JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 792782 10.1155/2013/792782 792782 Research Article Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation http://orcid.org/0000-0002-0659-4220 Ulukök Zübeyde Türkmen Ramazan Wang Baolin Department of Mathematics Science Faculty Selçuk University 42031 Konya Turkey selcuk.edu.tr 2013 28 11 2013 2013 22 05 2013 24 10 2013 2013 Copyright © 2013 Zübeyde Ulukök and Ramazan Türkmen. 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.

We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.

1. Introduction

In many areas of optimal control , robust control , robust stability , filter design , stability theory and analysis , control design  in control theory [10, 11] including optimization stability theory, and transient performance performance nonlinear systems , the algebraic Riccati and Lyapunov matrix equations play an important role.

For example, consider the following linear system such that  An×n,  Bn×m,  x0n  : (1)x.(t)=Ax(t)+Bu(t),x(0)=x0, with the state feedback control (2)u(t)=-Kx(t),K=BTP, and the performance index (3)J=0(xTQx+uTu)dt, where  R=BBTn×n,  Qn×n  is positive semidefinite matrix, and  P  is the positive semidefinite solution to the continuous algebraic Riccati matrix equation (CARE) (4)AP+PAT-PRP+Q=0.

When  B=0  and  A  is stable matrix, the CARE (4) becomes the continuous algebraic Lyapunov matrix equation (CALE) (5)ATP+PA=-Q.

It is assumed that the pair  (A,R1/2)  is stabilizable. Then the CARE (4) has a unique symmetric positive semidefinite stabilizing solution if the pair  (A,Q1/2)  is observable.

The problem of estimating solution bounds for the algebraic Riccati and Lyapunov matrix equations has widely been considered in the recent years, since these equations are widely used in many fields of control system analysis and design. A number of works have reported numerical algorithms to get the exact solution of the mentioned equations . However, we should note that the analytical solution of these equations has some complications and computational burdens, specially, when the dimensions of the system matrices increase. Thus, for some applications such as stability analysis , it is the only preferred solution matrix bounds for the exact solution that can be obtained without hard and complicated burdens. Moreover, as mentioned in , in practice, the solution matrix bounds can also be used as approximations of the exact solution or initial guesses in the numerical algorithms for the exact solution .

The existing results obtained during 1974–1994 have been summarized by Kwon et al.  only including all eigenvalue bounds such as the extreme eigenvalues, the summation, the trace, majorization inequalities, the product, and the determinant. Unfortunately, by this time, the upper matrix bounds for the solution of the CARE (4) have not been proposed in the literature. However, Lee in  has proposed upper and lower matrix bounds for the CARE (4) and henceforth many reports have been presented for the upper  and lower [18, 19, 21] bounds for the solution of the CARE (4). As matrix bounds include all eigenvalue bounds [14, 22, 23] particularly the minimum and maximum eigenvalues, trace [10, 24, 25], determinant , and norm  bounds, it is seen that they are the most general and useful. Therefore, this paper presents upper matrix bounds for the solution of the CARE (4) by utilizing various matrix identities and matrix inequalities.

Let  n×m  be the set of  n×m  real matrices. In this paper, we denote the eigenvalues of an  n×n  real matrix by  λi(X); if  Xn×n  is a symmetric matrix, then its eigenvalues are arranged in the nonincreasing order  λ1(X)λ2(X)λn(X). For  Xn×m, suppose that the singular values of  X  are ordered in nonincreasing form; that is,  s1(X)s2(X)sn(X). Also, let  tr(X),  XT,  X-1, and  det(X)  denote the trace, transpose, inverse, determinant, respectively. Additionally, the spectral condition number of any matrix  X  is defined by  κ2(X)=s1(X)s1(X-1). Write  X(>)0, if  X  is a positive semidefinite (positive definite) matrix. For the symmetric matrices of the same size  X  and  Y, if  X-Y  is positive semidefinite, we write  XY  or  YX. Then, Weyl’s monotonicity principle means that  YX  leads to  λi(Y)λi(X),  i=1,2,,n. The identity matrix in  n×n  is shown by  I.

The following lemmas are used to prove the main result of this paper.

Lemma 1 (see [<xref ref-type="bibr" rid="B3">27</xref>, <xref ref-type="bibr" rid="B7">28</xref>]).

Let  Xn×n  be symmetric matrix. Then the following inequality holds: (6)λn(X)IXλ1(X)I.

Lemma 2 (see [<xref ref-type="bibr" rid="B3">27</xref>, <xref ref-type="bibr" rid="B7">28</xref>]).

For any matrix  An×m  and any positive semidefinite matrices  X,Yn×n  such that  XY>()0, it holds that  ATXAATYA, with strict inequality if  X  and  Y  are positive definite and  A  is of full rank.

Lemma 3 (see [<xref ref-type="bibr" rid="B3">27</xref>, <xref ref-type="bibr" rid="B7">28</xref>]).

For any symmetric matrices  X,Yn×n, the following inequality holds: (7)λi+j-1(X+Y)λj(X)+λi(Y),lhhhhi+jn+1,1i,jn.

Lemma 4 (see [<xref ref-type="bibr" rid="B7">28</xref>]).

Let  X,Yn×n, for  i=1,2,,n, one has (8)si(X+Y)si(X)+s1(Y).

Lemma 5 (see [<xref ref-type="bibr" rid="B7">28</xref>]).

Let  Xm×n,  Yn×m, for  i=1,2,,n, one has (9)si(XY)si(X)s1(Y).

Lemma 6 (see [<xref ref-type="bibr" rid="B20">29</xref>]).

Let  Xn×n, for  i=1,2,,n, then (10)|λi(A+AT2)|si(A).

Lemma 7 (see [<xref ref-type="bibr" rid="B12">30</xref>]).

The following matrix inequality: (11)(WSSTV)>0, where  W=WT  and  V=VT, is equivalent to either (12)V>0,W-SV-1ST>0, or (13)W>0,V-STW-1S>0.

Lemma 8 (see [<xref ref-type="bibr" rid="B10">17</xref>]).

The positive semidefinite solution  P  of the CARE (4) has the following upper bound on its maximal eigenvalue: (14)λ1(P)λ1(DTD)λ1[(Q+KTK)DTD]λn(ZDTD)η, where  K  is any matrix stabilizing  A+BK  (i.e.,  Re(λi(A+BK))<0  for all  i) and the nonsingular matrix  D  and positive definite matrix  Z  are chosen to yield the LMI (15)(A+BK)TDTD+DTD(A+BK)-Z.

This eigenvalue upper bound (14) is always calculated if there exists a unique positive semidefinite solution of the CARE (4).

2. Main Results

Zhang and Liu in  obtained the lower and upper bounds for the solution of the CARE (4) which improve the results in . Also, Lee in  proposed upper and lower bounds for the solution of the CARE (4) by considering a different approach. In this section, we will present diverse upper matrix bounds for the solution matrix of the CARE (4) in the light of the reported results in [18, 19], by utilizing the above lemmas and generating some matrix identities.

Theorem 9.

Assume that  Q  is symmetric positive definite and there exists a unique symmetric positive semidefinite solution  P  to the CARE (4). Then  P  satisfies the following inequality: (16)P{11-λ1(I-εR)×{M+2φ1ελ1(I-εR)s1[A(I-εR)-1]I}11-λ1(I-εR)}1/2Pu1(ε,A,Q,R), where the positive semidefinite matrix  M1  and the positive constant  δ  are defined by (17)M=εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT,δ=ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR), where  ε  is any positive constant such that (18)0<ε<R+ATQ-1A-1, and positive constant  φ1  is defined by (19)φ111-λ1(I-εR)×{{[1-λ1(I-εR)]λ1(M)+δ2}1/2+δ}.

Proof.

By adding and subtracting  (1/ε)PP+A((1/ε)(I-R))-1AT  from (4), one gets (20)[P+A(1εI-R)-1](1εI-R)[P+A(1εI-R)-1]T-1εPP-A(1εI-R)-1AT+Q=0, therefore, (21)[P+A(1εI-R)-1](1εI-R)[P+A(1εI-R)-1]T=1εPP+A(1εI-R)-1AT-Q. Applying Lemmas 1 and 2 to (21) gives (22)1εPP+A(1εI-R)-1AT-Q=[P+A(1εI-R)-1](1εI-R)×[P+A(1εI-R)-1]Tλ1(1εI-R)[P+A(1εI-R)-1]×[P+A(1εI-R)-1]Tλ1(1εI-R)[P2+A(1εI-R)-2AThhhhhhhhhhhhhhl+P(1εI-R)-1AT+A(1εI-R)-1P]. For the part  P((1/ε)I-R)-1AT+A((1/ε)I-R)-1P  of (22), applying Lemmas 1, 6, and 5, respectively, shows that (23)P(1εI-R)-1AT+A(1εI-R)-1Pλ1(P(1εI-R)-1AT+A(1εI-R)-1P)I|λ1(P(1εI-R)-1AT+A(1εI-R)-1P)I|2s1(P(1εI-R)-1AT)I2λ1(P)s1(A(1εI-R)-1)I. Thus, in light of the fact (23), (22) becomes (24)1εPP+A(1εI-R)-1AT-Qλ1(1εI-R)[(A(1εI-R)-1)P2+A(1εI-R)-2AThhhhhhhhhhhhhhl+2λ1(P)s1(A(1εI-R)-1)I]. If  Q>0  and  ε  satisfies (18), then (25)I-εR-εATQ-1A>0,εQ>0. By the application of the Schur complement formula of Lemma 7 to (25), we can say that the above inequalities are satisfied if and only if (26)(εQεAεATI-εR)>0, which means that (27)εQ-ε2A(I-εR)AT>0,I-εR>0. Therefore, we say that (24) is equivalent to (28)P2λ1(I-εR)[P2+A(1εI-R)-2AT+2λ1(P)s1(A(1εI-R)-1)I]+εQ-ε2A(I-εR)-1AT. Since  1-λ1(I-εR)>0, (28) can be rewritten as (29)P211-λ1(I-εR)×{[A(I-εR)-1]εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]I}. Utilizing the relations in Lemmas 1 and 3, (29) becomes (30)λ1(P2)11-λ1(I-εR)λ1×{[εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT]+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]}11-λ1(I-εR)×{λ1[εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT]+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]}. Solving (30) according to  λ1(P)  gives (31)λ1(P){[ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)]211-λ1(I-εR)λ1×[εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT]+[ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)]2}1/2+ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)φ1. Substituting (31) into (29) results in the upper bound (32)P{11-λ1(I-εR)×{εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT+2φ1ελ1(I-εR)s1[A(I-εR)-1]I}11-λ1(I-εR)}1/2. This completes the proof.

Remark 10.

The inequality (3.5) in  is clearly as follows: (33)P2λ1(I-εR){λ12(P)+2λ1(P)εs1[A(I-εR)-1]+ε2s12[A(I-εR)-1]}+εQ-ε2A(I-εR)-1AT. Thus, when the inequality (28) is considered, from the facts (34)P2λ12(P),[A(I-εR)-1][A(I-εR)-1]Ts12[A(I-εR)-1], it is seen that the upper bound in Theorem 9 is always sharper than the result given by Theorem  3.1  in .

Remark 11.

It is well known that most of the studies in the literature have focused to derive the bounds for the maximum and minimum eigenvalues, the trace, and the determinant for the solution of the CARE (4); however, the matrix solution bounds are quite restriction. Among the mentioned bounds, the matrix solution bounds are the most useful and efficient because other bounds that are dependent on eigenvalue can be derived directly from matrix solution bounds via monotonicity.

By using Theorem 9, we can derive the following result immediately.

Corollary 12.

Assume that  Q  is symmetric positive definite and there exists a unique symmetric positive semidefinite solution  P  to the CARE (4). Then  P  satisfies the following upper eigenvalue bounds: (35)λi(P)minελi(Pu1(ε,A,Q,R))=Pu11λi(Pu1(ε,A,Q,R)),tr(P)i=1nPu11minεtr(Pu1(ε,A,Q,R))tr(Pu1(ε,A,Q,R)),det(P)i=1nPu11minεdet(Pu1(ε,A,Q,R))det(Pu1(ε,A,Q,R)), where  ε  satisfies (18) and  Pu1  is defined by (16).

By establishing the more general form than the matrix identity used in Theorem 9 for the CARE (4), one gets the following upper bounds.

Theorem 13.

Let  X  be any symmetric positive definite matrix. Then the unique symmetric positive semidefinite solution  P  to the CARE (4) has the following upper bound (36)PM1-1/2{M11/2[[ηs1(A)+λ1(X1)]2Q-X1+λ1(X1-1)×[ηs1(A)+λ1(X1)]2I]×M11/2}1/2M1-1/2Pu2(η,X1,A,Q,R), where the positive definite matrix  X1  is chosen so that (37)M1ATX1-1A+R>0, and  η  is defined by (14).

Proof.

By adding and subtracting  X1+PATX1-1AP  to the CARE (4), we can get (38)(PAT+X1)X1-1(PAT+X1)T-PRP+Q-X1-PATX1-1AP=0 which is equivalent to (39)0<P(R+ATX1-1A)P<=Q-X1+(PAT+X1)X1-1(PAT+X1)T. Introducing Lemmas 1, 2, 4, 5, and 8, respectively, to (39) gives (40)0<P(R+ATX1-1A)P=Q-X1+(PAT+X1)X1-1(PAT+X1)TQ-X1+λ1(X1-1)[(PAT+X1)(PAT+X1)T]Q-X1+λ1(X1-1)[s12(PAT+X1)]IQ-X1+λ1(X1-1)[s1(PAT)+λ1(X1)]2IQ-X1+λ1(X1-1)[λ1(P)s1(A)+λ1(X1)]2IQ-X1+λ1(X1-1)[ηs1(A)+λ1(X1)]2I. By the definition (37) of  M1  and pre- and postmultiplying  M1  to (40) yields (41)(M11/2PM11/2)2=M11/2PM1PM11/2M11/2[Q-X1+λ1(X1-1)[ηs1(A)+λ1(X1)]2I]M11/2. Solving this inequality for  P  shows the upper bound (36).

This builds the proof.

Remark 14.

Note that for the upper bound (36), the matrices  R  and  Q  don not have to be nonsingular. This means that the upper bound proposed by Theorem 13 can always be computed without any condition for positive definite matrix  X1  which arbitrarily is selected.

From Theorem 13, we have the following corollaries.

Corollary 15.

The positive semidefinite solution  P  to the CARE (4) has (42)P{1λn(M1)[Q-X1{1λn(M1)[Q+λ1(X1-1)[ηs1(A)+λ1(X1)]2I]1λn(M1)}1/2Pu21, where  η  and  M1  for the positive definite matrix  X1  are defined by (14) and (37), respectively.

Proof.

Applying Lemma 1 to the right side of (41) and solving it with regard to  P  give the upper bound  Pu21.

Corollary 16.

The solution  P  to the CARE (4) satisfies the following upper eigenvalue bounds: (43)λi(P)minλi(Pu2(η,X1,A,Q,R))=Pu2*λi(Pu2(η,X1,A,Q,R)),tr(P)i=1nPu2*mintr(Pu2(η,X1,A,Q,R))tr(Pu2(η,X1,A,Q,R)),det(P)i=1nPu2*mindet(Pu2(η,X1,A,Q,R))det(Pu2(η,X1,A,Q,R)), where  η  is defined by (14) and the positive matrix  X1  is selected so as to satisfy the definition (37), respectively.

Theorem 17.

Let  P  be the positive semidefinite solution of the CARE (4). Then  P  has the upper bound (44)PM2-1/2{M21/2[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)X22]M21/2}1/2M2-1/2Pu3(φ2,X2,A,Q,R), where the positive definite matrix  X2  is chosen so that (45)M2R-AT[λ1(X2-1)I-X2-1]A>0, and  φ2  is defined by (46)φ2{1λn(M2)λ1[Q-X2+λ1(X2-1)X22]+κ22(X2)s12(A)λn2(M2)}1/2+κ2(X2)s1(A)λn(M2).

Proof.

By the use of the equality (39), from Lemmas 1 and 2, we can write (47)P(R+ATX2-1A)P=Q-X2+(PAT+X2)X2-1(PAT+X2)TQ-X2+λ1(X2-1)[PATAP+PATX2+X2AP+X22]. Having applied Lemmas 1, 6, and 5, respectively, to the part of  PATX2+X2AP  in (47), since the following inequalities hold: (48)PATX2+X2APλ1(PATX2+X2AP)I|λ1(PATX2+X2AP)|I2s1(PATX2)2λ1(P)s1(A)λ1(X2)I via the definition of  M2  from (47), we arrive at (49)P[R-AT(λ1(X2-1)I-X2-1)A]P=PM2PQ-X2+λ1(X2-1)X22+2κ2(X2)s1(A)λ1(P)I. Applying Lemmas 1 and 3 to (49), we have (50)P21λn(M2)[Q-X2+λ1(X2-1)X22]+2κ2(X2)s1(A)λn(M2)λ1(P)I1λn(M2)λ1[Q-X2+λ1(X2-1)X22]I+2κ2(X2)s1(A)λn(M2)λ1(P)I. Then, (51)λ1(P2)1λn(M2)λ1[Q-X2+λ1(X2-1)X22]+2κ2(X2)s1(A)λn(M2)λ1(P). Solving (51) with respect to  λ1(P)  gives (52)λ1(P){κ22(X2)s12(A)λn2(M2)1λn(M2)λ1[Q-X2+λ1(X2-1)X22]+κ22(X2)s12(A)λn2(M2)}1/2+κ2(X2)s1(A)λn(M2)φ2. Substituting  φ2  into (49), we get (53)PM2PQ-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I. Pre- and postmultiplying  M21/2  to (53) leads to (54)(M21/2PM21/2)2M21/2[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)]M21/2. Therefore, by the nonsingularity of  M2, the upper matrix bound (44) is directly obtained by solving (54) with respect to  P.

The proof is finished.

According to Theorem 17, we can propose the following corollaries.

Corollary 18.

The positive semidefinite solution  P  to the CARE (4) satisfies (55)P{1λn(M2)[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)]1λn(M2)}1/2Pu31, where the positive definite matrices  X2, and  M2  and the positive constant  φ2  are defined by (45) and (46), respectively.

Proof.

Substituting  φ2  into (50), having solved (50) regard to  P, we obtain the upper bound (55).

Corollary 19.

The positive semidefinite solution  P  to the CARE (4) has the following eigenvalue upper bounds: (56)λi(P)minλi(Pu3(φ2,X2,A,Q,R))=Pu3*λi(Pu3(φ2,X2,A,Q,R))tr(P)i=1nPu3*mintr(Pu3(φ2,X2,A,Q,R))tr(Pu3(φ2,X2,A,Q,R))det(P)i=1nPu3*mindet(Pu3(φ2,X2,A,Q,R))det(Pu3(φ2,X2,A,Q,R)), where the positive definite matrices  X2  and  M2  and the positive constant  φ2  are defined by (45) and (46), respectively.

As considered a diverse matrix identity, in the case that the matrix  R  is nonsingular, we can derive the following alternative upper bounds for the solution of the CARE (4).

Theorem 20.

If the positive definite matrix  P  is a unique solution matrix of the CARE (4), then (57)P{1λn(M3)[[η+s1(ATX3)]2Q-ATX3A+λ1(X3-1)×[η+s1(ATX3)]2I]1λn(M3)}1/2Pu4(η,X3,A,Q,R), where  X3  is a positive constant matrix such that  M3R+X3-1>0  and  η  is defined by (14).

Proof.

When the term  PX3-1P+ATX3A  is added and subtracted from the CARE (4), we can write (58)(P+ATX3)X3-1(P+ATX3)T-PRP+Q-ATX3A-PX3-1P=0 which is equivalent to (59)  0<P(R+X3-1)P=Q-ATX3A+(P+ATX3)X3-1(P+ATX3)T. By the use of Lemmas 1, 2, 4, and 8 for the right side of the above equation, respectively, we obtain (60)0<P(R+X3-1)PQ-ATX3A+λ1(X3-1)(P+ATX3)(P+ATX3)TQ-ATX3A+λ1(X3-1)s12(P+ATX3)IQ-ATX3A+λ1(X3-1)[λ1(P)+s1(ATX3)]2IQ-ATX3A+λ1(X3-1)[η+s1(ATX3)]2I and by the application of Lemma 1 to the term  P(R+X3-1)P  of (60), we can write (61)P21λn(R+X3-1)×[Q-ATX3A+λ1(X3-1)[η+s1(ATX3)]2I]. Therefore, if the above inequality is solved with respect to  P, we arrive at the upper bound  Pu4.

Thus, the proof is established.

Theorem 21.

Let  P  be the positive semidefinite solution of the CARE (4). Then (62)P{1λn(M4)[Q-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)φ3I(X4-1)]1λn(M4)}1/2Pu5(φ3,X4,A,Q,R), where the positive definite matrix  X4  is selected such that (63)M4R-[λ1(X4-1)I-X4-1]>0 and the nonnegative constant  φ3  is defined by (64)φ3{κ22(X4)s12(A)λn2(M4)1λn(M4)λ1[Q-ATX4A+λ1(X4-1)ATX42A]+κ22(X4)s12(A)λn2(M4)}1/2+κ2(X4)s1(A)λn(M4).

Proof.

Consider (58). From Lemma 1, we can easily write (65)0<P(R+X4-1)PQ-ATX4A+λ1(X4-1)(P+ATX4)(P+ATX4)T and then via the inequality obtained by using Lemmas 1, 6, and 5, respectively, (66)PX4A+ATX4P2λ1(P)s1(A)λ1(X4) and the definition (63) of  M4, from (65), we have (67)0<P[R-(λ1(X4-1)I-X4-1)]P=PM4PQ-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)λ1(P)I. By the use of Lemmas 1 and 2, it is obtained that (68)P21λn(M4)×[Q-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)λ1(P)I(X4-1)AT] and thus applying Lemma 3 to (68) yields (69)λ1(P2)1λn(M4)×{λ1[Q-ATX4A+λ1(X4-1)ATX42A]+2s1(A)κ2(X4)λ1(P)[Q-ATX4A+λ1(X4-1)ATX42A]}. As solving (69) according to  λ1(P), one can reach the nonnegative constant  φ3  is defined by (64). If it is substituted  φ3  into (68), then (70)P21λn(M4)[Q-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)φ3I(X4-1)]. Thus, solving the inequality (70) derives the upper bound (62) for the solution  P  of the CARE (4).

This concludes the proof of the theorem.

Corollary 22.

The solution  P  to the CARE (4) has the following eigenvalue bounds for  j=4,5: (71)λi(P)minλi(Puj)=Puj*λi(Puj),tr(P)i=1nPuj*mintr(Puj)tr(Puj),det(P)i=1nPuj*mindet(Puj)det(Puj).

Remark 23.

Chen and Lee in  indicated in it is hard or impossible to determine the best matrix bound among the parallel results. Since we find that it is difficult to compare the tightness of our results to the parallel result in , we will only make the comparisons on an example.

3. Numerical Example

In this section, we will give a numerical example to demonstrate the effectiveness of the proposed results of this paper.

Example 1. Consider the CARE (4) with (72)A=(-1010),Q=(1008),R=(94416). Choose  ε=0.002, then using (16) shows the following upper matrix bound: (73)P(0.8232-0.0001-0.00011.2853)Pu1,tr(Pu1)=2.1085,det(Pu1)=1.0580. The upper bound (55) gives (74)P(0.81730.00680.00681.2815)Pu31,tr(Pu31)=2.0988,det(Pu31)=1.0473, with  X2=(100.10.110), and the upper bound (62) gives (75)P(0.8070001.2750)Pu5,tr(Pu5)=2.082,det(Pu5)=1.0289, with  X4=(100.10.110).

Using Theorem  3.1  of , we obtain the following upper matrix bound: (76)P(1.23280.00020.00021.2884)Pu,tr(Pu)=2.5212,det(Pu)=1.5883. The bounds  (18)  and  (23)  proposed in  are (77)P(1.7135001.0627)Pu1,tr(Pu1)=2.7762,det(Pu1)=1.8209,P(1.25270.00500.00501.6025)Pu2,tr(Pu2)=2.8552,det(Pu2)=2.0074, for  R¯1=X4.

By a simple computation, we have (78)Pu5Pu31PuPu2,Pu5Pu1P,tr(Pu5)tr(Pu31)tr(Pu1)tr(Pu),tr(Pu1)tr(Pu2),det(Pu5)det(Pu31)det(Pu1)det(Pu),det(Pu1)det(Pu2) which means that our upper bounds give more precise solution estimates than the results given by Theorem  3.1  in  and Theorems  2  and  3  in  for this case.

4. Conclusion

In this paper, new upper matrix bounds for the solution of the CARE are improved by using some linear algebraic techniques and matrix inequalities. A numerical example is given to show that the solution upper bounds presented in this paper are sharper than some results in the literature.

Acknowledgments

The authors would like to thank the editor and the reviewers for the very helpful comments and suggestions to improve the presentation of this study. This study has been supported by the Coordinatorship of Selçuk University’s Scientific Research Projects (BAP) and The Scientific and Technical Research Council of Turkey (TUBITAK).

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