Basic information of the course
Course Number:202012420025
The English Name of the Course:Stability Theory of Differential Equation
The Chinese Name of the Course:微分方程的稳定性理论
In-class Hours and Allocation(Sample: total class hours:32, classroom teaching: 16 class hours, classroom discussion: 16 class hoursetc.)
total hours:32 hours, lectures:16 hours, class discussions:16 hours
credits:2
starting semester:1
Applicable Discipline (Professional Degree Category):mathematics
Course Object Oriented:academic masters, academic doctors
Evaluation method:evaluate in process
Teaching method:mixed teaching
Teaching unit:college of mathematical science
II. Prerequisite Course
Mathematical analysis, Advanced algebra, Ordinary differential equations, Complex functions, Analytic geometry
III. The Objectives and Requirements of the Course
The stability theory of differential equations is an important course for mathematical majors, especially for operation and control. The teaching objective of this course is to enable graduate students to understand the qualitative analysis methods of differential equations in modern control theory by studying the qualitative and stability theory of differential equations, and make students master the basic theories needed to study modern cybernetics. This course will enable students to apply relevant theories to solve specific problems in the field of control.
Students are required to master the basic contents and methods of the qualitative and stability theories of differential equations, and have the ability to use the basic methods to solve simple problems of system control.
IV. The Content of the Course
Part One:Qualitative analysis of differential equations
Chapter One: principal theorem
Existence and uniqueness of solutions, continuation of solutions, dependence of solutions on initial values or parameters, comparison theorem.
Chapter Two: Basic knowledge of dynamical systems
Homemade systems and non-homemade systems, limit sets for orbitals, limit sets for planes.
Chapter Three: Stability theory
The definition of stability (including the raise of stability problem, the differential equations of disturbance movement, the methods to solve the problem of stability, the practical significance of stability theory), the basic theorem of Liapunov method (homemade or non-homemade system are discussed for the stability of the zero solution, including the definition and theorem of stability by Liapunov, criteria for the definiteness and variational properties of functions, Liapunov theorem about unstable, H.r.qeTaeB theorem of unstable and its applications, the stability of the system and its disturbance, etc.), the structure of the Liapunov function, and the comparative method in stability.
Chapter Four: Singularities of planar systems
Elementary singularities, determination of center and focus, higher-order singularities, number of rotations and exponents.
Chapter Five: Limit cycle
The basic concept of limit cycles; the existence theory of limit cycles, the following functions and the multiplicity and stability of limit cycles, the rotation vector field, and the uniqueness of limit cycles.
Chapter Seven: Singularity analysis of high-dimensional dynamical systems
Singularities of linear systems, stable manifold theorem, topological equivalence and Hartman-Grobman theorem, central form theorem, stability analysis of singularities in critical cases.
Part Two:Applications of stability theory in system control
(1) Discriminant method of linear matrix inequality for system stability. (2) System performance analysis. (3) Analysis and synthesis of unstable systems.
Part Three:Analysis of motion stability of dynamic system with time delay
(1) Ordinary differential equation and time-delay differential equation. (2) Basic theory of difference equation and its stability. (3) The fundamental theorem of the direct method. (4) Motion stability of one-dimensional system. (5) Stability analysis of the dynamic system with tiny time delay.
The ideological and political content of the course are integrated into the whole teaching process as far as possible. It is intended to introduce the progress of national science and technology in introducing the application examples of theories and methods in the courses. For example, when explaining the stability theory, the achievements of Chinese aerospace technology are introduced, such as Chang 'e spacecraft landed on the far side of the moon steadily and so on.
V. Reference Books, Reference Literatures, and Reference Materials
1.马知恩等.常微分方程的定性与稳定性方法.科学出版社，2001。
2.秦元勋.带有时滞的动力系统的稳定性.科学出版社，1984。
3.俞立.鲁棒控制.清华大学学出版社，2002。
4.张芷芬等.微分方程的定性理论.科学出版社,2001。
5.许凇庆.常微分方程稳定性理论.上海科学技术出版社,2003。
6.Min Wu,Yong He,Jin-Hua she.Stability Analysis and Robust Control of Time-Delay systems.Science Press,2009.
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