New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation

Volume 2009, Article ID 620758,18pages doi:10.1155/2009/620758

Research Article

New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation

Jianzhou Liu and Juan Zhang

Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China

Correspondence should be addressed to Jianzhou Liu,liujz@xtu.edu.cn Received 25 September 2008; Accepted 19 February 2009

Recommended by Panayiotis Siafarikas

By using singular value decomposition and majorization inequalities, we propose new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further, we give their application in the algebraic Riccati equation. Finally, numerical examples have illustrated that our results are effective and superior.

Copyrightq2009 J. Liu and J. Zhang. 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.

1. Introduction

In the analysis and design of controllers and filters for linear dynamical systems, the Riccati equation is of great importance in both theory and practicesee1–5. Consider the following linear systemsee4:

xt ˙ Axt But, x0 x0, 1.1

with the cost

J _∞

0

x^TQxu^Tu

dt. 1.2

Moreover, the optimal control rateu^∗and the optimal costJ^∗of1.1and1.2are u^∗P x, PB^TK,

J^∗x^T₀Kx0,

1.3

where x0 ∈ Rⁿ is the initial state of the systems1.1 and1.2,K is the positive definite solution of the following algebraic Riccati equationARE:

A^TKKA−KRK−Q, 1.4

with R BB^T andQ are symmetric positive definite matrices. To guarantee the existence of the positive definite solution to1.4, we shall make the following assumptions: the pair A, Ris stabilizable, and the pairQ, Ais observable.

In practice, it is hard to solve the ARE, and there is no general method unless the system matrices are special and there are some methods and algorithms to solve1.4, however, the solution can be time-consuming and computationally difficult, particularly as the dimensions of the system matrices increase. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of theARE 6–12.

In addition, from2,6, we know that an interpretation of trKis that trK/nis the average value of the optimal costJ^∗asx0varies over the surface of a unit sphere. Therefore, consider its applications, it is important to discuss trace bounds for the product of two matrices.

Most available results are based on the assumption that at least one matrix is symmetric 7,8,11,12. However, it is important and difficult to get an estimate of the trace bounds when any matrix in the product is nonsymmetric in theory and practice. There are some results in13–15.

In this paper, we propose new trace bounds for the product of two general matrices.

The new trace bounds improve the recent results. Then, for their application in the algebraic Riccati equation, we get some upper and lower bounds.

In the following, letR^n×ndenote the set ofn×nreal matrices. Letx x1, x2, . . . , xnbe a real n-element array which is reordered, and its elements are arranged in nonincreasing order. That is, x1 ≥ x2 ≥ · · · ≥ xn. Let |x| |x1|,|x2|, . . . ,|xn|. For A aij ∈ R^n×n, let dA d1A, d2A, . . . , dnA, λA λ1A, λ2A, . . . , λnA, σA σ1A, σ2A, . . . , σnAdenote the diagonal elements, the eigenvalues, the singular values ofA, respectively, Let trA, A^T denote the trace, the transpose ofA, respectively. We define A_iiaiidiA,A AA^T/2. The notationA >0A≥0is used to denote thatAis a symmetric positive definitesemidefinitematrix.

Letα, βbe two realn-element arrays. If they satisfy k

i1

αi≤^k

i1

βi, k1,2, . . . , n, 1.5

then it is said thatαis controlled weakly byβ, which is signed byα≺wβ.

Ifα≺wβand

n i1

αiⁿ

i1

βi, 1.6

then it is said thatαis controlled byβ, which is signed byα≺β.

Therefore, considering the application of the trace bounds, many scholars pay much attention to estimate the trace bounds for the product of two matrices.

Marshall and Olkin in16have showed that for anyA, B∈R^n×n,then

−ⁿ

i1

σiAσ_iB≤trAB≤ⁿ

i1

σiAσ_iB. 1.7

Xing et al. in13have observed another result. LetA, B∈R^n×nbe arbitrary matrices with the following singular value decomposition:

BUdiag

σ1B, σ2B, . . . , σnB

V^T, 1.8

whereU, V ∈R^n×nare orthogonal. Then

λnASⁿ

i1

σiB≤trAB≤λ1ASⁿ

i1

σiB, 1.9

whereSUV^T is orthogonal.

Liu and He in14have obtained the following: letA, B∈R^n×nbe arbitrary matrices with the following singular value decomposition:

BUdiag

σ1B, σ2B, . . . , σnB

V^T, 1.10

whereU, V ∈R^n×nare orthogonal. Then

1≤i≤nmin

V^TAU

ii

n i1

σiB≤trAB≤max

1≤i≤n

V^TAU

ii

n i1

σiB. 1.11

F. Zhang and Q. Zhang in15have obtained the following: letA, B∈R^n×nbe arbitrary matrices with the following singular value decomposition:

BUdiag

σ1B, σ2B, . . . , σnB

V^T, 1.12

whereU, V ∈R^n×nare orthogonal. Then n

i1

σ_iBλ_n−i1AS≤trAB≤ⁿ

i1

σ_iBλ_iAS, 1.13

whereSUV^T is orthogonal. They show that1.13has improved1.9.

2. Main Results

The following lemmas are used to prove the main results.

Lemma 2.1see16, page 92, H.2.c. Ifx1 ≥ · · · ≥ xn, y1 ≥ · · · ≥ yn andx ≺ y, then for any real arrayu1≥ · · · ≥un,

n i1

x_iu_i≤ⁿ

i1

y_iu_i. 2.1

Lemma 2.2see16, page 95, H.3.b. Ifx1 ≥ · · · ≥ xn, y1 ≥ · · · ≥ynandx≺wy, then for any real arrayu₁≥ · · · ≥u_n≥0,

n i1

xiui≤ⁿ

i1

yiui. 2.2

Remark 2.3. Note that ifx≺wy, then fork 1,2, . . . , n,x₁, . . . , x_k≺wy₁, . . . , y_k. Thus fromLemma 2.2, we have

k i1

xiui≤^k

i1

yiui, k1,2, . . . , n. 2.3

Lemma 2.4see16, page 218, B.1. LetAA^T ∈R^n×n, then

dA≺λA. 2.4

Lemma 2.5see16, page 240, F.4.a. LetA∈R^n×n, then

λ

AA^T 2

≺w

λ

AA^T 2

≺wσA. 2.5

Lemma 2.6see17. Let 0< m1≤ak≤M1,0< m2≤bk≤M2, k1,2, . . . , n,1/p1/q1.

Then

n k1

akbk≤ ⁿ

k1

a^p_k

_{1/p n}

k1

b^q_{k 1/q}

≤cp,q

n k1

akbk, 2.6

where

cp,q M^p₁M₂^q−m^p₁m^q₂ p

M1m2M^q₂−m1M2m^q₂1/p q

m1M2M^p₁−M1m2m^p₁1/q. 2.7

Note that if m1 0, m2/0 or m2 0, m1/0, obviously,2.6 holds. If m1 m2 0, choose cp,q ∞, then2.6also holds.

Remark 2.7. Ifpq2, then we obtain Cauchy-Schwartz inequality

n k1

akbk≤ ⁿ

k1

a²_k

_{1/2 n}

k1

b²_{k 1/2}

≤c2

n k1

akbk, 2.8

where

c2 M1M2

m1m2

m1m2

M1M2

. 2.9

Remark 2.8. Note that

p→ ∞lim

a^p₁a^p₂· · ·a^pn

1/p

max

1≤k≤n

ak

,

limp→ ∞

q→1

cp,qlim

p→ ∞

q→1

M^p₁M₂^q−m^p₁m^q₂ p

M1m2M^q₂−m1M2m^q₂1/p q

m1M2M^p₁−M1m2m^p₁1/q

lim

p→ ∞

q→1

M^p₁

M^q₂−m1/M1^pm^q₂ M^1/p₁

p

m2M₂^q−m1/M1M2m^q₂1/p

M^q/p₁ q

m1M2−M1m2m1/M1^p1/q

lim

p→ ∞

q→1

M2

M^1/pp/q−p₁ m1M2

lim

p→ ∞

q→1

1 M₁^1/p−1m1

M1

m1.

2.10

Letp → ∞, q → 1 in2.6, then we obtain

m1

n k1

bk≤ⁿ

k1

akbk≤M1

n k1

bk. 2.11

Lemma 2.9. Ifq≥1,ai≥0 i1,2, . . . , n, then

1 n

n i1

ai

_q

≤ 1 n

n i1

a^q_i. 2.12

Proof. 1Note thatq1, orai0i1,2, . . . , n, 1

n n

i1

ai

_q 1

n n

i1

a^q_i. 2.13

2Ifq > 1,ai > 0, forx > 0, choosefx x^q, thenfx qx^q−1 > 0 and f x qq−1x^q−2 > 0. Thus,fxis a convex function. Asai > 0 and1/n_n

i1ai > 0, from the property of the convex function, we have

1 n

n i1

ai

_q

f 1 n

n i1

ai

≤ 1 n

n i1

fai

1 n

n i1

a^q_i. 2.14

3Ifq >1, without loss of generality, we may assumeai 0 i1, . . . , r, ai>0i r1, . . . , n. Then from2, we have

1 n−r

_{q n}

i1

ai

_q

1 n−r

n i1

ai

_q

≤ 1 n−r

n i1

a^q_i. 2.15

Sincen−r/n^q≤n−r/n, thus 1

n n

i1

ai

_q

n−r n

_q 1 n−r

_{q n}

i1

ai

_q

≤ n−r n

1 n−r

n i1

a^q_i 1 n

n i1

a^q_i. 2.16

This completes the proof.

Theorem 2.10. Let A, B ∈ R^n×n be arbitrary matrices with the following singular value decomposition:

BUdiagσ1B, σ2B, . . . , σnBV^T, 2.17

whereU, V ∈R^n×nare orthogonal. Then n

i1

σ_iBd_n−i1

V^TAU

≤trAB≤ⁿ

i1

σ_iBd_i

V^TAU

. 2.18

Proof. By the matrix theory we have trAB tr

AUdiag

σ1B, σ2B, . . . , σnB V^T tr

V^TAUdiag

σ1B, σ2B, . . . , σnB ⁿ

i1

σiB

V^TAU

ii.

2.19

Sinceσ1B≥σ2B≥ · · · ≥σnB≥0, without loss of generality, we may assumeσB σ₁B, σ₂B, . . . , σ_nB. Next, we will prove the left-hand side of2.18:

n i1

σiBd_n−i1

V^TAU

≤ⁿ

i1

σiBdi

V^TAU

. 2.20

If

d

V^TAU

dn

V^TAU , dn−1

V^TAU

, . . . , d1

V^TAU

, 2.21

we obtain the conclusion. Now assume that there exists j < k such that djV^TAU >

dkV^TAU,then σjBdk

V^TAU

σkBdj

V^TAU

−σjBdj

V^TAU

−σkBdk

V^TAU

σ_jB−σ_kB dk

V^TAU

−dj

V^TAU

≤0.

2.22

We usedV ^TAUto denote the vector ofdV^TAUafter changingdjV^TAUanddkV^TAU, then

n i1

σiBdi

V^TAU

≤ⁿ

i1

σiBdi

V^TAU

. 2.23

After limited steps, we obtain the the left-hand side of2.18. For the right-hand side of2.18, n

i1

σiBdi

V^TAU

≤ⁿ

i1

σiBdi

V^TAU

. 2.24

If

d

V^TAU

d₁V^TAU , d₂

V^TAU

, . . . , d_n

V^TAU

, 2.25

we obtain the conclusion. Now assume that there exists j > k such that djV^TAU <

dkV^TAU,then σ_jBdk

V^TAU

σ_kBdj

V^TAU

−σ_jBdj

V^TAU

−σ_kBdk

V^TAU

σjB−σkB dk

V^TAU

−dj

V^TAU

≥0.

2.26

We usedV ^TAUto denote the vector ofdV^TAUafter changingdjV^TAUanddkV^TAU, then

n i1

σiBdi

V^TAU

≤ⁿ

i1

σiBdi

V^TAU

. 2.27

After limited steps, we obtain the right-hand side of2.18. Therefore, n

i1

σiBd_n−i1

V^TAU

≤trAB≤ⁿ

i1

σiBd_i

V^TAU

. 2.28

This completes the proof.

Since trAB trBA,applying2.18withBin lieu ofA,we immediately have the following corollary.

Corollary 2.11. Let A, B ∈ R^n×n be arbitrary matrices with the following singular value decomposition:

APdiagσ1A, σ2A, . . . , σnAQ^T, 2.29

whereP, Q∈R^n×nare orthogonal. Then n

i1

σiAdn−i1 Q^TBP

≤trAB≤ⁿ

i1

σiAdiQ^TBP. 2.30

Now using2.18and2.30, one finally has the following theorem.

Theorem 2.12. Let A, B ∈ R^n×n be arbitrary matrices with the following singular value decompositions, respectively:

APdiag

σ1A, σ2A, . . . , σnA Q^T, BUdiag

σ1B, σ2B, . . . , σnB V^T,

2.31

whereP, Q, U, V ∈R^n×nare orthogonal. Then

max _n

i1

σiAdn−i1 Q^TBP

, n

i1

σiBdn−i1

V^TAU

≤trAB≤min _n

i1

σiBdi

V^TAU ,

n i1

σiAdi

Q^TBP .

2.32

Remark 2.13. We point out that2.18improves1.11. In fact, it is obvious that

1≤i≤nmin

V^TAU

ii

n i1

σiB≤ⁿ

i1

σiBd_n−i1

V^TAU

≤trAB≤ⁿ

i1

σiBdi

V^TAU

≤max

1≤i≤n

V^TAU

ii

n i1

σiB.

2.33

This implies that2.18improves1.11.

Remark 2.14. We point out that2.18improves1.13. Since for i 1, . . . , n,σiB ≥ 0 and diV^TAU diV^TAU V^TAU^T/2, from Lemmas2.1and2.4, then2.18implies

n i1

σ_iBλ_n−i1 V^TAU

V^TAUT

2

≤ⁿ

i1

σiBdn−i1 V^TAU

V^TAUT

2

≤trAB

≤ⁿ

i1

σiBd_i V^TAU

V^TAUT

2

≤ⁿ

i1

σ_iBλ_i V^TAU

V^TAUT

2

.

2.34

In fact, fori1,2, . . . , n, we have

λi V^TAU V^TAU^T 2

λi

V^TAUV^T AUV^TT

2 V

λi AUV^T AUV^TT 2

λiAS.

2.35

Then2.34can be rewritten as n i1

σiBλn−i1AS≤ⁿ

i1

σiBdn−i1

V^TAU

≤trAB

≤ⁿ

i1

σ_iBd_i

V^TAU

≤ⁿ

i1

σiBλ_iAS.

2.36

This implies that2.18improves1.13.

Remark 2.15. We point out that1.13improves1.7. In fact, fromLemma 2.5, we have

λAS≺wσAS. 2.37

SinceSis orthogonal,σAS σA. Then2.37is rewritten as follows:λAS≺wσA.By usingσ₁B≥σ₂B≥ · · · ≥σ_nB≥0 andLemma 2.2, we obtain

n i1

σiBλiAS≤ⁿ

i1

σiBσiA. 2.38

Note thatλi−AS −λ_n−i1AS, fromLemma 2.2and2.38, we have

−ⁿ

i1

σiBλn−i1AS ⁿ

i1

σiBλi−AS

≤ⁿ

i1

σiBλiAS≤ⁿ

i1

σiBσiA.

2.39

Thus, we obtain

−ⁿ

i1

σ_iBσ_iA≤ⁿ

i1

σ_iBλ_n−i1AS. 2.40

Both2.38and2.40show that1.13is tighter than1.7.

3. Applications of the Results

Wang et al. in6have obtained the following: letKbe the positive semidefinite solution of the ARE1.4. Then the trace of matrixKhas the lower and upper bounds given by

λnA

λnA2

λ1RtrQ

λ₁R ≤trK≤ λ1A

λ1A2

λnR/ntrQ

λ_nR/n .

3.1

In this section, we obtain the application in the algebraic Riccati equation of our results including3.1. Some of our results and3.1cannot contain each other.

Theorem 3.1. If 1/p1/q1 andKis the positive semidefinite solution of the ARE1.4, then

1the trace of matrixKhas the lower and upper bounds given by

λnA

λnA2

_n

i1λ^p_iR1/p

trQ _n

i1λ^p_iR1/p

≤trK≤ λ1A

λ1A2

1/cp,qn^2−1/q_n

i1λ^p_iR1/p trQ 1/cp,qn^2−1/q_n

i1λ^p_iR1/p .

3.2

2IfAA^T/2≥0, then the trace of matrixKhas the lower and upper bounds given by

1/c_p,qn^1−1/q_n

i1λ^p_iA1/p

1/cp,qn^1−1/q_n

i1λ^p_iA1/p_{2 n}

i1λ^p_iR1/p

trQ _n

i1λ^p_iR1/p

≤trK

≤ _n

i1λ^p_iA1/p

_n

i1λ^p_iA2/p

1/cp,qn^2−1/q_n

i1λ^p_iR1/p trQ 1/cp,qn^2−1/q_n

i1λ^p_iR1/p .

3.3

3IfAA^T/2≤0, then the trace of matrixKhas the lower and upper bounds given by

−_n

i1λn−i1A^p1/p

_n

i1λn−i1A^p2/p

1/cp,qn^2−1/q_n

i1λ^p_iR1/p trQ 1/cp,qn^2−1/q_n

i1λ^p_iR1/p

≤trK

≤

−1/c_p,qn^1−1/q_n

i1λn−i1A^p1/p

_n

i1λ^p_iR1/p

1/cp,qn^1−1/q_n

i1λ_iA^p1/p2_n

i1λ^p_iR1/p

trQ _n

i1λ^p_iR1/p ,

3.4 where

cp,q M^prM^q_k−m^prm^q_k p

MrmkM_k^q−mrMkm^q_k1/p q

mrMkM^pr −Mrmkm^pr

1/q,

Mr λ1R, mrλnR, Mkλ1K, mkλnK,

c_p,q M^p₁M^q_k−m^p₁m^q_k p

M1mkM^q_k−m1Mkm^q_k1/p q

m1MkM^p₁−M1mkm^p₁1/q, M1λ1A, m1λnA.

3.5

Proof. 1Take the trace in both sides of the matrix ARE1.4to get

tr A^TK

trKA−trKRK −trQ. 3.6

SinceK is symmetric positive definite matrix,λK σK, trK _n

i1σiK,and from Lemma 2.9, we have

trK n^1−1/q ≤

tr K^q_1/q

≤trK, 3.7

n i1

σiKK ⁿ

i1

σ_i² K≤ _n

i1

σiK 2

trK₂

. 3.8

By the Cauchy-Schwartz inequality2.8, it can be shown that n

i1

σ_iKK ⁿ

i1

σ_i² K≥ _n

i1σiK₂

n

trK n

2

. 3.9

Note that

K²Udiagλ²₁K, λ²₂K, . . . , λ²_nKU^T, 3.10

K, Q, R > 0, λ_iU^TRU λ_iR i 1, . . . , n,then by2.34, use2.6, considering3.7 and3.9, we have

trKRK tr K²R

≥ⁿ

i1

λiRσi K²

≥ 1 cp,q

_n

i1

λ^p_iR _{1/p n}

i1

σ_i^q

K²_1/q

≥ 1

cp,qn^2−1/q _n

i1

λ^p_iR _1/p

trK2

.

3.11

From2.34, note thatλiU^TAU λiAandλiU^TA^TU λiA^T i 1, . . . , n,then we obtain

trAK≤ⁿ

i1

λiAσiK≤λ1Aⁿ

i1

σiK,

trA^TK≤ⁿ

i1

λ_iA^Tσ_iK≤λ₁A^Tn

i1

σ_iK.

3.12

It is easy to see that

trA^TK trKA≤ λ1

A^T

trK λ1AtrK trK 2λ1

A^TA 2

trK 2λ1AtrK.

3.13

Combine3.11and3.13, we obtain 1

cp,qn^2−1/q _n

i1

λ^p_iR _1/p

trK2−2trKλ_nA−trQ≤0. 3.14

Solving3.14for trKyields the right-hand side of the inequality3.2. Similarly, we can obtain the left-hand side of the inequality3.2.

2 Note that when A A^T/2 ≥ 0, λiU^TAU λiA and λiU^TA^TU λ_iA^T i1, . . . , n,by2.34,2.6and3.7, we have

tr A^TK

≤ _n

i1

λ^p_i

A^T_1/p trK,

trKA≤ _n

i1

λ^p_iA 1/p

trK.

3.15

Thus,

tr A^TK

trKA≤ _n

i1

λ^p_i

A^T_1/p

_n

i1

λ^p_iA

trK

≤2 _n

i1

λ^p_i

A^TA 2

^1/p trK

2 _n

i1

λ^p_iA 1/p

trK.

3.16

From3.11and3.16, with similar argument to1, we can obtain3.3easily.

3 Note that when A A^T/2 ≤ 0, by 3.3, we obtain 3.4 immediately. This completes the proof.

Remark 3.2. FromRemark 2.7andTheorem 3.1, letp2, q2 in3.2, then we obtain

λ_nA

λ_nA2

_n

i1λ²_iR1/2trQ _n

i1λ²_iR1/2

≤trK≤ λ₁A

λ₁A2

1/c1n^3/2_n

i1λ²_iR1/2 trQ 1/c1n^3/2_n

i1λ²_iR1/2 ,

3.17

wherec1

MrMk/mrmk

mrmk/MrMk.

Remark 3.3. FromRemark 2.7andTheorem 3.1, letp → ∞, q → 1 in3.2, then we obtain 3.1immediately.

4. Numerical Examples

In this section, firstly, we will give two examples to illustrate that our new trace bounds are better than the recent results. Then, to illustrate the application in the algebraic Riccati

equation of our results will have different superiority if we choose differentpandq, we will give two examples whenp2, q2,andp → ∞, q → 1.

Example 4.1see13. Now let

A

⎛

⎜⎜

⎜⎝

0.9140 0.6989 0.6062 0.2309 0.0169 0.04501 0.3471 0.5585 0.0304

⎞

⎟⎟

⎟⎠,

B

⎛

⎜⎜

⎜⎝

0.9892 0.1140 0.1233 0.0410 0.3096 0.5125 0.0476 0.7097 0.0962

⎞

⎟⎟

⎟⎠.

4.1

NeitherAnorBis symmetric. In this case, the results of6–12are not valid.

Using1.9we obtain

0.78≤trAB≤1.97. 4.2

Using1.11yields

0.8611≤trAB≤1.9090. 4.3

By2.18, we obtain

1.0268≤trAB≤1.7524, 4.4

where both lower and upper bounds are better than those of4.2and4.3.

Example 4.2. Let

A

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

0.0624 0.8844 0.2782 0.0389 0.7163 0.6565 0.2923 0.5980 0.5502 0.2660 0.5486 0.3376 0.1134 0.5739 0.3999 0.2792

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ ,

B

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

1.7205 0.6542 1.3030 0.8813 0.6542 0.0631 0.6191 0.2696 1.3030 0.6191 0.4715 0.7551 0.8813 0.2696 0.7551 0.4584

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ .

4.5

NeitherAnorBis symmetric. In this case, the results of6–12are not valid.

Using1.7yields

−6.1424≤trAB≤6.1424. 4.6

From1.9we have

−1.5007≤trAB≤5.0110. 4.7

Using1.11yields

−3.1058≤trAB≤6.0736. 4.8

By1.13, we obtain

−0.7267≤trAB≤4.3399. 4.9

The bound in2.18yields

−0.5375≤trAB≤4.2659. 4.10

Obviously,4.10is tighter than4.6,4.7,4.8and4.9.

Example 4.3. Consider the systems1.1,1.2with

A

⎛

⎜⎜

⎜⎝

−5 −2 4 2 3 −1 1 0 −3

⎞

⎟⎟

⎟⎠, BB^T

⎛

⎜⎜

⎜⎝ 8 2 3 2 7 4 3 4 9

⎞

⎟⎟

⎟⎠, Q

⎛

⎜⎜

⎜⎝

538 440 266 440 441 321 266 321 296

⎞

⎟⎟

⎟⎠. 4.11

Moreover, the corresponding ARE1.4with R BB^T,A, Ris stabilizable and Q, Ais observable.

Using3.17yields

8.5498≤trK≤47.9041. 4.12

Using3.1we obtain

9.0132≤trK≤19.0099, 4.13

where both lower and upper bounds are better than those of4.12.

Example 4.4. Consider the systems 1.1,1.2with

A

⎛

⎜⎜

⎜⎝

−6.0 1.5 2.0 0.0 −2.0 −3.0 2.5 4.0 −1.5

⎞

⎟⎟

⎟⎠, BB^T

⎛

⎜⎜

⎜⎝

4.0 1.0 2.0 1.0 2.0 0.5 2.0 0.5 2.5

⎞

⎟⎟

⎟⎠, Q

⎛

⎜⎜

⎜⎝

17.5 7.45 3.465 7.45 9.7 7.845 3.465 7.845 9.905

⎞

⎟⎟

⎟⎠.

4.14 Moreover, the corresponding ARE1.4with R BB^T,A, Ris stabilizable and Q, Ais observable.

Using3.1we obtain

1.6039≤trK≤5.6548. 4.15

Using3.17yields

1.6771≤trK≤5.5757, 4.16

where both lower and upper bounds are better than those of4.15.

5. Conclusion

In this paper, we have proposed lower and upper bounds for the trace of the product of two arbitrary real matrices. We have showed that our bounds for the trace are the tightest among the parallel trace bounds in nonsymmetric case. Then, we have obtained the application in the algebraic Riccati equation of our results. Finally, numerical examples have illustrated that our bounds are better than the recent results.

Acknowledgments

The author thanks the referee for the very helpful comments and suggestions. The work was supported in part by National Natural Science Foundation of China10671164, Science and Research Fund of Hunan Provincial Education Department06A070.

References

1 K. Kwakernaak and R. Sivan, Linear Optimal Control Systems, John Wiley & Sons, New York, NY, USA, 1972.

2 D. L. Kleinman and M. Athans, “The design of suboptimal linear time-varying systems,” IEEE Transactions on Automatic Control, vol. 13, no. 2, pp. 150–159, 1968.

3 R. Davies, P. Shi, and R. Wiltshire, “New upper solution bounds for perturbed continuous algebraic Riccati equations applied to automatic control,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 487–495, 2007.

4 M.-L. Ni, “Existence condition on solutions to the algebraic Riccati equation,” Acta Automatica Sinica, vol. 34, no. 1, pp. 85–87, 2008.

5 K. Ogata, Modern Control Engineering, Prentice-Hall, Upper Saddle River, NJ, USA, 3rd edition, 1997.

6 S.-D. Wang, T.-S. Kuo, and C.-F. Hsu, “Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation,” IEEE Transactions on Automatic Control, vol. 31, no. 7, pp. 654–656, 1986.

7 J. B. Lasserre, “Tight bounds for the trace of a matrix product,” IEEE Transactions on Automatic Control, vol. 42, no. 4, pp. 578–581, 1997.

8 Y. Fang, K. A. Loparo, and X. Feng, “Inequalities for the trace of matrix product,” IEEE Transactions on Automatic Control, vol. 39, no. 12, pp. 2489–2490, 1994.

9 J. Saniuk and I. Rhodes, “A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations,” IEEE Transactions on Automatic Control, vol. 32, no. 8, pp. 739–740, 1987.

10 T. Mori, “Comments on “A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equation”,” IEEE Transactions on Automatic Control, vol. 33, no. 11, pp. 1088–1091, 1988.

11 J. B. Lasserre, “A trace inequality for matrix product,” IEEE Transactions on Automatic Control, vol. 40, no. 8, pp. 1500–1501, 1995.

12 P. Park, “On the trace bound of a matrix product,” IEEE Transactions on Automatic Control, vol. 41, no.

12, pp. 1799–1802, 1996.

13 W. Xing, Q. Zhang, and Q. Wang, “A trace bound for a general square matrix product,” IEEE Transactions on Automatic Control, vol. 45, no. 8, pp. 1563–1565, 2000.

14 J. Liu and L. He, “A new trace bound for a general square matrix product,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 349–352, 2007.

15 F. Zhang and Q. Zhang, “Eigenvalue inequalities for matrix product,” IEEE Transactions on Automatic Control, vol. 51, no. 9, pp. 1506–1509, 2006.

16 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.

17 C.-L. Wang, “On development of inverses of the Cauchy and H ¨older inequalities,” SIAM Review, vol.

21, no. 4, pp. 550–557, 1979.