Lectures on qualitative theory of differential equations

196 Pages·2018·2.311 MB·English

by  Айсагалиев С.

AL-FARABI KAZAKH NATIONAL UNIVERSITY
S. Aisagaliev
LECTURES ON QUALITATIVE THEORY
OF DIFFERENTIAL EQUATIONS
Educational manual
Almaty
«Qazaq University»
2018
UDC 51 (075)
LBC 22.1 я 73
A 28
Recommended for publication by the decision
of the Academic Council of the Faculty of Mechanics and Mathematics,
Editorial and Publishing Council
of Al-Farabi Kazakh National University
(Protocol №6 dated 04.04.2018)
Reviewers:
Doctor of Physical and Mathematical Sciences, Academician T.Sh. Kalmenov
Doctor of Physical and Mathematical Sciences, Professor M.T. Jenaliev
Doctor of Physical and Mathematical Sciences, Professor S.Ya. Serovaysky
Aisagaliev S.
A 28 Lectures on qualitative theory of differential equations: educational manual /
S. Aisagaliev. ‒ Almaty: Qazaq University, 2018. ‒ 196 p.
ISBN 978-601-04-3485-1
The book is written on the basis of lectures delivered at the Mechanics and Mathematics
Faculty of KazNU named after al-Farabi, as well as scientific works on the qualitative theory
of differential equations. It describes the solvability and construction of solutions of integral
equations, boundary-value problems of ordinary differential equations with phase and integral
constraints, and a constructive method for solving the boundary-value problem of a linear
integro-differential equation. The book is intended for undergraduates and doctoral students in
the specialty «Mathematics».
Published in authorial release.
UDC 51 (075)
LBC 22.1 я 73
ISBN 978-601-04-3485-1 © Aisagaliev S., 2018
© Al-Farabi KazNU, 2018
2
CONTENTS

Preface ............................................................................................................................. 5

Chapter (cid:44)(cid:44). Integral equations ........................................................................................ 8
Lecture 1. Integral equations solvable for all right hand sides ......................................... 8
Lecture 2. Solvability of an integral equation with fixed right hand side ........................ 12
Lecture 3. Solvability of the first kind Fredholm integral equation ................................. 18
Lecture 4. An approximate solution of the first kind Fredholm integral equation ........... 23
Lecture 5. Integral equation with a parameter .................................................................. 28
Lecture 6. Integral equation for a function of several variables........................................ 32
Comments ......................................................................................................................... 37

Chapter (cid:44)(cid:44). Boundary value problems of linear ordinary differential
equations ......................................................................................................................... 40
Lecture 7. Two-point boundary value problem. A necessary and sufficient
condition for existence of a solution ................................................................................ 41
Lecture 8. Construction of a solution to two-point boundary value problem №1 ........... 46
Lecture 9. Boundary value problems with phase constraints ............................................ 55
Lecture 10. Boundary value problems with phase and integral constraints ..................... 59
Comments ......................................................................................................................... 69

Chapter (cid:44)(cid:44)(cid:44). Boundary value problems for nonlinear ordinary differential
equations ......................................................................................................................... 71
Lecture 11. Two-point boundary value problem .............................................................. 72
Lecture 12. Boundary value problems with state variable constraints ............................. 81
Lecture 13. Boundary value problems with state variable constraint
and integral constraints...................................................................................................... 85
Comments ......................................................................................................................... 93

Chapter (cid:44)V. Boundary value problem with a parameter for ordinary
differential equations ..................................................................................................... 95
Lecture 14. Statement of the problem. Imbedding principle ............................................ 95
Lecture 15. Optimization problem ................................................................................... 102
Lecture 16. Solution to the Sturm-Liouville problem ...................................................... 105
Lecture 17. Boundary value problems for linear systems with a parameter .................... 112
Lecture 18. Boundary value problems for nonlinear systems with a parameter ............... 121
Lecture 19. Boundary value problems with a parameter with pure state variable
constraints.......................................................................................................................... 126
Comments ......................................................................................................................... 129

Chapter V. Periodic solutions of autonomus dynamical systems .............................. 131
Lecture 20. Statement of the problem ............................................................................... 131
3
Lecture 21. A periodic solution of a nonlinear autonomous dynamical system.
Transformation ................................................................................................................. 135
Lecture 22. A necessary and sufficient condition for existence of
a periodic solution ............................................................................................................ 140
Lecture 23. An optimization problem ............................................................................. 145
Lecture 24. Construction of periodic solutions ................................................................ 148
Lecture 25. Periodic solutions to linear autonomous dynamical systems ........................ 156
Lecture 26. The case without constraints ......................................................................... 161
Lecture 27. The case with state variable constraints ........................................................ 168
Comments ......................................................................................................................... 171

Chapter V(cid:44)(cid:44). Constructive method for solving boundary value problems
of linear integro-differential equations ........................................................................ 174
Lecture 28. Statement of the problem. Transformation ................................................... 174
Lecture 29. Linear controllable system. Optimization problem ....................................... 177
Lecture 30. Gradient of the functional. Minimizing sequence ......................................... 181
Lecture 31. Existence of a solution. Constructing a solution to the boundary
value problem ................................................................................................................... 185
Comments ......................................................................................................................... 194

4
PREFACE

These notes build upon a course I taught at the department of
mechanics and mathematics of al-Farabi Kazakh National Uni-
versity during the recent years as well as upon scientific work of
the author on the qualitative theory of differential equations.
This is a course for undergraduates and doctoral students
specializing in mathematical control theory on specialty "Mathe-
matics". Mathematical control theory occurred at the junction of
differential equations, calculus of variations and analytical mecha-
nics is a new direction in qualitative theory of differential equa-
tions.
For solving topical problems of natural sciences, technology,
economy, ecology and other new mathematical methods are requi-
red to solve complex boundary value problems.
Mathematical models for processes of control of nuclear and
chemical reactors, power systems, robotic systems and economics
and other are complex boundary value problems of ordinary diffe-
rential equations and integro-differential equations.
Boundary value problems are said to be complex if in addition
to boundary conditions phase constraints and integral constraints
are posed on state variables of the system. The main problems are
to provide a necessary and sufficient condition for existence of a
solution to the boundary value problem and to develop methods for
constructing solutions to complex boundary value problems.
The existing system of training of mathematicians: bachelor's,
master's, and doctoral programs require new textbooks and teaching
aids for each level of education.
Graduate and doctoral students in mathematical control theory
must have fundamental knowledge on the theory of integral equa-
tions, controllability theory, the theory of extremal problems, the
theory of boundary value problems for differential and integro-
differential equations. Therefore there is a need for the creation of
textbooks in these areas with the results of new fundamental
research.
The book contains results on solvability and constructing a so-
lution for integral equations, boundary value problems of ordinary
5
differential equations with phase and integral constraints, the
constructive method for solving boundary value problem of linear
integro-differential equation.
In Chapter I solvability and construction of a solution for
integral equations at any right hand side as well as the case of fixed
right hand side are presented. A necessary and sufficient condition
of solvability is obtained and a constructing a general solution is
presented. The new method for studying solvability and construc-
ting a solution to the first kind Fredholm integral equation has been
developed. The first kind Fredholm integral equation of the
unknown functions of one variable and of several variables are
considered separately (lectures 1-6).
Chapter II covers the research results on two-point boundary
value problems, boundary value problems with phase constraints,
boundary value problems with phase and integral constraints for
linear system of ordinary differential equations. A necessary and
sufficient condition of solvability has been obtained for the men-
tioned problems and methods for solving them have been developed
(lectures 7-10).
Chapter III deals with solving boundary value problems for
nonlinear systems of ordinary differential equations. The boundary
value problems with boundary conditions on given convex closed
sets, boundary value problems with phase and integral constraints
are considered. A necessary and sufficient condition for an exis-
tence of a solution to the mentioned boundary value problems has
been derived and constructing solutions to them are presented
(lectures 11-13).
In Chapter IV the method for studying solvability and cons-
tructing a solution of boundary value problem with a parameter in
the presence of phase and integral constraints is considered. The
basics of the imbedding principle and reducing to the free end point
optimal control problem as well as solutions to the Sturm-Liouville
problem are presented (lectures 14-19).
Chapter V treats an existence of periodic solutions in autono-
mous dynamical systems and the methods of constructing them for
linear and nonlinear systems. A periodic solution to the Duffing
equation is considered (lectures 20-27).
Chapter VI deals with studying a solvability and constructing
a solution to linear integro-differential equations with phase and
integral constraints. A necessary and sufficient condition for
solvability and construction of a solution by generating minimizing
sequences have been derived. The basics for the proposed method of
solving boundary value problem is the imbedding principle (lectures
28-31).
6
This book presumes that a reader masters the course of func-
tional analysis and optimal control in the amount of the book
«Lectures on optimal control», – Almaty: Kazakh universiteti,
2007. – 278 p. (in russian) by the author.
The author is grateful to the reviewers – academician Kalme-
nov T.Sh., prof. doctor of phys.-math.sc. Dzhenaliev M.T., doctor
of phys.-math.sc. Serovajsky S.Ya.
The author expresses deep gratitude to the associate professors
cand. of phys.-math.sc. Zhunusova Zh.Kh., cand. of phys.-math.sc.
Kabidoldanova A.A. for translating this book into English, and also
grateful to the scientific employee Ayazbaeva A.A. for assisting in
the preparation of the manuscript for publication.
The author would like to thank the staff of the Department of
differential equations and control theory of the faculty of mechanics
and mathematics of al-Farabi Kazakh National University for help
in preparing the manuscript to edition and would be grateful for all
who send their feedback and comments on this book.

Aisagaliev S.

7
Chapter I

INTEGRAL EQUATIONS

Research results on solvability and construction of a solution to integral
equations subject to one-variable functions and multi-variable functions are
presented in this chapter. The first kind Fredholm equation and other different
types of integral equations are considered.

Lecture 1. Integral equations solvable for all right hand sides

A solving controllability problems, optimal control problems for processes
described by ordinary differential equations and boundary value problems of
ordinary differential equations with state variable constraints and integral
constraints involves the study of the integral equation :
b
Ku(cid:32)(cid:179)K(t,(cid:87))u((cid:87))d(cid:87)(cid:32) f(t), t(cid:143)[t ,t ], (1.1)
0 1
a
A special case of (1.1) is the integral equation
b
K w(cid:32)(cid:179)K(t ,(cid:87))w((cid:87))d(cid:87)(cid:32)(cid:69), t (cid:143)[t ,t ], (1.2)
1 * * 0 1
a
where K(t ,(cid:87))(cid:32)K((cid:87))(cid:32)||K ((cid:87))||, i(cid:32)1,n, j(cid:32)1,m is a given matrix with elements
* ij
from L , t (cid:143)[t ,t]is a fixed point, K ((cid:87))(cid:143)L (I ,R1),w((cid:87))(cid:143)L (I ,R1) is unknown
2 * 0 1 ij 2 1 2 1
function, (cid:69)(cid:143)Rn.
Problem 1. Provide a necessary and sufficient condition for existence of a
solution to the integral equation (1.2) for any (cid:69)(cid:143)Rn.
Problem 2. Find a general solution to the integral equation (1.2) for any
(cid:69)(cid:143)Rn.

The following theorem provides a necessary and sufficient condition for
existence of a solution to the integral equation (1.2).
Theorem 1. A necessary and sufficient condition for existence of a solution
to the integral equation (1.2) for any (cid:69)(cid:143)Rn is that the n(cid:117)n matrix
b
C=(cid:179)K((cid:87))K*((cid:87))d(cid:87) (1.3)
a
be positive definite for all a, b, b>a, where the superscript (*) means
transposed.
8
Proof. Sufficiency. Let the matrix C be positive definite. Show that
integral equation (1.2) has a solution for any (cid:69)(cid:143)Rn. Choose w((cid:87))=K*((cid:87))C(cid:16)1(cid:69),
(cid:87)(cid:143)I =[a,b]. Then
1
b
K w(cid:32)(cid:179)K((cid:87))K*((cid:87))d(cid:87)C(cid:16)1(cid:69)(cid:32)(cid:69).
2
a
Consequently in the case C>0, the integral equation (1.2) has at least one
solution w((cid:87))=K*((cid:87))C(cid:16)1(cid:69), (cid:87)(cid:143)I , here (cid:69)(cid:143)Rn is an arbitrary vector. This
1
concludes the sufficiency.
Necessity. Let us assume that the integral equation (1.2) has a solution for
any fixed (cid:69)(cid:143)Rn. Show that the matrix C>0. Since C(cid:116)0, it is sufficient to show
that the matrix C is nonsingular.
Assume the converse. Then the matrix C is singular. Therefore there exists
a vector c(cid:143)Rn, c(cid:122)0 such that c*Cc=0. Define the function v((cid:87))=K*((cid:87))c, (cid:87)(cid:143)I ,
1
v((cid:152))(cid:143)L (I ,Rm). Note that
2 1
b b
(cid:179)v*((cid:87))v((cid:87))d(cid:87)(cid:32)c*(cid:179)K*((cid:87))K((cid:87))d(cid:87)c(cid:32)c*Cc(cid:32)0.
a a
This means that the function v(t)=0, (cid:5)(cid:87), (cid:87)(cid:143)I. Since the integral equation
1
(1.2) has a solution for any (cid:69)(cid:143)Rn, in particular, there exists a function
w((cid:152))(cid:143)L (I ,Rm) such that ((cid:69)=c)
2 1
b
(cid:179)K((cid:87))w((cid:87))d(cid:87)(cid:32)c.
a
Then we have
b b
0(cid:32)(cid:179)v*((cid:87))w((cid:87))d(cid:87)(cid:32)c*(cid:179)K((cid:87))w((cid:87))d(cid:87)(cid:32)c*c.
a a

This contradicts the fact that c(cid:122)0. This finishes the proof of a necessity.
The proof of the theorem is complete.
The following theorem provides a general solution to the integral equation
(3).
Theorem 2. Let the matrix C defined by (1.3) be positive definite. Then for
any (cid:69)(cid:143)Rn
b
w((cid:87))=K*((cid:87))C(cid:16)1(cid:69)(cid:14) p(t)(cid:16)K*((cid:87))C(cid:16)1(cid:179)K((cid:75))p((cid:75))d(cid:75), (cid:87)(cid:143)I =[a,b], (1.4)
1
a
is a general solution to the integral equation (1.2), where p((cid:152))(cid:143)L (I ,Rm) is an
2 1
arbitrary function, (cid:69)(cid:143)Rn is an arbitrary vector.
Proof. Let us introduce the sets
b
W (cid:32){w((cid:152))(cid:143)L (I ,Rm)/ (cid:179)K((cid:87))w((cid:87))d(cid:87)(cid:32)(cid:69)}, (1.5)
2 1
a
9
Q(cid:32){w((cid:152))(cid:143)L (I ,Rm)/ w((cid:87))(cid:32)K*((cid:87))C(cid:16)1(cid:69)(cid:14) p(t)(cid:16)K*((cid:87))C(cid:16)1(cid:117)
2 1
b (1.6)
(cid:117)(cid:179)K((cid:75))p((cid:75))d(cid:75), (cid:5)p((cid:152)), p((cid:152))(cid:143)L (I ,Rm)}.
2 1
a
The set W contains all solutions of the integral equation (1.2) under the
condition C>0. The theorem states that the function w((cid:152))(cid:143)L (I ,Rm) belongs to the
2 1
set W if and only if it is contained in Q, i.e. W =Q. Show that W =Q. In order
to prove this it is sufficient to show that Q(cid:141)W and W (cid:141)Q.
Show that Q(cid:141)W. Indeed, if w((cid:87))(cid:143)Q , then as it follows from (7), the
following equality holds

b b b b
(cid:179)K((cid:87))w((cid:87))d(cid:87)(cid:32)(cid:179)K((cid:87))K*((cid:87))d(cid:87)C(cid:16)1(cid:69)(cid:14)(cid:179)K((cid:87))p((cid:87))d(cid:87)(cid:16)(cid:179)K((cid:87))K*((cid:87))d(cid:87)C(cid:16)1(cid:117)
a a a a
b b b
(cid:117)(cid:179)K((cid:75))p((cid:75))d(cid:75)(cid:32)(cid:69)(cid:14)(cid:179)K((cid:87))p((cid:87))d(cid:87)(cid:16)(cid:179)K((cid:75))p((cid:75))d(cid:75)(cid:32)(cid:69).
a a a

This implies that w((cid:87))(cid:143)W.
Show that W (cid:141)Q. Let w((cid:87))(cid:143)W, i.e. the following equality holds for the
*
function w(t)(cid:143)W (see (1.5)):
*
b
(cid:179)K((cid:87))w ((cid:87))d(cid:87)(cid:32)(cid:69).
*
a
Note that the function p(t)(cid:143)L (I ,Rm) is an arbitrary in the relation (1.4). In
2 1
particular, we can choose p(t)=w((cid:87)), (cid:87)(cid:143)I. Now the function w((cid:87))(cid:143)Q can be
* 1
rewritten in the form

b b
w((cid:87))(cid:32)K*((cid:87))C(cid:16)1(cid:69)(cid:14)w ((cid:87))(cid:16)K*((cid:87))C(cid:16)1(cid:179)K((cid:87))w ((cid:87))d(cid:87)(cid:32)K*(t)C(cid:16)1[(cid:179)K((cid:87))w ((cid:87))d(cid:87)](cid:14)
* * *
a a
b
(cid:14)w ((cid:87))(cid:16)K*((cid:87))C(cid:16)1(cid:179)K((cid:87))w ((cid:87))d(cid:87)(cid:32)w ((cid:87)), (cid:87)(cid:143)I .
* * * 1
a

Consequently w(t)=w(t)(cid:143)Q. This yields that W (cid:141)Q. It follows from the
*
inclusions Q(cid:141)W, W (cid:141)Q that W =Q. The theorem is proved.
The main properties of solutions of the integral equation (1.2):
1. The function w((cid:87)), (cid:87)(cid:143)I can be represented in the form
1
w((cid:87))=w((cid:87))(cid:14)w ((cid:87)), where w((cid:87))=K*((cid:87))C(cid:16)1(cid:69) is a particular solution of the integral
1 2 1
b
equation (1.2), w ((cid:87))= = p(t)(cid:16)K*((cid:87))C(cid:16)1(cid:179)K((cid:75))p((cid:75))d(cid:75), (cid:87)(cid:143)I , is a solution of the
2 1
a
b
homogeneous integral equation (cid:179)K((cid:87))w ((cid:87))d(cid:87)=0, where p(t)(cid:143)L (I ,Rm) is an
2 2 1
a
arbitrary function.

10