Trace Inequalities for Matrix Products and Trace Bounds for the Solution of the Algebraic Riccati Equations
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Research Article Trace Inequalities for Matrix Products and Trace Bounds for the Solution of the Algebraic Riccati Equations Jianzhou Liu,1, 2 Juan Zhang,2 and Yu Liu1 1
Department of Mathematic Science and Information Technology, Hanshan Normal University, Chaozhou, Guangdong 521041, China 2 Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China Correspondence should be addressed to Jianzhou Liu, [email protected] Received 25 February 2009; Revised 20 August 2009; Accepted 6 November 2009 Recommended by Jozef Banas By using diagonalizable matrix decomposition and majorization inequalities, we propose new trace bounds for the product of two real square matrices in which one is diagonalizable. These bounds improve and extend the previous results. Furthermore, we give some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions. Finally, numerical examples have illustrated that our results are effective and superior. Copyright q 2009 Jianzhou Liu 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.
1. Introduction As we all know, the Riccati equations are of great importance in both theory and practice in the analysis and design of controllers and filters for linear dynamical systems see 1–5. For example, consider the following linear system see 5: xt ˙ Axt But,
x0 x0 ,
1.1
with the cost J
∞
xT Qx uT u dt.
0
1.2
2
Journal of Inequalities and Applications
The optimal control rate u∗ the optimal cost J ∗ of 1.1 and 1.2 are u∗ P x,
P BT K,
J ∗ x0T Kx0 ,
1.3
where x0 ∈ Rn is the initial state of system 1.1 and 1.2 and K is the positive semidefinite solution of the following algebraic Riccati equation ARE: AT K KA − KRK −Q,
1.4
with R BBT and Q being positive definite and positive semidefinite matrices, respectively. To guarantee the existence of the positive definite solution to 1.4, we will make the following assumptions: the pair A, R is stabilizable, and the pair Q, A is observable. In practice, it is hard to solve the ARE, and there is no general method unless the system matrices are special and there are some methods and algorithms to solve 1.4; however, the solution can be time-consuming and computationally difficult, particularly as the dimensions of the system matrices increase. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of the ARE see 6– 16. Moreover, in terms of 2, 6, we know that an interpretation of trK is that trK/n is the average value of the optimal cost J ∗ as x0 varies over the surface of a unit sphere. Therefore, considering its applications, it is important to discuss trace bounds for the product of two matrices. In symmetric case, a number of works
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