On the Perturbation Bounds of Projected Generalized Continuous-Time Sylvester Equations

Abstract

This paper is devoted to the perturbation analysis for a projected generalized continuous-time Sylvester equation. Perturbation bounds of the solution based on the Euclidean norm are presented.

1. Introduction

In this paper we study the sensitivity of and derive perturbation bounds for the projected generalized continuous-time Sylvester equation 𝐴𝑋𝐶+𝐷𝑋𝐵+𝑃_𝑙,1𝐸𝑃_𝑟,2=0,𝑋=𝑃_𝑟,1𝑋𝑃_𝑙,2, (1.1) where 𝐴, 𝐷 are 𝑛×𝑛 matrices, 𝐵,𝐶 are 𝑚×𝑚 matrices, 𝐸 is an 𝑛×𝑚 matrix, and 𝑋 is the unknown 𝑛×𝑚 matrix, respectively, with real entries. Here, 𝑃_𝑟,1 and 𝑃_𝑟,2 are the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively, while 𝑃_𝑙,1 and 𝑃_𝑙,2 are the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively. We assume that the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 are regular, that is, det(𝜆𝐷−𝐴) and det(𝜆𝐶−𝐵) are not identically zero. Under the assumption, the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 have the Weierstrass canonical forms [1]: there exist nonsingular 𝑛×𝑛 matrices 𝑊₁, 𝑇₁ and 𝑚×𝑚 matrices 𝑊₂, 𝑇₂ such that 𝐷=𝑊₁𝐼00𝑁^(𝐴)𝑇₁,𝐴=𝑊₁𝐽^(𝐴)0𝑇0𝐼¹,𝐶=𝑊₂𝐼00𝑁^(𝐵)𝑇₂,𝐵=𝑊₂𝐽^(𝐵)0𝑇0𝐼², (1.2) where 𝐽^(𝐴), 𝐽^(𝐵), 𝑁^(𝐴), and 𝑁^(𝐵) are block diagonal matrices with each diagonal block being the Jordan block. The eigenvalues of 𝐽^(𝐴) and 𝐽^(𝐵) are the finite eigenvalues of the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively. 𝑁^(𝐴) and 𝑁^(𝐵) correspond to the eigenvalue at infinity. Using (1.2), 𝑃_𝑙,1, 𝑃_𝑙,2, 𝑃_𝑟,1, and 𝑃_𝑟,2 can be expressed as 𝑃_𝑙,1=𝑊₁𝑊𝐼000¹⁻¹,𝑃_𝑙,2=𝑊₂𝑊𝐼000²⁻¹,𝑃_𝑟,1=𝑇₁⁻¹𝑇𝐼000¹,𝑃_𝑟,2=𝑇₂⁻¹𝑇𝐼000². (1.3)

If 𝐷 and 𝐶 are nonsingular, then 𝑃_𝑟,1=𝑃_𝑙,1=𝐼_𝑛, 𝑃_𝑟,2=𝑃_𝑙,2=𝐼_𝑚, and (1.1) reduces to the generalized Sylvester equation 𝐴𝑋𝐶+𝐷𝑋𝐵+𝐸=0. As 𝑚=𝑛, 𝐵=𝐴^𝑇, and 𝐶=𝐷^𝑇, (1.1) is referred to as the projected generalized continuous-time Lyapunov equation. The projected generalized continuous-time Lyapunov equation plays an important role in stability analysis and control design problems for descriptor systems including the characterization of controllability and observability properties, computing ℋ₂ and Hankel norms, determining the minimal and balanced realizations as well as balanced truncation model order reduction; see [2–6] and the references therein. If the pencil 𝜆𝐷−𝐴 is 𝑐-stable, that is, all its finite eigenvalues have negative real parts, then the projected generalized Lyapunov equation has a unique solution for each 𝐸, and if, additionally, 𝐸 is symmetric and positive semidefinite, then the solution 𝑋 is symmetric and positive semidefinite see, for example, [4] for details. In [7], the generalized Bartels-Stewart method and the generalized Hammarling method are presented for solving the projected generalized Lyapunov equation. The generalized Hammarling method is designed to obtain the Cholesky factor of the solution. These two methods are based on the generalized real Schur factorization (generalized Schur factorization if the matrix entries are complex) of the pencil 𝜆𝐷−𝐴.

Zhou et al. [8] considered the projected generalized continuous-time Sylvester equation (1.1). They firstly presented one sufficient condition for the existence and uniqueness of the solution of this equation. Then, several numerical methods were proposed for solving (1.1). Finally, they shew that the solution of this equation is useful for computing the ℋℒ₂ inner product of two descriptor systems.

The perturbation analysis for Sylvester-type equations has been considered by several authors. Higham [9] presented a perturbation analysis of the standard Sylvetser equation 𝐴𝑋−𝑋𝐵=𝐶. By taking into account its specific structure, he derived expressions for the backward error and a normwise condition number which measures the worst-case sensitivity of a solution to small perturbations in the data 𝐴,𝐵, and 𝐶. In [10], a complete perturbation analysis of the nonsingular general Lyapunov equation was presented. Stykel [7] discussed the perturbation theory for the projected generalized continuous-time algebraic Lyapunov equation. Konstantinov et al. considered the perturbation analysis for several types of matrix equations in their monograph [11]. They presented the framework of the perturbation analysis and derived condition numbers, first-order homogeneous bounds, componentwise bounds, and nonlocal normwise and componentwise bounds for the general Sylvester equation and the general Lyapunov equation.

In this paper, we study the perturbation theory for the projected generalized continuous-time Sylvester equation (1.1) and derive a perturbation bound for its solution.

Throughout this paper, we adopt the following notation. 𝐼_𝑙 denotes the 𝑙×𝑙 identity matrix and 0 denotes the zero vector or zero matrix. If the dimension of 𝐼_𝑙 is apparent from the context, we drop the index and simply use 𝐼. The space of 𝑚×𝑛 real matrices is denoted by ℝ^𝑛×𝑚. The Euclidean norm for a vector or its associated induced matrix norm is denoted by ‖⋅‖₂. The superscript 𝑇 denotes the transpose of a vector or a matrix and 𝐴⁻¹ is the inverse of nonsingular 𝐴.

The remainder of the paper is organized as follows. In Section 2, we present the perturbation results for the projected generalized continuous-time Sylvester equation and the generalized continuous-time Sylvester equation. Conclusions are given in Section 3.

2. Perturbation Results for the Projected Generalized Continuous-Time Sylvester Equation

In this section, we firstly review and introduce one important theorem, for example, [12], which gives sufficient conditions for the existence, uniqueness, and analytic formula of the solution of the projected generalized continuous-time Sylvester equation (1.1).

Theorem 2.1. Let 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 be regular pencils with finite eigenvalues {𝜇₁,𝜇₂,…,𝜇_𝑛1} and {𝜈₁,𝜈₂,…,𝜈_𝑚1} counted according to their multiplicities, respectively. Let 𝑃_𝑟,1 and 𝑃_𝑟,2 be the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively, and let 𝑃_𝑙,1 and 𝑃_𝑙,2 be the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively. Then, the projected generalized continuous-time Sylvester equation (1.1) has a unique solution for every 𝐸 if 𝜇_𝑖+𝜈_𝑗≠0 for any 𝑖=1,2,…,𝑛₁ and 𝑗=1,2,…,𝑚₁.
Moreover, if 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 are c-stable, that is, all their finite eigenvalues have negative real part, then 𝑋 can be expressed as 1𝑋=2𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝐸𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑑𝜔. (2.1)

Now let us define a linear operator 𝒮⁻∶ℝ^𝑛×𝑚→ℝ^𝑛×𝑚, (2.2) which satisfies the following: for a matrix 𝐸∈ℝ^𝑛×𝑚, 𝑋=𝒮⁻(𝐸) is the unique solution of the projected generalized continuous-time Sylvester equation (1.1), that is, 𝑋=𝒮⁻(𝐸)=12𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝐸𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑑𝜔. (2.3)

The following result shows that the linear operator 𝒮⁻ is bounded and is very useful for the perturbation analysis of the projected generalized continuous-time Sylvester equation. Although the proof is similar to that of Lemma  3.6 in [7], we include it in this paper for completeness.

Lemma 2.2. Assume that the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 are c-stable. Then ‖𝒮⁻‖₂≤𝐻₁₂𝐻₂₂, (2.4) where 𝐻₁ and 𝐻₂ are the solutions of the projected generalized continuous-time Lyapunov equations 𝐴𝐻₁𝐷^𝑇+𝐷𝐻₁𝐴^𝑇+𝑃_𝑙,1𝑃^𝑇_𝑙,1𝐻=0,₁=𝑃_𝑟,1𝐻₁𝑃^𝑇_𝑟,1,𝐵^𝑇𝐻₂𝐶+𝐶^𝑇𝐻₂𝐵+𝑃^𝑇_𝑟,2𝑃_𝑟,2𝐻=0,₂=𝑃^𝑇_𝑙,2𝐻₂𝑃_𝑙,2, (2.5)(2.6) respectively.

Proof. Let 𝑢∈ℝ^𝑛 and 𝑣∈ℝ^𝑚 be the left and right singular vectors of unit length corresponding to the largest singular value of the solution 𝑋. Then, for any 𝐸∈ℝ^𝑛×𝑚, ‖𝒮⁻(𝐸)‖₂=‖𝑋‖₂=𝑢^𝑇=1𝑋𝑣2𝜋^∞_−∞𝑢^𝑇(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝐸𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹≤1𝑣𝑑𝜔‖𝐸‖2𝜋^2∞_−∞𝑢^𝑇(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,12𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑣₂≤1𝑑𝜔‖𝐸‖2𝜋^2∞_−∞𝑢^𝑇(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1²₂𝑑𝜔^1/2⋅^∞_−∞𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑣²₂𝑑𝜔^1/2. (2.7)
Here, we have used the Cauchy-Schwarz inequality. It holds that ^∞_−∞𝑢^𝑇(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1²₂𝑑𝜔=^∞_−∞𝑢^𝑇(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝑃^𝑇_𝑙,1𝑖𝜔𝐷^𝑇−𝐴^𝑇−1≤𝑢𝑑𝜔_∞−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝑃^𝑇_𝑙,1𝑖𝜔𝐷^𝑇−𝐴^𝑇−1𝑑𝜔₂=2𝜋𝐻₁₂, (2.8) where 𝐻₁=12𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝑃^𝑇_𝑙,1𝑖𝜔𝐷^𝑇−𝐴^𝑇−1𝑑𝜔 (2.9) is the unique solution of (2.5).
Similarly, we obtain ^∞_−∞𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑣²₂𝑑𝜔≤2𝜋𝐻₂₂, (2.10) where 𝐻₂=12𝜋^∞_−∞−𝑖𝜔𝐶^𝑇−𝐵^𝑇−1𝑃_𝑙,1𝑃^𝑇_𝑙,1(𝑖𝜔𝐶−𝐵)⁻¹𝑑𝜔 (2.11) is the unique solution of (2.6).
From (2.7), (2.8), and (2.10), it follows that for any 𝐸∈ℝ^𝑛×𝑚, ‖𝒮⁻(𝐸)‖₂≤‖𝐸‖₂𝐻₁₂𝐻₂₂. (2.12)
Hence, ‖𝒮⁻‖₂=sup_{𝐸∈ℝ𝑛×𝑚,𝐸≠0}‖𝒮⁻(𝐸)‖₂‖𝐸‖₂≤𝐻₁₂𝐻₂₂. (2.13)

Let 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 be slightly perturbed to 𝐴=𝐴+Δ𝐴,𝐵=𝐵+Δ𝐵,𝐶=𝐶+Δ𝐶,𝐷=𝐷+Δ𝐷,𝐸=𝐸+Δ𝐸, (2.14) respectively, where Δ𝐴,Δ𝐷∈ℝ^𝑛×𝑛, Δ𝐵,Δ𝐶∈ℝ^𝑚×𝑚, Δ𝐸∈ℝ^𝑛×𝑚 with ‖Δ𝐴‖₂≤𝜀‖𝐴‖₂,‖Δ𝐵‖₂≤𝜀‖𝐵‖₂,‖Δ𝐶‖₂≤𝜀‖𝐶‖₂,‖Δ𝐷‖₂≤𝜀‖𝐷‖₂,‖Δ𝐸‖₂≤𝜀‖𝐸‖₂. (2.15)

Then the projected generalized continuous-time Sylvester equation (1.1) is perturbed to: 𝐴𝑋𝐶+𝐷𝑋𝐵+𝑃_𝑙,1𝐸𝑃_𝑟,2𝑃=0,𝑋=_𝑟,1𝑋𝑃_𝑙,2, (2.16) where 𝑋=𝑋+Δ𝑋 with Δ𝑋∈ℝ^𝑛×𝑚, 𝑃_𝑟,1 and 𝑃_𝑟,2 are the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively, and 𝑃_𝑙,1 and 𝑃_𝑙,2 are the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵, respectively.

We assume in this paper that the spectral projectors of the pencils 𝜆𝐷−𝐴, 𝜆𝐶−𝐵 and the perturbed pencils 𝜆𝐷−𝐴, 𝜆𝐶−𝐵 satisfy Ker𝑃_𝑟,𝑖=Ker𝑃_𝑟,𝑖,Ker𝑃^𝑇_𝑙,𝑖=Ker𝑃^𝑇_𝑙,𝑖,𝑖=1,2. (2.17)

Such an assumption is reasonable in some applications; see, for example, [13]. We further assume that 𝑃_𝑙,𝑖−𝑃_𝑙,𝑖2≤𝜀𝐾,𝑃_𝑟,𝑖−𝑃_𝑟,𝑖2≤𝜀𝐾,𝑖=1,2, (2.18) where 𝐾 is a constant.

From (2.17), it follows that for 𝑖=1,2, 𝑃_𝑟,𝑖𝑃_𝑟,𝑖=𝑃_𝑟,𝑖,𝑃_𝑟,𝑖𝑃_𝑟,𝑖=𝑃_𝑟,𝑖,𝑃_𝑙,𝑖𝑃_𝑙,𝑖=𝑃_𝑙,𝑖,𝑃_𝑙,𝑖𝑃_𝑙,𝑖=𝑃_𝑙,𝑖. (2.19)

Moreover, it is not difficult to verify that 𝑃_𝑙,1𝐷=𝑃_𝑙,1𝐷𝑃_𝑟,1=𝐷𝑃_𝑟,1,𝑃_𝑙,1𝐴=𝑃_𝑙,1𝐴𝑃_𝑟,1=𝐴𝑃_𝑟,1,𝑃_𝑙,2𝐶=𝑃_𝑙,2𝐶𝑃_𝑟,2=𝐶𝑃_𝑟,2,𝑃_𝑙,2𝐵=𝑃_𝑙,2𝐵𝑃_𝑟,2=𝐵𝑃_𝑟,2,𝑃_𝑙,1𝐷=𝑃_𝑙,1𝐷𝑃_𝑟,1=𝐷𝑃_𝑟,1,𝑃_𝑙,1𝐴=𝑃_𝑙,1𝐴𝑃_𝑟,1=𝐴𝑃_𝑟,1,𝑃_𝑙,2𝐶=𝑃_𝑙,2𝐶𝑃_𝑟,2=𝐶𝑃_𝑟,2,𝑃_𝑙,2𝐵=𝑃_𝑙,2𝐵𝑃_𝑟,2=𝐵𝑃_𝑟,2. (2.20)(2.21)(2.22)(2.23)

We reformulate the first equation of the perturbed equation (2.16) as 𝐴𝑋𝐶+𝐷𝑋𝐵+𝑃_𝑙,1𝐸𝑃_𝑟,2+𝐹𝑋=0, (2.24) where 𝐹(𝑋)∈ℝ^𝑛×𝑚 is defined by 𝐹𝑋=𝐴𝑋Δ𝐶+Δ𝐴𝑋𝐶+𝐷𝑋Δ𝐵+Δ𝐷𝑋𝐵. (2.25)

Then we have the following lemma.

Lemma 2.3. The following relation holds 𝑃_𝑙,1𝐸𝑃_𝑟,2+𝐹𝑋=𝑃_𝑙,1𝑃_𝑙,1𝐸𝑃_𝑟,2+𝐹𝑋𝑃_𝑟,2. (2.26)

Proof. The equality 𝑃_𝑙,1𝐸𝑃_𝑟,2=𝑃_𝑙,1𝑃_𝑙,1𝐸𝑃_𝑟,2𝑃_𝑟,2 follows directly from (2.19).
Since 𝑋=𝑃_𝑟,1𝑋𝑃_𝑙,2, we have 𝑃_𝑟,1𝑋=𝑃_𝑟,1𝑃_𝑟,1𝑋𝑃_𝑙,2=𝑃_𝑟,1𝑋𝑃_𝑙,2=𝑋. (2.27) Similarly, 𝑋𝑃_𝑙,2=𝑋.
By using (2.19), (2.20), (2.21), and (2.23), we obtain 𝑋Δ𝐶𝑃_𝑟,2=𝑋𝑃_𝑙,2𝐶−𝐶𝑃_𝑟,2=𝑋𝑃_𝑙,2𝐶𝑃_𝑟,2−𝑋𝑃_𝑙,2𝐶=𝑋𝐶𝑃_𝑟,2𝑃_𝑟,2−𝑋𝑃_𝑙,2𝐶=𝑋𝐶𝑃_𝑟,2−𝑋𝑃_𝑙,2𝐶=𝑋𝑃_𝑙,2𝐶−𝑋𝑃_𝑙,2𝑃𝐶=𝑋Δ𝐶,_𝑙,1𝐴𝑋=𝑃_𝑙,1𝐴𝑃_𝑟,1𝑋=𝑃_𝑙,1𝑃_𝑙,1𝐴𝑋=𝑃_𝑙,1𝐴𝑋=𝐴𝑃_𝑟,1𝑋=𝐴𝑋. (2.28)
Hence, 𝑃_𝑙,1𝐴𝑋Δ𝐶𝑃_𝑟,2=𝐴𝑋Δ𝐶𝑃_𝑟,2=𝐴𝑋Δ𝐶. (2.29)
Using the similar manipulation, we can show that 𝑃_𝑙,1𝐷𝑋Δ𝐵𝑃_𝑟,2=𝐷𝑋Δ𝐵,𝑃_𝑙,1Δ𝐴𝑋𝐶𝑃_𝑟,2=Δ𝐴𝑋𝐶,𝑃_𝑙,1Δ𝐷𝑋𝐵𝑃_𝑟,2=Δ𝐷𝑋𝐵. (2.30)
Then, the equality (2.26) follows.

The following theorem provides the result on the relative error bound for the solution of the projected generalized continuous-time Sylvester equation (1.1).

Theorem 2.4. Assume that the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 are c-stable. Let 𝑋 be the unique solution of the projected generalized continuous-time Sylvester equation (1.1). Define 𝛼=‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂𝐻₁₂𝐻₂₂,𝛽=‖𝐸‖₂𝐻₁₂𝐻₂₂. (2.31)
If 𝜀(2+𝜀)𝛼<1, then the perturbed equation (2.16) has a unique solution 𝑋. Moreover, the relative error satisfies the following inequality: 𝑋−𝑋₂‖𝑋‖₂≤𝜀(2+𝜀)𝛼()𝛼+)𝐾𝑃1−𝜀2+𝜀𝜀𝛽(1+𝜀𝜀𝐾+_𝑟,22+(1+𝜀)𝐾𝑃_𝑙,12+𝑃_𝑙,12𝑃_𝑟,22(1−𝜀(2+𝜀)𝛼)‖𝑋‖₂. (2.32)

Proof. By (2.26), the perturbed equation can be rewritten as 𝐴𝑋𝐶+𝐷𝑋𝐵+𝑃_𝑙,1𝑃_𝑙,1𝐸𝑃_𝑟,2+𝐹𝑋𝑃_𝑟,2𝑃=0,𝑋=_𝑟,1𝑋𝑃_𝑙,2. (2.33)
It follows from (2.33) and Theorem 2.1 that the perturbed solution 𝑋 can be expressed as 1𝑋=2𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝑃_𝑙,1𝐸𝑃_𝑟,2+𝐹𝑋𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑑𝜔≡Φ𝑋. (2.34)
For any 𝑋∈ℝ^𝑛×𝑚, we have 𝐹𝑋₂≤𝐴₂‖Δ𝐶‖₂+‖Δ𝐴‖₂‖𝐶‖₂+𝐷₂‖Δ𝐵‖₂+‖Δ𝐷‖₂‖𝐵‖₂𝑋₂≤𝜀(2+𝜀)‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂𝑋₂. (2.35)
According to the definition of the linear operator Φ, by making use of Lemma 2.2 and (2.35), we get, for any 𝑋₁,𝑋₂∈ℝ^𝑛×𝑚, Φ𝑋₁−Φ𝑋₂₂≤12𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝐹𝑋₁−𝑋₂𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹𝑑𝜔₂≤𝐹𝑋₁−𝑋₂₂𝐻₁₂𝐻₂₂≤𝜀(2+𝜀)‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂𝐻₁₂𝐻₂₂𝑋₁−𝑋₂₂. (2.36)
It shows that if 𝜀(2+𝜀)(‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂)‖𝐻₁‖₂‖𝐻₂‖₂<1, then Φ is a contractive linear operator. At this moment, by the fixed point theorem [14], Φ(𝑋)=𝑋 has a unique solution.
It holds that 𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,22=𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,2+𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,2+𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,22≤𝑃_𝑙,1−𝑃_𝑙,12𝐸₂𝑃_𝑟,22+𝑃_𝑙,12𝐸₂𝑃_𝑟,2−𝑃_𝑟,22+𝑃_𝑙,12‖Δ𝐸‖₂𝑃_𝑟,22≤𝜀𝐾(1+𝜀)‖𝐸‖₂𝜀𝐾+𝑃_𝑟,22+𝜀(1+𝜀)𝐾𝑃_𝑙,12‖𝐸‖₂+𝜀𝑃_𝑙,12‖𝐸‖₂𝑃_𝑟,22=𝜀(1+𝜀)𝐾𝜀𝐾+𝑃_𝑟,22+(1+𝜀)𝐾𝑃_𝑙,12+𝑃_𝑙,12𝑃_𝑟,22‖𝐸‖₂. (2.37)
It follows from (2.35) that 𝐹𝑋₂≤𝜀(2+𝜀)‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂‖𝑋‖₂+𝑋−𝑋₂. (2.38)
By using (2.1), (2.34), (2.37), (2.38), and Lemma 2.2, we obtain 𝑋−𝑋₂=12𝜋^∞_−∞(−𝑖𝜔𝐷−𝐴)⁻¹𝑃_𝑙,1𝐹𝑋+𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,2𝑃_𝑟,2(𝑖𝜔𝐶−𝐵)⁻¹1𝑑𝜔₂≤𝐹𝑋+𝑃_𝑙,1𝐸𝑃_𝑟,2−𝑃_𝑙,1𝐸𝑃_𝑟,22𝐻₁₂𝐻₂₂≤𝜀(2+𝜀)𝛼‖𝑋‖₂+𝜀(2+𝜀)𝛼𝑋−𝑋₂+𝜀𝛽(1+𝜀)𝐾𝜀𝐾+𝑃_𝑟,22+(1+𝜀)𝐾𝑃_𝑙,12+𝑃_𝑙,12𝑃_𝑟,22, (2.39) where 𝛼 and 𝛽 are defined in (2.31). Then, the inequality (2.32) of the relative error bound results from the above inequality.

If 𝐷 and 𝐶 are also nonsingular, then 𝑃_𝑟,1=𝑃_𝑙,1=𝐼_𝑛 and 𝑃_𝑟,2=𝑃_𝑙,2=𝐼_𝑚. In this case, the projected generalized continuous-time Sylvester equation (1.1) reduces to the generalized continuous-time Sylvester equation 𝐴𝑋𝐶+𝐷𝑋𝐵+𝐸=0. (2.40)

The corresponding perturbed equation is 𝐴𝑋𝐶+𝐷𝑋𝐵+𝐸=0. (2.41)

Let 𝐻₁ and 𝐻₂ be the solutions of the generalized continuous-time Lyapunov equations 𝐴𝐻₁𝐷^𝑇+𝐷𝐻₁𝐴^𝑇+𝐼_𝑛𝐵=0,_𝑇𝐻₂𝐶+𝐶^𝑇𝐻₂𝐵+𝐼_𝑚=0, (2.42) respectively.

Note that (2.37) reduces to 𝐸−𝐸₂=‖Δ𝐸‖₂≤𝜀‖𝐸‖₂. (2.43)

Hence, we have 𝑋−𝑋₂≤𝐹𝑋+𝐸−𝐸₂𝐻₁₂𝐻₂₂≤𝜀(2+𝜀)𝛼‖𝑋‖₂+𝜀(2+𝜀)𝛼𝑋−𝑋₂+𝜀‖𝐸‖₂𝐻₁₂𝐻₂₂≤𝜀(2+𝜀)𝛼‖𝑋‖₂+𝜀(2+𝜀)𝛼𝑋−𝑋₂+𝜀𝛼‖𝑋‖₂=𝜀(3+𝜀)𝛼‖𝑋‖₂+𝜀(2+𝜀)𝛼𝑋−𝑋₂. (2.44)

From the above inequality, we obtain the relative error bound for the generalized continuous-time Sylvester equation (2.40).

Theorem 2.5. Assume that 𝐷 and 𝐶 are nonsingular, and the pencils 𝜆𝐷−𝐴 and 𝜆𝐶−𝐵 are c-stable. Let 𝑋 be the unique solution of the generalized continuous-time Sylvester equation (2.40). Let 𝛼=(‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂)‖𝐻₁‖₂‖𝐻₂‖₂, where 𝐻₁ and 𝐻₂ are the solutions of the generalized continuous-time Lyapunov equation (2.42), respectively. If 𝜀(2+𝜀)𝛼<1, then the perturbed equation (2.41) has a unique solution 𝑋. Moreover, the relative error satisfies the following inequality: 𝑋−𝑋₂‖𝑋‖₂≤𝜀(3+𝜀)𝛼()𝛼.1−𝜀2+𝜀 (2.45)

This result shows that (‖𝐴‖₂‖𝐶‖₂+‖𝐷‖₂‖𝐵‖₂)‖𝐻₁‖₂‖𝐻₂‖₂ may be used to measure the sensitivity of the solution of the generalized continuous-time Sylvester equation (2.40).

3. Conclusions

In this paper, we have studied the perturbation analysis for the projected generalized continuous-time Sylvester equation and the generalized continuous-time Sylvester equation. By making use of solutions of two special projected generalized continuous-time Sylvester equations, we obtain the perturbation bounds based on the Euclidean norm for their solutions.

Acknowledgments

The authors would like to thank Professor Mohammed Chadli and the referees for their useful comments. Y. Zhou is supported by the Science and Technology Planning Project of Hunan Province under Grant 2010JT4042. L. Bao is supported by the National Natural Science Foundation of China under Grants 10926150 and 11101149. Y. Lin is supported by the National Natural Science Foundation of China under Grant 10801048, the Natural Science Foundation of Hunan Province under Grant 11JJ4009, the Scientific Research Foundation of Education Bureau of Hunan Province for Outstanding Young Scholars in University under Grant 10B038, and the Chinese Postdoctoral Science Foundation under Grant 2012M511386.