Verified computation for the Hermitian positive definite solution of the conjugate discrete-time algebraic Riccati equation

2019 ◽  
Vol 350 ◽  
pp. 80-86 ◽  
Author(s):  
Shinya Miyajima
Keyword(s):  
Riccati Equation ◽  
Discrete Time ◽  
Positive Definite ◽  
Algebraic Riccati Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Verified Computation ◽  
Definite Solution

scholarly journals Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Wenling Zhao ◽  
Hongkui Li ◽  
Xueting Liu ◽  
Fuyi Xu
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Nonsingular Matrix ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Necessary And Sufficient ◽  
Definite Solution

We study the Hermitian positive definite solutions of the nonlinear matrix equationX+A∗X−2A=I, whereAis ann×nnonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations ofX+A∗X−2A=Iare presented while the matrix equation has a Hermitian positive definite solution.

scholarly journals Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Equation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

The nonlinear matrix equation,X-∑i=1mAi*XδiAi=Q,with-1≤δi<0is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

scholarly journals On the Hermitian Positive Definite Solutions of Nonlinear Matrix EquationXs+A∗X−t1A+B∗X−t2B=Q

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Aijing Liu ◽  
Guoliang Chen
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

Nonlinear matrix equationXs+A∗X−t1A+B∗X−t2B=Qhas 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 equationXs+A∗X−t1A+B∗X−t2B=Qare considered, whereQis a Hermitian positive definite matrix,A,Bare nonsingular complex matrices,sis a positive number, and0<ti≤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.

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