Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics 9783030503017, 9783030503024

Table of contents :
International Editorial Board of this Volume
Editors-in-Chief
Associate Editors
Editors
Preface
Contents
Contributors
Part I Modern Mechanics and Its Applications
1 Saddle Singularities in Integrable Hamiltonian Systems: Examples and Algorithms
1.1 Introduction
1.2 Basic Notions
1.2.1 Liouville Theorem
1.2.2 Equivalences of Non-degenerate Singularities
1.2.3 Classification of Local Singularities
1.2.4 Semi-local Singularities of IHS
1.2.5 Topology of Liouville Foliations on Invariant Submanifolds
1.3 Invariants of Semi-local Equivalence
1.3.1 f-Graph or Pair of Permutations Determine 2-Atom
1.3.2 General Saddle Singularities
1.4 A.T. Fomenko Conjecture on Loop Molecules of Non-Degenerate Singularities
1.5 Stability and Splittability of Singularities
1.6 Saddle-Saddle Singularities in Physical, Mechanical and Geometrical Systems
1.7 Foliations and Singularities of Integrable Billiards
1.7.1 Brief Description of Integrable Billiards
1.7.2 Realization of 3-Atoms (rk = 1 Singularities)
1.7.3 Realization of 4-Dimensional Singularities ofrk0 by Integrable Billiards with Hooke Potential
1.7.4 Loop Molecules of Complexity 1 Saddle Singularities Are Realized in Billiards
References
2 Reduction of the Lamé Tensor Equations to the System of Non-Coupled Tetraharmonic Equations
2.1 Introduction
2.2 The Lamé Vector Equations and the Galerkin Representation in Isotropic Elasticity
2.3 Linear Differential Systems of the Forth Order for the Tensors of the Second Rank
2.4 Fundamental Solutions of Tetraharmonic Equations in Many-Dimensional Spaces
References
3 Junction Flow Around Cylinder Group on Flat Platee
3.1 Introduction
3.2 Problem Statement, Experimental Setup and Numerical Procedure
3.3 Research Results
3.4 Conclusion
References
4 Accounting for Shear Deformation in the Problem of Vibrations and Dissipative Heating of Flexible Viscoelastic Structural Element with Piezoelectric Sensor and Actuator
4.1 Introduction
4.2 Problem Statement
4.3 Solution of the Problem
4.4 Results and Discussion
4.5 Conclusions
References
5 A Stochastic Theory of Scale-Structural Fatigue and Structure Durability at Operational Loading
5.1 Introduction
5.2 The Main Properties of Metal Fatigue at Microscopic, Mesoscopic and Macroscopic Scale-Structural Levels
5.3 Modern Theoretical Approaches to Metal Fatigue Description
5.4 Scale-Structural Fatigue Theory at Complex Stress State
5.5 Practical Implementation of the Scientific Results
References
6 On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with p-Laplace Operator and L1-type of Non-linearity
6.1 Introduction
6.2 Preliminaries
6.3 On Tikhonov Regularization of the Original OCP
6.4 Asymptotic Analysis of Regularized Optimal Control Problem
References
7 Symmetries and Conservation Laws of the Equations of Two-Dimensional Shallow Water Over Uneven Bottom
7.1 Introduction and Main Result
7.2 Symmetries of the System of Equations of Two-Dimensional Shallow Water Over Uneven Bottom
7.2.1 System of Determining Equations
7.2.2 Analysis of the Classifying Equation
7.2.3 Cases of Extension of the Kernel of Symmetry Operators
7.2.4 Nonlinearity of the System of Equations of Two-Dimensional Shallow Water Over Uneven Bottom
7.3 Conservation Laws of the Equations of Two-Dimensional Shallow Water Over Uneven Bottom
7.3.1 Determining System of Equations
7.3.2 Analysis of the Classifying Equation
7.3.3 Cases of Additional Conservation Laws
7.4 Conclusion
References
8 Existence and Stability Analysis of Solutions for an Ultradian Glucocorticoid Rhythmicity and Acute Stress Model
8.1 Introduction
8.2 Stability Analysis of the Model Without a Time Delay
8.2.1 Stability for Different Parameter Values
8.2.2 Lyapunov Stability Analysis
8.3 Analysis of the Model with a Time Delay
8.3.1 Stability Analysis with Respect to a Time Delay
8.3.2 Global in Time Existence of Solutions
8.3.3 Existence of Periodic Solutions
8.4 Discussion
8.5 Appendix
References
9 Mixed Dirichlet-Transmission Problems in Non-smooth Domains
9.1 Introduction
9.2 Functional Setting and Notations
9.3 The Main Results
9.4 Solvability of Transmission Problem (9.1)–(9.3) in G1pmG2pm
9.4.1 Integral Representation of (v10,v20)
9.4.2 Coercive Estimates of (v10,v20)
9.4.3 Solvability of Problem (9.8) with f0iequiv0, i=1,2, and Coercive Estimates
9.4.4 Solvability of Problem (9.7)
9.5 The Proof of the Main Results
9.5.1 Proof of Theorems 9.1 and 9.2
9.5.2 More Regular Solutions of (9.1)–(9.3)
9.6 Appendix: Proof of Proposition 9.4
References
Part II Determinism and Stochasticity in Modeling in Real Phenomena
10 Convergence Rate of Random Attractors for 2D Navier–Stokes Equation Towards the Deterministic Singleton Attractor
10.1 Introduction
10.2 Singleton Attractor of 2D Navier–Stokes Equation with Small Forcing Intensity
10.2.1 Preliminaries
10.2.2 Small Grashof Number and the Singleton Attractor
10.3 Additive Noise Case
10.3.1 Preliminaries
10.3.2 Uniform Estimates of Solutions
10.3.3 Perturbation Radius of the Singleton Attractor Under Additive Noise
10.4 Multiplicative Noise Case
10.4.1 Preliminaries
10.4.2 Uniform Estimates of Solutions
10.4.3 Perturbation Radius of the Singleton Attractor Under Multiplicative Noise
References
11 The Dynamics of Periodic Switching Systems
11.1 Introduction
11.2 One-Dimensional Dynamics Background
11.3 Applications
11.3.1 The Periodic Logistic Map
11.3.2 The Jonzén-Lundberg Model
11.4 Conclusions
References
12 Co-jumps and Markov Counting Systems in Random Environments
12.1 Introduction
12.2 Markov Counting Systems Without External Noise
12.3 Markov Counting Systems with External Noise
12.4 Markov Counting Systems with Correlated External Noise
12.4.1 Correlated External Noise in Bivariate Death Markov Counting Systems
12.4.2 Correlated External Noise in General Markov Counting Systems
12.4.3 Infinitesimal Covariance of Markov Counting Systems
12.5 Transition Rates of SIR-type Models Subjected to External Correlated Noises
References
13 On Fractal Dimension of Global and Exponential Attractors for Dissipative Higher Order Parabolic Problems in mathbbRN with General Potential
13.1 Introduction
13.2 Some Known Results Concerning Global Attractor for (13.1)
13.3 Estimate of Fractal Dimension of the Attractor
13.3.1 Estimates for the Semigroup on the Attractor and the Absorbing Set
13.3.2 Compactness of Associated Seminorm
13.3.3 Estimate from Above of Fractal Dimension of the Attractor
13.4 Existence of Exponential Attractor
13.5 Appendix: Abstract Estimate of Fractal Dimension
13.6 Appendix: Boundedness of Absorbing Set in Supremum Norm
References
14 Ergodicity of Stochastic Hydrodynamical-Type Evolution Equations Driven by α-Stable Noise
14.1 Introduction
14.2 Preliminaries
14.3 Well-Posedness of the Mild Solution
14.4 The Existence of the Invariant Measure
14.5 The Uniqueness of the Invariant Measure
14.5.1 Strong Feller Property
14.5.2 The Accessibility
14.6 Applications
14.6.1 Stochastic 2D Boussinesq Equation
14.6.2 Stochastic 2D Magneto-Hydrodynamic Equation
References
Part III Advances in Control and Optimization
15 Uniform Global Attractor for a Class of Nonautonomous Evolution Hemivariational Inequalities with Multidimensional ``Reaction-Velocity'' Law
15.1 Introduction
15.2 Main Result
15.3 Proof of Theorem 15.1
15.4 Auxiliary Properties of the Global Attractor in the Autonomous Case
15.5 Applications
References
16 On a Lyapunov Characterization of Input-To-State Stability for Impulsive Systems with Unstable Continuous Dynamics
16.1 Introduction
16.2 Input-To-State Stability for Impulsive Control Systems
16.3 Lyapunov Characterization of the ISS Property
16.4 Concluding Remarks
References
17 Practical Stability of Discrete Systems: Maximum Sets of Initial Conditions Concept
17.1 Introduction
17.2 Internal Practical Stability
17.3 Practical Stability and Lyapunov Functions
17.4 Maximum Sets of External Practical Stability
References
18 Optimal Control for Systems of Differential Equations on the Infinite Interval of Time Scale
18.1 Introduction
18.2 Preliminaries, Statement of the Problem and Main Results
18.3 Proof of the Theorems
18.3.1 Proof of Theorem 18.1
18.3.2 Proof of Theorem 18.2
References
19 Approximate Feedback Control for Hyperbolic Boundary-Value Problem with Rapidly Oscillating Coefficients in the Case of Non-convex Objective Functional
19.1 Introduction
19.2 Setting of the Problem
19.3 Main Results
References
20 Decomposition of Intersections with Fuzzy Sets of Operands
20.1 Introduction
20.2 Preliminaries
20.2.1 Intersection of Sets with a Fuzzy Set of Operands
20.2.2 Type-2 Fuzzy Sets
20.3 An Example of Intersection with a Fuzzy Set of Operands
20.4 Decomposition of Intersections with Fuzzy Sets of Operands
20.5 Conclusion
References
21 Distribution of Values of Cantor Type Fractal Functions with Specified Restrictions
21.1 Introduction
21.2 Singular Functions as Solutions of Systems of Functional Equations
21.3 Constructive Generalization of a Class of Continuous Functions
21.4 Properties of Monotonicity
21.5 Singular Functions of Cantor type
21.6 Cantor-Type Functions That Does not Have Intervals of Monotonicity, Except the Intervals Where They Are Constant
21.6.1 Images of Cantor Type Functions
21.6.2 The Distribution of Values of f for a Given Argument Distribution
21.7 Conclusion
References
22 Solvability Issue for Optimal Control Problem in Coefficients for Degenerate Parabolic Variational Inequality
22.1 Introduction
22.2 Notations and Preliminaries
22.2.1 Weighted Sobolev Spaces
22.2.2 Parabolic Variational Inequalities
22.2.3 Compensated Compactness Lemma in Variable Lebesgue and Sobolev Spaces
22.3 Setting of the Optimal Control Problem (OCP)
22.4 Existence of H-Optimal Solutions
References
23 Group Pursuit Differential Games with Pure Time-Lag
23.1 Introduction
23.2 Statement of the Problem
23.3 Main Results
23.4 Examples
23.5 Conclusion
References
24 An Indirect Approach to the Existence of Quasi-optimal Controls in Coefficients for Multi-dimensional Thermistor Problem
24.1 Introduction and Setting of the Optimal Control Problem
24.1.1 Relaxation of the Original OCP
24.1.2 Main Results
24.2 Preliminaries and Some Auxiliary Results
24.2.1 On Orlicz Spaces
24.2.2 On the Weak Convergence of Fluxes to Flux
24.3 On Approximated Optimal Control Problems in Coefficients and Their Properties
24.4 Asymptotic Analysis of the Approximated OCP (24.19)–(24.22) as εto0
References