Bất đẳng thức Halanay và bất đẳng thức Gronwall trong nghiên cứu định tính các phương trình sai phân

Luận văn: Bất đẳng thức Halanay và bất đẳng thức Gronwall trong nghiên cứu định tính các phương trình sai phân : Luận văn ThS. Khoa học: 60 46 01 02
Nhà xuất bản: ĐHKHTN
Ngày: 2014
Miêu tả: 48 tr.
Luận văn ThS. Toán học -- Trường Đại học Khoa học tự nhiên. Đại học Quốc gia Hà Nội, 2014
Mục lục

Spec. code : 60 46 01 02
Supervisor : Ass. Prof. Nguyen Sinh Bay
In qualitative study of differential equations or of difference equations often to assess the estimation of solutions or experimental error between the
exact solutions and approximate test. The differential, integro-differential
inequalities are often used in the stage of evaluation. There are many inequalities known, but in this thesis we only mention two very well-known
type of those inequalities. We would like to mention on the Gronwall inequality and Halanay inequality. After the presentation of the original formula of each inequality we will present some results as classical extensions
and some recent expansions. The inequality expand mainly in the form of
difference relations to serve the main goal - to investigate the stability of solutions of difference equations. By different ways of various discretization,
the differential equations are often taken to be the difference equation with
the admissible errors. The difference equations are the relationship containing the independent variables and unknown function of this variable with
some level of differences. Expansion of those inequalities and apply them
to the study of the properties of solutions for the equations above are an
actual need. Therefore, we choose topics for essays under the title above.
The thesis consists of the introduction, two chapters , conclusion and
list of references.
Chapter one presents the Gronwall inequality. Starting from the original
inequality, said for the case of continuous time, we outlined some of the
results of major expansion for discrete-time case.
Chapter two said about Halanay inequality. The problem was interpreted
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in the diagram in chapter one. Because this content is difficult so the main
results are presented the proof. In each chapter, the results of the survey
stability is given to illustrate the application value of the inequality raised.
Outstanding results in this application are criteria for stabilization of nonlinear systems with levels of growth lower than linear with a class of linear
differential equation special permission received by discrete approximation
of the two sides.
In total, the thesis presented method uses two types of inequality dedicated to the study some class of differential equations and difference equations. Thesis stated and proved the original inequality Gronwall ’s and
Halanay’s type, then extended to some cases with continuous time or discrete time. Applying the inequality has expanded research and dissertation
done examining the stability for a few class - equation special form and on
some specific examples.6
B£ng c¡c k‰ hi»u
R+ := [0; +1)
Z+ := f0; 1; 2; 3; :::g
Z := f0; ±1; ±2; ±3; :::g
C[R+; R+] - t“p c¡c hàm li¶n tục tł R+ vào R+:
Rn - không gian v†c tơ n- chi•u
Z+
n0 := fn0; n0 + 1; n0 + 2; :::g
Z(m;n) := fm; m + 1; m + 2; :::; ng
∆x(n) = x(n + 1) − x(n) tương đương với ∆xn = xn+1 − xn.
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