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E-raamat: Continuous and Distributed Systems: Theory and Applications

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Continuous and Distributed Systems: Theory and ApplicationsSuurem pilt
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In this volume, the authors close the gap between abstract mathematical approaches, such as abstract algebra, number theory, nonlinear functional analysis, partial differential equations, methods of nonlinear and multi-valued analysis, on the one hand, and practical applications in nonlinear mechanics, decision making theory and control theory on the other.

Readers will also benefit from the presentation of modern mathematical modeling methods for the numerical solution of complicated engineering problems in hydromechanics, geophysics and mechanics of continua. This compilation will be of interest to mathematicians and engineers working at the interface of these field. It presents selected works of the open seminar series of Lomonosov Moscow State University and the National Technical University of Ukraine “Kyiv Polytechnic Institute”. The authors come from Germany, Italy, Spain, Russia, Ukraine, and the USA.



This book closes the gap between abstract mathematical approaches such as number theory, nonlinear functional analysis, PDEs, nonlinear and multi-valued analysis, and practical applications in nonlinear mechanics, decision making theory and control theory.
Part I Abstract Algebra and Applications
1 Algebra and Geometry Through Hamiltonian Systems
3(20)
Anatoly T. Fomenko
Andrei Konyaev
1.1 Introduction
3(1)
1.2 Atoms and Their Symmetries
4(3)
1.3 Integer Lattices of Action Variables for "Spherical Pendulum" System
7(3)
1.4 Billiards in Confocal Quadrics
10(4)
1.5 Bertrand's Manifolds and Their Properties
14(3)
1.6 Lie Algebras with Generic Coadjoint Orbits of Dimension Two
17(6)
References
19(4)
2 On Hyperbolic Zeta Function of Lattices
23(40)
L. P. Dobrovolskaya
M. N. Dobrovolsky
N. M. Dobrovolski
N. N. Dobrovolsky
2.1 Introduction
23(22)
2.1.1 Lattices
24(2)
2.1.2 Exponential Sums of Lattices
26(3)
2.1.3 Multidimensional Quadrature Formulas and Hyperbolic Zeta Function of a Grid
29(5)
2.1.4 Hyperbolic Zeta Function of Lattices
34(6)
2.1.5 Generalised Hyperbolic Zeta Function of Lattices
40(5)
2.2 Functional Equation for Hyperbolic Zeta Function of Integer Lattices
45(7)
2.2.1 Periodized in the Parameter b Hurwitz Zeta Function
46(1)
2.2.2 Dirichlet Series with Periodical Coefficients
47(3)
2.2.3 Functional Equation for Hyperbolic Zeta Function of Integer Lattices
50(2)
2.3 Functional Equation for Hyperbolic Zeta Function of Cartesian Lattices
52(7)
2.4 On Some Unsolved Problems of the Theory of Hyperbolic Zeta Function of Lattices
59(4)
References
60(3)
3 The Distribution of Values of Arithmetic Functions
63(4)
G. V. Fedorov
References
66(1)
4 On the One Method of Constructing Digital Control System with Minimal Structure
67(6)
V. V. Palin
4.1 The Statement of Problem and Some Familiar Results
67(1)
4.2 Definitions and Some Preliminary Transformations
68(1)
4.3 The Method to Obtain the Characteristic of Completely Controllable
69(1)
4.4 Auxiliary Statements
69(1)
4.5 The Absence of Associated Vectors Case
70(1)
4.6 The Case of General Position
71(2)
Reference
71(2)
5 On Norm Maps and "Universal Norms" of Formal Groups over Integer Rings of Local Fields
73(8)
Nikolaj M. Glazunov
5.1 Introduction
73(2)
5.2 Norm Maps
75(3)
5.3 Results
78(3)
References
80(1)
6 Assignment of Factors Levels for Design of Experiments with Resource Constraints
81(8)
S. A. Smirnov
A. A. Glushchenko
E. A. Ilchuk
I. L. Makeenko
N. A. Oriekhova
6.1 Introduction
81(1)
6.2 Hansel Method
82(1)
6.3 Modification
83(2)
6.4 Example
85(1)
6.5 Conclusions
86(3)
References
86(3)
Part II Mechanics and Numerical Methods
7 How to Formulate the Initial-Boundary-Value Problem of Elastodynamics in Terms of Stresses?
89(8)
D. V. Georgievskii
7.1 The Classic Formulation of the Dynamic Problem and Its Peculiarities
89(2)
7.2 Ignaczak--Nowacki' Formulation
91(1)
7.3 Konovalov' Formulation
92(1)
7.4 Pobedria' Formulation
93(1)
7.5 One More Possible Formulation
93(4)
References
95(2)
8 Finite-Difference Method of Solution of the Shallow Water Equations on an Unstructured Mesh
97(18)
G. M. Kobelkov
A. V. Drutsa
8.1 Introduction
97(1)
8.2 Formulation of the Problem
97(1)
8.3 Mesh and Mesh Operators
98(2)
8.4 Finite-Dimensional Problem
100(1)
8.5 Convergence
101(3)
8.6 Results of Numerical Experiments
104(11)
8.6.1 Estimation of Convergence Order
104(1)
8.6.2 Computation of the Real Geographic Domain
105(8)
References
113(2)
9 Dynamics of Vortices in Near-wall Flows with Irregular Boundaries
115(16)
I. M. Gorban
O. V. Homenko
9.1 Introduction
115(2)
9.2 Model of Standing Vortex
117(2)
9.3 Standing Vortex in Cross Groove
119(2)
9.4 Standing Vortex in an Angular Region
121(2)
9.5 Resonant Properties of Standing Vortices and Their Behavior in Perturbed Flow
123(5)
9.6 Summary
128(3)
References
128(3)
10 Strongly Convergent Algorithms for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems
131(18)
Vladimir V. Semenov
10.1 Introduction
131(3)
10.2 Preliminaries
134(1)
10.3 Convergence Analysis
135(10)
10.4 Concluding Remarks
145(4)
References
145(4)
Part III Long-time Forecasting in Multidisciplinary Investigations
11 Multivalued Dynamics of Solutions for Autonomous Operator Differential Equations in Strongest Topologies
149(14)
Mikhail Z. Zgurovsky
Pavlo O. Kasyanov
11.1 Introduction: Statement of the Problem
149(2)
11.2 Additional Properties of Solutions
151(7)
11.3 Attractors in Strongest Topologies
158(2)
11.4 Application
160(1)
11.5 Conclusions
161(2)
References
161(2)
12 Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System
163(18)
Oleksiy V. Kapustyan
Pavlo O. Kasyanov
Jose Valero
Mikhail Z. Zgurovsky
12.1 Introduction
163(1)
12.2 Setting of the Problem
164(1)
12.3 Multivalued Processes and Uniform Attractors
165(9)
12.4 Uniform Global Attractor for RD-System
174(7)
References
180(1)
13 Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations
181(8)
Pavlo O. Kasyanov
Luisa Toscano
Nina V. Zadoianchuk
13.1 Introduction
181(2)
13.2 Topological Properties of Strong Solutions
183(1)
13.3 Proof of Theorem 13.2
184(1)
13.4 Proof of Theorem 13.1
185(4)
References
187(2)
14 Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation
189(16)
Natalia Chalkina
14.1 Introduction
189(2)
14.2 Statement of the Problem and Spectrum of the Linear Operator
191(2)
14.3 Sufficient Conditions for the Existence of Inertial Manifolds
193(4)
14.4 Proof of Theorem 14.3
197(8)
14.4.1 New Norm in the Spaces Hk, k = 1, ... k1
197(1)
14.4.2 New Norm in the Spaces Hk, k = k1 + 1, ..., k2
198(2)
14.4.3 New Norm in the Space H∞
200(2)
14.4.4 End of the Proof of Theorem 14.3
202(1)
References
203(2)
15 On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain
205(16)
Nataliia V. Gorban
Pavlo O. Kasyanov
15.1 Introduction
205(3)
15.2 On Compact Global Attractor for Reaction-Diffusion Inclusion in Unbounded Domain
208(9)
15.3 Regularity of All Weak Solutions and Their Attractors
217(4)
References
219(2)
16 On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity
221(20)
Nataliia V. Gorban
Oleksiy V. Kapustyan
Pavlo O. Kasyanov
Liliia S. Paliichuk
16.1 Introduction
221(1)
16.2 Setting of the Problem
222(1)
16.3 Preliminaries
223(2)
16.4 Properties of Solutions
225(6)
16.5 The Existence of a Global Attractor
231(1)
16.6 Global Attractors for Typically Discontinuous Interaction Functions
232(9)
References
237(4)
Part IV Control Theory and Decision Making
17 On the Regularities ofMass Random Phenomena
241(10)
Victor I. Ivanenko
Valery A. Labkovsky
17.1 Introduction
241(2)
17.2 Theorem of Existence of Statistical Regularities
243(3)
17.3 The Proof
246(1)
17.4 Applications in Decision Theory
247(2)
17.5 Concluding Remarks
249(2)
References
249(2)
18 Optimality Conditions for Partially Observable Markov Decision Processes
251(14)
Eugene A. Feinberg
Pavlo O. Kasyanov
Mikhail Z. Zgurovsky
18.1 Introduction
251(1)
18.2 Model Description
252(4)
18.3 Reduction of POMDPs to COMDPs and Optimality Results
256(6)
18.4 Example
262(1)
18.5 Conclusions
263(2)
References
264(1)
19 On Existence of Optimal Solutions to Boundary Control Problem for an Elastic Body with Quasistatic Evolution of Damage
265(22)
Peter I. Kogut
Gunter Leugering
19.1 Introduction
265(1)
19.2 Notation and Preliminaries
266(4)
19.3 Radon Measures and Convergence in Variable Spaces
270(3)
19.4 The Model of Quasistatic Evolution of Damage in an Elastic Material
273(5)
19.5 Setting of the Optimal Control Problems and Existence Theorem for Optimal Traction
278(9)
References
286(1)
20 On Existence and Attainability of Solutions to Optimal Control Problems in Coefficients for Degenerate Variational Inequalities of Monotone Type
287(16)
Olga P. Kupenko
20.1 Introduction
287(2)
20.2 Notation and Preliminaries
289(5)
20.3 Setting of the Optimal Control Problem
294(1)
20.4 Compensated Compactness Lemma in Variable Lebesgue and Sobolev Spaces
295(1)
20.5 Existence of H-Optimal Solutions
296(1)
20.6 Attainability of H-Optimal Solutions
297(6)
References
300(3)
21 Distributed Optimal Control in One Non-Self-Adjoint Boundary Value Problem
303(10)
V. O. Kapustyan
O. A. Kapustian
O. K. Mazur
21.1 Introduction
303(1)
21.2 Setting of the Problem
304(1)
21.3 Main Results
305(6)
21.4 Conclusions
311(2)
References
312(1)
22 Guaranteed Safety Operation of Complex Engineering Systems
313(14)
Nataliya D. Pankratova
Andrii M. Raduk
22.1 Introduction
314(1)
22.2 Information Platform of Engineering Diagnostics of the Complex Object Operation
315(6)
22.3 Diagnostic of Reanimobile's Functioning
321(4)
22.4 Conclusion
325(2)
References
326(1)
Appendix A To the Arithmetics of the Bose--Maslov Condensate Statistics 327(2)
Appendix B Numerical Algorithms for Multiphase Flows and Applications 329(2)
Appendix C Singular Trajectories of the First Order in Problems with Multidimensional Control Lying in a Polyhedron 331(2)
Appendix D The Guaranteed Result Principle in Decision Problems 333
Viktor A. Sadovnichiy, Lomonosov Moscow State University; Michael Z. Zgurovsky, National Technical University of Ukraine "Kiev Polytechnic Institute".
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