DSpace at VNU: New criteria for exponential stability of nonlinear time-varying differential systems

DOI: 10.1002/rnc.2885 New criteria for exponential stability of nonlinear time-varying differential systems Pham Huu Anh Ngoc*,† International University, Vietnam National University-HCM

Int J Robust Nonlinear Control (2012)

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/rnc.2885

New criteria for exponential stability of nonlinear time-varying

differential systems Pham Huu Anh Ngoc*,†

International University, Vietnam National University-HCMC, Thu Duc, Ho Chi Minh City, Vietnam

SUMMARY General nonlinear time-varying differential systems are considered An explicit criterion for exponential stability is presented Furthermore, an explicit robust stability bound for systems subjected to nonlinear time-varying perturbations is given In particular, it is shown that the generalized Aizerman conjecture holds for positive linear systems Some examples are given to illustrate obtained results Copyright © 2012 John Wiley & Sons, Ltd

Received 25 April 2012; Revised 2 July 2012; Accepted 10 July 2012

KEY WORDS: nonlinear differential system; time-varying; exponential stability; perturbation

1 INTRODUCTION AND PRELIMINARIES Stability analysis of time-varying differential systems is always a central issue of control theory of dynamical systems Problems of stability and robust stability of time-varying differential systems have attracted much attention from researchers and have been studied intensively during the past decades (see, e.g [1–20] and references therein) In this paper, we investigate exponential stability

of general nonlinear time-varying differential systems of the form

P x.t / D f t , x.t //, t >  > 0 (1) Stability analysis of the nonlinear time-varying differential system (1) is, in general, hard Several approaches have been proposed in the literature, and most of them are based on the classical Lyapunov method and its variants (see, e.g [1, 2, 10–15]) In the present paper, we propose a new approach to problems of stability and robust stability of the nonlinear time-varying differential system (1) Our approach is based on the celebrated Perron–Frobenius theorem and ideas of the comparison principle We first present a new explicit criterion for exponential stability of the non-linear time-varying differential system (1) Then, we give an explicit robust stability bound for (1) subjected to nonlinear time-varying perturbations In particular, we show that the generalized Aizerman conjecture holds for positive linear systems

Let N be the set of all natural numbers For given m 2 N, let us denote mWD ¹1, 2, : : : , mº Let

KD C or R where C and R denote the sets of all complex and all real numbers, respectively For integers l, q > 1, Kl denotes the l-dimensional vector space over K, and Klqstands for the set of all l  q-matrices with entries in K Inequalities between real matrices or vectors will be understood componentwise; that is, for two real matrices A D aij/ and B D bij/ in Rłq, we write A > B if

aij >bij for i D 1,    , l, j D 1,    , q In particular, if aij > bij for i D 1,    , l, j D 1,    , q, then we write A  B instead of A > B We denote by RlqC the set of all nonnegative matrices

*Correspondence to: Pham Huu Anh Ngoc, International University, Vietnam National University-HCMC, Thu Duc,

Ho Chi Minh City, Vietnam.

† E-mail: phangoc@hcmiu.edu.vn

A > 0 Similar notations are adopted for vectors For x 2 Knand P 2 Klq, we define jxj D jxij/ and jP j D 

jpijj

A norm k  k on Kn is said to be monotonic if kxk 6 kyk whenever x,

y 2 Kn, jxj 6 jyj Every p-norm on Kn(kxkp D jx1jpC jx2jpC    C jxnjp/p1, 1 6 p < 1 and kxk1 D maxi D1,2,:::,njxij), is monotonic Throughout the paper, if otherwise not stated, the norm of vectors on Kn is monotonic, and the norm of a matrix P 2 Klq is understood as its operator norm associated with a given pair of monotonic vector norms on Kl and Kq, that is

kP k D max¹kP yk W kyk D 1º Note that

P 2 Klq, Q 2 RlqC , jP j 6 Q ) kP k 6 k jP j k 6 kQk, (2) see, e.g [21] In particular, if Kn is endowed with k  k1 or k  k1, then kAk D kjAjk for any

A D aij/ 2 Knn More precisely, one has

kAk1D kjAjk1D max

16j 6n

n

X

i D1

jaijjI kAk1D kjAjk1D max

16i6n

n

X

j D1

jaijj

Let BrWD ¹x 2 Rn W kxk 6 rº for given r > 0 For any matrix M 2 Cnn, the spectral abscissa of

M is denoted by .M / D max¹< W  2  M /º, where  M / WD ¹´ 2 C W det.´In M / D 0º is the spectrum of M A matrix A 2 Rnnis called Hurwitz stable if .A/ < 0

A matrix M 2 Rnnis called a Metzler matrix if all off-diagonal elements of M are nonnegative.

We now summarize in the following theorem some properties of Metzler matrices

Theorem 1.1 ([21])

Suppose that M 2 Rnnis a Metzler matrix Then

(i) (Perron–Frobenius) .M / is an eigenvalue of M , and there exists a nonnegative eigenvector

x ¤ 0 such that M x D .M /x

(ii) Given ˛ 2 R, there exists a nonzero vector x > 0 such that M x > ˛x if and only if

.M / > ˛

(iii) tIn M /1exists and is nonnegative if and only if t > .M /

(iv) Given B 2 RnnC , C 2 Cnn Then

jC j 6 B H) .M C C / 6 .M C B/

The following is immediate from Theorem 1.1 and is used in what follows

Theorem 1.2

Let M 2 Rnnbe a Metzler matrix Then the following statements are equivalent

(i) .M / < 0;

(ii) Mp  0 for some p 2 Rn, p  0I

(iii) M is invertible and M160;

(iv) For given b 2 Rn, b  0, there exists x 2 RnC, such that M x C b D 0

(v) For any x 2 RnCn ¹0º, the row vector xTM has at least one negative entry

2 CRITERIA FOR EXPONENTIAL STABILITY Consider a nonlinear time-varying differential system of the form (1), where f W RC Rn! Rnis continuous and is locally Lipschitz in the second argument, uniformly in t on compact intervals of

RCand f t , 0/ D 0, for all t 2 RC

It is well known that for a fixed  > 0 and a given x02 Rn, there exists a unique local solution

of (1) satisfying the initial condition

This solution is continuously differentiable on Π,  / for some  >  and satisfies (1) for every

t 2 Π,  / (see, e.g [6]) It is denoted by x.I  , x0/ Furthermore, if the interval Π,  / is the maximum interval of existence of x.I  , x0/, then x.I  , x0/ is said to be noncontinuable The

existence of a noncontinuable solution follows from Zorn’s lemma, and the maximum interval of existence must be open

Definition 2.1

The zero solution of (1) is said to be exponentially stable if there exist positive numbers r, K, ˇ such

that for each  2 RCand each x02 Br, the solution x.I  , x0/ of (1) and (3) exists on Π, 1/ and furthermore satisfies

kx.t I  , x0/k 6 Keˇ t  /kx0k, 8t >  When the zero solution of (1) is exponentially stable, we also say that (1) is exponentially stable

We are now in the position to state the main result of this note

Theorem 2.2

Suppose that for each t 2 RC, f t , / is continuously differentiable on Rn Let

J.t , x/ WD

@f

i

@xj

.t , x/



2 Rnn, t 2 RC, x 2 Rn,

be the Jacobian matrix of f t , / at x Assume that there exists a Hurwitz stable Metzler matrix

A WD aij/ 2 Rnnsuch that for any t > 0 and any x 2 Rn,

@fi

@xi

.t , x/ 6 ai i, i 2 nI

ˇ

ˇ@fi

@xj

.t , x/

ˇ

ˇ6aij, i 6D j , i , j 2 n (4) Then, (1) is exponentially stable

Proof

Let x0 2 Rnbe given and let x.t / WD x.t I  , x0/, t 2 Œ ,  / be a noncontinuable solution of (1) and (3) We first show that there exists ˇ > 0 such that for any  > 0 and any r > 0 and any

x02 Br, we have

kx.t I  , x0/k 6 Keˇ t  /, 8t 2 Œ ,  /, (5) where K 2 R is independent of t ,  and x0

Because A is a Hurwitz stable Metzler matrix, there exists p WD ˛1, ˛2, : : : , ˛n/T, ˛i> 0, 8i 2 n such that

by (ii) of Theorem 1.2 Furthermore, (6) implies that

Ap  ˇp D ˇ.˛1, : : : , ˛n/T, (7) for some sufficiently small ˇ > 0 Fix r > 0 and choose K > 0 such that jx0j  Kp for any

x0 2 Br Define u.t / WD Keˇ t  /p, t 2 Œ , 1/ Set x.t / WD x.t I  , x0/, t 2 Œ ,  / Note that jx. /j D jx0j  u. / D Kp We claim that jx.t /j 6 u.t / for any t 2 Œ ,  /

Assume on the contrary that there exists t0>  such that jx.t0/j 66 u.t0/ Set t1WD inf¹t 2  ,  / W jx.t /j 66 u.t /º By continuity, t1>  , and there is i02 n such that

jx.t /j 6 u.t /, 8t 2 Π, t1/I jxi0.t1/j D ui0.t1/, jxi0.t /j > ui0.t /, 8t 2 t1, t1C /, (8) for some  > 0 By the mean value theorem [22], we have for each t 2 R and for each i 2 n

P

xi.t / D fi.t , x.t // D fi.t , x/  fi.t , 0// D

n

X

j D1

Z 1 0

@fi

@xj

.t , sx.t //ds



xj.t /

d

dtjxi.t /j D sgn.xi.t // Pxi.t / D sgn.xi.t //

n

X

j D1

Z 1 0

@fi

@xj

.t , sx.t //ds



xj.t /

D

Z 1

0

@fi

@xi

.t , sx.t //ds



jxi.t /j C sgn.xi.t //

n

X

j D1,j 6Di

Z 1 0

@fi

@xj

.t , sx.t //ds



xj.t /

for almost any t 2 Π,  / Then, (4) implies

d

dtjxi.t /j 6 ai ijxi.t /j C

n

X

j D1,j 6Di

for almost any t 2 Π,  / It follows that for any t 2 Π,  /

DCjxi.t /j WD lim sup

h!0 C

jxi.t C h/j  jxi.t /j

h!0 C

1 h

Z t Ch t

d

dsjxi.s/jds

6ai ijxi.t /j C

n

X

j D1,j 6Di

aijjxj.t /j,

where DCdenotes the Dini upper-right derivative In particular, it follows from (7) and (8) that

DCjxi 0.t1/j.8/6ai 0 i 0Keˇ t1  /

˛i 0C

n

X

j D1,j 6Di 0

ai 0 jKeˇ t1  /

˛j D Keˇ t1  /

n

X

j D1

ai 0 j˛j

.7/

< K.ˇ/eˇ t1  /˛i0D DCui0.t1/

However, this conflicts with (8) Hence,

jx.t I  , x0/j 6 u.t / D Keˇ t  /p, 8 > 0I 8x02 BrI 8t 2 Œ ,  /

By the monotonicity of vector norms, this yields

kx.t I  , x0/k 6 K1eˇ t  /, 8 > 0I 8x02 BrI 8t 2 Œ ,  /, for some K1> 0

Finally, we show that  D 1, and so (1) is exponentially stable Seeking a contradiction, we assume that  < 1 Then it follows from (5) that x.I  , x0/ is bounded on Π,  / Furthermore, this together with (1) implies that Px./ is bounded on Π,  / Thus, x./ is uniformly continuous

on Π,  / Therefore, limt ! x.t / exists, and x./ can be extended to a continuous function on Π,   Then, one can find a solution of (1) through  , x. // to the right of  This contradicts the noncontinuability hypothesis on x./ Thus,  must be equal to 1, and this completes the proof  The following is immediate from Theorem 2.2

Corollary 2.3

Suppose f W Rn! Rnis continuously differentiable Then, the system

P x.t / D f x.t //, t >  > 0,

is exponentially stable provided that there exists a Hurwitz stable Metzler matrix A WD aij/ 2 Rnn such that for any x 2 Rn,

@fi

@xi

.x/ 6 ai i, i 2 nI

ˇ

ˇ@fi

@xj

.x/

ˇ

ˇ6aij, i 6D j , i , j 2 n

To state the next result, we now consider a linear time-varying differential system of the form

P x.t / D A.t /x.t /, t >  > 0, (10) where A./ W RC! Rnn, k 2 m is a given continuous vector function

Corollary 2.4

Let A.t / WD aij.t //, t > 0 Suppose there exists a Hurwitz stable Metzler matrix A WD aij/ 2 Rnn

so that for any t > 0

ai i.t / 6 ai i, i 2 nI jaij.t /j 6 aij, 8i 6D j , i , j 2 n

Then, (10) is exponentially stable

We illustrate the obtained results by a couple of examples

Example 2.5

Consider the nonlinear time-varying differential equation

P x.t / D 3

2x.t / C sin

 2t

t2C 1x.t /



Clearly, (11) is of the form (1) with f t , x/ WD 32x Csin

2t

t 2 C1x Furthermore, f and@f@x.t , x/ D

32Ct22tC1cos



2t

t 2 C1x are continuous on RCR Because@f@x.t , x/ 6 12, 8t > 0, 8x 2 R, (11) is exponentially stable, by Theorem 2.2

Example 2.6

Consider the nonlinear time-varying differential system

8 ˆ ˆ

P

x1.t / D 4et2x1.t / C x1 t /

1Cx 2 t /sin x2.t /

P

x2.t / D sin

2tx1.t /

t 2 C1



 2et2x2.t /

(12)

where t >  > 0

Then, (12) can be represented in the form (1) with

f W RC R2! R2I f t , x/ D t , x/ 7!

0 B

@

4et2x1C x1

1Cx 2sin x2

sin

2tx1

t 2 C1



 2et2x2,

1 C

A , x D x1, x2/T 2 R2

(13)

It is clear that f is continuous on RC R2and f t , / is continuously differentiable on R2for each

t 2 RCand f t , 0/ D 0, 8t 2 RC Furthermore, the Jacobian matrix of f t , / is given by

J.t , x/ D

0 B

@

4et2C 1x2

.1Cx 2/2sin x2

x1

x 2 C1cos x2 2t

t 2 C1cos

2tx1

t 2 C1



2et2

1 C

A , t > 0, x D x1, x2/T 2 R2

It is easy to check that the matrix A WD



3 12

1 2



is Hurwitz stable and satisfies (4) with n D 2 Thus, (12) is exponentially stable, by Theorem 2.2

3 STABILITY OF PERTURBED SYSTEMS Suppose all hypotheses of Theorem 2.2 hold Thus, (1) is exponentially stable Consider a perturbed system of the form

P

x.t / D f t , x.t // C

N

X

kD1

Dk.t , x.t //Pk.t ,Ek.t , x.t /// , t >  > 0, (14)

where N is a given positive integer andDkW RC Rn! Rnlk, EkW RC Rn! Rqk, (k 2 N ) are given continuous functions andPkW RC Rqk! Rlk (k 2 N ) are unknown continuous functions.

Furthermore, we assume that

(H1) for each k 2 N ,Dk,Pk and Ek (k 2 N ) are locally Lipschitz in the second argument uniformly in t on compact intervals of RCandPk.t , 0/ D 0,Ek.t , 0/ D 0 for all t 2 RC; (H2) there exist Dk2 Rnlk

C , Ek2 Rqk n

C and Pk2 Rlk qk

C (k 2 N ) such that

jDk.t , x/j 6 Dk, 8t 2 RC, x 2 Rn (15) and

jEk.t , x/j 6 Ekjxj, 8t 2 RC, 8x 2 RnI jPk.t , y/j 6 Pkjyj, 8t 2 RC, 8y 2 Rqk (16) Note that the preceding assumptions imply that the right hand side of (14) is continuous on RCRn and is locally Lipschitz in the second argument Thus, (14) always has a unique local solution satisfying the initial condition (3)

The main problem here is to find a positive number  such that an arbitrary perturbed system of

the form (14) remains exponentially stable whenever the size of perturbations is less than

Remark 3.1

In particular, if

Dk.t , x/ WD Dk.t /I Ek.t , x/ WD Ek.t /xI Pk.t , y/ WD Pk.t /y, t > 0I x 2 RnI y 2 Rqk, then perturbation termPN

kD1Dk.t , x.t //Pk.t ,Ek.t , x.t /// becomesPN

kD1Dk.t /Pk.t /Ek.t /x.t / The robust stability of linear time-varying system (10) under the time-varying multi-perturbations

A.t / ,! A.t / C

N

X

kD1

has been analyzed in [7], and an abstract stability bound was given in terms of input–output operators

Theorem 3.2

Assume that all hypotheses of Theorem 2.2 hold and A 2 Rnnis as in Theorem 2.2 If (H1)–.H2) hold and

N

X

kD1

kPkk < 1

maxi ,j 2NkEiA1Djk, (18) then (14) remains exponentially stable

Proof

We divide the proof into two steps

Step 1

We claim that 



A CPN kD1DkPkEk

< 0

Because A is a Metzler matrix and Dk, Ek, Pkare nonnegative for any k 2 N , A C

N

P

kD1

DkPkEk

is a Metzler matrix We show that 0WD  A C

N

P

kD1

DkPkEk

!

< 0 Assume on the contrary that

0>0 By the Perron–Frobenius theorem (Theorem 1.1 (i)), there exists x 2 RnC, x 6D 0 such that

A C

N

X

kD1

DkPkEk

!

x D 0x

Let Q.t / D tIn A, t 2 R Because .A/ < 0, Q.0/ is invertible It follows that

Q.0/1

N

X

kD1

Let i0 be an index such that kEi0xk D maxk2NkEkxk It follows from (19) that kEi0xk > 0 Multiply both sides of (19) from the left by Ei0to obtain

N

X

kD1

Ei0Q.0/1DkPkEkx D Ei0x

It follows that

N

X

kD1

kEi0Q.0/1DkkkPkkkEkxk > kEi0xk

Thus,

max

i ,j 2NkEiQ.0/1Djk

N

X

kD1

kPkk

!

kEi0xk > kEi0xk,

or equivalently,

max

i ,j 2NkEiQ.0/1Djk

N

X

kD1

On the other hand, the resolvent identity gives

Q.0/1 Q.0/1D 0Q.0/1Q.0/1 (21) Because A is a Metzler matrix with .A/ < 0 and 0 >0, Theorem 1.1 (iii) yields Q.0/1>0 and Q.0/1 > 0 Then, (21) implies that Q.0/1 > Q.0/1 > 0 Hence, EiQ.0/1Dj >

EiQ.0/1Dj >0, for any i , j 2 N By (2), kEiQ.0/1Djk > kEiQ.0/1Djk, for any i ,

j 2 N Thus, (20) implies that

N

X

kD1

maxi ,j 2NkEiQ.0/1Djk. However, this conflicts with (18)

Step 2

The proof of Step 2 is similar to that of Theorem 2.2

Let x02 Rnbe given, and let x.t / WD x.t I  , x0/, t 2 Π,  / be a noncontinuable solution of (14) satisfying the initial condition (3)

We first show that there exists ˇ > 0 such that for any  > 0 and any r > 0 and any x0 2 Br,

we have

kx.t I  , x0/k 6 Keˇ t  /, 8t 2 Œ ,  /, (22) where K is independent of t ,  , x0

By (ii) of Theorem 1.2, there exists p WD ˛1, ˛2, : : : , ˛n/T, ˛i> 0, 8i 2 n such that

A C

N

X

kD1

DkPkEk

!

Furthermore, (23) implies that

A C

N

X

kD1

DkPkEk

!

p  ˇp D ˇ.˛1, : : : , ˛n/T, (24)

for some sufficiently small ˇ > 0 Fix r > 0 and choose K > 0 such that jx0j  Kp for any

x0 2 Br Define u.t / WD Keˇ t  /p, t 2 Œ , 1/ Set x.t / WD x.t I  , x0/, t 2 Œ ,  / Note that jx. /j D jx0j  u. / D Kp We show that jx.t /j 6 u.t / for any t 2 Œ ,  /

LetPN

kD1DkPkEk WD bij/ 2 Rnn Taking (15)–(16) into account, we obtain the following estimate

d

dtjxi.t /j 6 ai ijxi.t /j C

n

X

j D1,j 6Di

jaijjjxj.t /j C

n

X

j D1

bijjxj.t /j, (25)

for almost any t 2 Π,  / The remainder of the proof is similar to that of the proof of

Suppose all hypotheses of Corollary 2.4 hold Thus, (10) is exponentially stable Consider a perturbed system of the form

P

x.t / D A.t /x.t / C

N

X

kD1

Dk.t , x.t //Pk.t ,Ek.t , x.t /// , t >  > 0, (26)

whereDk,PkandEk, (k 2 N ) are as previously

The following is immediate from Theorem 3.2

Corollary 3.3

Let A 2 Rnn be as in Corollary 2.4 Suppose H1)–.H2) hold If (18) holds, then (26) is exponentially stable

Corollary 3.4

Let A 2 Rnnbe a Hurwitz stable Metzler matrix Suppose DkW RC! Rnlk, EkW RC! Rqk n

.k 2 N / are given continuous functions and PkW RC! Rlk qk

is an unknown continuous function

If there exist Dk2 Rnlk

C , Ek2 Rqk n

C and Pk2 Rlk qk

C k 2 N / such that

jDk.t /j 6 DkI jEk.t /j 6 EkI jPk.t /j 6 Pk, 8t > 0, and (18) holds, then the perturbed system

P x.t / D A C

N

X

kD1

Dk.t /Pk.t /Ek.t /

! x.t /, t >  > 0, (27)

is exponentially stable

Remark 3.5

When A 2 Rnnis a Metzler matrix, the system

P

is positive That is, for any initial state x0 2 Rn

C, the corresponding trajectory of the system x.t , x0/ remains in Rn

C for all t > 0 Positive dynamical systems play an important role in modeling of dynamical phenomena whose variables are restricted to be nonnegative They are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modeling transport and accumulation phenomena of substances (see, e.g [5, 23])

In particular, the problem of robust stability of the positive linear differential system (28) under the time-invariant structured perturbations

A ,! A C DE has been studied in [9, 21, 24] For example, it has been shown in [9, Theorem 5] that if (28) is exponentially stable and positive and D, E are given nonnegative matrices, then a perturbed system

of the form

P x.t / D A C DE/x.t /, t > 0, remains exponentially stable whenever

kk < 1 kEA1Dk. This result has been extended to various classes of positive differential systems such as positive linear time-delay differential systems, positive linear functional differential systems, positive linear Volterra integro-differential systems, and so on (see, e.g [25–31]) Furthermore, the problem of robust stability of the positive system (28) under the time-invariant multi-perturbations

A ,! A C

N

X

i D1

DiiEi,

has been analyzed in [9] by techniques of -analysis

Although there are many works devoted to the study of robust stability of differential systems,

to the best of our knowledge, the problem of robust stability of the positive system (28) under the time-varying multi-perturbations

A ,! A C

N

X

kD1

Dk.t /Pk.t /Ek.t /, has not yet been studied, and a result like Corollary 3.4 cannot be found in the literature

We illustrate the obtained results by a couple of examples

Example 3.6

We now reconsider (11) As shown in Example 2.5, (11) is exponentially stable Consider a perturbed equation given by

P

x.t / D



3

2C bet2

 x.t / C sin

 2t

t2C 1x.t /



C sin.ax.t //, t >  > 0 (29) where a, b 2 R are parameters

Note that jbet2xj 6 jbjjxj and j sin.ax/j 6 jajjxj, for all t > 0, x 2 R Thus, by Theorem (3.2), (29) is exponentially stable if jaj C jbj 612

Example 3.7

Consider a linear differential equation in R2defined by

P

where

A WD



1 2

 Clearly, (30) is positive and exponentially stable Consider a perturbed system given by

P x.t / D A C D1.t /P1.t /E1.t / C D2.t /P2.t /E2.t // x.t /, (31) where

D1.t / WD



 sin t 1

 , t > 0I D2.t / WD

1 cos2t C1

 , t > 0I

E1.t / WD e

t 2

0

1 t22tC1

! , t > 0I E2.t / WD

1 1Ct 2 0

0 t C11

! , t > 0,

and P1.t / WD a.t /, b.t // 2 R12, t > 0 and P2.t / WD c.t /, d.t // 2 R12, t > 0 are unknown Note that for any t > 0, we have

jD1.t /j 6 D1WD

 1 1



I jD2.t /j 6 D2WD

 0 1

 I

jE1.t /j 6 E1WD



1 0

1 1



I jE2.t /j 6 E2WD



1 0

0 1



and

E1A1D1D



1 0

1 1

 

  1 1

 D



3

5

 I

E1A1D2D



1 0

1 1

 

  0 1

 D



1

2

 I

E2A1D1D



1 0

0 1

 

  1 1

 D



3

2

 I

E2A1D2D



1 0

0 1

 

  0 1

 D



1

1



Let R2be endowed with 2-norm By Corollary 3.4, (31) is exponentially stable if a./, b./, c./, d./ are continuous, bounded and satisfy

r

sup

t 2RC

ja.t /j/2C sup

t 2RC

jb.t /j/2Cr

sup

t 2RC

jc.t /j/2C sup

t 2RC

jd.t /j/2<p1

34.