Hardcover
Published: 31st January 1999
ISBN: 9780792355045
Number Of Pages: 648
At the beginning of this century Emil Picard wrote: "Les equations differentielles de la mecanique classique sont telles qu 'il en resulte que le mouvement est determine par la simple connaissance des positions et des vitesses, c 'est-a-dire par l 'etat a un instant donne et a ['instant infiniment voison. Les etats anterieurs n'y intervenant pas, l'heredite y est un vain mot. L 'application de ces equations ou le passe ne se distingue pas de l 'avenir, ou les mouvements sont de nature reversible, sont done inapplicables aux etres vivants". "Nous pouvons rever d'equations fonctionnelles plus compliquees que les equations classiques parce qu 'elles renfermeront en outre des integrates prises entre un temps passe tres eloigne et le temps actuel, qui apporteront la part de l'heredite". (See "La mathematique dans ses rapports avec la physique, Actes du rv congres international des Mathematiciens, Rome, 1908. ) Many years have passed since this publication. These years have seen substantial progress in many aspects of Functional Differential Equations (FDEs ). A distinguishing feature of the FDEs under consideration is that the evolution rate of the proc{lsses described by such equations depends on the past history. The discipline of FDEs has grown tremendously, and publication of literature has increased perhaps twofold over publication in the previous decade. Several new scientific journals have been introduced to absorb this increased productivity. These journals reflect the broadening interests of scientists, with ever greater attention being paid to applications.
Preface | p. 1 |
Modelling by Functional Differential Equations | |
Theoretical preliminaries | p. 11 |
Functional differential equations (FDEs) | p. 11 |
Some classes of FDEs | p. 11 |
Solution concept for a FDE | p. 12 |
FDE with retardation | p. 14 |
A little bit of philosophy | p. 19 |
Models | p. 23 |
Viscoelasticity | p. 23 |
Aftereffect in mechanics | p. 25 |
Motion of a particle in a liquid | p. 25 |
Controlled motion of a rigid body | p. 26 |
Models of polymer crystallization | p. 28 |
Stretching of a polymer filament | p. 28 |
Hereditary phenomena in physics | p. 30 |
Dynamics of oscillation | p. 30 |
Relativistic dynamics | p. 30 |
Nuclear reactors | p. 31 |
Distributed networks (long line with tunnel diode) | p. 32 |
Heat flow in materials with memory | p. 34 |
Models of lasers | p. 35 |
Neural network | p. 35 |
Models with delays in technical problems | p. 36 |
Infeed grinding and cutting | p. 36 |
Technological delay | p. 38 |
Car chasing | p. 39 |
Ship course stabilization | p. 39 |
Process of combustion in small rockets | p. 39 |
Delay-differential equations in engineering applications | p. 40 |
Aftereffect in biology | p. 61 |
Evolution equations of a single species | p. 61 |
Interaction of two species | p. 65 |
Population dynamics model of N interacting species | p. 66 |
Coexistence of competitive micro-organisms | p. 67 |
Control problems in ecology | p. 67 |
Control problems in microbiology (chemostat models) | p. 68 |
Nicholson blowflies model | p. 70 |
Helical movement of tips of growing plants | p. 70 |
Grazing system | p. 70 |
Aftereffect in medicine | p. 71 |
Mathematical models of the sugar quantity in blood | p. 71 |
Model of arterial blood pressure regulation | p. 72 |
Cancer chemotherapy | p. 74 |
Mathematical models of learning | p. 74 |
Mathematical models in immunology and epidemiology | p. 75 |
Model of the human immunodeficiency virus (HIV) epidemic | p. 75 |
Model of survival of red blood cells | p. 78 |
Vision process in the compound eye | p. 78 |
Human respiratory system | p. 78 |
Regulation of glucose-insulin system | p. 79 |
A disease transmission model | p. 79 |
Aftereffect in economy and other sciences | p. 80 |
Optimal skill with retarded controls | p. 80 |
Optimal advertising policies | p. 81 |
Commodity price fluctuations | p. 82 |
Model of the fishing process | p. 82 |
River pollution control | p. 83 |
Control of financial management | p. 83 |
Theoretical Background of Functional Differential Equations | |
General theory | p. 87 |
Introduction. Method of steps | p. 87 |
Notation | p. 87 |
Cauchy problem for FDEs | p. 88 |
Steps method for FDEs of retarded type (RDEs) | p. 89 |
Steps methods for FDEs of neutral type (NDEs) | p. 91 |
Problem for a process with aftereffect renewal | p. 92 |
Cauchy problem for RDEs | p. 94 |
Basic solvability theorem | p. 94 |
Variants | p. 96 |
Semigroup relation | p. 98 |
Absolutely continuous solutions | p. 100 |
RDEs with infinite delay | p. 101 |
Features of the Cauchy problem for RDEs | p. 104 |
Cauchy problem for NDEs | p. 107 |
Smooth solutions | p. 107 |
NDEs with a functional of integral type | p. 111 |
Application of the steps method | p. 114 |
Transition to the operator equation | p. 116 |
Hale's form of NDEs | p. 119 |
Differential inclusions of retarded type (RDIs) | p. 120 |
Introduction | p. 120 |
Multimaps | p. 121 |
Solvability of the Cauchy problem for RDIs | p. 122 |
Generalized solutions of RDEs and RDIs | p. 126 |
General linear FDEs with aftereffect | p. 131 |
Cauchy problem for linear RDEs | p. 131 |
Generalization | p. 136 |
Integral representation for the solution of the Cauchy problem (variation of constants formula) | p. 139 |
Adjoint equation. Periodic solutions | p. 141 |
Linear NDEs | p. 142 |
Simplest nonautonomous RDEs of the first and second orders | p. 146 |
Linear autonomous FDEs | p. 163 |
Exponential solutions of linear autonomous RDEs | p. 163 |
Solution of the Cauchy problem | p. 167 |
Example of a showering person | p. 170 |
Linear autonomous NDEs | p. 174 |
Hopf bifurcation of FDEs | p. 176 |
Introduction | p. 176 |
Example | p. 177 |
General case | p. 182 |
Variants | p. 186 |
Example of an RDE with constant delay: intraspecific struggle for a common food | p. 187 |
Example of an RDE with autoregulative delay: combustion in the chamber of a turbojet engine | p. 189 |
Example NDE: auto-oscillation in a long line with tunnel diod | p. 191 |
Stochastic retarded differential equations (SRDEs) | p. 191 |
Initial value problem | p. 192 |
Existence and uniqueness of solution | p. 194 |
Some characteristics of solutions of linear equations | p. 195 |
Stability | |
Stability of retarded differential equations | p. 199 |
Liapunov's direct method | p. 199 |
Stability definitions | p. 199 |
Stability theorems for equations with bounded delay | p. 204 |
Stability of equations with unbounded delay | p. 211 |
Stability of linear nonautonomous RDEs | p. 216 |
Stability of linear periodic RDEs | p. 217 |
Application of comparison theorems | p. 222 |
Stability in the first approximation | p. 223 |
Case of non-positive derivative | p. 224 |
Linear autonomous RDEs | p. 226 |
Laplace transformation | p. 226 |
Stability conditions | p. 228 |
Stability investigation methods for linear autonomous RDEs | p. 232 |
Introduction | p. 232 |
Mikhailov criterium | p. 232 |
Scalar n-th order equations | p. 233 |
Equations with discrete delays | p. 236 |
Razumikhin's method | p. 247 |
Introduction | p. 247 |
Guiding functions for systems without delay | p. 248 |
Guiding functionals for RDEs | p. 249 |
Direct application of Liapunov functions to RDEs | p. 251 |
The main idea of B.S. Razumikhin | p. 251 |
"Impossibility of the first breakdown" | p. 252 |
Precize formulation | p. 253 |
Connection between Razumikhin's method and Liapunov functionals | p. 254 |
Asymptotic stability | p. 254 |
Refinement of estimations | p. 255 |
Example | p. 255 |
Transformation of RDEs | p. 258 |
Other applications of Razumikhin's method | p. 260 |
Stability of RDEs with autonomous linear part | p. 263 |
Notations | p. 263 |
L[superscript 2]-stability | p. 266 |
Estimates for the Green function | p. 270 |
A bound for a region of attraction | p. 274 |
Liapunov functionals for concrete FDEs | p. 279 |
Statement of the problem | p. 279 |
Formal description of the procedure | p. 280 |
Dissipative systems | p. 284 |
Stability | p. 284 |
Exponential contractivity | p. 291 |
Stability in the first approximation | p. 293 |
Exponentially stable linear part | p. 294 |
Smooth coefficients | p. 294 |
Scalar RDEs | p. 296 |
Scalar equations of n-th order | p. 296 |
Scalar equations of second order | p. 299 |
Stability of chemostat | p. 304 |
Riccati type stability conditions of some linear systems with delay | p. 307 |
Introduction | p. 307 |
Special case | p. 309 |
The stability condition for this case | p. 309 |
An application of a form of NDE | p. 310 |
One more stability condition | p. 312 |
Discrete delay-independent stability conditions | p. 313 |
Delay-dependent stability conditions for equations with discrete delays | p. 316 |
The stability condition | p. 317 |
An application of a form of NDE | p. 318 |
One more stability condition | p. 319 |
Distributed delay | p. 321 |
The stability condition | p. 322 |
An application of a form of NDE | p. 323 |
One more stability condition | p. 324 |
Stability of neutral type functional differential equations | p. 329 |
Direct Liapunov's method | p. 329 |
Degenerate Liapunov functionals | p. 329 |
Stability in a first approximation | p. 335 |
The use of functionals depending on derivatives | p. 336 |
Instability of NDEs | p. 337 |
Stability of linear NDEs | p. 343 |
Linear autonomous NDEs | p. 343 |
Scalar NDEs | p. 346 |
Stability of NDEs with discrete delays | p. 349 |
The influence of small delays on stability | p. 351 |
Linear inhomogeneous NDEs | p. 352 |
Boundedness of derivatives for linear NDEs | p. 352 |
Boundedness of derivatives for nonlinear NDEs | p. 353 |
Linear periodic NDEs | p. 355 |
Application of the direct Liapunov method | p. 359 |
Description of the procedure | p. 359 |
Scalar NDEs of n-th order | p. 362 |
Linear NDEs | p. 367 |
The stability condition | p. 367 |
Another stability condition | p. 368 |
The summarizing result | p. 370 |
Nonlinear NDEs | p. 371 |
Stability of the second order NDEs | p. 375 |
An illustrative example for dimension n = 3 | p. 380 |
Matrix Riccati equations in stability of NDEs | p. 386 |
Stability of stochastic functional differential equations | p. 387 |
Statement of the problem | p. 387 |
Definitions of stability | p. 387 |
Ito's formula | p. 389 |
Liapunov's direct method | p. 389 |
Asymptotic stability | p. 389 |
Examples | p. 390 |
Exponential stability | p. 394 |
Stability in the first approximation | p. 395 |
Stability under persistent disturbances | p. 396 |
Boundedness of moments of solutions | p. 397 |
General conditions for boundedness of moments | p. 397 |
Scalar SRDE | p. 398 |
Second order SRDE | p. 401 |
Construction of Liapunov functionals for SNDEs | p. 402 |
Statement of the problem | p. 402 |
Description of the procedure | p. 404 |
Scalar SNDE | p. 405 |
Nonlinear example | p. 407 |
Riccati matrix equations in stability of linear SRDEs | p. 415 |
Boundary Value Problems and Periodic Solutions of Differential Equations | |
Boundary value problems for functional differential equations | p. 443 |
Boundary value problems for FDEs of evolutionary type | p. 443 |
Introduction | p. 443 |
Problems with a finite defect | p. 443 |
Halanay's boundary value problem | p. 446 |
Periodic problem | p. 448 |
Boundary value problems for FDEs of nonevolutionary type | p. 449 |
FDEs with unique principal term | p. 450 |
FDEs with nonunique principal term | p. 453 |
Fredholm alternative for periodic solutions of linear FDEs | p. 459 |
Existence of periodic solutions | p. 459 |
Statement of the problem | p. 459 |
Conditions of the Fredholm alternative validity | p. 461 |
Connection between boundedness and periodicity | p. 467 |
Periodic solution of linear difference equations (DcEs) | p. 468 |
Statement of the problem | p. 468 |
Stationary case. Commensurable shifts of the argument | p. 468 |
Arbitrary delays | p. 470 |
Variable coefficients, delays divisible by a period | p. 471 |
Variable coefficients, delays commensurable with a period | p. 474 |
Variable coefficients, arbitrary constant delays | p. 478 |
NDEs with small nonlinearities | p. 479 |
Periodic solutions of autonomous FDEs with small parameter | p. 481 |
Generalized periodic solutions of Functional Differential Equations | p. 489 |
Some prerequisites | p. 489 |
Conditions of existence of periodic solutions | p. 491 |
Relation between stability and periodicity | p. 494 |
Application of the direct Liapunov method | p. 494 |
Stability of periodic solutions | p. 498 |
Periodic solutions of concrete classes of equations | p. 499 |
The case of quasilinear deterministic equation | p. 499 |
Linear equations | p. 502 |
Periodic solutions of the Ito's SFDEs | p. 503 |
Existence of periodic solutions | p. 503 |
Scalar SRDEs | p. 506 |
Method of Liapunov functionals | p. 510 |
Uniqueness of periodic solutions | p. 517 |
Control and Estimation in Hereditary Systems | |
Problems of control for deterministic FDEs | p. 523 |
The dynamic programming method for deterministic RDEs. Bellman's equation | p. 523 |
Statement of the problem | p. 523 |
Optimality conditions | p. 525 |
Linear quadratic problems | p. 526 |
Optimal control synthesis | p. 526 |
Exact solution | p. 528 |
Systems with delays in the control | p. 529 |
Effects of delays in regulators | p. 532 |
NDE | p. 533 |
Optimal control of bilinear hereditary systems | p. 534 |
Optimality conditions | p. 534 |
Construction of the optimal control synthesis | p. 535 |
Model of optimal feedback control for microbial growth | p. 538 |
Control problems with phase constraint formula | p. 538 |
General optimality conditions | p. 538 |
Equations with discrete delays | p. 540 |
Necessary optimality conditions | p. 543 |
Systems with state delays | p. 543 |
Systems with delays in the control | p. 545 |
Systems with distributed delays | p. 546 |
Linear systems with discrete and distributed delays | p. 547 |
Neutral type systems | p. 549 |
Adaptive control of FDEs | p. 550 |
Scalar equations | p. 550 |
Delay identification | p. 553 |
Multidimensional systems | p. 554 |
Optimal control of stochastic delay systems | p. 557 |
Dynamic programming method for controlled stochastic hereditary processes | p. 557 |
The linear quadratic problem | p. 558 |
Bellman functional and optimal control | p. 558 |
Approximate solution | p. 560 |
Some generalizations | p. 564 |
Approximate optimal control for equations with small parameters | p. 564 |
Formal algorithm | p. 564 |
Quasilinear systems with quadratic cost | p. 566 |
Another approach to the problem of optimal synthesis control | p. 568 |
Admissible functionals | p. 568 |
Quasilinear quadratic problems | p. 569 |
State estimates of stochastic systems with delay | p. 573 |
Filtering of Gaussian processes | p. 573 |
Problem statement | p. 573 |
Integral representation for the optimal estimate | p. 574 |
The fundamental filtering equation | p. 575 |
Dual optimal control problem | p. 578 |
Particular cases | p. 580 |
Dependence of the error of the optimal estimate on the delay | p. 581 |
Some generalizations | p. 587 |
Filtering of solutions of Ito's equations with delay | p. 589 |
Problem statement | p. 589 |
Dual control problem | p. 590 |
Minimax filtering in systems with delay | p. 592 |
Statement of the problem | p. 592 |
Approximate solution | p. 595 |
Bibliography | p. 601 |
Index | p. 643 |
Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780792355045
ISBN-10: 0792355040
Series: Mathematics and Its Applications
Audience:
Professional
Format:
Hardcover
Language:
English
Number Of Pages: 648
Published: 31st January 1999
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5
x 4.45
Weight (kg): 2.44