Numerical Methods For The Minimum Nonnegative Solution Of A Class Of Nonsymmetric Algebraic Riccati Equations

The non-symmetric algebraic Riccati equation and the coupled non-symmetric algebraic Riccati equation have important applications in transport theory,traffic flow problems and optimization control.Such as the determination of scattering function in transport theory and wiener-hopf decomposition of Markov chain in traffic problems and many other problems can be transformed into the study of properties and solutions of non-symmetric algebraic Riccati equation;the stability and controllability of the control system can be transformed into the solution of the corresponding coupled algebraic Riccati equation.Therefore,it is of great scientific significance to study the numerical algorithm for a class of non-symmetric algebraic Riccati equations.In this paper,we firstly consider the non-symmetric algebraic Riccati equation whose four coefficient matrices form a regular M-matrix,obtain the modified inexact Newton method,the modified structure-preserving doubling method and the convergence analysis.Secondly,For the general expression of the coupled non-symmetric algebraic Riccati equation,we consider the coupling term as a whole and give the existence theory of the minimal non-negative solution of the equation,obtain the coupled inexact Newton iteration method,the coupled alternating linear implicit iteration method and the convergence analysis.In chapter one,we introduce some theoretical knowledge,recent works of this kind of non-symmetric algebraic Riccati equations.The related sign,definition and lemma involved in this paper are given.In chapter two,we consider the non-symmetric algebraic Riccati equation whose four coefficient matrices form a regular M-matrix,based on Newton iteration method and structure-preserving doubling method,we introduce parameter,using Cayley transformation and double transformation to construct the standard pair to obtain the modified inexact Newton method and the modified structure-preserving doubling method;further,according to the flexibility of parameter value the matrix approximate inverse method is given;then,the convergence of the methods are proved by the properties of the nonsingular M-matrix and double transformation;finally,numerical examples show the effectiveness of the methods.In chapter three,we consider the general expression of the coupled non-symmetric algebraic Riccati equations whose four coefficient matrices form nonsingular Mmatrix or singular M-matrix,and combined with fixed point iteration obtain the existence theorem of the minimal non-negative solution of the equation,it's shown theoretically that the theorem assumption conditions generalizes some existing results;then,we considering coupling terms as a whole,based on the Newton iteration method introduce Cayley transformation and matrix alternating decomposition to obtain the coupled inexact Newton iteration method and the coupled alternating linear implicit iteration method,the convergence of the methods are proved by the properties of the nonsingular M-matrix and monotonic convergence;finally,the validity of the numerical methods presented is illustrated by extending the coupling terms in numerical examples.