Stability of MotionSpringer Science & Business Media, 2012/12/06 - 448 ページ The theory of the stability of motion has gained increasing signifi cance in the last decades as is apparent from the large number of publi cations on the subject. A considerable part of this work is concerned with practical problems, especially problems from the area of controls and servo-mechanisms, and concrete problems from engineering were the ones which first gave the decisin' impetus for the expansion and modern development of stability theory. In comparison with the many single publications, which are num bered in the thousands, the number of books on stability theory, and especially books not \\Titten in Russian, is extraordinarily small. Books which giw the student a complete introduction into the topic and which simultaneously familiarize him with the newer results of the theory and their applications to practical questions are completely lacking. I hope that the book which I hereby present will to some extent do justice to this double task. I haw endeavored to treat stability theory as a mathe matical discipline, to characterize its methods, and to prove its theorems rigorollsly and completely as mathematical theorems. Still I always strove to make reference to applications, to illustrate the arguments with examples, and to stress the interaction between theory and practice. The mathematical preparation of the reader should consist of about two to three years of university mathematics. |
目次
19 | 25 |
56 | 62 |
93 | 96 |
The Principal Theorems of the Direct Method for Autonomous Differential Equations | 102 |
26 Supplements to the Principal Theorems | 108 |
Construction of a Liapunov Function for a Linear Equation | 115 |
Liapunov Functions for Perturbed Linear Equations | 120 |
An Example for Instability | 164 |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 | |
Homogeneous Right Side | 17 |
General Systems of the Second Order | 18 |
Second Order Systems with Homogeneous Right Sides | 19 |
Second Order Linear Systems | 20 |
Perturbed Second Order Linear Systems | 21 |
Conservative Second Order Systems | 22 |
The Direct Method of Liapunov 23 Geometric Interpretation | 23 |
Some Subsidiary Considerations X 1 15 1 5 9 9 | 24 |
Liapunov Functions | 181 |
Applications and Examples I Differential and Difference | 204 |
System Stability and Stability of Invariant Sets | 219 |
The Converse of the Stability Theorems | 225 |
Stability Properties of Ordinary Differential | 257 |
Perturbed Equations | 271 |
Equations with Homogeneous Right Side | 278 |
Linear Differential Equations | 285 |
The Liapunov Expansion Theorem | 330 |
The Critical Cases for Differential Equations | 342 |
Periodic and Almost Periodic Motions | 353 |
Bibliography | 432 |
443 | |
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多く使われている語句
a₁ arbitrarily small assume asymptotically stable autonomous bounded c₁ c₂ characteristic roots coefficients comparison functions components constant defined denote depends difference equation differential equation differential inequality domain of attraction equa equilibrium equilibrium is stable estimate example exists exponentially finite fixed follows function v(x h₁ Hence Hurwitz polynomial hypothesis implies inequality initial values initial vector integral interval Liapunov function linear matrix negative definite nonlinear obtain order numbers origin parameter partial derivatives periodic solution perturbed phase curves polynomial positive definite proof of Theorem resp respect right side saddle point satisfies scalar equation sequence singular point stability behavior sufficiently small t₁ t₂ tends to zero Theorem tion trajectory transfer unit transformation uniformly unperturbed motion unstable variable vector x₁