книги / Матричное исчисление с приложениями в теории динамических систем
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101.Т и х о н о в А. А. К вопросу об устойчивости движения на конечном интер вале времени//Вестн. ЛГУ.—1968. — № 19.
102.Т и х о н о в А. А. К задаче об устойчивости движения при постоянно дейст вующих возмущениях//Вестн. ЛГУ.—1969.— № 19.—С. 116—122.
103.Т р у б и ц и н Н. Ф. Признак устойчивости линейных систем с переменными коэффициентами//Диф. ур.—1968,—Т. 4, № 11.—С. 1994—2001.
104.X а ч а т р я н С. Т. Некоторые условия равномерной устойчивости движения на заданном интервале времени//Изв. АН АрмССР.—1973.—Т. 26, № 2.—С. 18—27.
105.Х а ч а т р я н С. Т. Об устойчивости движения на заданном интервале времени//Изв. АН АрмССР. Механика.— 1974.—Т. 27, № 3.—С. 47—55.
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Ч е т а ев |
Н. Г Об одной мысли Пуанкаре//Сб. науч. тр. Казанск. авиац. |
ин-та.—1935.—№ 3. |
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107. |
Ч е та ев |
Н. Г. О наименьшем характеристическом члене//ПММ.—1945.— |
Т. 9, № |
3.—С. 193—196. |
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108. |
Ч е т а ев |
Н. Г. Примечания в книге А. М. Ляпунова. Общая задача об ус |
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109. |
Ч е т а ев |
Н. Г. О выборе параметров устойчивой механической систе- |
мы//ПММ.—1951.—Т. 15, № 3.—371—372. |
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110. |
Ч е т а ев |
Н. Г. К вопросу об оценках приближенных интегрирова- |
НИЙ//ПММ.—1957.—Т. 21, N2 3.—С. 419—421; 1958 —№ 6.— С. 63—70. |
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Ш . Ч е т а е в |
Н. Г. О некоторых вопросах, относящихся к задаче об устойчивости |
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неустановившихся движений//ПММ.—1960.— Т. 24, № 1.—С. 6—19; 1960.—№ 11. 112. Ч е т а ев Н. Г. Устойчивость движения.—М.: Наука, 1990.
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114.Ш т о к ал о И. 3. Критерий устойчивости и неустойчивости решений ли нейных дифференциальных уравнений в квазипериодическими коэффициентами//Математический сборник— 1946.—Т. 19 (61), № 2.
115.B a i l e y F. N. The application of Lyapunov’s Second Method to Interconected Systems//.!. Soc. Industr. and Appl. Math. (USA).—1965 (1966).—A3, 3.—P. 443—462.
116.B o g u s z W. Technical Stability of Nonlinear Systems//Strojnicky Cos.—1968.— V. 19, № 5.—P. 529—533.
117.B o g u s z W. Technical Stability of Vibrations Excited by Shocks//Zesz. Nauk. Acad. Gom-Hutn.—1969.— V.204.— P. 7—11.
118. B o g u s z W. Technical Stability of Nonlinear Systems Disturbed Randomly and by Constantly acting Perturbations//Tp. 5-й Межд. конф. по нелин. колеб., 1969.—Ки
ев: Наукова думка, 1970.—С. 108—113. |
drgan. |
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119. B o g u s z |
W- Rozwoj i zastosowanie statecznosci techniczney//Zag. |
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nielin.—1975.— V. 16—P. 185—193. |
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120. G r u i j d |
L. T. On practical Stability//Int. J. Contr.—1973.—V. 17, |
№ 4.— |
P.881—887.
121.G r u i j d L. T. Non-Lyapunov Stability of Large-Scale Systems on Time-Varying
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122. |
G u n d e r s o n R. W. On Stability over Afinite Interval//IEEE Trans. Automat. |
Contr. —1967.—AC-12, № ‘5.—P 634—635. |
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123. |
H al lam T. G., Komkov V. Application af Liapunov’s Functions to Finite |
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126. L a k s h m i k a n t h a m V., Lee la S. Differential and Integral Inegualltyes. Theory and Appl. Vol. 1. —New York: Acad. Press, 1969.
127. La S a l l e |
J., L e f s c h e t z |
S. S. Stability by Liapunov’s direct Method.—New |
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York—London: Acad. Press, 1961. (Пер. с англ.: Л а - С а л л ь Ж., Л е ф ш е ц |
С. Ис |
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следование устойчивости прямым методом Ляпунова.—М.: Мир, 1964, 1965.) |
der |
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128. M a g n u s |
К. Zur |
Entwicklung des Stabilitatsbegriffes in |
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Mechanik//Naturwissenschaften.—1969.—B. 46, № 21.— S. 590—595.
129.M a s s e r a J. L. Contributions to Stability Theoiy//Ann. Math.—1956.—V. 64,
№1. —P. 182—206.
130.M a s s e r a J. L. Contributions to stability Theory//Ann. Math.— 1956.—V. 64,
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131. M o h a n a |
R ao |
Rama. Upper and Lower Bounds of the Norm of Solutions of |
Nonlinear Volterra Integral Equations//Proc. Nat. Acad. Sci. India.—1963. —V. 33. —P. |
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263—266. |
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132. M b r t o n |
M. D. A macroscopic Condition for Stability//AIChE Journal.— |
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1970.—V. 16, № |
4.—P. 670—672. |
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133. O ls z e w s k i J. |
On the practical Stability of the Triangular Points in the |
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Restricted Three-Body Problem//Celest. Mech.—1971.—V, 4, № 1. —P. 3—14.
134.R e i s s i n g R. Stabilitat einer Bewegung Verschiedene Begriffsdefinitioncn und ihre praktische Bedeutung//Forsch und Forlschr. —1959.—B. 33, № 2. —S. 38—41.
135.S t o k e r J. J. On the Stability of Mechanical Systems//Communs. Pure and Appl. Math. —1955. —V. 8, № 1. —P. 133—141.
136. |
T s o k o s |
С. P., Le e Ia |
S. Control |
Systems and Finite Time Stability//Notes |
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Commun. Mat.—1969,—N° 22.—P. 1—11. |
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137. |
T s o k o s |
C. O . ,M o h a n a R ao |
Rama. Finite Time Stability of Control Systems |
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and Integral Inegualities//Bul. Inst. |
Politechn. Lasi.—1969.—V. 15 (19), № 1—2.— |
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P. 105—112. |
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138. |
W a t s o n |
I. W., S t u b b e r u d A.R. Stability of Systems Operating in a Finite |
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Time Interval//IEEE Trans. Automat. |
Contr.—1967.—AC-12, № 1.—P. 116. |
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139. |
W a z e w s k i |
T. Sur |
la |
Limitation |
des Integrates des Systems d’Equations |
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Diffdrentieles Lindaires Ordinaires//Stud. Math.—1948.—V. 10.—P. 48—59. |
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140. |
W e i s s |
L., |
I n f a n t e |
E. F. On the stability of Systems Defined over a Finite |
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Time Interval//Proc. Nat. Acad. Sci. |
USA.—1965.—V. 54, № 1.—P. 44—48. |
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141. W e i s s |
L., I n f a n t e |
E. F. Finite Time Stability Under Perturbing Forces and |
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on Product Spaces//IEEE. Trans. Automat. Contr.—1967.—AC-12, № 1.—P .5 4 —59.
142.А б г а р я н К. А. Асимптотическое расщепление уравнений регулируемого процесса при медленном изменении параметра регулируемого объекта и системы регулирования//ДАН СССР.— 1964.—Т. 158, № 3.
143.А б г а р я н К. А. Асимптотическое расщепление уравнений линейной систе мы автоматического управления//ДАН СССР.— 1966.—Т. 166, № 2.—С. 301—304.
144.А б г а р я н К. А. Об асимптотическом интегрировании уравнений регулиру емого процесса//ДАН СССР.—1967.—Т. 177, № 4.
145.А б г а р я н К. А. Канонические преобразования уравнений нестационарной системы регулирования//А&Т.—1969.—№ 2.
146.А б г а р я н К. А. Асимптотическое преобразование уравнений нестационар ной линейной системы и критерии устойчивости//Теория и проектирование систем автоматического управления летательными аппаратами/Под ред. Б. Н. Петрова.— М.: Машиностроение, 1970.
147.А б г а р я н К. А. К теории нестационарных систем автоматического управления//ДАН СССР.—1970.—Т. 194, № 2.
148.А б г а р я н К. А. Об устойчивости движения на заданном промежутке времени//Изв. АН АрмССР. Механика.—1972.—Т. 25, № 5.
149.А н д р е е в Ю. Н. Управление конечномерными линейными объектами.— М.: Наука, 1976.
150. А б г а р я н К. А., Х р у с т е л е в М. М., Ж и р н о в а Э. Б. Управляемость и наблюдаемость линейных систем. Учебное пособие по курсу «Теория управле ния»,—М : МАИ, 1977.
151. Б е с е к е р с к и й В. А., Попов Е. П. Теория систем автоматического регу лирования.—М.: Наука, 1966.
152.Б о р с к и й В. О свойствах импульсных переходных функций систем с пере менными параметрами//А&Т.—1959.—Т. 20, № 7.
153.Д ’ А н ж е л о Г. Линейные системы с переменными параметрами.—М.: Ма шиностроение, 1972.
154.З у б о в В. И. Математические методы исследования систем автоматического регулирования//Л.: Судпромгиз, 1959.
155.Л е т о в А. М. Устойчивость нелинейных регулируемых систем.—М.: Физматгиз, 1962.
156. Ли Э. Б., М а р к у с |
Л. Основы теории оптимального управления.—М.: На |
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ука, 1972. |
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157. |
Л ур ье |
А. Е. Некоторые нелинейные задачи теории автоматического peiy- |
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лирования.—М.: Гостехиздат, 1951. |
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158. |
Основы |
автоматического управления/Под ред. Пугачева В. С.—М.: Наука, |
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1968. |
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159. |
Р а п о п о р т И. М. |
Об устойчивости ре!улируемых процессов//ДАН |
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СССР.—1964.—Т. 158, № 2.
160.Р о й т е н б е р г Я. Н. Автоматическое управление.—М.: Наука, 1971.
161.Современная теория систем управления/ Под ред. Леондеса.—М.: Наука,
1970.
162. |
С о л од о в А. В., П е т р о в Ф . С . Линейные автоматические системы с пе |
ременными параметрами.—М.: Наука, 1971. |
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163. |
Ф е л ь д б а у м А. А., Б у т к о в с к и й А. Г. Методы теории автоматического |
управления.—М.: Наука, 1971. |
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164. |
Ко рн Г., К орн Т. Справочник по математике для научных работников и |
инженеров.—М.: Наука, 1968.
K a r l e n A. A b g a r y a n . «The matrix calculous with applications in the theory of dynamic systems».
Moscow, Fizmatlit Publishing Company, 1994.
The author: Prof. Karlen A. Abgaryan, Corr. Member of the USSR Acad. Sci., Dr. Sci. (Techn.) Moscow Aviation Institute, member of the Soviet National Committee on Theoretical and Applied Mechanics, member of the Soviet National Committee CODATA, a renouned specialist in mechanics, control theory, theory of the stability of motion at the finite interval of time.
The book: This book is a result of the radical overhaul (additions, amendments) of the previous author’s monograph «Matrix and asymptotic methods in the theory of linear systems» (Moscow, Nauka, SSSR, 1973). The chapters covering the theory of matrix, asymptotic decomposition integral and integro-differential systems transformations have been remained mainly intact (Chapter 1 to 9). This part includes now an additional chapter (Chapter 11) on singularly perturbed multi-tempo systems. Chapters 14, 15 and 16 on the stability of processes have been completely reworked. The book contains a new full section of 6 chapters covering problems of stability. It is a first rendering of author's research results obtained within the last decade. A large new section of the book is on the control theory. It includes some chapters (10...11) from the above cited book «The matrix methods» as well as material from some research papers.
The final part of the book is a large bibliography (167 items) including titles of both Soviet and foreign authors.
The author seeks to distribute and expose his ideas in an optimal way. Firstly, the reader would not have to do any serious preparatory work in matrix calculus, special fields of mechanics and in the control theory. Secondly, the reading of the book would not require the study of other sources, since all the information needed to comprehend any chapter is there in the foregoing part of the book. Thus given consistent and concentrated study of the chapters one by one, the reader will not have to turn to any other source. In our view this manner of exposition will make the book accessible for a wide range of specialists in the matematics, mechanics and control theory.
Audience: The manual is intended for senior University students majoring in mathematics, mechanics and physics, for post-graduate students of technical colleges as well as for experts in mathematics, mechanics and automatic control.
Contents:
Foreword/The Matrix/Vectors, Vectors Space. Linear Operators and Matrix/Linear Operators in n-dimensional Vectors Space/The Decomposition of Space into Invariant Subspaces. The Normal Matrix Forms/The Transformation of the Matrix to-a quasiDiagonai Form and its Decomposition into Components/On Square and Hermitian Forms/The General Properties of Linear Differential Equations and Systems of Equations/The Vector-Matrix Unear Equation with Constant Matrix Coefficients/The Asymptotic Decomposition of the Vector-Matrix Equation with Variable Matrix Coefficients/The Asymptotic Decomposition of the Vector-Matrix Equation with Variable Coeffecients (The second method)/The Decomposition of a Singularly Perturbed multiTempo System/Some Properties of Linear Differential Equation Systems/The Evaluation of the Solutions of a Linear Homogeneous Systems of Differential Equations/The Development of the Motion Stability Concept. A Short Historical Survey/The Concept of Ад-Stability. The Statement of a Task/The General Theorems of Ад-Stability/The
Construction of the Lapunov Functions/ Ад-Stability on an Unlimited Time Interval/ АдStability at the Finite Time Interval and Technical Stability/Controlled Systems/The Con trollability of Linear Systems/The Observability of Linear Systems/The Dynamic Characteristics of Unear Systems/The Approximate Integration of Equations of the Con trolled Process/Some Canonical Forms of Linear Process Equations/The Asymptotic Decomposition of the Equations of a multi-Dimensional non-Stationary Linear System of Optional Automatic Control/The Synthesis of Linear Controls of a multi-Dimensional Unear non-Stationary System with a Diagonal Matrix/Literature/Index.
CONTENTS |
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Foreword.......................................................................................................... |
11 |
S e c t i o n I . T h e m a t r i x t h e o r y ...................................... |
13 |
Chapter 1. Matrices........................................................................................ |
13 |
1.1. Basic definitions and designations................................................... |
13 |
1.2. Addition of matrices and multiplication of a matrix by a number...... |
15 |
1.3. Multiplication of square matrices..................................................... |
16 |
1.4. The determinant ofa matrix product................................................ |
18 |
1.5. Transposition of a matrix and transition to the conjugate matrix....... |
20 |
1.6. The adjoint matrix........................................................................... |
21 |
1.7. The reciprocal matrix...................................................................... |
22 |
1.8. The block matrix............................................................................. |
23 |
1.9. Linear transformations and matrices................................................ |
27 |
1.10. Tasks and exercises....................................................................... |
30 |
Chapter 2. Vectors, vector spaces, linear operators and matrices................. |
33 |
2.1. Vectors and vector spaces................................................................ |
33 |
2.2. Linear dependence ofvectors........................................................... |
34 |
2.3. The dimension and the basis of a vector space.................................. |
36 |
2.4. Isomorphism ofл-dimensional spaces.............................................. |
38 |
2.5. Vector space subspaces.................................................................... |
39 |
2.6. Linear operators in vector spaces..................................................... |
40 |
2.7. The matrix as a linear operator in numerical spaces......................... |
42 |
2.8. The Sylvester inequalities................................................................ |
46 |
2.9. Rectangular factorisation of a matrix............................................... |
48 |
2.10. Tasks and exercises....................................................................... |
49 |
Chapter 3. Linear operators in //-dimensional vector space........................... |
51 |
3.1. The ring of linear operators............................................................. |
51 |
3.2. Matrices of a linear operator in various bases.................................. |
52 |
3.3. The inverse operator........................................................................ |
53 |
3.4. Eigenvectors and eigenvalues of a linear operator and a square |
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matrix............................................................................................... |
54 |
3.5. Linear operators and matrices of a simple structure.......................... |
56 |
3.6. Decomposition of/7-dimensional space............................................. |
58 |
3.7. Projective operators and matrices...................................................... |
59 |
Chapter 4. Decomposition of a space into invariant subspaces. The normal |
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forms of matrices................................................................... |
51 |
4.1. The minimal polynomials of a vector, a vector space, a matrix......... |
65 |
4.2. Invariant subspaces of a vector space................................................ |
68 |
4.3. Decomposition of a vector space into invariant subspaces with |
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mutually disjoint minimal polynomials.............................................. |
70 |
4.4. Comparisons. The space of comparable vector classes...................... |
73 |
4.5. Cyclical subspaccs of a vector space................................................... |
77 |
4.6. The normal form of a matrix............................................................. |
81 |
4.6.1. The natural normal forms (82). 4.6.2. The Jordan normal |
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form (83)
4.7. Invariant polynomials. Uniqueness of the normal form of a linear
operator.............................................................................................. |
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85 |
Chapter 5. Transformation of a matrix to the quasi-diagonal form and its |
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decomposition into components....................................................... |
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90 |
5.1. The defect of a matrix polynomial..................................................... |
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90 |
5.2. The Hamilton-Caylcy theorem.......................................................... |
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92 |
5.3. Construction of a matrix, transforming a square matrix to the |
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quasi-diagonal form.......................................................... |
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93 |
5.4. Eigenvalues of a submatrix of the transformed quasi-diagonal |
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matrix................................................................................................ |
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96 |
5.5. The general form of the transformation matrix.................................. |
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97 |
5.6. Construction of the Jordan form of a matrix...................................... |
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99 |
5.7. The case of matrix of a simple structure............................................. |
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103 |
5.8. Decomposition of a square matrix into components.......................... |
105 |
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5.9. Matrices of the orthogonal projection............................................... |
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5.10. On reduction of a special matrix to the quasi-diagonal form and |
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its decomposition into components..................................................... |
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115 |
5.11. Analytical functions of matrices...................................................... |
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117 |
Chapter 6, Quadratic and Hcrmitian forms.................................................... |
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120 |
6.1. Matrisation of a vector space............................................................ |
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120 |
6.2. Orthonormalised bases in unitary and Euclidean spaces.................... |
124 |
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6.3. Linear operators in the unitary space................................................. |
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125 |
6.3.1. The conjugated operator (125). 6.3.2. |
Eigenvectors |
and |
eigenvalues of the Hermilian operator (127). 6.3.3. The unitary |
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operator (130). 6.3.4. Transformation of the Hermitian matrix to |
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the diagonal form with the help of the unitary matrix (131). |
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6.4. Linear operators in the Euclidean space............................................. |
132 |
6.4.1.The transposed operator. The symmetric operator. (132).
6.4.2.The orthogonal operator (135). 6.4.3. Transformation of a symmetric matrix to the diagonal form with the help of the orthogonal matrix (136).
6.5. Quadratic forms................................................................................ |
137 |
6.5.1. The variables substitution (138). 6.5.2. The law of inertia |
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(138). 6.5.3. Reduction of a quadratic form to the principal axes |
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(140). |
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6.6. Hcrmitian forms............................................................................... |
142 |
6.6.1.The variables substitution (143). 6.6.2. The law of inertia
(143). 6.5.3. Reduction of an Hcrmitian form to the principal axes
(144).
C hapter |
7. The general properties of |
linear differential equations and |
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equation system s................................................................................................ |
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146 |
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7.1. The derivative and the integral of a matrix...................................... |
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146 |
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7.2. The vector-matrix notation of linear differential equations............... |
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147 |
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7.3. The matrix norm............ |
,............................................................... 153 |
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7.4. The matrix series............................................................................. |
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154 |
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7.5. The theorem on existence and uniqueness of solution of a |
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homogeneous vector-matrix equation................................................. |
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155 |
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7.5.1. Existence of the solution (155). 7.5.2. Uniqueness of the |
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solution (157). |
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7.6. The fundamental matrix of a system................................................. |
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159 |
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7.6.1. Solution of a matrix differential equation (159). 7.6.2. The |
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Ostrogradskii-Liouville formula (160). 7.6.3. |
The fundamental |
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matrix (161). |
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7.7. The matriciant................................................................................. |
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162 |
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7.8. The conjugate equation.................................................................... |
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163 |
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7.9. Non-homogeneous vector-matrix equations...................................... |
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164 |
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7.9.1. The Lagrange method of arbitrary constants variation (164). |
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7.9.2. Another method (165). |
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7.10. Solution of a certain vector-matrix equation................................... |
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166 |
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C hapter |
8. A vector-matrix linear |
equation with |
constant |
m atrix |
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coefficients............................................................................................................ |
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168 |
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8.1. The exponential of a matrix.............................................................. |
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168 |
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8.2. Solution of an equation in the form of exponential............................ |
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170 |
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8.3. The Euler method............................................................................ |
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170 |
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8.4. Solution of an equation by means ofthe Laplace transformation....... |
172 |
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8.5. Integration of an equation by variables substitution........................... |
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175 |
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8.6. Decomposition of a system into independent subsystems of smaller |
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order. 177 |
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8.6.1. Transformation of a square matrix of a system to the quasi |
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diagonal form (177). 8.6.2. Decomposition of a system (178). 8.6.3. |
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The case of matrix of a simple structure (179). 8.6.4. Complete |
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decomposition in the general case (179). 8.6.5. Decomposition of |
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the conjugated system. Biorthogonality(181). |
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8.7. The theory of disturbances.............................................................. |
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182 |
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8.7.1. The method of successive approximations for a homogeneous |
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system (182). 8.7.2. On solution of a certain matrix equation. The |
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principal lemma (182). |
8.7.3. |
The asymptotic method |
for |
a |
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homogeneous equation (184). 8.7.4. The asymptotic method for a non-homogeneous equation (187).
C h ap ter 9. T he asym ptotic decomposition of a vector-m atrix |
equation |
w ith variable m atrix coefficients (the first m ethod)................................... |
189 |
9.1. Differentiability of a matrix, transforming a square matrix to the |
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quasi-diagonal form .......................................................................................... |
189 |
9.2. Construction of a formal process for decomposition of a vector- |
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matrix equation.................................................................................................. |
194 |
9.3. Some properties of the matrices, participating in the formal |
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process of decomposition................................................................................ |
201 |
9.4. Recursion correlations in some particular cases...................................... |
206 |
9.5. Conditions for preservation of the equation solution norm in case |
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of variables substitution.................................................................................. |
207 |
9.6. The case of complete decomposition of the system ................................ |
211 |
9.7. Equations with constant matrix coefficients.............................................. |
211 |
9.8. Decomposition of the conjugated vector-matrix equation...................... |
213 |
9.9. The approximate solution of the system ..................................................... |
221 |
9.10. The asymptotic character of the approximate solution........................ |
223 |
C hapter 10. The asymptotic decomposition of a vector-m atrix |
equation |
with variable coefficients (the second m ethod)............................................ |
233 |
10.1. Two lemmas................................................................................................... |
233 |
10.2. Transformation of a homogeneous linear differential system with |
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constant coefficients to a system of independent equations.................... |
236 |
10.2.1. Transformation to a decomposed second-order equation |
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system (236). 10.2.2. The general case (238). 10.2.3. The case of |
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matrix of a simple structure (240) |
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10.3. Transformation of a homogeneous non-stationary system of |
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differential equations to a decomposed system .......................................... |
241 |
10.3.1.Transformation to a decomposed second-order system (241).
10.3.2.The general case (246). 10.3.3. The case of simple eigenvalues of the system matrix (251). 10.3.4. Approximate solution of the system (252)
10.4. Decomposition of a non-homogencous system ...................................... |
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252 |
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10.5. The asymptotic estimates of the approximate solution of the |
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system |
.......................................................................................................... |
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260 |
C h ap ter 11. Decomposition of a singularly perturbed m ulti-tempo system ....262 |
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11.1. The task form ulation................................................................................... |
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262 |
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11.2. A method of successive separation of mono-tempo subsystems |
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from a multi-tempo system............................................................................. |
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265 |
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11.3. An algorithm for formal decomposition of the system (11.1.3)......... |
267 |
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11.3.1. Construction of К ^ and AjJJ (270). 11,3.2. Construction |
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of |
and |
(273). 11.3.3. Construction of |
(f> ^) |
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11.4. Approximate solution of the equation (11.3.1)...................................... |
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278 |
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11.4.1. The asymptotic estimates of the approximate solution (279) |
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12.1. The problem of transformation of a linear differential equations |
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system into another one with a pre-assigned m atrix............................... |
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287 |
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12.2. The matrix differential equation of kinematic |
similarity. |
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Kinematically similar matrices and kinematically equivalent |
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systems............................................................................................................ |
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289 |
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12.2.1. The general solution of a matrix differential equation of |
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kinematic similarity (289). 12.2.2. Solution of the problem of |
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transformation of a linear system into another one with a pre |
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assigned matrix (290). 12.2.3. Conclusions (291). |
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C hapter 13. |
Estimates of |
solutions of a |
linear homogeneous |
system |
of |
differential equations........................................................................ |
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292 |
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13.1. The Vazhcvsky inequalities..................................................................... |
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292 |
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13.2. A more precise definition of the Vazhevsky inequalities.................... |
293 |
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13.3. Evaluation of co-ordinates and norms of solutions of linear |
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homogeneous systems................................................................................... |
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296 |
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13.3.1. Evaluation of co-ordinates of solutions |
of linear |
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homogeneous systems (296). 13.3.2. Evaluation of |
norms |
of |
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solutions of linear homogeneous systems (302). |
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13.4. The Tchetayev inequalities....................................................................... |
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303 |
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13.5. On selection of the function V (t ,x ) in the Tchetayev inequalities....306 |
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13.6. Precise evaluation by means of the Tchetayev inequalities................ |
307 |
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13.7. The bundle of solutions of a linear system............................................. |
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311 |
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S e c t i o n |
I I I . T h e |
t h e o r y |
o f m o t i o n |
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s t a b i l i t y |
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C hapter 14. The development of the concept of motion stability. A brief |
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survey |
..................................................................................................................... |
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312 |
14.1. Stability of motion according to A.M.Lyapunov.................................. |
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313 |
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14.1.1.The definition of motion stability according to Lyapunov
(313). 14.1.2. Methods of solution of stability tasks (315).
14.2. The Lyapunov general theorems of the direct method......................... |
317 |
14.3. On the fundamental concepts of the Lyapunov theory......................... |
318 |
14.4. On the ways of further development of the motion stability |
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concept............................................................................................................. |
320 |
14.5. The general formulation of the task of technical stability................... |
321 |
14.5.1.Perturbed and non-perturbed motion. The general form of perturbed motion equations in the vector-matrix notation (321).
14.5.2.Linearisation of perturbed motion equations. The first approximation equations (323). 14.5.3. The main definitions of the concept of technical stability (324).
14.6. Stability |
of |
motion at |
a |
finite time interval |
according |
to |
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G.V.Kamenkov................................................................................................. |
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326 |
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14.6.1. The task formulation (326). 14.6.2. Geometrical semantics |
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of the concept of stability at a finite time interval (327). 14.6.3. The |
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main theorems on stability and unstability of motion at a finite time |
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interval (328). |
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C h ap ter 15. The concept of .KT^-stability. The task form ulation........................ |
329 |
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15.1. Construction of a domain of admissible states and the definition |
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of stability |
.......................................................................................................... |
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329 |
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15.2. The geometrical semantics of the concept o f K £ -stability................. |
331 |
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15.2.1. Definition of the area (15.2.1) diameter (333). |
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15.3. On the method of studying of K ®-stability........................................... |
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336 |
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C hapter 16. The m ain theorem s of |
-stability .................................................... |
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338 |
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16.1. On the skeletal decomposition of m atrices............................................. |
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338 |
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16.2. The main theorems of K ®-stability and K ®-unstability................... |
342 |
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C hapter 17. Construction of the Lyapunov functions............................................ |
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348 |
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17.1. Diagonalisation of a linear system .......................................................... |
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348 |
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17.2. The bundle of solutions of a linear system. |
Conditions |
for |
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stability of a linear system............................................................................... |
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349 |
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17.3. Criteria of stability of the trivial solution of a linear system .............. |
351 |
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17.4. Non-linear systems...................................................................................... |
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353 |
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17.5. One more method to construct the Lyapunov function........................ |
355 |
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C h ap ter 18. .K^-stability at an infinite tim e interval............................................. |
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358 |
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18.1. Introductory notes......................................................................................... |
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358 |
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18.2. Stability according to Lyapunov and jfi^-stability at an infinite |
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time interval..................................................................................................... |
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360 |
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18.3. |
The |
main |
theorems |
of |
K ®-stability and |
AT^-unstability |
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(A |
= ['o ,° ° ]).................................................................................................... |
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365 |
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18.4. Stability of autonomous system s............................................................... |
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371 |
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18.4.1. A linear homogeneous system (371). 18.4.2. Criteria of |
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stability of a non-linear system by the first approximation (375). |
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18.5.Equivalence, reducibility and stability of non-autonomous
system s............................................................................................................... |
377 |
18.5.1. Equivalent systems (377). |
18.5.2. On the problem of |
reducibility of linear systems (381). 18.5.3. On stability of linear |
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systems (382). |
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