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AL-FARABI KAZAKH NATIONAL UNIVERSITY
S.A. Aisagaliev
LECTURES ON THE STABILITY OF THE SOLUTION OF AN EQUATION
WITH DIFFERENTIAL INCLUSIONS
Educational manual
Almaty
«Qazaq University»
2020
Lectures on the stability of the solution of an equation with differential inclusions
UDС 517.9
LBC 22.17
A 28
Recommended for publication by the Academic Council
of the Faculty of Mechanics and Mathematics and Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol No.4 dated 19.06.2020)
Reviewers:
Doctor of Physical and Mathematical Sciences, Professor M.T. Jenaliev
Doctor of Physical and Mathematical Sciences, Professor S.Ya. Serovaysky
Aisagaliev S.A.
A 28 Lectures on the stability of the solution of an equation with differential inclusions: educational manual / S.A. Aisagaliev. – Almaty: Qazaq University, 2020. – 229 p.
ISBN 978-601-04-4743-1
The book is written based on lectures, read by the author at the Mechanics and Mathematics Faculty of al-Farabi Kazakh National University. It presents the results of the study of the author on the theory of absolute stability of one-dimensional and multidimensional regulated systems, solution of Aizerman’s problem for the systems with limited resources. The results of fundamental research on the theory of dynamic systems with cylindrical phase space are presented.
The book is intended for undergraduates and Ph.D. students specializing in the specialty «Mathematics».
UDС 517.9
LBC 22.17
ISBN 978-601-04-4743-1 |
© Aisagaliev S.A., 2020 |
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© Al-Farabi KazNU, 2020 |
2
Foreword
CONTENTS
FOREWORD ......................................................................................................................... |
5 |
Chapter І. ABSOLUTE STABILITY AND ISERMAN’S PROBLEM |
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OF ONE-DIMENSIONAL REGULATED SYSTEMS ....................................................... |
7 |
Lecture 1. Absolute stability of equilibrium state in the general case. |
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Problem statement. Nonsingular transformation ...................................................................... |
7 |
Lecture 2. Solution properties in general case ......................................................................... |
12 |
Lecture 3. Improper integrals in general case .......................................................................... |
16 |
Lecture 4. Absolute stability and Iserman’s problem in general case ...................................... |
19 |
Lecture 5. The solution to the model problem for the general case .......................................... |
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Lecture 6. Absolute stability and Iserman’s problem in a simple critical case. |
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Problem statement. Nonsingular transformation ...................................................................... |
30 |
Lecture 7. Solution properties in a simple critical case ............................................................ |
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Lecture 8. Improper integrals and absolute stability in a simple critical case .......................... |
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Lecture 9. Iserman’s problem. A solution of the model problem for the simply critical case ... |
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Lecture 10. Absolute stability and Iserman’s problem in a critical case. |
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Problem statement. Nonsingular transformation ...................................................................... |
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Lecture 11. Solution properties in a critical case ..................................................................... |
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Lecture 12. Improper integrals and absolute stability in a critical case .................................... |
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Lecture 13. Iserman’s problem. The solution to the model problem in a critical case ............. |
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Comments ................................................................................................................................ |
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Literature ................................................................................................................................. |
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Chapter ІІ. ABSOLUTE STABILITY AND ISERMAN’S PROBLEM |
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MULTIDIMENSIONAL REGULATED SYSTEMS .......................................................... |
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Lecture 14. Absolute stability and Iserman’s problem in the general case. |
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Problem statement. Nonsingular transformation ...................................................................... |
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Lecture 15. Solution properties in the general case and improper integrals in general case ..... |
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Lecture 16. Absolute stability and Iserman’s problem in general case .................................... |
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Lecture 17. The solution of model problems in general case ................................................... |
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Lecture 18. Absolute stability and Iserman’s problem of multidimensional systems |
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in a simple critical case. Problem statement. Nonsingular transformation ............................... |
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Lecture 19. Solution properties in the general case and improper integrals in a simple |
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critical case .............................................................................................................................. |
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Lecture 20. Absolute stability and Iserman’s problem of multidimensional systems |
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in a simple critical case ............................................................................................................ |
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Lecture 21. The solution of model problems in a simple critical case .................................... |
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Lecture 22. Absolute stability and Iserman’s problem of multidimensional systems in |
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a critical case. Problem statement. Nonsingular transformation .............................................. |
140 |
Lecture 23. Solution properties and improper integrals in a critical case ................................. |
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Lecture 24. Absolute stability and Iserman’s problem of multidimensional systems |
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in a critical case ....................................................................................................................... |
151 |
Lecture 25. The solution of model problems in a critical case ................................................. |
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Comments ................................................................................................................................ |
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Literature ................................................................................................................................. |
163 |
3
Lectures on the stability of the solution of an equation with differential inclusions |
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Chapter ІІІ. GLOBAL ASYMPTOTIC STABILITY OF THE SOLUTION |
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OF ONE DIMENSIONAL DYNAMIC SYSTEMS WITH CYLINDRICAL |
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PHASE SPACE ...................................................................................................................... |
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Lecture 26. Problem statement. Nonsingular transformation. Properties of solution ............... |
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Lecture 27. Improper integrals ................................................................................................. |
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Lecture 28. Global asymptotic stability ................................................................................... |
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Lecture 29. The problem of phase synchronization ................................................................. |
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Lecture 30. Study on synchronization theory of complex dynamic systems ............................ |
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Lecture 31. Global asymmetric stability of complex dynamic systems ................................... |
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Comments ................................................................................................................................ |
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Literature ................................................................................................................................. |
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Chapter IV. GLOBAL ASYMPTOTIC STABILITY OF MULTIDIMENSIONAL |
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DYNAMIC SYSTEMS WITH CYLINDRICAL PHASE SPACE ..................................... |
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Lecture 32. Problem statement. Nonsingular transformation ................................................... |
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Lecture 33. Solution properties. Improper integrals ................................................................. |
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Lecture 34. Global asymptotic stability of multidimensional dynamic systems I. |
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Auxiliary lemmas ..................................................................................................................... |
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Lecture 35. Global asymptotic stability of multidimensional dynamic systems II. .................. |
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Lecture 36. The solution to the model problem of the phase system ....................................... |
225 |
Comments ................................................................................................................................ |
228 |
Literature ................................................................................................................................. |
228 |
4
FOREWORD
The book is written based on lectures, read by the author at the Mechanics and Mathematics Faculty of al-Farabi Kazakh National University, it presents the results of the study by the author on the theory of absolute stability of one-dimensional and multidimensional regulated systems, solution of Aizerman’s problem for the systems with limited resources. The results of studies on the theory of dynamic systems with cylindrical phase space are presented.
It is intended for undergraduates and Ph.D. students specializing in the specialty «Mathematics». The achievements of science and technology in recent years, the development of atomic energy, the launch of artificial Earth satellites, manned spacecraft, soft landing on the moon, control of nuclear and chemical reactors would not have been possible without the use of modern principles and methods of construction of regulated systems.
Mathematical model of pendulum systems in mechanics, navigation systems in radio engineering, synchronous machines in the energy sector, synchronization in electronics, vibration systems in engineering are dynamic systems with a cylindrical phase space.
In the theory of regulated systems a class of ordinary differential equations, the right part of which contains non-linear functions from a given set, is considered. Such an uncertainty on the right-hand side gives rise to non-uniqueness, which makes it necessary to study the group properties of the solutions of the system. One of such properties is absolute stability of a trivial solution, i.e. properties, at which all the solutions outgoing from any starting point for any non-linear function x from a given set over time tend to equilibrium.
The mathematical model of dynamic systems with cylindrical phase space is a class of ordinary differential equations, the right part of which contains periodic functions from a given set. As the set of states of the system is countable, then for the stability of the trivial solution of the system it is necessary that every solution asymptotically approached some equilibrium state from a countable set. This property of solutions of phase systems is called global stability.
It offers a completely new method of study of absolute stability of nonlinear regulated systems without the involvement of any Lyapunov’s functions and frequency theory, by evaluating improper integrals along the solution of the system.
This book presents new results of basic research on the global asymptotic stability of phase systems. A general theory of global asymptotic stability of phase systems with countable equilibrium state, based on a priori estimation of improper integrals along solutions of the system, was developed.
The first chapter sets out the new method of study of absolute stability of one-di- mensional regulated systems (regulated systems of any order with one nonlinear element) with limited resources based on an assessment of improper integrals along the solution of the system. A nonlinear transformation is given allowing the use of data on
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Lectures on the stability of the solution of an equation with differential inclusions
the properties of nonlinearity. The class of regulated systems, for which Iserman’s problem has a solution, is determined. For this class of regulated systems necessary and sufficient conditions of absolute stability are obtained.
It is shown that for the systems with limited resources, phase variables are limited and are uniformly continuous functions. These properties are used when obtaining conditions of stability and for evaluating improper integrals. The main, simple critical and critical cases are considered separately. The results of the study allow us to significantly expand the area of absolute stability in space of constructive parameters of the system as compared with the known results and, in some cases, we can get necessary and sufficient conditions of absolute stability.
The second chapter contains the results of the study of multidimensional regulated systems (regulated systems of any order with many nonlinearities). For the systems with limited resources estimates of phase variables and identities along the solution of the system are obtained. Assessments of improper integrals along the solution of the system and formulated conditions of absolute stability are found. The class of multidimensional nonlinear regulated systems, for which Iserman’s problem has a solution, is determined. For this class of regulated systems necessary and sufficient conditions for absolute stability are obtained. Key results and effectiveness of the created method are shown on examples. The study of multidimensional systems unlike one-dimensional requires special nonsingular transformation and estimates of improper integrals.
The third chapter contains the results of research on the global asymptotic stability of one-dimensional phase systems with a countable equilibrium state. A completely new approach to the study of the solution properties of dynamic systems with a countable equilibrium state with incomplete information on nonlinearity is proposed. Estimates of improper integrals along the solution of the system for two cases are obtained: when the value of an integral from periodic function in period is equal to zero; when the value of an integral in period is not equal to zero.
In the fourth chapter for multidimensional phase systems conditions of global asymptotic stability are obtained. The dynamic system with cylindrical phase space with a countable equilibrium state by nonsingular transformation is reduced to a special form, composed of two parts. The first part of the differential equations is solvable regarding the components of periodic functions, and the second part does not contain nonlinear functions. A distinctive feature of the proposed method of study of multidimensional phase systems from the known methods is that it is applicable for the systems of any order with any numbers of nonlinear periodic functions and for the study we do not use periodic functions of Lyapunov and frequency theorems. Noteworthy that proposed conditions of global asymptotic stability are easily verifiable compared to the frequency conditions and conditions obtained via Lyapunov’s periodic functions.
The author is grateful to the reviewers, professors Dr. Sc. M.T. Jenaliev, Doctor of Physics and Mathematics S.Ya. Serovaysky for reading the manuscript and making some very useful comments.
The author expresses deep gratitude to Ph.D. K. Imanberdiev, scientific researchers I. Sevryugin and A. Ayazbayeva, for translating this book into English, and for the great help in preparing the manuscript for publication.
The author is grateful to the staff of the Department of Differential Equations and Control Theory of the Mechanics and Mathematics Faculty of Al-Farabi Kazakh National University, for the assistance in preparing the manuscript for publication and will be grateful to everyone who will send his feedback and comments on this book.
S.A. Aisagaliev
6
Chapter 1. Absolute stability and Aizerman’s problem of one-dimensional regulated systems
Chapter I
ABSOLUTE STABILITY AND АIZERMAN'S PROBLEM OF ONE-DIMENSIONAL REGULATED SYSTEMS
There are two approaches to the study of absolute stability of regulated systems: the method of A.I. Lurie [1] and the method of V.M. Popov [2]. The relationship between these methods was established in the works of V.A. Yakubovich and his disciples [3]. Solutions of A.I. Lurie’s equations were obtained based on the second Lyapunov’s method by choosing the Lyapunov’s function in the form of «a quadratic form plus an integral of nonlinearities».
Special conditions for the absolute stability of V.M. Popov are necessary and sufficient conditions for the solvability of matrix inequalities of A.I. Lurie. The complexity of checking the frequency conditions and the need to highlight the region of absolute stability in the space of structural parameters of the system led to creation of algebraic conditions of absolute stability by reducing the frequency conditions to checking positivity of polynomials on the positive semiaxis [4, 5].
New results on new studies of the absolute stability of controlled systems based on the evaluation of improper integrals along the solution of the system are presented in papers [6-9].
In 1949, M.A. Aizerman formulated the following problem [10]: let the solutions of all linear systems of the form x = Ax B , 0 0 , = Sx be asymptotically
stable. Will the solutions of the system
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asymptotic stability in general?
Aizerman's problem was solved by I.G. Malkin, N.P. Yerugin, N.N. Krasovsky. Aiserman's solutions for the case when n > 2 remain unsolved.
This chapter presents the results of the study on absolute stability based on works [6-9, 16] and a solution of the Iserman problem for the case when n > 2.
Lecture 1.
Absolute stability of equilibrium state in the main case.
Statement of the problem. Nonsingular transformation
Statement of the problem. A differential equation of controlled systems in the main case has the form:
x = Ax B ( ), = Sx, x(0) = x0 , t I = [0, ), |
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Lectures on the stability of the solution of an equation with differential inclusions
where A, B, S are matrices of constant orders n n, n 1, |
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A nonlinear automatic control system with equation (1.1) is called a system with a limited resource if a function
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sion (1.2) contains all nonlinearities from the sector [0, 0 ]. Note that system (1.1) with nonlinearity
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x = Ax B Sx = A ( )x, x(0) = x , |
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8
Chapter 1. Absolute stability and Aizerman’s problem of one-dimensional regulated systems
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Definition 1. A trivial solution x(t) 0, t I |
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absolutely stable if: 1) matrices |
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Definition 2. Conditions for absolute stability of the system (1.1), (1.2) are called |
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the relations that connect the constructive parameters of the system |
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Problem 1. Find the condition for the absolute stability of an equilibrium position |
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Definition 3. We assume that in a sector [0, 0 ] Iserman's problem has a solution |
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a trivial solution of the system (1.1), (1.2) is absolutely stable. |
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Problem 2. Find a sector [0, |
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Note that, as follows from [11], the Iserman’s problem does not always have a |
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solution. The problem is to find such a feedback vector S R |
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Nonsingular transformation. Information about properties of nonlinearity is |
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contained in the inclusion ( ) |
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tions of motion (1.1) to explicitly represent the function ( (t)), t I |
through phase |
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variables of the system. As follows from the statement of the problem, it is required for A, A1(μ) to be a Hurwitz matrix. Therefore, in a nonsingular transformation, it must at least be taken into consideration that A is a Hurwitz matrix.
In addition, the nonsingular transformation must be such that in improper integrals related to inclusion ( ) 0 the integrand function can be represented as the
sum of two terms. The first term is a quadratic form reduced to a diagonal form, and the second term is the total differential of the function with respect to time. Such a representation of the integrand ultimately leads to easily verified conditions of absolute stability.
9
Lectures on the stability of the solution of an equation with differential inclusions
Below a nonlinear transformation of the original equation of motion of the adjustable system (1.1) satisfying the specified requirements is given. It should be noted that the existing nonsingular transformation of a linear system from [12; § 5, theorem 2], based on the controllability of pair ( A, B) does not satisfy the above requirements.
The characteristic polynomial of the matrix |
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A has a form: |
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A |= a |
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a a |
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where In is a unit matrix n n, |
ai , i = |
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Hurwitz matrix is easily determined through coefficients |
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from the Hamilton-Cayley theorem, ( A) = 0. Then |
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Lemma 1. Let the row vector = ( 1, 2 , , n ) be that |
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n 2 |
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B = 0, AB = 0, , A |
B = 0, A |
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Then the differential equation from (1.1) can be represented as
fact that A is a
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As follows |
(1.3)
where |
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Proof.
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x, y |
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Consider the first equation from (1.1). Multiplying on the left by
(1.4)
we get
x = Ax B ( ) = Ax, x(0) = x |
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(1.5)
by equality B = 0, where x = x(t) = y1(t), Ax(t) = y2 (t), Differentiating identity (1.5) by t , we get
y2 = x = Ax = A[ Ax B ( )] = A2 x = y3 , y2
t I. |
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(0) = Ax0 , |
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where AB = 0. wing system of
By analogy, taking into account relation (1.3), we obtain the follodifferential equations
y |
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= x = A2 x = |
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y (0) = A2 x , , |
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= An 2 x = An 1x = y |
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t I , |
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10