Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation

Abstract

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 [1–3], robust control [1], robust stability [4], filter design [5], stability theory and analysis [6–8], control design [9] in control theory [10, 11] including optimization stability theory, and transient performance performance nonlinear systems [12], the algebraic Riccati and Lyapunov matrix equations play an important role.

For example, consider the following linear system such that𝐴∈ℝ^𝑛×𝑛,𝐵∈ℝ^𝑛×𝑚,𝑥₀∈ℝ^𝑛[13]: 𝑥̇(𝑡)(𝑡)(𝑡),𝑥(0)=𝐴𝑥+𝐵𝑢=𝑥₀, (1) with the state feedback control 𝑢(𝑡)=−𝐾𝑥(𝑡),𝐾=𝐵^𝑇𝑃, (2) and the performance index 𝐽=^∞₀𝑥^𝑇𝑄𝑥+𝑢^𝑇𝑢𝑑𝑡, (3) where𝑅=𝐵𝐵^𝑇∈ℝ^𝑛×𝑛,𝑄∈ℝ^𝑛×𝑛is positive semidefinite matrix, and𝑃is the positive semidefinite solution to the continuous algebraic Riccati matrix equation (CARE) 𝐴𝑃+𝑃𝐴^𝑇−𝑃𝑅𝑃+𝑄=0. (4)

When𝐵=0and𝐴is stable matrix, the CARE (4) becomes the continuous algebraic Lyapunov matrix equation (CALE) 𝐴^𝑇𝑃+𝑃𝐴=−𝑄. (5)

It is assumed that the pair(𝐴,𝑅^1/2)is stabilizable. Then the CARE (4) has a unique symmetric positive semidefinite stabilizing solution if the pair(𝐴,𝑄^1/2)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 [7]. 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 [8], 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 [12], 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 [10].

The existing results obtained during 1974–1994 have been summarized by Kwon et al. [14] 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 [15] has proposed upper and lower matrix bounds for the CARE (4) and henceforth many reports have been presented for the upper [16–20] 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 [14], and norm [26] 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.

Letℝ^𝑛×𝑚be the set of𝑛×𝑚real matrices. In this paper, we denote the eigenvalues of an𝑛×𝑛real matrix by𝜆_𝑖(𝑋); if𝑋∈ℝ^𝑛×𝑛is a symmetric matrix, then its eigenvalues are arranged in the nonincreasing order𝜆₁(𝑋)≥𝜆₂(𝑋)≥⋯≥𝜆_𝑛(𝑋). For𝑋∈ℝ^𝑛×𝑚, suppose that the singular values of𝑋are ordered in nonincreasing form; that is,𝑠₁(𝑋)≥𝑠₂(𝑋)≥⋯≥𝑠_𝑛(𝑋). Also, lettr(𝑋),𝑋^𝑇,𝑋⁻¹, anddet(𝑋)denote the trace, transpose, inverse, determinant, respectively. Additionally, the spectral condition number of any matrix𝑋is defined by𝜅₂(𝑋)=𝑠₁(𝑋)𝑠₁(𝑋⁻¹). Write𝑋≥(>)0, if𝑋is a positive semidefinite (positive definite) matrix. For the symmetric matrices of the same size𝑋and𝑌, if𝑋−𝑌is positive semidefinite, we write𝑋≥𝑌or𝑌≤𝑋. Then, Weyl’s monotonicity principle means that𝑌≤𝑋leads to𝜆_𝑖(𝑌)≤𝜆_𝑖(𝑋),𝑖=1,2,…,𝑛. The identity matrix inℝ^𝑛×𝑛is shown by𝐼.

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

Lemma 1 (see [27, 28]). Let𝑋∈ℝ^𝑛×𝑛be symmetric matrix. Then the following inequality holds: 𝜆_𝑛(𝑋)𝐼≤𝑋≤𝜆₁(𝑋)𝐼. (6)

Lemma 2 (see [27, 28]). For any matrix𝐴∈ℝ^𝑛×𝑚and any positive semidefinite matrices𝑋,𝑌∈ℝ^𝑛×𝑛such that𝑋≥𝑌>(≥)0, it holds that𝐴^𝑇𝑋𝐴≥𝐴^𝑇𝑌𝐴, with strict inequality if𝑋and𝑌are positive definite and𝐴is of full rank.

Lemma 3 (see [27, 28]). For any symmetric matrices𝑋,𝑌∈ℝ^𝑛×𝑛, the following inequality holds: 𝜆_{𝑖+𝑗−1}(𝑋+𝑌)≤𝜆_𝑗(𝑋)+𝜆_𝑖(𝑌),𝑖+𝑗≤𝑛+1,1≤𝑖,𝑗≤𝑛. (7)

Lemma 4 (see [28]). Let𝑋,𝑌∈ℝ^𝑛×𝑛, for𝑖=1,2,…,𝑛, one has 𝑠_𝑖(𝑋+𝑌)≤𝑠_𝑖(𝑋)+𝑠₁(𝑌). (8)

Lemma 5 (see [28]). Let𝑋∈ℝ^𝑚×𝑛,𝑌∈ℝ^𝑛×𝑚, for𝑖=1,2,…,𝑛, one has 𝑠_𝑖(𝑋𝑌)≤𝑠_𝑖(𝑋)𝑠₁(𝑌). (9)

Lemma 6 (see [29]). Let𝑋∈ℝ^𝑛×𝑛, for𝑖=1,2,…,𝑛, then 𝜆_𝑖𝐴+𝐴^𝑇2≤𝑠_𝑖(𝐴). (10)

Lemma 7 (see [30]). The following matrix inequality: 𝑆𝑊𝑆_𝑇𝑉>0, (11) where𝑊=𝑊^𝑇and𝑉=𝑉^𝑇, is equivalent to either 𝑉>0,𝑊−𝑆𝑉⁻¹𝑆^𝑇>0, (12) or 𝑊>0,𝑉−𝑆^𝑇𝑊⁻¹𝑆>0. (13)

Lemma 8 (see [17]). The positive semidefinite solution𝑃of the CARE (4) has the following upper bound on its maximal eigenvalue: 𝜆₁(𝑃)≤𝜆₁𝐷^𝑇𝐷𝜆₁𝑄+𝐾^𝑇𝐾𝐷^𝑇𝐷𝜆_𝑛𝑍𝐷^𝑇𝐷≡𝜂, (14) where𝐾is any matrix stabilizing𝐴+𝐵𝐾(i.e.,Re(𝜆_𝑖(𝐴+𝐵𝐾))<0for all𝑖) and the nonsingular matrix𝐷and positive definite matrix𝑍are chosen to yield the LMI (𝐴+𝐵𝐾)^𝑇𝐷^𝑇𝐷+𝐷^𝑇𝐷(𝐴+𝐵𝐾)≤−𝑍. (15)
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 [19] obtained the lower and upper bounds for the solution of the CARE (4) which improve the results in [21]. Also, Lee in [18] 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 that𝑄is symmetric positive definite and there exists a unique symmetric positive semidefinite solution𝑃to the CARE (4). Then𝑃satisfies the following inequality: 1𝑃≤1−𝜆₁()×𝐼−𝜀𝑅𝑀+2𝜑₁𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹𝐼^1/2≡𝑃_𝑢1(𝜀,𝐴,𝑄,𝑅), (16) where the positive semidefinite matrix𝑀₁and the positive constant𝛿are defined by 𝑀=𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇,𝛿=𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹1−𝜆₁(),𝐼−𝜀𝑅 (17) where𝜀is any positive constant such that 0<𝜀<𝑅+𝐴^𝑇𝑄⁻¹𝐴⁻¹, (18) and positive constant𝜑₁is defined by 𝜑₁≡11−𝜆₁()×𝐼−𝜀𝑅1−𝜆₁(𝐼−𝜀𝑅)𝜆₁(𝑀)+𝛿^21/2+𝛿. (19)

Proof. By adding and subtracting(1/𝜀)𝑃𝑃+𝐴((1/𝜀)(𝐼−𝑅))⁻¹𝐴^𝑇from (4), one gets 1𝑃+𝐴𝜀𝐼−𝑅⁻¹1𝜀1𝐼−𝑅𝑃+𝐴𝜀𝐼−𝑅^−1𝑇−1𝜀1𝑃𝑃−𝐴𝜀𝐼−𝑅⁻¹𝐴^𝑇+𝑄=0, (20) therefore, 1𝑃+𝐴𝜀𝐼−𝑅⁻¹1𝜀1𝐼−𝑅𝑃+𝐴𝜀𝐼−𝑅^−1𝑇=1𝜀1𝑃𝑃+𝐴𝜀𝐼−𝑅⁻¹𝐴^𝑇−𝑄. (21) Applying Lemmas 1 and 2 to (21) gives 1𝜀1𝑃𝑃+𝐴𝜀𝐼−𝑅⁻¹𝐴^𝑇=1−𝑄𝑃+𝐴𝜀𝐼−𝑅⁻¹1𝜀×1𝐼−𝑅𝑃+𝐴𝜀𝐼−𝑅^−1𝑇≤𝜆₁1𝜀1𝐼−𝑅𝑃+𝐴𝜀𝐼−𝑅⁻¹×1𝑃+𝐴𝜀𝐼−𝑅^−1𝑇≤𝜆₁1𝜀𝐼−𝑅𝑃²1+𝐴𝜀𝐼−𝑅⁻²𝐴^𝑇1+𝑃𝜀𝐼−𝑅⁻¹𝐴^𝑇1+𝐴𝜀𝐼−𝑅⁻¹𝑃. (22) For the part𝑃((1/𝜀)𝐼−𝑅)⁻¹𝐴^𝑇+𝐴((1/𝜀)𝐼−𝑅)⁻¹𝑃of (22), applying Lemmas 1, 6, and 5, respectively, shows that 𝑃1𝜀𝐼−𝑅⁻¹𝐴^𝑇1+𝐴𝜀𝐼−𝑅⁻¹𝑃≤𝜆₁𝑃1𝜀𝐼−𝑅⁻¹𝐴^𝑇1+𝐴𝜀𝐼−𝑅⁻¹𝑃𝐼≤𝜆₁𝑃1𝜀𝐼−𝑅⁻¹𝐴^𝑇1+𝐴𝜀𝐼−𝑅⁻¹𝑃𝐼≤2𝑠₁𝑃1𝜀𝐼−𝑅⁻¹𝐴^𝑇𝐼≤2𝜆₁(𝑃)𝑠₁𝐴1𝜀𝐼−𝑅⁻¹𝐼. (23) Thus, in light of the fact (23), (22) becomes 1𝜀1𝑃𝑃+𝐴𝜀𝐼−𝑅⁻¹𝐴^𝑇−𝑄≤𝜆₁1𝜀𝐼−𝑅𝑃²1+𝐴𝜀𝐼−𝑅⁻²𝐴^𝑇+2𝜆₁(𝑃)𝑠₁𝐴1𝜀𝐼−𝑅⁻¹𝐼. (24) If𝑄>0and𝜀satisfies (18), then 𝐼−𝜀𝑅−𝜀𝐴^𝑇𝑄⁻¹𝐴>0,𝜀𝑄>0. (25) 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 𝜀𝑄𝜀𝐴𝜀𝐴^𝑇𝐼−𝜀𝑅>0, (26) which means that 𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)𝐴^𝑇>0,𝐼−𝜀𝑅>0. (27) Therefore, we say that (24) is equivalent to 𝑃²≤𝜆₁(𝐼−𝜀𝑅)𝑃²1+𝐴𝜀𝐼−𝑅⁻²𝐴^𝑇+2𝜆₁(𝑃)𝑠₁𝐴1𝜀𝐼−𝑅⁻¹𝐼+𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇. (28) Since1−𝜆₁(𝐼−𝜀𝑅)>0, (28) can be rewritten as 𝑃²≤11−𝜆₁()×𝐼−𝜀𝑅𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇+2𝜆₁(𝑃)𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹𝐼. (29) Utilizing the relations in Lemmas 1 and 3, (29) becomes 𝜆₁𝑃²≤11−𝜆₁()𝜆𝐼−𝜀𝑅¹×𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇+2𝜆₁(𝑃)𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹≤11−𝜆₁()×𝜆𝐼−𝜀𝑅₁𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇+2𝜆₁(𝑃)𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹. (30) Solving (30) according to𝜆₁(𝑃)gives 𝜆₁(𝑃)≤11−𝜆₁()𝜆𝐼−𝜀𝑅¹×𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇+𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹1−𝜆₁(𝐼−𝜀𝑅)^21/2+𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹1−𝜆₁(𝐼−𝜀𝑅)≡𝜑₁. (31) Substituting (31) into (29) results in the upper bound 1𝑃≤1−𝜆₁()×𝐼−𝜀𝑅𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜀²𝜆₁(𝐼−𝜀𝑅)𝐴(𝐼−𝜀𝑅)⁻²𝐴^𝑇+2𝜑₁𝜀𝜆₁(𝐼−𝜀𝑅)𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹𝐼^1/2. (32) This completes the proof.

Remark 10. The inequality (3.5) in [19] is clearly as follows: 𝑃²≤𝜆₁(𝐼−𝜀𝑅)𝜆²₁(𝑃)+2𝜆₁(𝑃)𝜀𝑠₁𝐴(𝐼−𝜀𝑅)⁻¹+𝜀²𝑠²₁𝐴(𝐼−𝜀𝑅)⁻¹+𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇. (33) Thus, when the inequality (28) is considered, from the facts 𝑃²≤𝜆²₁(𝑃),𝐴(𝐼−𝜀𝑅)⁻¹𝐴(𝐼−𝜀𝑅)^−1𝑇≤𝑠²₁𝐴(𝐼−𝜀𝑅)⁻¹, (34) it is seen that the upper bound in Theorem 9 is always sharper than the result given by Theorem3.1in [19].

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 that𝑄is symmetric positive definite and there exists a unique symmetric positive semidefinite solution𝑃to the CARE (4). Then𝑃satisfies the following upper eigenvalue bounds: 𝜆_𝑖(𝑃)≤min_𝜀𝜆_𝑖𝑃_𝑢1(𝜀,𝐴,𝑄,𝑅)=𝑃¹_𝑢1≤𝜆_𝑖𝑃_𝑢1(),(𝑃)≤𝜀,𝐴,𝑄,𝑅tr^𝑛_𝑖=1𝑃¹_𝑢1≤min_𝜀tr𝑃_𝑢1()𝑃𝜀,𝐴,𝑄,𝑅≤tr_𝑢1(),(𝑃)≤𝜀,𝐴,𝑄,𝑅det^𝑛_𝑖=1𝑃¹_𝑢1≤min_𝜀det𝑃_𝑢1()𝑃𝜀,𝐴,𝑄,𝑅≤det_𝑢1(𝜀,𝐴,𝑄,𝑅), (35) where𝜀satisfies (18) and𝑃_𝑢1is 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. Let𝑋be any symmetric positive definite matrix. Then the unique symmetric positive semidefinite solution𝑃to the CARE (4) has the following upper bound 𝑃≤𝑀₁^−1/2𝑀₁^1/2𝑄−𝑋₁+𝜆₁𝑋₁⁻¹×𝜂𝑠₁(𝐴)+𝜆₁𝑋₁²𝐼×𝑀₁^1/21/2𝑀₁^−1/2≡𝑃_𝑢2𝜂,𝑋₁,𝐴,𝑄,𝑅, (36) where the positive definite matrix𝑋₁is chosen so that 𝑀₁≡𝐴^𝑇𝑋₁⁻¹𝐴+𝑅>0, (37) and𝜂is defined by (14).

Proof. By adding and subtracting𝑋₁+𝑃𝐴^𝑇𝑋₁⁻¹𝐴𝑃to the CARE (4), we can get 𝑃𝐴^𝑇+𝑋₁𝑋₁⁻¹𝑃𝐴^𝑇+𝑋₁^𝑇−𝑃𝑅𝑃+𝑄−𝑋₁−𝑃𝐴^𝑇𝑋₁⁻¹𝐴𝑃=0 (38) which is equivalent to 0<𝑃𝑅+𝐴^𝑇𝑋₁⁻¹𝐴𝑃=𝑄−𝑋₁+𝑃𝐴^𝑇+𝑋₁𝑋₁⁻¹𝑃𝐴^𝑇+𝑋₁^𝑇. (39) Introducing Lemmas 1, 2, 4, 5, and 8, respectively, to (39) gives 0<𝑃𝑅+𝐴^𝑇𝑋₁⁻¹𝐴𝑃=𝑄−𝑋₁+𝑃𝐴^𝑇+𝑋₁𝑋₁⁻¹𝑃𝐴^𝑇+𝑋₁^𝑇≤𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝑃𝐴^𝑇+𝑋₁𝑃𝐴^𝑇+𝑋₁^𝑇≤𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝑠²₁𝑃𝐴^𝑇+𝑋₁𝐼≤𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝑠₁𝑃𝐴^𝑇+𝜆₁𝑋₁²𝐼≤𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝜆₁(𝑃)𝑠₁(𝐴)+𝜆₁𝑋₁²𝐼≤𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝜂𝑠₁(𝐴)+𝜆₁𝑋₁²𝐼. (40) By the definition (37) of𝑀₁and pre- and postmultiplying𝑀₁to (40) yields 𝑀₁^1/2𝑃𝑀₁^1/22=𝑀₁^1/2𝑃𝑀₁𝑃𝑀₁^1/2≤𝑀₁^1/2𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝜂𝑠₁(𝐴)+𝜆₁𝑋₁²𝐼𝑀₁^1/2. (41) Solving this inequality for𝑃shows the upper bound (36).
This builds the proof.

Remark 14. Note that for the upper bound (36), the matrices𝑅and𝑄don 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 matrix𝑋₁which arbitrarily is selected.

From Theorem 13, we have the following corollaries.

Corollary 15. The positive semidefinite solution𝑃to the CARE (4) has 1𝑃≤𝜆_𝑛𝑀₁𝑄−𝑋₁+𝜆₁𝑋₁⁻¹𝜂𝑠₁(𝐴)+𝜆₁𝑋₁²𝐼^1/2≡𝑃¹_𝑢2, (42) where𝜂and𝑀₁for the positive definite matrix𝑋₁are defined by (14) and (37), respectively.

Proof. Applying Lemma 1 to the right side of (41) and solving it with regard to𝑃give the upper bound𝑃¹_𝑢2.

Corollary 16. The solution𝑃to the CARE (4) satisfies the following upper eigenvalue bounds: 𝜆_𝑖(𝑃)≤min𝜆_𝑖𝑃_𝑢2𝜂,𝑋₁,𝐴,𝑄,𝑅=𝑃^∗_𝑢2≤𝜆_𝑖𝑃_𝑢2𝜂,𝑋₁,(𝑃)≤,𝐴,𝑄,𝑅tr^𝑛_𝑖=1𝑃^∗_𝑢2≤mintr𝑃_𝑢2𝜂,𝑋₁𝑃,𝐴,𝑄,𝑅≤tr_𝑢2𝜂,𝑋₁,(𝑃)≤,𝐴,𝑄,𝑅det^𝑛_𝑖=1𝑃^∗_𝑢2≤mindet𝑃_𝑢2𝜂,𝑋₁𝑃,𝐴,𝑄,𝑅≤det_𝑢2𝜂,𝑋₁,𝐴,𝑄,𝑅, (43) where𝜂is defined by (14) and the positive matrix𝑋₁is selected so as to satisfy the definition (37), respectively.

Theorem 17. Let𝑃be the positive semidefinite solution of the CARE (4). Then𝑃has the upper bound 𝑃≤𝑀₂^−1/2𝑀₂^1/2𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+2𝜅₂𝑋₂𝑠₁(𝐴)𝜑₂𝐼𝑀₂^1/21/2𝑀₂^−1/2≡𝑃_𝑢3𝜑₂,𝑋₂,𝐴,𝑄,𝑅, (44) where the positive definite matrix𝑋₂is chosen so that 𝑀₂≡𝑅−𝐴^𝑇𝜆₁𝑋₂⁻¹𝐼−𝑋₂⁻¹𝐴>0, (45) and𝜑₂is defined by 𝜑₂≡1𝜆_𝑛𝑀₂𝜆₁𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+𝜅²₂𝑋₂𝑠²₁(𝐴)𝜆²_𝑛𝑀₂^1/2+𝜅₂𝑋₂𝑠₁(𝐴)𝜆_𝑛𝑀₂. (46)

Proof. By the use of the equality (39), from Lemmas 1 and 2, we can write 𝑃𝑅+𝐴^𝑇𝑋₂⁻¹𝐴𝑃=𝑄−𝑋₂+𝑃𝐴^𝑇+𝑋₂𝑋₂⁻¹𝑃𝐴^𝑇+𝑋₂^𝑇≤𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑃𝐴^𝑇𝐴𝑃+𝑃𝐴^𝑇𝑋₂+𝑋₂𝐴𝑃+𝑋²₂. (47) Having applied Lemmas 1, 6, and 5, respectively, to the part of𝑃𝐴^𝑇𝑋₂+𝑋₂𝐴𝑃in (47), since the following inequalities hold: 𝑃𝐴^𝑇𝑋₂+𝑋₂𝐴𝑃≤𝜆₁𝑃𝐴^𝑇𝑋₂+𝑋₂𝐼≤𝜆𝐴𝑃₁𝑃𝐴^𝑇𝑋₂+𝑋₂𝐴𝑃𝐼≤2𝑠₁𝑃𝐴^𝑇𝑋₂≤2𝜆₁(𝑃)𝑠₁(𝐴)𝜆₁𝑋₂𝐼 (48) via the definition of𝑀₂from (47), we arrive at 𝑃𝑅−𝐴^𝑇𝜆₁𝑋₂⁻¹𝐼−𝑋₂⁻¹𝐴𝑃=𝑃𝑀₂𝑃≤𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+2𝜅₂𝑋₂𝑠₁(𝐴)𝜆₁(𝑃)𝐼. (49) Applying Lemmas 1 and 3 to (49), we have 𝑃²≤1𝜆_𝑛𝑀₂𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂𝜅+2²𝑋₂𝑠₁(𝐴)𝜆_𝑛𝑀₂𝜆₁(𝑃)𝐼≤1𝜆_𝑛𝑀₂𝜆₁𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂𝐼𝜅+2²𝑋₂𝑠₁(𝐴)𝜆_𝑛𝑀₂𝜆₁(𝑃)𝐼. (50) Then, 𝜆₁𝑃²≤1𝜆_𝑛𝑀₂𝜆₁𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂𝜅+2²𝑋₂𝑠₁(𝐴)𝜆_𝑛𝑀₂𝜆₁(𝑃). (51) Solving (51) with respect to𝜆₁(𝑃)gives 𝜆₁(𝑃)≤1𝜆_𝑛𝑀₂𝜆₁𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+𝜅²₂𝑋₂𝑠²₁(𝐴)𝜆²_𝑛𝑀₂^1/2+𝜅₂𝑋₂𝑠₁(𝐴)𝜆_𝑛𝑀₂≡𝜑₂. (52) Substituting𝜑₂into (49), we get 𝑃𝑀₂𝑃≤𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+2𝜅₂𝑋₂𝑠₁(𝐴)𝜑₂𝐼. (53) Pre- and postmultiplying𝑀₂^1/2to (53) leads to 𝑀₂^1/2𝑃𝑀₂^1/22≤𝑀₂^1/2𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+2𝜅₂𝑋₂𝑠₁(𝐴)𝜑₂𝐼𝑀₂^1/2. (54) Therefore, by the nonsingularity of𝑀₂, the upper matrix bound (44) is directly obtained by solving (54) with respect to𝑃.
The proof is finished.

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

Corollary 18. The positive semidefinite solution𝑃to the CARE (4) satisfies 1𝑃≤𝜆_𝑛𝑀₂𝑄−𝑋₂+𝜆₁𝑋₂⁻¹𝑋²₂+2𝜅₂𝑋₂𝑠₁(𝐴)𝜑₂𝐼^1/2≡𝑃¹_𝑢3, (55) where the positive definite matrices𝑋₂, and𝑀₂and the positive constant𝜑₂are defined by (45) and (46), respectively.

Proof. Substituting𝜑₂into (50), having solved (50) regard to𝑃, we obtain the upper bound (55).

Corollary 19. The positive semidefinite solution𝑃to the CARE (4) has the following eigenvalue upper bounds: 𝜆_𝑖(𝑃)≤min𝜆_𝑖𝑃_𝑢3𝜑₂,𝑋₂,𝐴,𝑄,𝑅=𝑃^∗_𝑢3≤𝜆_𝑖𝑃_𝑢3𝜑₂,𝑋₂(𝑃)≤,𝐴,𝑄,𝑅tr^𝑛_𝑖=1𝑃^∗_𝑢3≤mintr𝑃_𝑢3𝜑₂,𝑋₂𝑃,𝐴,𝑄,𝑅≤tr_𝑢3𝜑₂,𝑋₂(𝑃)≤,𝐴,𝑄,𝑅det^𝑛_𝑖=1𝑃^∗_𝑢3≤mindet𝑃_𝑢3𝜑₂,𝑋₂𝑃,𝐴,𝑄,𝑅≤det_𝑢3𝜑₂,𝑋₂,𝐴,𝑄,𝑅, (56) where the positive definite matrices𝑋₂and𝑀₂and the positive constant𝜑₂are defined by (45) and (46), respectively.

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

Theorem 20. If the positive definite matrix𝑃is a unique solution matrix of the CARE (4), then 1𝑃≤𝜆_𝑛𝑀₃𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹×𝜂+𝑠₁𝐴^𝑇𝑋₃²𝐼^1/2≡𝑃_𝑢4𝜂,𝑋₃,𝐴,𝑄,𝑅, (57) where𝑋₃is a positive constant matrix such that𝑀₃≡𝑅+𝑋₃⁻¹>0and𝜂is defined by (14).

Proof. When the term𝑃𝑋₃⁻¹𝑃+𝐴^𝑇𝑋₃𝐴is added and subtracted from the CARE (4), we can write 𝑃+𝐴^𝑇𝑋₃𝑋₃⁻¹𝑃+𝐴^𝑇𝑋₃^𝑇−𝑃𝑅𝑃+𝑄−𝐴^𝑇𝑋₃𝐴−𝑃𝑋₃⁻¹𝑃=0 (58) which is equivalent to 0<𝑃𝑅+𝑋₃⁻¹𝑃=𝑄−𝐴^𝑇𝑋₃𝐴+𝑃+𝐴^𝑇𝑋₃𝑋₃⁻¹𝑃+𝐴^𝑇𝑋₃^𝑇. (59) By the use of Lemmas 1, 2, 4, and 8 for the right side of the above equation, respectively, we obtain 0<𝑃𝑅+𝑋₃⁻¹𝑃≤𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹𝑃+𝐴^𝑇𝑋₃𝑃+𝐴^𝑇𝑋₃^𝑇≤𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹𝑠²₁𝑃+𝐴^𝑇𝑋₃𝐼≤𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹𝜆₁(𝑃)+𝑠₁𝐴^𝑇𝑋₃²𝐼≤𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹𝜂+𝑠₁𝐴^𝑇𝑋₃²𝐼 (60) and by the application of Lemma 1 to the term𝑃(𝑅+𝑋₃⁻¹)𝑃of (60), we can write 𝑃²≤1𝜆_𝑛𝑅+𝑋₃⁻¹×𝑄−𝐴^𝑇𝑋₃𝐴+𝜆₁𝑋₃⁻¹𝜂+𝑠₁𝐴^𝑇𝑋₃²𝐼. (61) Therefore, if the above inequality is solved with respect to𝑃, we arrive at the upper bound𝑃_𝑢4.
Thus, the proof is established.

Theorem 21. Let𝑃be the positive semidefinite solution of the CARE (4). Then 1𝑃≤𝜆_𝑛𝑀₄𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+2𝑠₁(𝐴)𝜅₂𝑋₄𝜑₃𝐼^1/2≡𝑃_𝑢5𝜑₃,𝑋₄,𝐴,𝑄,𝑅, (62) where the positive definite matrix𝑋₄is selected such that 𝑀₄≡𝑅−𝜆₁𝑋₄⁻¹𝐼−𝑋₄⁻¹>0 (63) and the nonnegative constant𝜑₃is defined by 𝜑₃≡1𝜆_𝑛𝑀₄𝜆₁𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+𝜅²₂𝑋₄𝑠²₁(𝐴)𝜆²_𝑛𝑀₄^1/2+𝜅₂𝑋₄𝑠₁(𝐴)𝜆_𝑛𝑀₄. (64)

Proof. Consider (58). From Lemma 1, we can easily write 0<𝑃𝑅+𝑋₄⁻¹𝑃≤𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝑃+𝐴^𝑇𝑋₄𝑃+𝐴^𝑇𝑋₄^𝑇 (65) and then via the inequality obtained by using Lemmas 1, 6, and 5, respectively, 𝑃𝑋₄𝐴+𝐴^𝑇𝑋₄𝑃≤2𝜆₁(𝑃)𝑠₁(𝐴)𝜆₁𝑋₄ (66) and the definition (63) of𝑀₄, from (65), we have 0<𝑃𝑅−𝜆₁𝑋₄⁻¹𝐼−𝑋₄⁻¹𝑃=𝑃𝑀₄𝑃≤𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+2𝑠₁(𝐴)𝜅₂𝑋₄𝜆₁(𝑃)𝐼. (67) By the use of Lemmas 1 and 2, it is obtained that 𝑃²≤1𝜆_𝑛𝑀₄×𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+2𝑠₁(𝐴)𝜅₂𝑋₄𝜆₁(𝑃)𝐼 (68) and thus applying Lemma 3 to (68) yields 𝜆₁𝑃²≤1𝜆_𝑛𝑀₄×𝜆₁𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+2𝑠₁(𝐴)𝜅₂𝑋₄𝜆₁(𝑃). (69) As solving (69) according to𝜆₁(𝑃), one can reach the nonnegative constant𝜑₃is defined by (64). If it is substituted𝜑₃into (68), then 𝑃²≤1𝜆_𝑛𝑀₄𝑄−𝐴^𝑇𝑋₄𝐴+𝜆₁𝑋₄⁻¹𝐴^𝑇𝑋²₄𝐴+2𝑠₁(𝐴)𝜅₂𝑋₄𝜑₃𝐼. (70) Thus, solving the inequality (70) derives the upper bound (62) for the solution𝑃of the CARE (4).
This concludes the proof of the theorem.

Corollary 22. The solution𝑃to the CARE (4) has the following eigenvalue bounds for𝑗=4,5: 𝜆_𝑖(𝑃)≤min𝜆_𝑖𝑃_𝑢𝑗=𝑃^∗_𝑢𝑗≤𝜆_𝑖𝑃_𝑢𝑗,tr(𝑃)≤^𝑛_𝑖=1𝑃^∗_𝑢𝑗≤mintr𝑃_𝑢𝑗≤tr𝑃_𝑢𝑗,det(𝑃)≤^𝑛_𝑖=1𝑃^∗_𝑢𝑗≤mindet𝑃_𝑢𝑗≤det𝑃_𝑢𝑗. (71)

Remark 23. Chen and Lee in [16] 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 [18], 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 𝐴=−1010,𝑄=1008,𝑅=94416. (72) Choose𝜀=0.002, then using (16) shows the following upper matrix bound: 𝑃≤0.8232−0.0001−0.00011.2853≡𝑃_𝑢1,tr𝑃_𝑢1=2.1085,det𝑃_𝑢1=1.0580. (73) The upper bound (55) gives 𝑃≤0.81730.00680.00681.2815≡𝑃¹_𝑢3,tr𝑃¹_𝑢3𝑃=2.0988,det¹_𝑢3=1.0473, (74) with𝑋₂=^100.1_0.110, and the upper bound (62) gives 𝑃≤0.8070001.2750≡𝑃_𝑢5,tr𝑃_𝑢5𝑃=2.082,det_𝑢5=1.0289, (75) with𝑋₄=^100.1_0.110.

Using Theorem3.1of [19], we obtain the following upper matrix bound: 𝑃≤1.23280.00020.00021.2884≡𝑃_𝑢[19],tr𝑃_𝑢[19]𝑃=2.5212,det_𝑢[19]=1.5883. (76) The bounds(18)and(23)proposed in [18] are 𝑃≤1.7135001.0627≡𝑃_𝑢1[18],tr𝑃_𝑢1[18]𝑃=2.7762,det_𝑢1[18]=1.8209,𝑃≤1.25270.00500.00501.6025≡𝑃_𝑢2[18],tr𝑃_𝑢2[18]𝑃=2.8552,det_𝑢2[18]=2.0074, (77) for𝑅₁=𝑋₄.

By a simple computation, we have 𝑃_𝑢5≤𝑃¹_𝑢3≤𝑃_𝑢[19]≤𝑃_𝑢2[18],𝑃_𝑢5≤𝑃_𝑢1≤𝑃_[19],tr𝑃_𝑢5≤tr𝑃¹_𝑢3≤tr𝑃_𝑢1≤tr𝑃_𝑢[19],≤tr𝑃_𝑢1[18]≤tr𝑃_𝑢2[18],det𝑃_𝑢5≤det𝑃¹_𝑢3≤det𝑃_𝑢1≤det𝑃_𝑢[19],≤det𝑃_𝑢1[18]≤det𝑃_𝑢2[18] (78) which means that our upper bounds give more precise solution estimates than the results given by Theorem3.1in [19] and Theorems2and3in [18] 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).