DSpace at VNU: Floquet theorem for linear implicit nonautonomous difference systems

Floquet theorem for linear implicit nonautonomousdifference systems Pham Ky Anh∗, Ha Thi Ngoc Yen Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Han

Floquet theorem for linear implicit nonautonomous

difference systems

Pham Ky Anh∗, Ha Thi Ngoc Yen

Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

Received 14 June 2005 Available online 13 October 2005 Submitted by S.R Grace

Abstract

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS) It

is proved that any index-1 LIDS can be transformed into its Kronecker normal form Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established As

an immediate consequence, the Lyapunov reduction theorem is proved Some applications of the obtained results are discussed

©2005 Elsevier Inc All rights reserved

Keywords: Implicit difference equations; Differential algebraic equations; Floquet theorem; Lyapunov reduction

theorem

1 Introduction

Implicit difference systems (IDS) arise in many applications, such as the Leontief dynamic

model of multisector economy, the Leslie population growth model, singular discrete optimal control problems and so forth On the other hand, IDS can be considered as discrete analogues

of differential–algebraic equations (DAEs) which have already found various applications and

attracted much attention of researchers

In this paper we show that the Floquet theory, first established for regular ordinary differential equations (ODEs), and later for difference equations (see [1]) and recently for DAEs [4], can

be developed for index-1 LIDS The results of this paper are discrete analogues of those in [4]

* Corresponding author.

E-mail address: anhpk@vnu.edu.vn (P.K Anh).

0022-247X/$ – see front matter © 2005 Elsevier Inc All rights reserved.

doi:10.1016/j.jmaa.2005.08.075

The paper is organized as follows Section 2 is devoted to index-1 LIDS and their properties Section 3 deals with the Floquet theorem on representation of the fundamental solution and the Lyapunov reduction theorem for index-1 periodic LIDS Finally, in Section 4 some illustrative examples are given and further applications of obtained results are discussed

2 Linear implicit difference systems and their properties

Consider a linear difference system

where A n , B n∈ Rm ×m and q

n∈ Rmare given Throughout this paper, we assume that the

singu-lar matrices A n have the same rank for all n  0, i.e rank A n ≡ r (1  r  m − 1).

Together with (2.1) we consider a linear DAE

where A, B ∈ C(J, R m ×m ) and q ∈ C(J, R m ) According to [3], Eq (2.2) is called transferable

or index-1 tractable if there is a smooth projection Q ∈ C1(J,Rm ×m ) onto ker A(t), such that

the matrix G(t) = A(t) + B(t)Q(t) is nonsingular for all t ∈ J The transferability of Eq (2.2) does not depend on the choice of smooth projections Q(t), and it is equivalent to the condition S(t ) ∩ ker A(t) = {0}, ∀t ∈ J , where S(t) = {ξ ∈ R m : B(t)ξ ∈ im A(t)}.

For LIDS we have introduced the following similar definition (cf [2,6])

Definition 2.1 The LIDS (2.1) is said to be of index-1 if

(i) rank A n = r (n  0),

(ii) S n ∩ ker A n−1= {0} (n  1),

where, as in the DAE case, S n = {ξ ∈ R m : B n ξ ∈ im A n}

In what follows we always assume that dim S0= r.

The main difference between index-1 LIDS and linear transferable DAEs is that the matrix pencil{A n , B n } is not necessarily of index-1 while the pencil {A(t), B(t)} is of index 1 for all

t ∈ J Indeed, let

A n=









in Eq (2.1) Then S n = ker A n = span{(−n − 1, 1)T}, hence Eq (2.1) is of index-1 since S n∩

ker A n−1= {0} (n  1) On the other hand, as S n ∩ ker A n = {0}, the index of the pencil {A n , B n} does not equal 1

Let Q n∈ Rm ×m be any projection onto ker A

n , i.e Q2= Q n and im Q n = ker A n Then there

exists a nonsingular matrix V n∈ Rm ×m such that Q n = V n QV˜ −1

n ,where ˜Q := diag(O r , I m −r ) and O r , I m −r are r ×r zero and (m−r)×(m−r) identity matrices, respectively Put ˜ P := I − ˜ Q, where I is m × m identity matrix.

We define the so-called connecting operators (see [2]) as follows: Q n −1,n := V n−1QV˜ −1

n and

Q n,n−1:= V n QV˜ −1

n−1. Clearly, Q n −1,n = Q n−1Q n −1,n = Q n −1,n Q n ; Q n −1,n Q n,n−1= Q n−1

and Q n,n−1Q n −1,n = Q n

The following lemma plays an important role in the theory of index-1 LIDS (see [2,6])

Lemma 2.2 The following assertions are equivalent:

(i) S n ∩ ker A n−1= {0};

(ii) the matrix G n := A n + B n Q n −1,n is nonsingular;

(iii) Rm = S n ⊕ ker A n−1.

Lemma 2.3 Suppose Eq (2.1) is of index-1 Let Q n−1= V n−1QV˜ −1

n−1be an arbitrary projection

onto ker A n−1(n  1) Then

(i) ˜Q n−1:= Q n −1,n G−1

n B n is the canonical projection onto ker A n−1along S n; (ii) ˜Q n−1 = ˜V n−1Q ˜˜V−1

n−1, where ˜ V n−1 = (s1

n , , s n r , h r+1

n−1, , h m n−1) is a matrix, whose

columns form certain bases of S n and ker A n−1, respectively, i.e S n = span({s i

n}r

i=1) and

ker A n−1= ({h j

n−1}m

j =r+1 ).

Proof (i) The nonsingularity of G n and ˜V n−1 are followed from Lemma 2.2 Letting P n=

I − Q n , we find G n P n = A n and G n Q n = B n Q n −1,n, therefore

G−1

n A n = P n and G−1

n B n Q n −1,n = Q n

Further, ˜Q2n−1 = Q n −1,n (G−1

n B n Q n −1,n )G−1

n B n = Q n −1,n G−1

n B n = ˜Q n−1 Since ˜Q n−1 =

Q n−1Q n −1,n G−1

n B n, it implies that im ˜Q n−1 ⊂ im Q n−1= ker A n−1 Conversely, let x ∈

ker A n−1, hence x = Q n−1x, then

˜

Q n−1x = Q n −1,n G−1

n B n Q n−1x = Q n −1,n

G−1

n B n Q n −1,n

Q n,n−1x = Q n−1x = x, therefore ker A n−1⊂ im ˜Q n−1 Thus ker A n−1= im ˜Q n−1 Now it is easy to show that x ∈ S nif

and only if Q n G−1

n B n x = 0, or V n−1V−1

n Q n G−1

n B n x= 0, which is equivalent to the relations

˜

Q n−1x = Q n −1,n G−1

n B n x = 0 The last equality means that ker ˜ Q n−1= S n

(ii) From ˜V−1

n−1˜V n−1 = I, it follows ˜V−1

n−1s n i = e i (i = 1, r) and ˜V n−1−1h j n−1 = e j (j =

r + 1, m), where e k = (0, , 1, , 0)T(k = 1, m) Observing that ˜ Q n−1s n i = ˜V n−1Q ˜˜V−1

n−1s n i =

˜V n−1Qe˜ i = 0 (i = 1, r) and ˜ Q n−1h j n−1= ˜V n−1Q ˜˜V−1

n−1h j n−1 = ˜V n−1Qe˜ j = ˜V n−1e j = h j

n−1

(j = r + 1, m) we come to the conclusion that ˜ Q n−1is the canonical projection onto ker A n−1

along S n Lemma 2.3 is proved 2

Now consider a linear system, obtained from (2.1) via scaling and transforming variables, namely, the following equation:

¯

where ¯A n = E n A n F n ; ¯B n = E n B n F n−1; ¯q n = E n q n and the matrices E n , F n are nonsingular

Here E n are scaling matrices, while the transformations of variables are defined by x n = F n−1¯x n

Since ¯S n∩ ker ¯A n−1= F n−1−1(S n ∩ ker A n−1),the index-1 property of LIDS is invariant under scaling and linear transformations

Theorem 2.4 Every index-1 LIDS can be reduced to the Kronecker normal form

Proof Suppose Eq (2.1) is of index-1 First we show that the matrix ˜G n := A n ˜V n + B n ˜V n−1Q˜

is nonsingular This fact is followed from Lemma 2.2, however it can be verified directly In-deed, suppose ˜G n x = 0, then B n ˜V n−1Qx˜ = −A n ˜V n x,hence ˜V n−1Qx˜ ∈ S n On the other hand,

˜V n−1Qx˜ = ˜Q n−1˜V n−1x ∈ ker A n−1 This implies the conclusion ˜V n−1Qx˜ ∈ S n ∩ ker A n−1=

{0}, or ˜ Qx = 0, therefore A n ˜V n x = 0 The last relation ensures that x ∈ ˜V−1

n ker A n =

span( {e j}m

j =r+1 ). Thus x= ˜Qx = 0, hence ˜G nis nonsingular

Following the DAE case (see [4]), we use the scaling E n = ˜G−1

n and the transformation

of variables F n = ˜V n , i.e ¯A n = ˜G−1

n A n ˜V n and ¯B n = ˜G−1

n B n ˜V n−1. Observing that ˜G n Q˜ =

A n ˜V n Q˜ + B n ˜V n−1Q˜ = A n Q˜n ˜V n + B n ˜V n−1Q˜ = B n ˜V n−1Q,˜ we find ˜G−1

n B n ˜V n−1Q˜ = ˜Q. Sim-ilarly, since ˜G n P˜ = A n ˜V n P˜ + B n ˜V n−1Q ˜˜P = A n ˜V n , it follows ˜G−1

n A n ˜V n = ˜P Thus ¯A n =

˜G−1

n A n ˜V n= ˜P = diag(I r , O m −r ).From ¯B n Q˜= ˜G−1

n B n ˜V n−1Q˜ = ˜Qwe get

On the other hand, if z∈ im ˜Q then z= ˜Qzand ˜Q ¯ B n z= ˜Q( ˜ G−1

n B n ˜V n−1Q)z˜ = ˜Qz = z Sim-ilarly, for any z∈ im ˜P , ˜ V n−1z ∈ S n , hence B n ˜V n−1z = A n ζ for some ζ∈ Rm A further com-putation gives ˜Q ¯ B n z= ˜Q ˜ G−1

n B n ˜V n−1z= ˜Q ˜ G−1

n A n ζ = ˜Q ˜ P ˜ V−1

n ζ = 0 Thus for every x ∈ R m,

˜

Q ¯ B n x= ˜Q ¯ B n ( ˜ P x)+ ˜Q ¯ B n ( ˜ Qx)= ˜Qx.It leads to the relation

˜

Combining relations (2.5), (2.6) and taking into account that ˜Q = diag(O r , I m −r )we come

to the representation ¯B n = diag(W n , I m −r ), where W n∈ Rr ×r are certain matrices The proof of

Theorem 2.4 is complete 2

3 Floquet theorem for index-1 periodic LIDS

We begin this section with some definitions

Definition 3.1 System (2.1) is called periodic of period N∈ N if

A n +N = A n , B n +N = B n , and q n +N = q n ∀n  0.

For an N -periodic difference system we define A−1:= A N−1.

Definition 3.2 The matrix X n∈ Rm ×msatisfying the initial-value problem (IVP)

where P−1= P N−1 is a projection onto S N−1 along ker A−1= ker A N−1,will be called the fundamental matrix of Eq (2.1)

Theorem 3.3 There exist an N -periodic nonsingular matrix F n and a nonsingular constant ma-trix R∈ Cr ×r such that the fundamental matrix of a index-1 periodic LIDS (2.1) with nonsingular

matrices B n can be represented as

X n = F n−1diag

R n , O m −r

F−1

Proof Using the transformation X n = ˜V n−1¯X nand scaling Eq (3.1) by ˜G−1

n as in the proof of Theorem 2.4, we get

˜

where ¯B n = diag(W n , I m −r ).Performing ˜P and ˜Qon both sides of Eq (3.3), respectively, and taking into account relations (2.5), (2.6) we find ˜P ¯ X n+1+ ˜P ¯ B n P ¯˜X n= 0 and ˜Q ¯ X n = 0 Further,

let ˜V−1:= ˜V N−1, from P−1(X0− I) = 0, it implies ˜V−1P ˜˜V−1

−1(X0− I) = 0 or ˜ P ¯ X0= ˜P ˜ V−1

−1.

Since ˜Q ¯ X n = 0 (n  0), it follows ¯X n= ˜P ¯ X nand ¯X0= ˜P ˜ V−1

−1.Thus we come to the IVP:

¯X n+1+ ˜P ¯ B n ¯X n= 0; ¯X0= ˜P ˜ V−1

Equation (3.4) has a unique solution ¯X n = (−1) n+1P ¯˜B n P ¯˜B n−1· · · ˜P ¯ B0P ˜˜V−1

−1 Using

rela-tion (2.5) and taking into account the fact that ˜P = diag(I r , O m −r ), ¯ B n = diag(W n , I m −r ),we can rewrite the fundamental matrix ¯X n+1as

¯X n+1= diag(Z n+1, O m −r ) ˜ V−1

−1, where Z n+1= (−1) n+1W

n · · · W0.

Thus

X n = ˜V n−1¯X n = ˜V n−1diag(Z n , O m −r ) ˜ V−1

Since S n and ker A n−1are periodic, we can choose periodic bases{s i

n}r

i=1, {h j

n−1}m

j =r+1in

S n and ker A n−1,respectively The periodicity of ˜V n and ˜G−1

n implies the periodicity of ¯B n=

˜G−1

n B n ˜V n−1, hence W n is periodic If B nare all nonsingular then ¯B nare nonsingular too, so are

W n and Z n Using the periodicity of ˜V nand the relation ˜V−1= ˜V N−1we find

X n +N = ˜V n +N−1diag

( −1) n +N W

n +N−1 · · · W N W N−1· · · W0, O m −r ˜V−1

−1

= ˜V n−1diag

( −1) n W n−1· · · W0, O m −r ˜V−1

−1 ˜V N−1

× diag( −1) N W N−1· · · W0, O m −r ˜V−1

−1

= X n X N

From (3.5) and the last relation, it follows Z n +N = Z n Z N In particular, Z0= I r

Since the matrix Z N is nonsingular, there exists a nonsingular matrix R such that Z N = R N,

hence Z n +N = Z n R N Defining F n−1= ˜V n−1diag(Z n R −n , I

m −r ) (n  0), we have F−1=

˜V−1diag(Z0, I m −r ) = ˜V−1and F n−1is nonsingular Further,

F n +N−1 = ˜V n +N−1diag

Z n +N R −n−N , I

m −r

= ˜V n−1diag

Z n R N R −N R −n , I

m −r

= ˜V n−1diag

Z n R −n , I

m −r

= F n−1.

Thus the decomposition (3.2) follows The Floquet theorem on representation of the fundamental matrix of a index-1 periodic LIDS is proved 2

Theorem 3.4 Every index-1 periodic LIDS (2.1) with nonsingular matrices B n can be reduced

to the Kronecker normal form with constant coefficients.

Proof We shall use the transformations

E n= diagZ−1

n+1R n+1, I m −r ˜G−1

n and F n = ˜V ndiag

Z n+1R −n−1 , I

m −r

for Eq (2.1) Let X n = F n−1˜X n ,then by Theorem 3.3, ˜X n = F n−1−1X n = diag(R n , O m −r )F−1

−1.

A simple calculation shows that

˜

A n = E n A n F n= diagZ−1

n+1R n+1, I m −r ˜G−1

n A n ˜V ndiag

Z n+1R −n−1 , I

m −r

= diag(I r , O m −r ).

Further,

˜B n = E n B n F n−1= diagZ−1

n+1R n+1, I m −r ˜G−1

n B n ˜V n−1diag

Z n R −n , I

m −r

= diagZ−1

n+1R n+1W n Z n R −n , I

m −r

= diag(C n , I m −r ), where C n = Z−1n+1R n+1W

n Z n R −n .Since ˜A

n ˜X n+1+ ˜B n ˜X n = 0, it follows R n+1+ C n R n= 0 or

C n = −R Thus the representation (3.6) with ˜q n = E n q nis established The proof of Theorem 3.4

is complete 2

4 Some further applications and examples

In this section we discuss on possible applications of results obtained in Sections 2 and 3 and give some illustrative examples

First we consider a linear delay equation with periodic coefficient matrices

where A n +N = A n ; B n +N = B n ; C n +N = C n ; and γ i (i = 0, n0)are given data A particular case

of (4.1), (4.2), where A n , B n , and C nare constant matrices has been studied recently in [5] Let the corresponding Eq (2.1) be of index-1 Using the periodic transformations given in Theo-rem 2.4 we can reduce problem (4.1), (4.2) to the form

diag(I r , O m −r ) ¯x n+1+ diag(W n , I m −r ) ¯x n + ¯C n ¯x n −n0= ¯q n (n  0), (4.3)

where ¯C n = ˜G−1

n C n ˜V n −n0−1; ¯q n = ˜G−1

n q n (n  0) and ¯γ i = ˜V i−1−n0−1γ i (i = 0, n0) Thanks to the

periodicity of ˜V n ,the matrices ¯C n (n  0) and the vectors ¯γ i (i = 0, n0)are well defined Then decomposing ¯C nin (4.3) into blocks

¯C n= ¯C 1n ¯C 2n

¯C 3n ¯C 4n



,

we can easily derive certain conditions for existence and uniqueness of solutions of problem (4.1), (4.2)

Further, we describe shortly how to use the Lyapunov reduction theorem to study the stability

of trivial solutions of a nonlinear periodic index-1 implicit difference system

where f n:Rm×Rm→ Rm is a continuously differentiable function, f n (0, 0) = 0, f n +N (y, x)=

f n (y, x) ∀n  0, y, x ∈ R m

Assume that Eq (4.5) is of index-1 (see [2]), i.e.

(i) ker∂f n

∂y (y, x) = N n , dim N n = m − r, for some 1  r  m − 1, ∀n  0, ∀y, x ∈ R m

(ii) S n (y, x) ∩ N n−1= {0}, where S n (y, x) = {ξ ∈ R m: ∂f n

∂x (y, x)ξ∈ im∂f n

∂y (y, x) }.

Let A n:=∂f n

∂y (0, 0); B n:=∂f n

∂x (0, 0) We rewrite (4.5) as

where h n (y, x) := f n (y, x) − A n y − B n x Assuming that B nare nonsingular matrices and using the periodic transformations described in Theorem 3.4 we can reduce (4.6) to a simpler system

diag(I r , O m −r ) ¯x n+1+ diag(−R, I m −r ) ¯x n + ¯h n ( ¯x n+1, ¯x n ) = 0,

where the nonlinear part ¯h n satisfies conditions ¯h n (0, 0)= 0; ∂ ¯ h n

∂y (0, 0)= 0 and ∂ ¯ h n

∂x (0, 0) = 0.

If all eigenvalues of R have modulus less one then the trivial solution of (4.5) is exponentially

asymptotically stable, i.e.x n  c ˜ P−1x0e −αn , for some positive constants α and c, where

˜

P−1= ˜P N−1—the canonical projection along ker A N−1,provided ˜P−1x0 is sufficiently small

Example 1 Consider the IVP (3.1) with the data

A n=



cos2π n3 sin2π n3

− sin2π n

3 − sin2π n

3 tg2π n3



1+ tg2π n

3 tg2 2π n3 + tg2π(n −1)

3

1− tg2π n

3 tg2π n3 (1− tg2π(n −1)



and P−1= P2is a projection along ker A−1= ker A2.

We get ker A n = S n = {t(tg 2π n

3 , −1)T: t ∈ R} Now, choosing

tg2π n3 − tg2π(n −1)

3



tg2π n3 − tg2π(n −1)

3



we can find

˜G n= cos2π n3 tg2π n3

− sin2π n

 and ¯B n=cos (2π n/3)1 0



.

We get

Z n = (−1) n

n−1

i=0

1 cos2π(n −i)

3

hence R= −41 By Theorem 3.3, X n = F n−1diag(R n , 1)F−1

−1 and observing the fact that

F−1

−1 = F2−1= −1

4



3



,

− tg2π(n −1)

3 + tg2π n

3



4−n/3sin2π n

3 ω n − tg2π(n −1)

3

−4−n/3cos2π n



with ω n = (−1) nn

i=0cos (2π(n1−i)/3), we get

X n= 1

4(tg 2π(n −1)

3 − tg2π n

3 )

 sin2π n

3 sin2π n3 ω n

− cos2π n

3 ω n −√3 cos2π n3 ω n



.

Consider system (2.1) with the same coefficients A n , B n as above Using the scaling and the periodic transformations given in Theorem 3.4:

E n= cos2π n

3



( −1) n4n+13 ω n ( −1) n+14n+1

3 tg2π n3 ω n



and the above transformations F n−1we can reduce (2.1) to the normal form





˜x n+1+





˜x n = ˜q n

Example 2 Consider system (4.5) with the data

f n (y, x)=



f 1,n (y1, y2, x1, x2)

f 2,n (y1, y2, x1, x2)



,

where

f 1,n (y1, y2, x1, x2)

= y1cos2π(n + x1+ x2)

2)

3

+ (−1) n



x2cosπ n

3 − x1cosπ(n − 1)

3

 cos2π(n − y2

1− y2

2)

2x2sin

2π n

3 ,

f 2,n (y1, y2, x1, x2)

= −1

2y1sin

2π(n − x2

1)

3 + y2sinπ(n + x2

2)

3 +( −1) n

2



x1cosπ(n − 1)

 sin2π(n − y2

1)

3 + x2cos2π(n + y2)

Further,

A n=∂f n

∂y (0, 0)=

 cos2 π n3 − cosπ n

3

−1

2sin2π n3 sinπ n3



,

B n=∂f n

∂x (0, 0)=

( −1) n+1cos2 π n

3 cosπ(n −1)

3 ( −1) ncos2 π n

3 +1

2sin2π n3

( −1) n

2 sin2π n3 cosπ(n −1)

3

( −1) n+1

2 sin2π n3 + cosπ n3



and ker A n = {t(1, cos π n

3 )T: t ∈ R}, S n = {t(cos π n

3 , 0)T: t∈ R} Choosing

cosπ n3 cosπ(n −1)

3

 cosπ n3 0

1 cosπ n3



we find

˜G n=



cosπ n3 sinπ n3

− sinπ n

3 cosπ n3

 and ¯B n=



( −1) ncosπ n3 0



.

We get Z n = (−1) nn−1

k=0( −1) kcosπ k3, Z0= 1, Z6= 1

16, hence R= 1

1/3 < 1.

Now, using the scaling and the periodic transformations given in Theorem 3.4, we can reduce (4.5) to the form





¯x n+1+



41/3 0



¯x n + ¯h n ( ¯x n+1, ¯x n ) = 0.

Because R < 1 the trivial solution of (4.5) is exponentially asymptotically stable.

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[5] Y Li, X Zhang, Y Liu, Basic theory of linear singular discrete system with delay, Appl Math Comput 108 (2000) 33–46.

[6] L.C Loi, N.H Du, P.K Anh, On linear implicit non-autonomous system of difference equations, J Difference Equ Appl 8 (2002) 1085–1105.

[3] E Griepentrog, R März, Differential–Algebraic Equations and Their... 9

Now, using the scaling and the periodic transformations given in Theorem 3.4, we can reduce (4.5) to the form





¯x... (2000) 33–46.

[6] L.C Loi, N.H Du, P.K Anh, On linear implicit non-autonomous system of difference equations, J Difference Equ Appl (2002) 1085–1105.