控制系统的约束矩阵方程及其高效数值算法

在控制系统中，探讨其可控性和稳定性常常需要转化为研究其对应的线性和非线性矩阵方程。本项目将着重讨论控制系统中的矩阵方程的求解（约束解）及其解的性质，进而设计高效稳定的数值算法，促进矩阵计算与控制理论的发展。. 我们将重点研究以下问题：1）控制系统中的线性和非线性矩阵方程的求解（约束解）和解的数值迭代算法。2）基于控制系统的大型矩阵方程的降阶处理。3）控制系统中的矩阵方程和矩阵微分方程约束解的矩阵界和约束解的数值界估计。. 上述问题在工程领域应用广泛，并且涉及到矩阵理论和矩阵计算中的深刻问题。本项目将利用构造性方法、压缩映像原理、算子理论等思想方法，结合矩阵Schur补、矩阵分解和特殊矩阵等丰富的矩阵理论知识和技巧，研究和解决这些问题。

Discussing many important characteristics of control systems such as controllability and stability can often be converted into studying related linear and nonlinear matrix equations. We will focus on discussing solving the solution (constraint solution) and its properties of matrix equations in control systems, and try to design highly efficient stable numerical algorithms to promote the development of matrix computation and control theory.. The project will focus on researching the following questions: 1) Solving the solution (constraint solution) and numerical iteration algorithms of the solution for linear and nonlinear matrix equations in control systems. 2) Reduced order processing of large matrix equations based on control systems. 3) The matrix and numerical bounds of constraint solution for matrix equations and matrix differential equations in control systems. . These problems have wide applications in engineering fields, and involve the deep problems in matrix theory and matrix computation. We will use constructive method, contraction mapping principle and operator theory, combining with rich matrix theory knowledge and skills, such as matrix Schur complement, matrix decomposition and special matrix, to study and solve the above problems.