The Existence Of The Solution For One Kind Of Algebraic Riccati Equation

The matrix equation A^TX + XA + XRX + Q = O is called algebraic Riccatiequation, which is very important in the field of automatic control and other engineering applications.Many researcher have studied the solutions of various algebraic Riccati equation, most of them mainly applied the matrix methods, few used the functional analysis theories. This paper mainly studies the existence of the solutions of the following kind of algebraic Riccati equation from the functional viewpoint:Here, X, A,R,Q∈R^n×n , Q is the symmetric matrix, and R is positivesemi-definite or semi-negative definite matrix,λis arbitrary constants. This paper uses functional approach such as fixed-point theorem and thinking of contraction mapping, in order to gives two sufficient condition for solvability about this kind of Riccati equation in given two different conditions,and some conclusions are given .This article is divided into four chapters .In chapter 1, we briefly introduce the research background and motivation and a general solution of Lyapunov equations and Riccati equations of general form, as well as some of the relevant definitions and symbols in this article.In chapter 2, we introduce the fixed-point theorem and give proof of existence of Riccati type equation solutions under the first condition.In chapter 3, we present thinking of the contraction mapping and gives proof of existence of Riccati type equation solutions under the second condition.In chapter 4, we simply sum up the work in this article .