100-LuanVanThacSi-ChuaPhanLoai (189).pdf
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INFORMATION ON MASTER’S THESIS
1. Full name: Nguyen Duy Khanh
2. Sex: Male
3. Date of birth: 21/06/1989
4. Place of birth: Nam Dinh
5. Changes in academic process: No
6. Thesis title: Stability problem of nonlinear differential equations and
applications.
7. Major: Mathematical analysis
8. Code: 60 46 01 02
9. Supervisors: Professor Dr. Vu Ngoc Phat
10. Theis summary
In practice, many problems involved technical and control problems are described by the following differential equations
x(t)
˙
= f (t, x(t), u(t)),
t ≥ 0,
where x(t) is the state output, u(t) is the control input. The input data have
important impacts which may affect on the operation of the system output.
Thus, we can understand the control system as a mathematical model described the input - output relationship.
One of the main purposes of control problem is to find the control input
so that the output receives the desired system properties. In particular, the
stability is one of the important research topics in the qualitative theory of
dynamical systems, and it is extensively used in many applicable areas, e.g. in
mechanics, physics, applied mathematics, control engineering, economics, etc.
The research on stability problems has began from the 19-th century by
famous Russian mathematician V. Lyapunov, who introduced foundation for
stability theory, especially he proposed two methods for investigating stability
1
problem. First method is based on stability exponent or on linear approximation. If the right-hand side function is smooth enough, for example, is
continuously differentiable function, then the stabiliuty of nonlinear system
can be derived from the stability system of approximated linear system. The
second method is based on the existence of a non-negative function such that
the stability of the system is verified directly via the sign of derivative of this
function along the solution of the system. Each method has its advantage and
disadvantage. The first method requires the smooth enough property of the
right-hand side function, while the second method requires the existence of a
specific tested function, which is, in general, very difficult to find out. So far
no effective method to find Lyapunov function.
Thesis Stability problem of nonlinear differential equations and applications presents some results of stability, asymptotic stability, exponential
stability and some related applications.
The thesis consists of two chapters
Chapter I: Mathematical basic backrounds.
The chapter includes three issues:
- Presentation of basic knowledge on differential equations, control systems,
existence of solutions.
- Presentation on some concepts, fundamental theorems on the stability of
the nonlinear differential equations. Introduction of Lyapunov function method
to consider the stability of the system, and some basic theorems on the stability.
- Presentation of basic concepts of stabilization problem of control systems
and some basic results on stabilization.
Chapter II: Stability of nonlinear differential equations and applications.
The chapter includes two sections.
1) Asymptotic stability of nonlinear differential equations:
+ Theorem on asymptotic stability of the semilinear autonomous system
x(t)
˙
= Ax(t) + f (x(t)),
where A is a given constant matrix, f (x) is a small nonlinear perturbation.
+ Theorem on asymptotic stability of semilinear non-autonomous systems
of the form
x(t)
˙
= A(t)x(t) + g(t, x(t))
2
where A(t) is a given matrix function, g(t, x) is a small nonlinear nonautonomous perturbation.
+ Introduction of the concepts of Lyapunov-like and generalized Lyapunovlike functions. Some results on the stability of nonlinear differential equations based on choosing the Lyapunov-like and generalized Lyapunov-like
functions are presented.
2) Stabilization of nonlinear control system.
Based on the results obtained in previous section, we present some results
on stabilization for nonlinear control systems.
Ha Noi, 26th October 2015
Student
Nguyen Duy Khanh
3
