�;�8� ù ���º�*��� � ù àR����¡Á�Øo§W½/£N³h£6®¯ª¯µV©2³_´o®¯ªê¦¥µV®A³o´�Àh½/·�®¯ªa¦¥³�¸>³o©�µ/°Ò´_°%´_®A³o¦m´ �6ô ø ��� Ì Í/�V�TÝC�