Continuous and Distributed Systems II

Preface......Page 7
 Editors......Page 8
Acknowledgments......Page 9
Contents......Page 10
Contributors......Page 18
Part I Applied Methods of Modern Algebraand Analysis......Page 21
 1.1 Introduction......Page 22
 1.2 Main Results......Page 24
 1.3 Proofs......Page 25
 References......Page 29
 2.1 ``Atoms'' and Morse Functions......Page 30
 2.2 Complicated Atoms and Molecules......Page 34
 2.3 Topology of Integrable Hamiltonian Systems  with Two Degrees of Freedom......Page 35
 2.4 Geodesic Flows with Potential on the Surfaces  of Revolution......Page 39
 2.5 The Case of Gravitational Potential: Topological Classification......Page 40
 2.6 Topological Equivalence Between Different  Integrable Systems......Page 43
 References......Page 45
 3.1 Additive Problems......Page 47
 3.2 Multiplicative Problems......Page 51
 References......Page 54
4 Critical Analysis of Amino Acids  and Polypeptides Geometry......Page 55
 4.2 Protein Data Bank......Page 56
  4.2.1 Extracting Information from PDB......Page 57
  4.2.2 The First Steps in Polypeptides Visualization......Page 58
  4.2.3 Some Difficulties in PDB-Files Treatment......Page 61
  4.3.1 Estimation of Spread in Lengths of Covalent Bonds  in Amino Acids......Page 66
  4.3.2 Spread Estimation of Distances Between Consecutive Alpha Carbons (Beginning)......Page 71
  4.3.3 Spread Estimation of the Lengths of Peptide Bonds......Page 72
  4.3.4 Pauling Plane Law......Page 73
  4.3.5 Spread Estimation of Distances Between Consecutive Alpha Carbons (End)......Page 78
  4.3.6 Spread Estimation of Angles Between Covalent Bonds  in Polypeptides......Page 79
 4.4 Amino Acids' Mobility......Page 80
  4.4.1 Orientation of Amino Acids......Page 82
 4.5 Addendum (in collaboration with E.A. Vilkul)......Page 84
  4.5.1 The Number of Models' Distribution......Page 85
  4.5.2 ``Representativity'' of the First Model from the Pathologies Point of View......Page 87
  4.5.3 ``Representativity'' of the First Model from the Plane Law Point of View......Page 89
 4.6 Geometry of Planar and Space Polygonal Lines: Spirals Detecting......Page 92
 4.7 Torsion and Curvature of Space Curves......Page 97
 References......Page 99
 5.1 Setting up the Problem......Page 100
  5.2.1 Critical Manifold Σ+ : Condition of Jordanity......Page 101
  5.3.1 Stabilization Conditions: The Choice  of Parameters of Problem......Page 102
  5.3.3 The ODE System for Two-Front Solution......Page 103
  5.3.5 Lax's Condition : One-Front Solution......Page 104
  5.3.6 Lax's Condition for Two-Front Solution:  Condition of Monotonicity......Page 105
  5.4.1 The Existence of One-Front Solution then Moving  in the Noncritical Eigenvector Direction......Page 106
  5.4.2 One-Front Solution as the Traveling Wave  for ω=λ+|q=0......Page 107
 5.5 The Existence of Two-Front Solutions as ω=λ+|q=0......Page 108
  5.5.1 Different Forms of Two-Front Solutions......Page 111
  5.5.3 The Conditions for Two-Front Solutions  of Humped Kink Type in the ω Terms......Page 112
 5.6 Bifurcation of Rarefaction Waves  on the Critical Manifold Σ+......Page 113
  5.6.2 The Proof of the Existence of rs-Type Solution......Page 114
 References......Page 115
Part II Non-autonomous and StochasticDynamical Systems......Page 117
6 Dynamics of Nonautonomous Chemostat Models......Page 118
 6.1 Introduction......Page 119
 6.2 Preliminaries on Nonautonomous Dynamical Systems......Page 120
 6.3 Properties of Solutions......Page 122
  6.4.1 Chemostats with Wall Growth, Variable Delays, and Fixed Inputs......Page 125
  6.4.2 Chemostat with Wall Growth, Variable Inputs,  and No Delays......Page 126
  6.4.3 Chemostat with No Wall Growth or Delays......Page 130
 6.5 Random Chemostat Models......Page 131
 6.6 Overyield in Nonautonomous Chemostats......Page 132
 References......Page 134
 7.1 Introduction......Page 136
 7.2 Preliminaries on Random Dynamical Systems......Page 137
  7.3.1 Mathematical Settings......Page 140
  7.3.2 Selected Results for First-Order SLDEs......Page 142
  7.3.3 A Brief on Second-Order SLDEs......Page 146
  7.3.4 Mathematical Setting......Page 147
  7.3.5 Existence of Random Attractors......Page 148
 7.4 Closing Remarks......Page 149
 References......Page 150
 8.1 Introduction......Page 152
 8.2 Setting of the Problem......Page 154
 8.3 A Priori Estimates......Page 156
 8.4 The Continuous Semigroup......Page 159
 References......Page 162
 9.1 Introduction......Page 164
 9.2 Abstract Theory of Pullback calD-Attractors  for Multivalued Processes......Page 165
 9.3 Application......Page 171
 References......Page 180
 10.1 Introduction......Page 182
 10.2 The Construction of (ωotimesSω) for a Fractional Brownian motion......Page 186
 10.3 The Construction of (ωotimesSω) for a Brownian motion......Page 191
 10.4 Additional Properties of (ωotimesSω)......Page 200
 References......Page 202
 11.1 Introduction......Page 204
 11.2 Notation......Page 205
 11.3 Existence of Random Inertial Manifolds......Page 207
 11.4 Periodicity and Almost Periodicity of Inertial Manifolds......Page 221
 References......Page 223
 12.1 Introduction......Page 224
 12.2 Notations, Definitions, and Some Properties on the Multivalued Process......Page 225
 12.3 Pullback Attraction and Properties on ω-limit Sets......Page 228
 References......Page 233
 13.1 Introduction and Setting of the Problem......Page 235
 13.2 Preliminary Properties of Weak Solutions......Page 238
 13.3 Uniform Trajectory Attractor and Main Result......Page 240
 13.4 Proof of Theorem 13.1......Page 242
 References......Page 245
 14.1 Introduction and Regularity of All Weak Solutions......Page 247
 14.2 A Lyapunov Type Function and Strongest Convergence Results for All Weak Solutions......Page 249
 14.3 Structure Properties and Regularity of Global  and Trajectory Attractors......Page 252
 14.4 Faedo--Galerkin Approximation for the Global  and Trajectory Attractors......Page 253
 14.5 Applications......Page 254
 References......Page 255
Part III Optimization, Control and DecisionSciences for Continuum MechanicsProblems......Page 258
15 Robust Stability, Minimax Stabilization  and Maximin Testing in Problems  of Semi-Automatic Control......Page 259
 15.2 Robust Stability of Linear Systems......Page 260
 15.3 Minimax Stabilization and Antagonistic Game......Page 267
 15.4 Maximin Testing of Quality of Control Algorithm......Page 272
  15.4.1 Program Strategy of Testing......Page 273
  15.4.2 Closed-Loop Strategy of Testing......Page 274
 15.5 Conclusions......Page 276
 References......Page 277
16 Dynamics of Solutions for Controlled Piezoelectric Fields with Multivalued ``Reaction-Displacement'' Law......Page 278
 16.1 Introduction and the Main Problem......Page 279
 16.2 Setting of the Problem and the Main Results......Page 280
 References......Page 286
 17.1 Introduction......Page 288
  17.2.1 Problem Statement......Page 289
  17.2.2 Formal Solution of the Problem......Page 290
  17.3.1 Problem Statement......Page 299
  17.3.2 Formal Solution of the Problem......Page 300
 References......Page 307
 18.1 Introduction......Page 308
 18.2 Setting of the Problem......Page 309
 18.3 Some Facts from the Theory of Fourier-Bessel Series......Page 311
 18.4 Existence of Classical Solution of the Problem (??) with Fixed Control......Page 314
 18.5 The Optimal Control Problem (??) and (??)......Page 322
 References......Page 325
 19.1 Introduction......Page 326
 19.2 Notation and Preliminaries......Page 328
 19.3 mathbbSN-Valued Radon Measures and Weak  Convergence in Variable Lp-Spaces......Page 331
 19.4 Auxiliary Results......Page 338
 19.5 Setting of the Optimal Control Problem......Page 341
 19.6 Existence of Weak Optimal Solutions......Page 344
 References......Page 347
Part IV Fundamental and ComputationalMechanics......Page 349
 20.1 Postulates of Continuum Mechanics......Page 350
 20.2 Ideal Liquid and Gas......Page 351
 20.3 Newtonian Viscous Fluid......Page 354
 20.4 Linear Anisotropic Elastic Solid......Page 356
 References......Page 361
 21.1 Introduction......Page 362
 21.2 Mathematical Formulation of the Problem......Page 364
 21.3 Standing Vortex Within the Groove  in the Stationary Flow......Page 367
 21.4 Standing Vortex in the Groove in Periodically Perturbed Flow......Page 372
 21.5 Summary......Page 375
 References......Page 376
 22.1 Introduction......Page 377
 22.2 Problem Statement......Page 379
  22.3.1 General Principles......Page 381
  22.3.2 Free-Surface Modeling......Page 382
  22.3.3 The Vortex Method for 2-D Flows......Page 386
 22.4 Results and Disscussion......Page 387
 22.5 Conclusions......Page 393
 References......Page 394