Cesari / Hale / Lasalle Dynamical Systems

An International Symposium
1. Auflage 2014
ISBN: 978-1-4832-6203-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 366 Seiten, Web PDF

Dynamical Systems: An International Symposium, Volume 1 contains the proceedings of the International Symposium on Dynamical Systemsheld at Brown University in Providence, Rhode Island, on August 12-16, 1974. The symposium provided a forum for reviewing the theory of dynamical systems in relation to ordinary and functional differential equations, as well as the influence of this approach and the techniques of ordinary differential equations on research concerning certain types of partial differential equations and evolutionary equations in general. Comprised of 29 chapters, this volume begins with an introduction to some aspects of the qualitative theory of differential equations, followed by a discussion on the Lefschetz fixed-point formula. Nonlinear oscillations in the frame of alternative methods are then examined, along with topology and nonlinear boundary value problems. Subsequent chapters focus on bifurcation theory; evolution governed by accretive operators; topological dynamics and its relation to integral equations and non-autonomous systems; and non-controllability of linear time-invariant systems using multiple one-dimensional linear delay feedbacks. The book concludes with a description of sufficient conditions for a relaxed optimal control problem. This monograph will be of interest to students and practitioners in the field of applied mathematics.

Autoren/Hrsg.

Cesari, Lamberto

Hale, Jack K.

Lasalle, Joseph P.

Inhaltsverzeichnis

1;Front Cover;1
2;Dynamical Systems: An International Symposium;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;List of Contributors;12
7;Preface;16
8;SOLOMON LEFSCHETZ: A Memorial Address;18
9;Contents of Volume 2;24
10;PART 1: QUALITATIVE THEORY;30
10.1;Chapter 1. Some Aspects of the Qualitative Theory
of Differential Equations;30
10.1.1;REFERENCES;41
10.2;Chapter 2. The Lefschetz Fixed-Point Formula;
Smoothness and Stability;42
10.2.1;I. The Lefschetz Fixed-Point Formula and Smoothness;42
10.2.2;II. Structural Stability;46
10.2.3;III. The Lefschetz Fixed-Point Formula and Stability;52
10.2.4;REFERENCES;55
11;PART 2: GENERAL THEORY;58
11.1;Chapter 3. Nonlinear Oscillations in the Frame of Alternative
Methods;58
11.1.1;1. Introduction;58
11.1.2;2. An Alternative Scheme;59
11.1.3;3. Reduction to an Alternative Problem by Contraction Maps;61
11.1.4;4. Topological Degree Argument;62
11.1.5;5. Hale's Statement Concerning the Norm of H;63
11.1.6;6. Reduction to an Alternative Problem
by Monotone Operators;65
11.1.7;7. Considerations on the Positivity Property;68
11.1.8;8. A Strongly Nonlinear Oscillation Problem;69
11.1.9;9. Forced Oscillations in Liénard Systems;69
11.1.10;10. Nonlinear Non-Self-Adjoint Problems for Ordinary
Differential Equations. Local Analysis;70
11.1.11;11. Nonlinear, Non-Self-Adjoint Elliptic Problems of Order m
with Nontrivial Kernel—Local Analysis;74
11.1.12;12. The Theorem of Landesman and Lazer;76
11.1.13;REFERENCES;77
11.2;Chapter 4. Topology and Nonlinear Boundary Value Problems;80
11.2.1;1. Introduction;80
11.2.2;2. Coincidence Degree
and a Generalized Continuation Theorem;82
11.2.3;3. Mapping Theorems for Quasi-Bounded Perturbations
of Linear Fredholm Operators;84
11.2.4;4. Periodic Solutions of Ordinary
and Functional Differential Equations;91
11.2.5;5. Boundary Value Problems
for Some Semilinear Elliptic Partial Differential Equations;94
11.2.6;6. Coincidence Degree and Alternative Problems;102
11.2.7;7. Nonlinear Perturbations
of Fredholm Mappings of Nonzero Index;106
11.2.8;REFERENCES;108
11.3;Chapter 5. A Survey of Bifurcation Theory;112
11.3.1;I. Thermal Convection—The Bénard Problem;112
11.3.2;II. Rotating Fluids—The Taylor Problem;113
11.3.3;III. Buckling Phenomena in Elasticity—The Flat Plate;113
11.3.4;REFERENCES;123
11.4;Chapter 6. Generalized Linear Differential Systems
and Associated Boundary Problems;126
11.4.1;1. Introduction;126
11.4.2;2. Basic Properties;127
11.4.3;3. Applications;128
11.4.4;REFERENCES;131
11.5;Chapter 7. Some Stochastic Systems Depending on Small
Parameters;132
11.5.1;1. Introduction;132
11.5.2;2. Nearly Deterministic Systems;133
11.5.3;3. Nearly Linear Systems;134
11.5.4;4. An Application in Population Genetics Theory;137
11.5.5;5. Optimal Stochastic Control;141
11.5.6;REFERENCES;142
11.6;Chapter 8. Bifurcation;144
11.6.1;1. Introduction;144
11.6.2;2. A Selective Iteration Procedure;144
11.6.3;3. Positone Problems;149
11.6.4;4. Bifurcation;153
11.6.5;REFERENCES;157
12;PART 3: EVOLUTIONARY EQUATIONS;160
12.1;Chapter 9. An Introduction to Evolution Governed
by Accretive Operators;160
12.1.1;Introduction;160
12.1.2;1. Orientation;162
12.1.3;2. Accretive Operators, the Cauchy Problem, and Semigroups;164
12.1.4;3. Examples;169
12.1.5;4. Auxiliary Results: Perturbation and Continuous Dependence;175
12.1.6;5. Tangency Conditions;176
12.1.7;6. References and Comments;179
12.1.8;Comments on Appendix 1;184
12.1.9;Appendix 1;185
12.1.10;Appendix 2. Benilan's Uniqueness Theorem;189
12.1.11;REFERENCES;192
12.2;Chapter 10. Evolution Equations in Infinite
Dimensions;196
12.2.1;I. Introduction: The Linear Case;196
12.2.2;II. The Nonlinear Case: Approximation
by Finite-Dimensional Problems;198
12.2.3;III. The Nonlinear Case: Other Methods;203
13;PART 4: FUNCTIONAL DIFFERENTIAL
EQUATIONS;208
13.1;Chapter 11. Functional Differential Equations of Neutral Type;208
13.1.1;1. Introduction;208
13.1.2;2. Notation;209
13.1.3;3. Existence, Uniqueness, Continuous Dependence;210
13.1.4;4. Stable
D-Operators;212
13.1.5;5.
.-Limit Sets;214
13.1.6;6. Representation of Solution Operator;216
13.1.7;7. Linear Equations;218
13.1.8;8. Linear Periodic Systems;219
13.1.9;9. Periodic Solutions;220
13.1.10;REFERENCES;221
13.2;Chapter 12. Functional Differential Equations—Generic Theory;224
13.2.1;Introduction;224
13.2.2;I. Autonomous Functional Differential Equations;224
13.2.3;II. Autonomous Retarded Functional Differential Equations;227
13.2.4;III. Nonautonomous Retarded Functional Differential Equations;237
13.2.5;ACKNOWLEDGMENTS;237
13.2.6;REFERENCES;237
14;PART 5: TOPOLOGICAL DYNAMICAL SYSTEMS;240
14.1;Chapter 13. Stability Theory and Invariance Principles;240
14.1.1;1. Introduction;240
14.1.2;2. Abstract Dynamical Systems;241
14.1.3;3. Associating Dynamical Systems with Nonautonomous
Flows (Processes);246
14.1.4;REFERENCES;249
14.2;Chapter 14. Topological Dynamics and Its Relation to Integral
Equations and Nonautonomous Systems;252
14.2.1;I. Introduction;252
14.2.2;II. Local Dynamical Systems;253
14.2.3;III. Nonautonomous Ordinary Differential Equations;256
14.2.4;IV. Integral Equations and
Semiflows;259
14.2.5;V. Stability and Other Asymptotic Properties;260
14.2.6;VI. Periodic and Almost Periodic Solutions;263
14.2.7;VII. Generic Theory;264
14.2.8;VIII. Linear Theory: Dichotomies and Invariant Splittings;267
14.2.9;IX. Asymptotic Behavior of Linear Equations;270
14.2.10;X. Semigroups and Integrodinferential Equations;274
14.2.11;REFERENCES;275
15;PART 6: PARTIAL DIFFERENTIAL EQUATIONS;280
15.1;Chapter 15. Nonlinear Oscillations under Hyperbolic Systems;280
15.1.1;1. A Boundary Value Problem for Courant-Lax
Hyperbolic Systems;280
15.1.2;2. Extension of Theorem A to Schauder's Systems;283
15.1.3;3. A Counterexample;286
15.1.4;4. Reduction of Hyperbolic Systems in Two Independent
Variables to Schauder's Canonic Form;286
15.1.5;5. A Hyperbolic Problem in Nonlinear Optics;288
15.1.6;REFERENCES;290
15.2;Chapter 16. Liapunov Methods for a One-Dimensional Parabolic
Partial Differential Equation;292
15.2.1;REFERENCES;294
15.3;Chapter 17. Discontinuous Periodic Solutions of an Autonomous
Wave Equation;296
15.3.1;REFERENCES;299
15.4;Chapter 18. Continuous Dependence of Forced Oscillations for
ut = ...(|.u|).u;302
15.4.1;REFERENCE;304
15.5;Chapter 19. Partial Differential Equations and Nonlinear
Hydrodynamic Stability;306
15.5.1;1. Introduction;306
15.5.2;2. Wavy Taylor Vortices;306
15.5.3;3. Eccentric Taylor Vortices;308
15.5.4;4. Wave Systems in Shear Flows;309
15.5.5;ACKNOWLEDGMENT;310
15.5.6;REFERENCES;310
16;PART 7: CONTROL THEORY;312
16.1;Chapter 20. On Normal Control Processes;312
16.1.1;1;312
16.1.2;2;313
16.1.3;3;313
16.1.4;4;314
16.1.5;REFERENCES;315
16.2;Chapter 21. Projection Methods for Hereditary Systems;316
16.2.1;1. Introduction;316
16.2.2;2. Statement of Results;318
16.2.3;REFERENCES;319
16.3;Chapter 22. Lower Bounds for the Extreme Value of a Parabolic
Control Problem;320
16.3.1;1. Statement of the Problem and Basic Results;320
16.3.2;2. Lower Bounds for the Extreme Value
via Dualizing the Problem;321
16.3.3;REFERENCES;324
16.4;Chapter 23. Controllability for Neutral Systems of Linear
Autonomous Differential-Difference Equations;326
16.4.1;REFERENCES;330
16.5;Chapter 24. Local Controllability of a Hyperbolic Partial
Differential Equation;332
16.5.1;Introduction;332
16.5.2;Development of Theory;333
16.5.3;REFERENCES;335
16.6;Chapter 25. A Connection between Optimal Control
and Disconjugacy;336
16.6.1;REFERENCES;339
16.7;Chapter 26. Control for Linear Volterra Systems without Convexity;340
16.7.1;REFERENCES;344
16.8;Chapter 27. Noncontrollability of Linear Time-Invariant Systems Using Multiple One-Dimensional Linear
Delay Feedbacks;346
16.8.1;1. Introduction;346
16.8.2;2. Preliminaries;347
16.8.3;3. Representation for Solution of Eq. (2) for
a Special Classof Initial Functions;348
16.8.4;4. Necessary Conditions of Pointwise Degeneracy;349
16.8.5;5. Consequences of Bi = b1c1 Ton the Necessary Conditions of
Pointwise Degeneracy;349
16.8.6;REFERENCES;351
16.9;Chapter 28. A Perturbation Method for the Solution of an Optimal
Control Problem Involving Bang-Bang Control;354
16.9.1;REFERENCES;358
16.10;Chapter 29. Sufficient Conditions for a Relaxed Optimal
Control Problem;360
16.10.1;Introduction;360
16.10.2;Description of Results;360
16.10.3;REFERENCES;360
17;Author Index;362
18;Subject Index;366