Perturbation Analysis Of The Stochastic Algebraic Riccati Equation

Stochastic algebraic Riccati equation plays an important role in physics and engineering, such as the linear quadratic optimal control and robust control problems. In modern industrial production, many problems can be transformed into solve this type of equation and estimate of the approximate solution. This paper will take the stochastic algebraic Riccati equation as the main research object, discuss the upper bound of the solution to the perturbation equation.Firstly, in the introduction part, mainly introduced the background and historical source of the algebraic Riccati equation, then we show the main research work of domestic and international scholars have done on this problem, and gives the main object in this paper.The first chapter is a preliminary knowledge of this article, explains the symbol mark and the related theorem.The second chapter and the third chapter is the core part of this paper. The second chapter mainly discusses the perturbation analysis of the stochastic algebraic Riccati equation. First, Section2.1gives the coefficient matrices of the algebraic Riccati equation small perturbations, get the corresponding perturbation equation, through the equivalent deformation on the perturbation equation, deducing the relatively simple expression. By using the fixed point theorem, in Section2.2gives a perturbation boundary of the equation. The third chapter mainly studies the backward error of the stochastic algebraic Riccati equation. Section3.1presents the upper bound of the backward error estimate. Section3.2estimates the lower bound of the backward error, and by using Taylor expansion, the first-order approximate estimates of backward error is given.The fourth chapter is the numerical experiment, the sharpness and tightness of our perturbation bounds and estimation of backward error are illustrated by some numerical examples.