Research Article | Open Access

# Hermitian Positive Definite Solution of the Matrix Equation

**Academic Editor:**Asai Asaithambi

#### Abstract

We consider the Hermitian positive definite solution of the nonlinear matrix equation . Some new sufficient conditions and necessary conditions for the existence of Hermitian positive definite solutions are derived. An iterative method is proposed to compute the Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results.

#### 1. Introduction

In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation where are complex matrices; and are Hermitian positive definite matrices. Here, denotes the conjugate transpose of the matrix . The nonlinear matrix equation (1) plays an important role in linear optimal and robust control. For instance, the solvability of the discrete-time linear quadratic optimal control problem with , depends on the solvability of (1) in some special cases [1–4]. Due to the important applications in system and control theory, in the past decades, (1) with has been extensively studied, and the research results mainly concentrated on the following: (a)sufficient conditions and necessary conditions for the existence of an Hermitian solution [5–9];(b)numerical methods for computing the Hermitian solution [4, 10–13];(c)properties of the Hermitian solution [14, 15];(d)perturbation analysis for the discrete algebraic Riccati equation [16–18].(e)connection with symplectic matrix pencil [9, 19, 20];(f)connection with stochastic realization and spectral factorization [21–23].Nonetheless, (1) with has not been studied as far as we know.

In this paper we study the generalized nonlinear matrix equation (1). Firstly, we transform (1) into an equivalent nonlinear matrix equation. By Sherman-Woodbury-Morrison formula [24, Page 50], we have then

Set then That is, Therefore, the nonlinear matrix equation (1) can be equivalently rewritten as (7). So we first investigate the Hermitian positive definite solution of (7) in Section 2 and then derive some new results on the nonlinear matrix equation (1) by using the matrix transformations (5) in Section 3. Finally, we use an example to illustrate the correctness and application of the results of Section 3.

Throughout this paper, we write if the matrix is Hermitian positive definite (semidefinite). If is Hermitian positive definite (semidefinite), then we write . If an Hermitian positive definite matrix satisfies , we denote by . We use to denote all eigenvalues (each repeated as many times as its algebraic multiplicity) of an Hermitian matrix . The symbol denotes the spectral norm of the matrix .

#### 2. Hermitian Positive Definite Solution of (7)

In this section, we first give some sufficient conditions and necessary conditions for the existence of an Hermitian positive definite solution of (7), and then we construct an iterative method to compute the Hermitian positive definite solution. We begin with some lemmas.

Lemma 1 (see [25]). *If , then .*

Lemma 2. *If (7) has an Hermitian positive definite solution , then
*

*Proof. *Since (7) has an Hermitian positive definite solution , that is,
then
And from Lemma 1 it follows that
which leads to
that is,

Lemma 3. *Suppose that the matrices are nonsingular; if (7) has an Hermitian positive definite solution , then
*

*Proof. *Since (7) has an Hermitian positive definite solution , that is,
then
which implies
By (17) and Lemma 1 and noting that the matrices are nonsingular, then we have
which leads to
that is,
By (16) and (20), we have

Theorem 4. *Let
**
If
**
then (7) has an Hermitian positive definite solution and , where
*

*Proof. *Since
and noting that
then we have
Therefore, the quadratic equation
has a positive root , and the quadratic equation
has a positive root , where and are defined by (24).

Now we consider the map
Obviously, is a convex, closed, and bounded set and the map is continuous on . For arbitrary , since and are positive roots of (28) and (29), respectively, then we have
that is, , which implies that . By Brouwer’s fixed point theorem, the map has a fixed point , which is the positive definite solution of (7). The theorem is proved.

Theorem 5. *If
**
for all , then (7) has an Hermitian positive definite solution. Furthermore, if
**
then (7) has a unique Hermitian positive definite solution.*

*Proof. *Consider the map
Obviously, is a convex, closed, and bounded set and the map is continuous on . If
for all , then we have
that is,
Hence, . By Brouwer’s fixed point theorem, the map has a fixed point in , which is the Hermitian positive definite solution of (7).

For arbitrary , we have
Since , then is a contraction map in . By Banach’s fixed point theorem, we know that the map has a unique fixed point in , and this shows that (1) has a unique Hermitian positive definite solution in . Noting Lemma 3, we know that (1) has a unique Hermitian positive definite solution. The theorem is proved.

Nextly, we give an iterative method to compute the Hermitian positive definite solution of (7).

Theorem 6. *If there is a real number satisfying
**
then the matrix sequence converges the Hermitian positive definite solution of (7), where the sequence is generated by the following iterative method:
*

*Proof. *Consider the iterative method (40). From (39) it follows that

Since , then and
which implies that
Therefore,

Assume that when , we have ; then
which implies that
By mathematical induction, we obtain that the matrix sequence is monotone decreasing. Now we begin to use mathematical induction to show that the matrix sequence is bounded below by some Hermitian positive definite solution of (7).

When , by (39) we have
Assume that when , we have ; then , and for we have

Therefore, the matrix sequence is a monotonic decreasing sequence and bounded below by some Hermitian positive definite solution , and then it converges to an Hermitian positive definite solution of (1). The theorem is proved.

#### 3. Hermitian Positive Definite Solution of (1)

By the matrix transformations (5), Lemmas 2 and 3, and Theorems 4–6, we have the following theorems.

Theorem 7. *If (1) has an Hermitian positive definite solution , then
*

Theorem 8. *Suppose that the matrices are nonsingular; if (1) has an Hermitian positive definite solution , then
*

*Remark 9. *The upper bounds of Hermitian solution for the discrete algebraic Riccati equation have been extensively studied in the past three decades (see [14] and references therein). Compared with previous works, one gives a new upper bound of Hermitian solution but also gives a lower bound in Theorem 8.

Theorem 10. *Let
**
If
**
then (1) has an Hermitian positive definite solution , and , where
*

Theorem 11. *If
**
for all , then (1) has an Hermitian positive definite solution. Furthermore, if
**
then (1) has a unique Hermitian positive definite solution.*

Theorem 12. *If there is a real number satisfying
**
then the matrix sequence converges to the Hermitian positive definite solution of (1), where the matrix sequence is generated by the following iterative method:
*

#### 4. Numerical Experiments

In this section, we give an example to demonstrate the correctness and application of our results.

*Example 1. *Consider the following discrete-time LQR optimal control problem:
with the additional constraint that , where
, and both and are as follows:
The optimal control sequence that minimizes the performance index is , where and satisfies the discrete algebraic Riccati equation
It is easy to verify that the nonlinear matrix equation (62) satisfies the conditions of Theorem 10; hence, (62) has an Hermitian positive definite solution . Now we use iterative method (58) to compute the Hermitian positive definite solution of (62). Let , and after 6 iterations, we get the positive definite solution
and its residual error . Of course it is easy to verify that
which confirms the correctness of Lemma 3.

From
we have
hence, . Then taking we have

#### 5. Conclusion

In this paper, we consider the Hermitian positive definite solution of the nonlinear matrix equation , which plays an important role in linear optimal and robust control. By making use of Sherman-Woodbury-Morrison formula, we transform this equation into an equivalent nonlinear matrix equation. So we can investigate this nonlinear matrix equation by the aid of the equivalent nonlinear matrix equation. Some new sufficient conditions and necessary conditions for the existence of an Hermitian positive definite solution are derived. An iterative method is proposed to compute its Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was supported by the National Natural Science Foundation of China (nos. 11101100, 11301107, and 11261014) and the Natural Science Foundation of Guangxi Province (nos. 2012GXNSFBA053006 and 2013GXNSFBA019009).

#### References

- D. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for descriptor systems,”
*IEEE Transactions on Automatic Control*, vol. 32, no. 8, pp. 672–688, 1987. View at: Publisher Site | Google Scholar | MathSciNet - P. Lancaster and L. Rodman,
*Algebraic Riccati Equations*, Oxford University Press, Oxford, UK, 1995. View at: MathSciNet - B. D. Anderson and J. B. Moore,
*Linear Optimal Control*, Prentice Hall, Englewood Cliffs, NJ, USA, 1971. View at: MathSciNet - T. Pappas, A. J. Laub, and N. R. Sandell, “On the numerical solution of the discrete-time algebraic Riccati equation,”
*IEEE Transactions on Automatic Control*, vol. 25, no. 4, pp. 631–641, 1980. View at: Publisher Site | Google Scholar | MathSciNet - D. J. Clements and H. K. Wimmer, “Existence and uniqueness of unmixed solutions of the discrete-time algebraic Riccati equation,”
*Systems & Control Letters*, vol. 50, no. 5, pp. 343–346, 2003. View at: Publisher Site | Google Scholar | MathSciNet - J. C. Engwerda, A. C. Ran, and A. L. Rijkeboer, “Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+{A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and Its Applications*, vol. 186, pp. 255–275, 1993. View at: Publisher Site | Google Scholar | MathSciNet - A. Ferrante and B. C. Levy, “Hermitian solutions of the equation $X=Q+{NX}^{-1}{N}^{*}$,”
*Linear Algebra and its Applications*, vol. 247, pp. 359–373, 1996. View at: Publisher Site | Google Scholar | MathSciNet - G. Freiling, V. Mehrmann, and H. Xu, “Existence, uniqueness, and parametrization of Lagrangian invariant subspaces,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 23, no. 4, pp. 1045–1069, 2002. View at: Publisher Site | Google Scholar | MathSciNet - H. K. Wimmer, “Normal forms of symplectic pencils and the discrete-time algebraic Riccati equation,”
*Linear Algebra and Its Applications*, vol. 147, pp. 411–440, 1991. View at: Publisher Site | Google Scholar | MathSciNet - P. Benner,
*Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems*, Logos, Berlin, Germany, 1997. - P. Van Dooren, “A generalized eigenvalue approach for solving Riccati equations,”
*Society for Industrial and Applied Mathematics*, vol. 2, no. 2, pp. 121–135, 1981. View at: Publisher Site | Google Scholar | MathSciNet - L. Z. Lu and W. W. Lin, “An iterative algorithm for the solution of the discrete-time algebraic Riccati equation,”
*Linear Algebra and Its Applications*, vol. 188-189, pp. 465–488, 1993. View at: Publisher Site | Google Scholar | MathSciNet - A. J. Laub, “A Schur method for solving algebraic Riccati equations,”
*IEEE Transactions on Automatic Control*, vol. 24, no. 6, pp. 913–921, 1979. View at: Publisher Site | Google Scholar | MathSciNet - R. Davies, P. Shi, and R. Wiltshire, “Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations,”
*Automatica*, vol. 44, no. 4, pp. 1088–1096, 2008. View at: Publisher Site | Google Scholar | MathSciNet - I. G. Ivanov, “Properties of Stein (Lyapunov) iterations for solving a general Riccati equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 4, pp. 1155–1166, 2007. View at: Publisher Site | Google Scholar | MathSciNet - M. M. Konstantinov, P. H. Petkov, and D. W. Gu, “Improved perturbation bounds for general quadratic matrix equations,”
*Numerical Functional Analysis and Optimization*, vol. 20, no. 7-8, pp. 717–736, 1999. View at: Publisher Site | Google Scholar | MathSciNet - J. G. Sun, “Sensitivity analysis of the discrete-time algebraic Riccati equation,” in
*Proceedings of the 6th Conference of the International Linear Algebra Society*, vol. 275/276 of*Linear Algebra and its Applications*, pp. 595–615. View at: Publisher Site | Google Scholar | MathSciNet - L. Zhou, Y. Lin, Y. Wei, and S. Qiao, “Perturbation analysis and condition numbers of symmetric algebraic Riccati equations,”
*Automatica*, vol. 45, no. 4, pp. 1005–1011, 2009. View at: Publisher Site | Google Scholar | MathSciNet - A. Ferrante and B. C. Levy, “Canonical form of symplectic matrix pencils,”
*Linear Algebra and its Applications*, vol. 274, pp. 259–300, 1998. View at: Publisher Site | Google Scholar | MathSciNet - A. Ferrante and L. Ntogramatzidis, “A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems,”
*Linear Algebra and its Applications*, vol. 425, no. 2-3, pp. 242–260, 2007. View at: Publisher Site | Google Scholar | MathSciNet - A. Ferrante, “A parameterization of minimal stochastic realizations,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 10, pp. 2122–2126, 1994. View at: Publisher Site | Google Scholar | MathSciNet - A. Lindquist and G. Picci, “Realization theory for multivariate stationary Gaussian processes,”
*SIAM Journal on Control and Optimization*, vol. 23, no. 6, pp. 809–857, 1985. View at: Publisher Site | Google Scholar | MathSciNet - A. Lindquist and G. Picci, “A geometric approach to modelling and estimation of linear stochastic systems,”
*Journal of Mathematical Systems, Estimation, and Control*, vol. 1, no. 3, pp. 241–333, 1991. View at: Google Scholar | MathSciNet - G. H. Golub and C. F. Van Loan,
*Matrix Computations*, Johns Hopkins University, London, UK, 3rd edition, 1996. View at: MathSciNet - X. Zhan,
*Matrix Inequalities*, Springer, Berlin , Germany, 2002. View at: MathSciNet

#### Copyright

Copyright © 2014 Chun-Mei Li and Jing-Jing Peng. 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.