线性动力系统研究的是单个或者多个线性算子作用下点的轨道的渐进性质。本项目以点的轨道在整个相空间中的稠密性为主要研究对象，即动力系统的传递性。对于单个算子作用的系统我们关注非传递的线性算子(如酉算子) 在扰动后谱的变化，特别是有限秩的算子扰动，使得扰动后成为传递的线性算子。这有助于理解线性算子的稳定性，以及传递的线性算子构成的集合在全体有界线性算子的集合中的分布状况。对于多个算子作用的系统，我们关注有限个特殊矩阵为生成元的矩阵半群的传递性，这些矩阵与微分方程的差分格式有关。无论是单个还是多个算子作用的动力系统，都能与数学控制理论中的离散线性控制系统建立联系。 控制系统在反馈映射下成为单个算子作用下的动力系统，或者在限定控制信号的数目的条件下成为多个算子作用下的动力系统。最终通过动力系统的方法研究控制系统的性质，特别是通过动力系统的传递性得出控制系统能控性.

Linear dynamical systems study the asymptotic properties of the orbits of points under the action of single or multiple linear operators. In this project, the main research object is the density of point orbit in the whole phase space, that is, the transitivity of the dynamical system. For a system with a single operator, we pay attention to the spectral changes of nontransitive linear operators (such as unitary operators) after perturbations, especially the operator perturbations with finite rank, which make the perturbations become transitive linear operators. This is helpful to understand the stability of linear operators and the distribution of the set of transitive linear operators in the set of all bounded linear operators. For systems with multiple operators, we pay attention to the transitivity of matrix semigroups whose finite special matrices are generators, which are related to the difference schemes of differential equations. No matter the dynamical system acted by single or multiple operators, it can be connected with the discrete linear control system in mathematical control theory. The control system becomes a dynamical system for the action of a single operator under the feedback mapping, or becomes a dynamical system for the action of multiple operators under the condition of limiting the number of control signals. Finally, the property of control systems is studied by the method of dynamical systems, especially the controllability of control systems is obtained by the transitivity of dynamical systems.