Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 107059, 11 pages

doi:10.1155/2012/107059

Research Article

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

Yujian Zhou,¹ Liang Bao,² and Yiqin Lin¹

¹Department of Mathematics and Computational Science, Institute of Computational Mathematics, Hunan University of Science and Engineering, Yongzhou 425100, China

²Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 28 February 2012; Revised 19 September 2012; Accepted 7 October 2012

Academic Editor: Mohammed Chadli

Copyright © 2012 Yujian Zhou et al. 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.

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

where , are matrices, are matrices, is an matrix, and is the unknown matrix, respectively, with real entries. Here, and are the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils and , respectively, while and are the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of and , respectively. We assume that the pencils and are regular, that is, and are not identically zero. Under the assumption, the pencils and have the Weierstrass canonical forms [1]: there exist nonsingular matrices , and matrices , such that

where , , , and are block diagonal matrices with each diagonal block being the Jordan block. The eigenvalues of and are the finite eigenvalues of the pencils and , respectively. and correspond to the eigenvalue at infinity. Using (1.2), , , , and can be expressed as

If and are nonsingular, then , , and (1.1) reduces to the generalized Sylvester equation . As , , and , (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 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 is -stable, that is, all its finite eigenvalues have negative real parts, then the projected generalized Lyapunov equation has a unique solution for each , and if, additionally, is symmetric and positive semidefinite, then the solution 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 .

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 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 . 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 , and . 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. denotes the identity matrix and denotes the zero vector or zero matrix. If the dimension of is apparent from the context, we drop the index and simply use . The space of real matrices is denoted by . The Euclidean norm for a vector or its associated induced matrix norm is denoted by . The superscript denotes the transpose of a vector or a matrix and is the inverse of nonsingular .

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 and be regular pencils with finite eigenvalues and counted according to their multiplicities, respectively. Let and be the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils and , respectively, and let and be the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of and , respectively. Then, the projected generalized continuous-time Sylvester equation (1.1) has a unique solution for every if for any and .

Moreover, if and are c-stable, that is, all their finite eigenvalues have negative real part, then can be expressed as

Now let us define a linear operator

which satisfies the following: for a matrix , is the unique solution of the projected generalized continuous-time Sylvester equation (1.1), that is,

The following result shows that the linear operator 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 and are c-stable. Then

where and are the solutions of the projected generalized continuous-time Lyapunov equations

respectively.

Proof. Let and be the left and right singular vectors of unit length corresponding to the largest singular value of the solution . Then, for any ,

Here, we have used the Cauchy-Schwarz inequality. It holds that

where

is the unique solution of (2.5).

Similarly, we obtain

where

is the unique solution of (2.6).

From (2.7), (2.8), and (2.10), it follows that for any ,

Hence,

Let , , , , and be slightly perturbed to

respectively, where , , with

Then the projected generalized continuous-time Sylvester equation (1.1) is perturbed to:

where with , and are the spectral projectors onto the right deflating subspaces corresponding to the finite eigenvalues of the pencils and , respectively, and and are the spectral projectors onto the left deflating subspaces corresponding to the finite eigenvalues of and , respectively.

We assume in this paper that the spectral projectors of the pencils , and the perturbed pencils , satisfy

Such an assumption is reasonable in some applications; see, for example, [13]. We further assume that

where is a constant.

From (2.17), it follows that for ,

Moreover, it is not difficult to verify that

We reformulate the first equation of the perturbed equation (2.16) as

where is defined by

Then we have the following lemma.

Lemma 2.3. The following relation holds

Proof. The equality follows directly from (2.19).

Since , we have

Similarly, .

By using (2.19), (2.20), (2.21), and (2.23), we obtain

Hence,

Using the similar manipulation, we can show that

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 and are c-stable. Let be the unique solution of the projected generalized continuous-time Sylvester equation (1.1). Define

If , then the perturbed equation (2.16) has a unique solution . Moreover, the relative error satisfies the following inequality:

Proof. By (2.26), the perturbed equation can be rewritten as

It follows from (2.33) and Theorem 2.1 that the perturbed solution can be expressed as

For any , we have

According to the definition of the linear operator , by making use of Lemma 2.2 and (2.35), we get, for any ,

It shows that if , then is a contractive linear operator. At this moment, by the fixed point theorem [14], has a unique solution.

It holds that

It follows from (2.35) that

By using (2.1), (2.34), (2.37), (2.38), and Lemma 2.2, we obtain

where and are defined in (2.31). Then, the inequality (2.32) of the relative error bound results from the above inequality.

If and are also nonsingular, then and . In this case, the projected generalized continuous-time Sylvester equation (1.1) reduces to the generalized continuous-time Sylvester equation

The corresponding perturbed equation is

Let and be the solutions of the generalized continuous-time Lyapunov equations

respectively.

Note that (2.37) reduces to

Hence, we have

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

Theorem 2.5. Assume that and are nonsingular, and the pencils and are c-stable. Let be the unique solution of the generalized continuous-time Sylvester equation (2.40). Let , where and are the solutions of the generalized continuous-time Lyapunov equation (2.42), respectively. If , then the perturbed equation (2.41) has a unique solution . Moreover, the relative error satisfies the following inequality:

This result shows that 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.

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