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Introduction to the Theory and Applications of Functional Differential Equations : Mathematics and Its Applications - V. Kolmanovskii

Introduction to the Theory and Applications of Functional Differential Equations

Mathematics and Its Applications

Hardcover Published: 31st January 1999
ISBN: 9780792355045
Number Of Pages: 648

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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.

Prefacep. 1
Modelling by Functional Differential Equations
Theoretical preliminariesp. 11
Functional differential equations (FDEs)p. 11
Some classes of FDEsp. 11
Solution concept for a FDEp. 12
FDE with retardationp. 14
A little bit of philosophyp. 19
Modelsp. 23
Viscoelasticityp. 23
Aftereffect in mechanicsp. 25
Motion of a particle in a liquidp. 25
Controlled motion of a rigid bodyp. 26
Models of polymer crystallizationp. 28
Stretching of a polymer filamentp. 28
Hereditary phenomena in physicsp. 30
Dynamics of oscillationp. 30
Relativistic dynamicsp. 30
Nuclear reactorsp. 31
Distributed networks (long line with tunnel diode)p. 32
Heat flow in materials with memoryp. 34
Models of lasersp. 35
Neural networkp. 35
Models with delays in technical problemsp. 36
Infeed grinding and cuttingp. 36
Technological delayp. 38
Car chasingp. 39
Ship course stabilizationp. 39
Process of combustion in small rocketsp. 39
Delay-differential equations in engineering applicationsp. 40
Aftereffect in biologyp. 61
Evolution equations of a single speciesp. 61
Interaction of two speciesp. 65
Population dynamics model of N interacting speciesp. 66
Coexistence of competitive micro-organismsp. 67
Control problems in ecologyp. 67
Control problems in microbiology (chemostat models)p. 68
Nicholson blowflies modelp. 70
Helical movement of tips of growing plantsp. 70
Grazing systemp. 70
Aftereffect in medicinep. 71
Mathematical models of the sugar quantity in bloodp. 71
Model of arterial blood pressure regulationp. 72
Cancer chemotherapyp. 74
Mathematical models of learningp. 74
Mathematical models in immunology and epidemiologyp. 75
Model of the human immunodeficiency virus (HIV) epidemicp. 75
Model of survival of red blood cellsp. 78
Vision process in the compound eyep. 78
Human respiratory systemp. 78
Regulation of glucose-insulin systemp. 79
A disease transmission modelp. 79
Aftereffect in economy and other sciencesp. 80
Optimal skill with retarded controlsp. 80
Optimal advertising policiesp. 81
Commodity price fluctuationsp. 82
Model of the fishing processp. 82
River pollution controlp. 83
Control of financial managementp. 83
Theoretical Background of Functional Differential Equations
General theoryp. 87
Introduction. Method of stepsp. 87
Notationp. 87
Cauchy problem for FDEsp. 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 renewalp. 92
Cauchy problem for RDEsp. 94
Basic solvability theoremp. 94
Variantsp. 96
Semigroup relationp. 98
Absolutely continuous solutionsp. 100
RDEs with infinite delayp. 101
Features of the Cauchy problem for RDEsp. 104
Cauchy problem for NDEsp. 107
Smooth solutionsp. 107
NDEs with a functional of integral typep. 111
Application of the steps methodp. 114
Transition to the operator equationp. 116
Hale's form of NDEsp. 119
Differential inclusions of retarded type (RDIs)p. 120
Introductionp. 120
Multimapsp. 121
Solvability of the Cauchy problem for RDIsp. 122
Generalized solutions of RDEs and RDIsp. 126
General linear FDEs with aftereffectp. 131
Cauchy problem for linear RDEsp. 131
Generalizationp. 136
Integral representation for the solution of the Cauchy problem (variation of constants formula)p. 139
Adjoint equation. Periodic solutionsp. 141
Linear NDEsp. 142
Simplest nonautonomous RDEs of the first and second ordersp. 146
Linear autonomous FDEsp. 163
Exponential solutions of linear autonomous RDEsp. 163
Solution of the Cauchy problemp. 167
Example of a showering personp. 170
Linear autonomous NDEsp. 174
Hopf bifurcation of FDEsp. 176
Introductionp. 176
Examplep. 177
General casep. 182
Variantsp. 186
Example of an RDE with constant delay: intraspecific struggle for a common foodp. 187
Example of an RDE with autoregulative delay: combustion in the chamber of a turbojet enginep. 189
Example NDE: auto-oscillation in a long line with tunnel diodp. 191
Stochastic retarded differential equations (SRDEs)p. 191
Initial value problemp. 192
Existence and uniqueness of solutionp. 194
Some characteristics of solutions of linear equationsp. 195
Stability
Stability of retarded differential equationsp. 199
Liapunov's direct methodp. 199
Stability definitionsp. 199
Stability theorems for equations with bounded delayp. 204
Stability of equations with unbounded delayp. 211
Stability of linear nonautonomous RDEsp. 216
Stability of linear periodic RDEsp. 217
Application of comparison theoremsp. 222
Stability in the first approximationp. 223
Case of non-positive derivativep. 224
Linear autonomous RDEsp. 226
Laplace transformationp. 226
Stability conditionsp. 228
Stability investigation methods for linear autonomous RDEsp. 232
Introductionp. 232
Mikhailov criteriump. 232
Scalar n-th order equationsp. 233
Equations with discrete delaysp. 236
Razumikhin's methodp. 247
Introductionp. 247
Guiding functions for systems without delayp. 248
Guiding functionals for RDEsp. 249
Direct application of Liapunov functions to RDEsp. 251
The main idea of B.S. Razumikhinp. 251
"Impossibility of the first breakdown"p. 252
Precize formulationp. 253
Connection between Razumikhin's method and Liapunov functionalsp. 254
Asymptotic stabilityp. 254
Refinement of estimationsp. 255
Examplep. 255
Transformation of RDEsp. 258
Other applications of Razumikhin's methodp. 260
Stability of RDEs with autonomous linear partp. 263
Notationsp. 263
L[superscript 2]-stabilityp. 266
Estimates for the Green functionp. 270
A bound for a region of attractionp. 274
Liapunov functionals for concrete FDEsp. 279
Statement of the problemp. 279
Formal description of the procedurep. 280
Dissipative systemsp. 284
Stabilityp. 284
Exponential contractivityp. 291
Stability in the first approximationp. 293
Exponentially stable linear partp. 294
Smooth coefficientsp. 294
Scalar RDEsp. 296
Scalar equations of n-th orderp. 296
Scalar equations of second orderp. 299
Stability of chemostatp. 304
Riccati type stability conditions of some linear systems with delayp. 307
Introductionp. 307
Special casep. 309
The stability condition for this casep. 309
An application of a form of NDEp. 310
One more stability conditionp. 312
Discrete delay-independent stability conditionsp. 313
Delay-dependent stability conditions for equations with discrete delaysp. 316
The stability conditionp. 317
An application of a form of NDEp. 318
One more stability conditionp. 319
Distributed delayp. 321
The stability conditionp. 322
An application of a form of NDEp. 323
One more stability conditionp. 324
Stability of neutral type functional differential equationsp. 329
Direct Liapunov's methodp. 329
Degenerate Liapunov functionalsp. 329
Stability in a first approximationp. 335
The use of functionals depending on derivativesp. 336
Instability of NDEsp. 337
Stability of linear NDEsp. 343
Linear autonomous NDEsp. 343
Scalar NDEsp. 346
Stability of NDEs with discrete delaysp. 349
The influence of small delays on stabilityp. 351
Linear inhomogeneous NDEsp. 352
Boundedness of derivatives for linear NDEsp. 352
Boundedness of derivatives for nonlinear NDEsp. 353
Linear periodic NDEsp. 355
Application of the direct Liapunov methodp. 359
Description of the procedurep. 359
Scalar NDEs of n-th orderp. 362
Linear NDEsp. 367
The stability conditionp. 367
Another stability conditionp. 368
The summarizing resultp. 370
Nonlinear NDEsp. 371
Stability of the second order NDEsp. 375
An illustrative example for dimension n = 3p. 380
Matrix Riccati equations in stability of NDEsp. 386
Stability of stochastic functional differential equationsp. 387
Statement of the problemp. 387
Definitions of stabilityp. 387
Ito's formulap. 389
Liapunov's direct methodp. 389
Asymptotic stabilityp. 389
Examplesp. 390
Exponential stabilityp. 394
Stability in the first approximationp. 395
Stability under persistent disturbancesp. 396
Boundedness of moments of solutionsp. 397
General conditions for boundedness of momentsp. 397
Scalar SRDEp. 398
Second order SRDEp. 401
Construction of Liapunov functionals for SNDEsp. 402
Statement of the problemp. 402
Description of the procedurep. 404
Scalar SNDEp. 405
Nonlinear examplep. 407
Riccati matrix equations in stability of linear SRDEsp. 415
Boundary Value Problems and Periodic Solutions of Differential Equations
Boundary value problems for functional differential equationsp. 443
Boundary value problems for FDEs of evolutionary typep. 443
Introductionp. 443
Problems with a finite defectp. 443
Halanay's boundary value problemp. 446
Periodic problemp. 448
Boundary value problems for FDEs of nonevolutionary typep. 449
FDEs with unique principal termp. 450
FDEs with nonunique principal termp. 453
Fredholm alternative for periodic solutions of linear FDEsp. 459
Existence of periodic solutionsp. 459
Statement of the problemp. 459
Conditions of the Fredholm alternative validityp. 461
Connection between boundedness and periodicityp. 467
Periodic solution of linear difference equations (DcEs)p. 468
Statement of the problemp. 468
Stationary case. Commensurable shifts of the argumentp. 468
Arbitrary delaysp. 470
Variable coefficients, delays divisible by a periodp. 471
Variable coefficients, delays commensurable with a periodp. 474
Variable coefficients, arbitrary constant delaysp. 478
NDEs with small nonlinearitiesp. 479
Periodic solutions of autonomous FDEs with small parameterp. 481
Generalized periodic solutions of Functional Differential Equationsp. 489
Some prerequisitesp. 489
Conditions of existence of periodic solutionsp. 491
Relation between stability and periodicityp. 494
Application of the direct Liapunov methodp. 494
Stability of periodic solutionsp. 498
Periodic solutions of concrete classes of equationsp. 499
The case of quasilinear deterministic equationp. 499
Linear equationsp. 502
Periodic solutions of the Ito's SFDEsp. 503
Existence of periodic solutionsp. 503
Scalar SRDEsp. 506
Method of Liapunov functionalsp. 510
Uniqueness of periodic solutionsp. 517
Control and Estimation in Hereditary Systems
Problems of control for deterministic FDEsp. 523
The dynamic programming method for deterministic RDEs. Bellman's equationp. 523
Statement of the problemp. 523
Optimality conditionsp. 525
Linear quadratic problemsp. 526
Optimal control synthesisp. 526
Exact solutionp. 528
Systems with delays in the controlp. 529
Effects of delays in regulatorsp. 532
NDEp. 533
Optimal control of bilinear hereditary systemsp. 534
Optimality conditionsp. 534
Construction of the optimal control synthesisp. 535
Model of optimal feedback control for microbial growthp. 538
Control problems with phase constraint formulap. 538
General optimality conditionsp. 538
Equations with discrete delaysp. 540
Necessary optimality conditionsp. 543
Systems with state delaysp. 543
Systems with delays in the controlp. 545
Systems with distributed delaysp. 546
Linear systems with discrete and distributed delaysp. 547
Neutral type systemsp. 549
Adaptive control of FDEsp. 550
Scalar equationsp. 550
Delay identificationp. 553
Multidimensional systemsp. 554
Optimal control of stochastic delay systemsp. 557
Dynamic programming method for controlled stochastic hereditary processesp. 557
The linear quadratic problemp. 558
Bellman functional and optimal controlp. 558
Approximate solutionp. 560
Some generalizationsp. 564
Approximate optimal control for equations with small parametersp. 564
Formal algorithmp. 564
Quasilinear systems with quadratic costp. 566
Another approach to the problem of optimal synthesis controlp. 568
Admissible functionalsp. 568
Quasilinear quadratic problemsp. 569
State estimates of stochastic systems with delayp. 573
Filtering of Gaussian processesp. 573
Problem statementp. 573
Integral representation for the optimal estimatep. 574
The fundamental filtering equationp. 575
Dual optimal control problemp. 578
Particular casesp. 580
Dependence of the error of the optimal estimate on the delayp. 581
Some generalizationsp. 587
Filtering of solutions of Ito's equations with delayp. 589
Problem statementp. 589
Dual control problemp. 590
Minimax filtering in systems with delayp. 592
Statement of the problemp. 592
Approximate solutionp. 595
Bibliographyp. 601
Indexp. 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