Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 163585, 18 pages
http://dx.doi.org/10.1155/2011/163585
Research Article

On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 𝟏 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 𝟐 𝐵 = 𝑄

Aijing Liu1,2 and Guoliang Chen1

1Department of Mathematics, East China Normal University, Shanghai 200062, China
2School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Received 18 April 2011; Accepted 11 July 2011

Academic Editor: Mohammad Younis

Copyright © 2011 Aijing Liu and Guoliang Chen. 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

Nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 are considered, where 𝑄 is a Hermitian positive definite matrix, 𝐴 , 𝐵 are nonsingular complex matrices, 𝑠 is a positive number, and 0 < 𝑡 𝑖 ≤ 1 , 𝑖 = 1 , 2 . Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.