Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 620758 18 pages doi 2009 620758 Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation Jianzhou Liu and Juan Zhang Department of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411105 China Correspondence should be addressed to Jianzhou Liu liujz@ Received 25 September 2008 Accepted 19 February 2009 Recommended by Panayiotis Siafarikas By using singular value decomposition and majorization inequalities we propose new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further we give their application in the algebraic Riccati equation. Finally numerical examples have illustrated that our results are effective and superior. Copyright 2009 J. Liu and J. Zhang. 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. 1. Introduction In the analysis and design of controllers and filters for linear dynamical systems the Riccati equation is of great importance in both theory and practice see 1-5 . Consider the following linear system see 4 x t Ax f Bu f x 0 x0 with the cost J f xTQx uTu dt. 0 Moreover the optimal control rate u and the optimal cost J of and are u Px P BtK J xTKx0 2 Journal of Inequalities and Applications where x0 6 Rn is the initial state of the systems and K is the positive definite solution of the following algebraic Riccati equation ARE ATK KA - KRK -Q with R BBt and Q are symmetric positive definite matrices. To guarantee the existence of the positive definite solution to we shall make the following assumptions the pair A R is stabilizable and the pair Q A is observable. In .
