The Hermite Positive Definite Solutions Of The Nonlinear Matrix Equation X^s+ A^*X^-t A=Q

Studying nonlinear matrix equations is one of the most important areas inthe control theory all the time,in the meantime it is widely used in other fieldssuch as numerical algebra,statistics,dynamic programming,stochastic filtering,ladder networks and queueing theory In many optimal control problems,it isnecessary to solve the discrete coupled algebraic Riccati equations Actually,these problems are equivalent to solving the matrix equation X^S+A*X^-tA=QSo studying nonlinear matrix equations is very helpful to tackle many complexcontrol problems Because the equations are nonlinear,it is becoming one of themost difficult problems in recent years Many important conclusions have beenobtained by scholars at home and abroadIn this thesis,we quest the Hermitian positive definite solutions of the non—linear matrix equationX^S+A*X^-tA=Q. On one hand,we study the existenceand the uniqueness of the HPD solutions by using the fixed point theorems Onthe other hand,we contact the equationX^S+A*X^-tA=Q and the equation Y+A*Y^-qA=Q firmly by a kind of transformation,then more precise intervalestimates and iterative methods are given by this transformationIn chapter one,the development course,the application background and theresearch status of the equation X^S+A*X^-tA=Q are introduced firstly Thenthe main work about this paper is given,as well as sorfle useful symbolsIn chapter two,the solutions'existence of the equation X^S+A*X^-tA=Q is studied First we verify the sufficient condition of the HPD solutions'uniquenessby using the fixed point theorems,then analyze that this condition'S range isdifferent from the previous conclusions,and following construct iterative methodto solve this unique solution At the same time,two numerical iterative methodsare given while the equation'S coefficient matrices sarisfy some special inequal—ity,and the methods'convergence is verified The difference between the twouniqueness theorems and the validity of the numerical algorithms are illustratedby examplesIn chapter three,the equation X^S+A*X^-tA=Q is transformed into theequation Y+A*Y^-qA=Q by a kind of transformation,then we could findthe internal relation between these two equations．and obtain a new solutions'boundary of equation X^S+A*X^-tA=Q which is more precise than the previous interval estimates．a new uniqueness theorem is also given in this new boundestimate by use of the fixed point theorems At the sarile time,we acquire nu—merical algorithms to research the maximum and the minimum HPD solutionsin the general cases,which reriloves the restriction that coefficient matrices mustsatisfy some special conditions Examples explain the effectiveness...