Matrix Bounds for the Solution of the Continuous Algebraic Riccati Equation

Abstract

We propose new upper and lower matrix bounds for the solution of the continuous algebraic Riccati equation (CARE). In certain cases, these lower bounds improve and extend the previous results. Finally, we give a corresponding numerical example to illustrate the effectiveness of our results.

1. Introduction

In many areas of optimal control, filter design, and stability analysis, the continuous algebraic Riccati equation plays an important role (see [1–5]). For example, consider the following linear system (see [5]): 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑥(0)=𝑥₀, (1.1) where 𝐴∈𝑅^𝑛×𝑛, 𝐵∈𝑅^𝑛×𝑚, 𝑥₀∈𝑅^𝑛 is the initial state. The state feedback control and the performance index of the system (1.1), respectively, are 𝑢(𝑡)=−𝐾𝑥(𝑡),𝐾=𝐵^𝑇𝑃,𝐽=^∞₀(𝑥^𝑇𝑄𝑥+𝑢^𝑇𝑢)𝑑𝑡, (1.2) where 𝑃 is the symmetric positive semidefinite solution of the continuous algebraic Riccati equation (CARE) 𝐴𝑃+𝑃𝐴^𝑇−𝑃𝑅𝑃=−𝑄, (1.3) with 𝑅=𝐵𝐵^𝑇∈𝑅^𝑛×𝑛 and 𝑄∈𝑅^𝑛×𝑛 are symmetric positive semidefinite matrices. Assume that the pair (𝐴^𝑇,𝑅^1/2) is stabilizable. Then the above CARE has a unique symmetric positive semidefinite stabilizing solution if the pair (𝐴^𝑇,𝑄^1/2) is observable (detectable).

Besides, from [1, 6], we know that in the optimal regulator problem, the optimal cost can be written as 𝐽^∗=𝑥^𝑇₀𝑃𝑥₀, (1.4) where 𝑥₀ is the initial state of the system (1.1) and 𝑃 is the symmetric positive semidefinite solution of CARE (1.3). An interpretation of tr(𝑃) is that tr(𝑃)/𝑛 is the average value of the cost 𝐽^∗ as 𝑥₀ varies over the surface of a unit sphere.

Considering these applications, deriving the solution of the CARE has become a heated topic in the recent years. However, as we all know, for one thing, the analytical solution of this equation is often computational difficult and time-consuming as the dimensions of the system matrices increase, and we can only solve some special Riccati matrix equations and design corresponding algorithms (see [7, 8]). For another, in practice, the solution bounds can also be used as approximations of the exact solution or initial guesses in the numerical algorithms for the exact solution (Barnett and Storey 1970 [9]; Patel and Toda 1984 [10]; Mori and Derese 1984 [11]; Kwon et al. 1996 [12]). Therefore, during the past two and three decades, many scholars payed attention to estimate the bounds for the solution of the continuous algebraic Riccati equation (Kwon and Pearson 1977 [13]; Patel and Toda 1978 [14]; Yasuda and Hirai 1979 [15]; Karanam 1983 [16]; Kwon et al. 1985 [17]; Wang et al. 1986 [6]; Saniuk and Rhodes 1987 [18]; Kwon et al. 1996 [12]; Lee, 1997 [19]; Choi and Kuc, 2002 [20]; Chen and Lee, 2009 [21]). The previous results during 1974-1994 have been summarized in Kwon et al. 1996 [12].

In this paper, we propose new upper and lower matrix bounds for the solution of the continuous algebraic Riccati equation. And, using the upper and lower matrix bounds we obtain the trace, the eigenvalue, and the determinant bounds. In certain cases, these lower bounds improve and extend the previous results. Finally, we give a numerical example to illustrate the effectiveness of our results.

In the following, let 𝑅^𝑛×𝑛 and 𝑅^𝑛 denote the set of 𝑛×𝑛 real matrices and 𝑛-dimensional column vector. Let 𝑋∈𝑅^𝑛×𝑛 be an arbitrary symmetric matrix, then we assume that the eigenvalues of 𝑋 are arranged so that 𝜆₁(𝑋)≥𝜆₂(𝑋)≥⋯≥𝜆_𝑛(𝑋). For 𝑋∈𝑅^𝑚×𝑛, we assume that the singular values of 𝑋 are arranged so that 𝜎₁(𝑋)≥𝜎₂(𝑋)≥⋯≥𝜎_{min{𝑚,𝑛}}(𝑋). If 𝑋∈𝑅^𝑛×𝑛, let tr(𝑋),𝑋^𝑇,𝑋⁻¹,det(𝑋),‖𝑋‖ denote the trace, the transpose, the inverse, the determinant and the spectral norm of 𝑋, respectively. The notation 𝑋>0 (𝑋≥0) is used to denote that 𝑋 is symmetric positive definite (semidefinite). For any symmetric matrices 𝑋,𝑌∈𝑅^𝑛×𝑛, 𝑋>(≥)𝑌) means that 𝑋−𝑌 is positive definite (semidefinite).

The following lemmas are used to prove the main results.

Lemma 1.1 (see [15]). The symmetric positive semidefinite solution 𝑃 to CARE (1.3) has the following lower bound on its minimum eigenvalue: 𝜆_𝑛(𝑃)≥𝑓(−𝜆_𝑛(𝐴𝑄⁻¹),𝜆₁(𝑅𝑄⁻¹),1)≡𝜂, (1.5) with 𝑓(𝑎,𝑏,𝑐)=(−𝑎+(𝑎²+𝑏𝑐)^1/2)/𝑏, 𝐴=(𝐴+𝐴^𝑇)/2.

Lemma 1.2 (see [22]). For any symmetric matrix 𝑋∈𝑅^𝑛×𝑛, the following inequality holds: 𝜆_𝑛(𝑋)𝐼≤𝑋≤𝜆₁(𝑋)𝐼. (1.6)

Lemma 1.3 (see [23, Chapter 9, A.1.a]). Let 𝐵,𝐶∈𝑅^𝑛×𝑛, for 𝑖=1,2,…,𝑛, then 𝜆_𝑖(𝐵𝐶)=𝜆_𝑖(𝐶𝐵). (1.7)

Lemma 1.4 (see [24, page 48]). Let 𝐴,𝐵∈𝑅^𝑛×𝑛 be symmetric positive semidefinite matrices and there exist an integer 𝑘 such that 1≤𝑘≤𝑛. Then for any index sequences 1≤𝑖₁≤𝑖₂≤⋯≤𝑖_𝑘≤𝑛, we have ^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴)𝜆_{𝑛−𝑡+1}(𝐵)≤^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴𝐵)≤^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴)𝜆_𝑡(𝐵). (1.8)

Lemma 1.5 (see [24, page 49]). Let 𝐴,𝐵∈𝑅^𝑛×𝑛 be symmetric matrices and there exist an integer 𝑘 such that 1≤𝑘≤𝑛. Then for any index sequences 1≤𝑖₁≤𝑖₂≤⋯≤𝑖_𝑘≤𝑛, we have ^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴)+^𝑘_𝑡=1𝜆_{𝑛−𝑡+1}(𝐵)≤^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴+𝐵)≤^𝑘_𝑡=1𝜆_𝑖𝑡(𝐴)+^𝑘_𝑡=1𝜆_𝑡(𝐵). (1.9)

Lemma 1.6 (see [25]). Let 𝐴,𝐵∈𝑅^𝑛×𝑛, for 𝑖=1,2,…,𝑛, then 𝜎_𝑖(𝐴+𝐵)≤𝜎_𝑖(𝐴)+𝜎₁(𝐵). (1.10)

Remark 1.7. From Lemma 1.6, for 𝑖=1,2,…,𝑛, we have 𝜎_𝑖(𝐴)=𝜎_𝑖[(𝐴±𝐵)+(∓𝐵)]≤𝜎_𝑖(𝐴±𝐵)+𝜎₁(∓𝐵)=𝜎_𝑖(𝐴±𝐵)+𝜎₁(B), (1.11) which is equivalent to 𝜎_𝑖(𝐴±𝐵)≥𝜎_𝑖(𝐴)−𝜎₁(𝐵). (1.12)

Lemma 1.8 (see [26]). The following matrix inequality: 𝑆𝑊𝑆_𝑇𝑉>0, (1.13) where 𝑊=𝑊^𝑇 and 𝑉=𝑉^𝑇, is equivalent to either 𝑉>0,𝑊−𝑆𝑉⁻¹𝑆^𝑇>0, (1.14) or 𝑊>0,𝑉−𝑆^𝑇𝑊⁻¹𝑆>0. (1.15)

2. Lower Matrix Bounds for the Continuous Algebraic Riccati Equation

Choi and Kuc in [20] obtained the following. Assume that 𝑄 is symmetric positive definite and there exists a unique symmetric positive semidefinite solution 𝑃 to CARE (1.3). Then 𝑃 satisfies the following inequality: 𝑃≥[𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇]^1/2=Ψ(𝜀,𝐴,𝑄,𝑅), (2.1) where 𝜀 is any positive constant such that 0<𝜀<‖𝑅+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹. (2.2)

In this section, we will give new lower matrix bounds for the solution of the continuous algebraic Riccati equation which improve (2.1).

Theorem 2.1. Assume that 𝑄 is symmetric positive definite and there exists a unique symmetric positive semidefinite solution 𝑃 to CARE (1.3). Then 𝑃 satisfies the following inequality: 𝑃≥{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+14𝜂²𝜆_𝑛(𝐼−𝜀𝑅)𝐼}^1/2=Φ_𝑙(𝜀,𝐴,𝑄,𝑅), (2.3) where 𝜀 is any positive constant such that 𝜂0<𝜀≤min2𝜎₁(𝐴)+𝜂𝜆₁(𝑅),‖𝑅+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹, (2.4) and 𝜂 is defined by Lemma 1.1.

Proof. By adding and subtracting (1/𝜀)𝑃𝑃+𝐴((1/𝜀)(𝐼−𝑅))⁻¹𝐴^𝑇 from (1.3), we can get 1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇−1𝜀1𝑃𝑃−𝐴(𝜀𝐼−𝑅)⁻¹𝐴^𝑇+𝑄=0, (2.5) which implies that 1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇=1𝜀1𝑃𝑃+𝐴(𝜀𝐼−𝑅)⁻¹𝐴^𝑇−𝑄. (2.6) Applying Lemmas 1.2, 1.3, and 1.4 to (2.6) gives 1𝜀1𝑃𝑃+𝐴(𝜀𝐼−𝑅)⁻¹𝐴^𝑇1−𝑄=[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇≥𝜆_𝑛1{[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇}𝐼=𝜆_𝑛1{(𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]}𝐼≥𝜆_𝑛(1𝜀𝐼−𝑅)𝜆_𝑛1{[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]}𝐼=𝜆_𝑛(1𝜀𝐼−𝑅)𝜎²_𝑛1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]𝐼. (2.7) If 𝑄>0 and 𝜀 satisfies (2.4), then 𝐼−𝜀𝑅−𝜀𝐴^𝑇𝑄⁻¹𝐴>0,𝜀𝑄>0. (2.8) Using the Schur complement formula of Lemma 1.8 we can see that the above inequalities are satisfied if and only if 𝜀𝑄𝜀𝐴𝜀𝐴^𝑇𝐼−𝜀𝑅>0, (2.9) that is, 𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇>0,𝐼−𝜀𝑅>0. (2.10) On the other hand, if 𝜀 satisfies (2.4), then 𝜂𝜀≤2𝜎₁(𝐴)+𝜂𝜆₁.(𝑅) (2.11) Further, 𝜀[2𝜎₁(𝐴)+𝜂𝜆₁(𝑅)]≤𝜂. (2.12) Hence, 2𝜀𝜎₁(𝐴)≤𝜂(1−𝜀𝜆₁(𝑅))=𝜂𝜆_𝑛(𝐼−𝜀𝑅). (2.13) In terms of (1.5), we have 𝜀≤𝜂𝜆_𝑛(𝐼−𝜀𝑅)2𝜎₁=𝜂(𝐴)2𝜎₁(𝐴)𝜆₁[(𝐼−𝜀𝑅)⁻¹]≤𝜂2𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]≤𝜆_𝑛(𝑃)2𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹], (2.14) which means that 𝜎_𝑛(𝑃)−𝜎₁1[𝐴(𝜀𝐼−𝑅)⁻¹]=𝜆_𝑛(𝑃)−𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹1]≥2𝜂≥0. (2.15) Applying (1.12) and (2.15) to (2.7) gives 1𝜀𝑃𝑃+𝜀𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇1−𝑄≥𝜀𝜆_𝑛(𝐼−𝜀𝑅){𝜎_𝑛1(𝑃)−𝐴(𝜀𝐼−𝑅)⁻¹}²𝐼=1𝜀𝜆_𝑛(𝐼−𝜀𝑅){𝜎_𝑛(𝑃)−𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼=1𝜀𝜆_𝑛(𝐼−𝜀𝑅){𝜆_𝑛(𝑃)−𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼≥1𝜂4𝜀²𝜆_𝑛(𝐼−𝜀𝑅)𝐼. (2.16) Thus, by (2.10), we can easily get that 𝑃²≥𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+14𝜂²𝜆_𝑛(𝐼−𝜀𝑅)𝐼>0. (2.17) After all, we obtain 𝑃≥{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+14𝜂²𝜆_𝑛(𝐼−𝜀𝑅)𝐼}^1/2>0. (2.18) This completes the proof.

By using Theorem 2.1, we can derive the following result immediately.

Corollary 2.2. Assume that 𝑄 is symmetric positive definite and there exists a unique symmetric positive semidefinite solution 𝑃 to CARE (1.3). Then 𝑃 satisfies the following lower eigenvalue bounds for any 𝜀 satisfying (2.4): 𝜆_𝑖(𝑃)≥max_𝜀𝜆_𝑖(Φ_𝑙(𝜀,𝐴,𝑄,𝑅))=Φ^∗_𝑖𝑙≥𝜆_𝑖(Φ_𝑙(𝜀,𝐴,𝑄,𝑅)),tr(𝑃)≥^𝑛_𝑖=1Φ^∗_𝑖𝑙≥max_𝜀tr(Φ_𝑙(𝜀,𝐴,𝑄,𝑅))≥tr(Φ_𝑙(𝜀,𝐴,𝑄,𝑅)),det(𝑃)≥^𝑛_𝑖=1Φ^∗_𝑖𝑙≥max_𝜀det(Φ_𝑙(𝜀,𝐴,𝑄,𝑅))≥det(Φ_𝑙(𝜀,𝐴,𝑄,𝑅)). (2.19)

2.1. Remarks and Comparisons to Results

Remark 2.3. For CARE (1.3), the condition that 𝑛/(2𝜎₁(𝐴)+𝜂𝜆₁(𝑅))≥‖𝑅+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹ often appears in the theory and practice. Then we can obtain simple choices of the tuning parameter 𝜀 to be 𝜀^∗_𝑖(𝑖=1,…,4) as in [20]. That is, 𝜀^∗₁=12‖𝑅+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹,𝜀^∗₂=argmax_𝜀𝜆_𝑛𝜀(Ψ(𝜀,𝐴,𝑄,𝑅)),_∗3=𝜆_n(𝑄)𝜆_𝑛(𝑄)‖𝑅‖+‖𝐴‖²+‖𝐴‖‖𝐴‖²+𝜆_𝑛,𝜀(𝑄)‖𝑅‖_∗4=argmax_𝜀tr(Ψ²=(𝜀,𝐴,𝑄,𝑅))tr(𝑄)tr(𝑄)‖𝑅‖+tr(𝐴𝐴^𝑇)+tr²(𝐴𝐴^𝑇)+tr(𝐴𝐴^𝑇.)tr(𝑄)‖𝑅‖ (2.20) The authors in [20] point out that, usually, 𝜀^∗₁ as well as 𝜀^∗₄ yields good bounds.

Remark 2.4. From Remark 2.3, since 𝜂𝐿=²4𝜆_𝑛(𝐼−𝜀𝑅)𝐼≥0, (2.21) then Φ²_𝑙=Ψ²+𝐿≥Ψ². (2.22) This implies that (2.3) improves (2.1). And the additional computational effort required for (2.3) in comparison to (2.1) is the calculation of (𝜂²/4)𝜆_𝑛(𝐼−𝜀𝑅)𝐼.

Remark 2.5. It is known to us that in most of the previous results much attention had been payed to derive the bounds for the maximum eigenvalue; the minimum eigenvalue; the trace; the determinant for the exact solution of the CARE, while there have been little work focusing on the matrix solution bounds. However, matrix bounds are the most general and desirable as they can infer all other types of bounds mentioned above. The matrix bounds yields less conservative results than eigenvalue bounds in the practical applications of the solution bounds (Mori and Derese 1984 [11], Lee 1997 [19]; Kwon et al. 1996 [12]).

Remark 2.6. From Section 1, it is easy to see that even though 𝜆_𝑛(𝑄)=0 CARE (1.3) has a unique positive definite solution. However, if 𝜆_𝑛(𝑄)=0, most of the previous results cannot be applied or satisfy the trivial lower bound 0 which is not so significant. And, if 𝜆_𝑛(𝑄)=0, as long as 𝐴 is in the range space of 𝑄^1/2, Lee 1997 [19], Choi and Kuc 2002 [20], and our method satisfy positive semidefinite matrix bounds of CARE (1.3). The lower matrix bounds for the CARE given in Lee 1997 [19], Choi and Kuc 2002 [20], and ours involve searching the optimal parameter values, which require much more computational efforts than the other methods.

Mori and Derese 1984 [11], Kwon et al. 1996 [12], and Chen and Lee 2009 [21] pointed out that a general comparison between any parallel bounds for the same measure is either difficult or actually impossible. However, we can make definite illustrations about the tightness in some cases as follows.

Comparison
Viewing the literatures, we know that lower matrix bounds for the solution of CARE (1.3) have been presented only in Kwon and Pearson 1977 [13], Lee 1997 [19], Choi and Kuc 2002 [20], and Chen and Lee 2009 [21]. Firstly, in Choi and Kuc 2002 [20] it has been shown that if we choose the parameter 𝜀 as 𝜀=(‖𝑅‖+𝛼)⁻¹ which yields the constraint 0<𝜀<‖‖𝑅‖+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹ (2.23) such that 𝛼𝑄>𝐴𝐴^𝑇, then the method of Choi and Kuc 2002 [20] can always obtain a sharper lower matrix bound than that of Lee 1997 [19], 1Ψ(𝜀,𝐴,𝑄,𝑅)≥[1‖𝑅‖+𝛼(𝑄−𝛼𝐴𝐴^𝑇)]^1/2=𝐸(𝛼). (2.24) Secondly, in Lee 1997 [19] it has been shown that if we take the parameter 𝛼 as 𝛼=𝑓(−‖𝐴‖²,𝜆_𝑛(𝑄),‖𝐴‖²𝜆₁(𝑅)) (2.25) such that 𝛼𝑄>𝐴𝐴^𝑇, then the method of Lee 1997 [19] can always get a sharper lower matrix bound than that of Kwon and Pearson 1977 [13], 𝑃≥𝑓(‖𝐴‖,‖𝑅‖,𝜆_𝑛(𝑄))𝐼=𝐺, (2.26) where 𝑓(𝑎,𝑏,𝑐)=(−𝑎+√𝑎²+𝑏𝑐)/𝑏,  𝑏≠0. Thirdly, as Chen and Lee 2009 [21] pointed out, it is hard to compare the sharpness of bounds of Lee 1997 [19] and Choi and Kuc 2002 [20]. Further, we will give a numerical example (Example 4.1) in Section 4 to illustrate our lower bound is tighter than Chen and Lee 2009 [21] under certain cases. Consequently, considering Remark 2.4, it is simple to see that our lower matrix bound are tighter than Chen and Lee 2009 [21] in certain cases.

3. Upper Matrix Bounds for the Continuous Algebraic Riccati Equation

In this section, we will give new upper matrix bounds for the solution of the continuous algebraic Riccati equation.

Theorem 3.1. Assume that 𝑄 is symmetric positive definite and there exists a unique symmetric positive semidefinite solution 𝑃 to CARE (1.3). Then 𝑃 satisfies the following inequality: 𝑃≤{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅){𝜏+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼}^1/2=Φ_𝑢(𝜀,𝐴,𝑄,𝑅), (3.1) where 𝜀 is any positive constant such that 0<𝜀<‖𝑅+𝐴^𝑇𝑄⁻¹𝐴‖⁻¹1,𝜏≡1−𝜆₁(𝐼−𝜀𝑅)×{𝜀𝜆₁(𝐼−𝜀𝑅)𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]+⟨𝜀²𝜆²₁(𝐼−𝜀𝑅)𝜎²₁[𝐴(𝐼−𝜀𝑅)⁻¹]−[1−𝜆₁(𝐼−𝜀𝑅)]×{𝜀²𝜆_𝑛[𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇]−𝜀𝜆₁(𝑄)−𝜀²𝜆₁(𝐼−𝜀𝑅)𝜎²₁[𝐴(𝐼−𝜀𝑅)⁻¹]}⟩^1/2}. (3.2)(3.3)

Proof. Applying Lemmas 1.2, 1.3, 1.4, and 1.6 to (2.6) gives 1𝜀1𝑃𝑃+𝐴(𝜀𝐼−𝑅)⁻¹𝐴^𝑇1−𝑄=[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇≤𝜆₁1{[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹1](𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇}𝐼=𝜆₁1{(𝜀1𝐼−𝑅)[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]}𝐼≤𝜆₁(1𝜀𝐼−𝑅)𝜆₁1{[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]^𝑇1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]}𝐼=𝜆₁(1𝜀𝐼−𝑅)𝜎²₁1[𝑃+𝐴(𝜀𝐼−𝑅)⁻¹]𝐼≤𝜆₁(1𝜀𝐼−𝑅){𝜎₁(𝑃)+𝜎₁1[𝐴(𝜀I−𝑅)⁻¹]}²𝐼=1𝜀𝜆₁(𝐼−𝜀𝑅){𝜆₁(𝑃)+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼, (3.4) which is equivalent to 𝑃²≤𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅){𝜆₁(𝑃)+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼. (3.5) If 𝑄>0 and 𝜀 satisfies (3.2), from (2.10), then 𝐼−𝜀𝑅>0 and 𝜀𝑄−𝜀²𝐴(I−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅){𝜆₁(𝑃)+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼≥0. (3.6) Hence, (3.5) changes to 𝑃≤{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅){𝜆₁(𝑃)+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼}^1/2. (3.7) Introducing Lemma 1.2 to (3.5) gives 𝑃²≤𝜆₁{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅)[𝜆₁(𝑃)+𝜀𝜎₁(𝐴(𝐼−𝜀𝑅)⁻¹)]²𝐼}𝐼. (3.8) Then 𝜆₁(𝑃²)≤𝜆₁[𝜆₁{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅)[𝜆₁(𝑃)+𝜀𝜎₁(𝐴(𝐼−𝜀𝑅)⁻¹)]²𝐼}𝐼]. (3.9) Applying Lemmas 1.4 and 1.5 to (3.9) gives 𝜆²₁(𝑃)≤𝜀𝜆₁(𝑄)−𝜀²𝜆_𝑛[𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇]+𝜆₁(𝐼−𝜀𝑅){𝜆₁(𝑃)+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}². (3.10) Solving (3.10) for 𝜆₁(𝑃) gives 𝜆₁1(𝑃)≤1−𝜆₁(𝐼−𝜀𝑅)×{𝜀𝜆₁(𝐼−𝜀𝑅)𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]+⟨𝜀²𝜆²₁(𝐼−𝜀𝑅)𝜎²₁[𝐴(𝐼−𝜀𝑅)⁻¹]−[1−𝜆₁(𝐼−𝜀𝑅)]×{𝜀²𝜆_𝑛[𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇]−𝜀𝜆₁(𝑄)−𝜀²𝜆₁(𝐼−𝜀𝑅)𝜎²₁[𝐴(𝐼−𝜀𝑅)⁻¹]}⟩^1/2}≡𝜏. (3.11) Substituting (3.11) into (3.7) gives 𝑃≤{𝜀𝑄−𝜀²𝐴(𝐼−𝜀𝑅)⁻¹𝐴^𝑇+𝜆₁(𝐼−𝜀𝑅){𝜏+𝜀𝜎₁[𝐴(𝐼−𝜀𝑅)⁻¹]}²𝐼}^1/2. (3.12) This completes the proof.

By using Theorem 3.1, we can derive the following result immediately.

Corollary 3.2. Assume that 𝑄 is symmetric positive definite and there exists a unique symmetric positive semidefinite solution 𝑃 to CARE (1.3). Then 𝑃 satisfies the following upper eigenvalue bounds for any 𝜀 satisfying (3.2): 𝜆_𝑖(𝑃)≤min_𝜀𝜆_𝑖(Φ_𝑢(𝜀,𝐴,𝑄,𝑅))=Φ^∗_𝑖𝑢≤𝜆_𝑖(Φ_𝑢(𝜀,𝐴,𝑄,R)),tr(𝑃)≤^𝑛_𝑖=1Φ^∗_𝑖𝑢≤min_𝜀tr(Φ_𝑢(𝜀,𝐴,𝑄,𝑅))≤tr(Φ_𝑢(𝜀,𝐴,𝑄,𝑅)),det(𝑃)≤^𝑛_𝑖=1Φ^∗_𝑖𝑢≤min_𝜀det(Φ_𝑢(𝜀,𝐴,𝑄,𝑅))≤det(Φ_𝑢(𝜀,𝐴,𝑄,𝑅)). (3.13)

Remark 3.3. As Chen and Lee 2009 [21] pointed out, to give a general comparison between any parallel upper bounds for the same measure is either difficult or actually impossible. We also find that it is hard to compare the sharpness of our upper bounds to the parallel results.

4. A Numerical Example

Consider the following example.

Example 4.1. Let 𝐴=−1010,𝑄=1008,𝑅=94416. (4.1)
Choose 𝜀=0.0417, then using (2.3) shows the following lower matrix bound: 𝑃≥Φ_𝑙(𝜀=0.0417=𝜀^∗₁,𝐴,𝑄,𝑅)=0.22790.05670.05670.5864, (4.2) leading to the following eigenvalue bounds: 𝜆_𝑛(𝑃)≥0.2192,𝜆₁(𝑃)≥0.5951,tr(𝑃)≥0.8143,det(𝑃)≥0.1304. (4.3)
Using Theorem 4 of Choi and Kuc 2002 [20] we obtain the following lower matrix bound: 𝑃≥Ψ(𝜀=0.0417,𝐴,𝑄,𝑅)=0.19620.05670.05670.5748, (4.4) leading to the following eigenvalue bounds: 𝜆_𝑛(𝑃)≥0.1879,𝜆₁(𝑃)≥0.5831,tr(𝑃)≥0.771,det(𝑃)≥0.1095. (4.5)

Using Theorem 2.1 of Chen and Lee 2009 [21], we have 𝑃≥𝑃_𝑙=0.14180.14790.14790.4385, (4.6) leading to the following eigenvalue bounds: 𝜆_𝑛(𝑃)≥0.0807,𝜆₁(𝑃)≥0.4996,tr(𝑃)≥0.5803,det(𝑃)≥0.1226. (4.7)

Using Theorem 4 of Lee 1997 [19], we have 𝑃≥𝐸(𝛼=6.5)=0.18650.07950.07950.5681, (4.8) leading to the following eigenvalue bounds: 𝜆_𝑛(𝑃)≥0.1706,𝜆₁(𝑃)≥0.5840,tr(𝑃)≥0.7546,det(𝑃)≥0.0996. (4.9)

By using the method of Kwon and Pearson 1977 [13], we can also obtain the following lower matrix bound: 𝑃≥𝐺=0.1705000.1705. (4.10)

By computation, it is obvious that Φ_𝑙−Ψ≥0,Φ_𝑙−𝑃_𝑙≥0,Φ_𝑙−𝐸≥0,Φ_𝑙−𝐺≥0, (4.11) that is, Φ_𝑙≥Ψ,Φ_𝑙≥𝑃_𝑙,Φ_𝑙≥𝐸,Φ_𝑙≥𝐺, (4.12) which implies that our lower bound is tighter than the parallel results for this case.

In Table 1, we summarize the above lower bounds together with the numerical results that can be obtained by other methods. From Table 1, we can see that our lower eigenvalue bounds of CARE (1.3) can be tighter than the previous results.

Table 1 

Numerical results of the lower eigenvalue bounds.

Using (3.1) yields to the following upper matrix bound: 𝑃≤Φ_𝑢(𝜀=0.0176,𝐴,𝑄,𝑅)=0.64930.01930.01930.7381. (4.13)

5. Conclusion

In this paper, we have proposed new lower and upper bounds for the solution of the continuous algebraic Riccati equation (CARE). The numerical example has illustrated that in certain cases our lower bounds are tighter than the previous results.

Acknowledgments

The authors would like to thank Professor John Burns and the referees for the very helpful comments and suggestions to improve the contents and presentation of this paper. The work was supported in part by the National Natural Science Foundation of China (10971176), young scientist project of National Natural Science Foundation of China (11001233) and the key project of Hunan Provincial Natural Science Foundation of China (10JJ2002).