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## Über dieses Buch

This book focuses on the latest approaches and methods in fundamental mathematics and mechanics, and discusses the practical application of abstract mathematical approaches, such as differential geometry, and differential and difference equations in solid mechanics, hydrodynamics, aerodynamics, optimization, decision-making theory and control theory. Featuring selected contributions to the open seminar series of Lomonosov Moscow State University and Igor Sikorsky Kyiv Polytechnic Institute by mathematicians from China, Germany, France, Italy, Spain, Russia, Ukraine and the USA, the book will appeal to mathematicians and engineers working at the interface of these fields

## Inhaltsverzeichnis

### Chapter 1. Saddle Singularities in Integrable Hamiltonian Systems: Examples and Algorithms

Abstract
Saddle or hyperbolic singularities of Liouville foliations of integrable Hamiltonian systems are discussed. We observe new and classical results on their classification, representation and invariants with respect to topological equivalence depending on number of degrees of freedom. Then criterion of their component-wise stability by A.A. Oshemkov and its application are reminded. At last, we discuss saddle singularities of famous dynamical and physical systems, particularly problem of realization (modeling) of Liouville foliations and their singularities (A.T. Fomenko billiard conjecture) by integrable billiards. New result is obtained: loop molecules of all saddle-saddle singularities with one equilibrium are modeled by billiard books, i.e. integrable billiards on CW-complexes introduced by V.V. Vedyushkina.
Anatoly T. Fomenko, Vladislav A. Kibkalo

### Chapter 2. Reduction of the Lamé Tensor Equations to the System of Non-Coupled Tetraharmonic Equations

Abstract
Reductions of the systems of equations in terms of displacements in 3D elasticity to the systems of higher orders based on the operators which are suitable for both numerical and analytical investigation better than the Lamé operator, are called representations of the solution. The Galerkin representation is one of such typical procedures in classical theory of elasticity. Below the Galerkin procedure is generalized in conformity to the systems generated by linear symmetric tensor (of the second rank) differential operator of the forth order acting on symmetric tensor field. Reduction of such systems to the systems of non-coupled tetraharmonic equations is realized. The fundamental solutions of the derived tetraharmonic equations in many-dimensional spaces are given.
D. V. Georgievskii

### Chapter 3. Junction Flow Around Cylinder Group on Flat Platee

Abstract
Results of numerical and physical simulations of a junction flow around a three-row cylinder group mounted on the rigid, hydraulically smooth flat surface are presented. Numerical model is based on the vortex method used to integrate 2D Navier–Stokes equations. Experimental researches are carried out in the open hydraulic channel where color inks and contrasting water-soluble coating are applied for visualization of the jet flows and vortex structures generating inside this group. Derived velocity fields demonstrate that the most intense lateral flows form between the first and second as well as between the penultimate and midday cylinders. The currents give rise to developing of large-scale coherent horseshoe vortices at the foot of the cylinders. The size, rotation frequency and angular velocity of the vortices were measured. The estimations show that the vortex generated between the first and second cylinders are more intense than the vortex locating before the group.
V. A. Voskoboinick, I. M. Gorban, A. A. Voskoboinick, L. N. Tereshchenko, A. V. Voskoboinick

### Chapter 4. Accounting for Shear Deformation in the Problem of Vibrations and Dissipative Heating of Flexible Viscoelastic Structural Element with Piezoelectric Sensor and Actuator

Abstract
Statement of the problem on forced resonance vibration, active control and dissipative heating of flexible viscoelastic beam containing piezoelectric sensor and actuator with taking account for shear deformation and quadratic geometrical nonlinearity is elaborated on the base of coupled electro-thermomechanics theory. Procedure of numerical solution of the problem is developed as well. For the most energy intensive first bending mode of beam vibration, influence of mechanical end fixing conditions, heat exchange, shear deformation and accounting for geometrical nonlinearity onto frequency characteristics of maximum deflection amplitude, level of dissipative heating temperature and electric parameter of sensor are investigated under monoharmonic loading. Possibility of active damping of beam vibration by means of piezoactuator with making use of electric parameter of sensor is studied in the case of unknown loading amplitude.
I. F. Kirichok, Y. A. Zhuk, S. Yu. Kruts

### Chapter 5. A Stochastic Theory of Scale-Structural Fatigue and Structure Durability at Operational Loading

Abstract
Here are discussed the results of numerous experimental and theoretical investigations of multicyclic fatigue multilevel processes at complex stress state of metals and alloys on solid state physics, metal science and solid mechanics. Proposed by the author on their basis the scale-structural fatigue theory describing the evolution of fatigue defects and allowing to find the metal durability on a certain level of accumulated defects are presented. A new method for evaluation the durability of long structures, including criteria of structural reliability and technogenic safety of operation and calculation of structural element durability on the offered theory, is briefly considered.
E. B. Zavoychinskaya

### Chapter 6. On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with p-Laplace Operator and -type of Non-linearity

Abstract
We discuss the existence of solutions to an optimal control problem for the Dirichlet boundary value problem for strongly non-linear p-Laplace equations with $$L_1$$-type of nonlinearity and $$p\ge 2$$. The control variable u is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution $$y_d\in L^p(\varOmega )$$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of “defect”. We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.
Peter I. Kogut, Olha P. Kupenko

### Chapter 7. Symmetries and Conservation Laws of the Equations of Two-Dimensional Shallow Water Over Uneven Bottom

Abstract
A system of equations of two-dimensional shallow water over uneven bottom is considered. Overdetermined systems of equations for determining the symmetries and the conservation laws are obtained. The compatibilities of this overdetermined systems of equations are investigated. The general forms of the solutions of the overdetermined systems are found. The kernels of the symmetry operators and conservation laws are found. Cases of kernels extensions of symmetry operators and conservation laws are presented. The corresponding classifying equations are given. The results of the group classification have indicated that the system of equations of two-dimensional shallow water over uneven bottom cannot be linearized by point transformation in contrast to the system of equations of one-dimensional shallow water in the cases of horizontal and inclined bottom profiles.
A. V. Aksenov, K. P. Druzhkov

### Chapter 8. Existence and Stability Analysis of Solutions for an Ultradian Glucocorticoid Rhythmicity and Acute Stress Model

Abstract
The hypothalamic pituitary adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol (CORT) level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor (GR) but does not take into account the system response delay, we analyze linear and non-linear stability of stationary solutions. For a second mathematical model that describes the mechanism of the HPA axis self-regulatory activities and takes into account a delay of system response, we prove existence of periodic solutions under certain assumptions on ranges of parameter values and analyze stability of these solutions with respect to the time delay value.
Casey Johnson, Roman M. Taranets, Nataliya Vasylyeva, Marina Chugunova

### Chapter 9. Mixed Dirichlet-Transmission Problems in Non-smooth Domains

Abstract
In this paper, we consider the Poisson equations in two dimensional non-smooth domains on the boundaries of which inhomogeneous Dirichlet and transmission conditions are imposed. Under certain assumptions on the parameter in the model, we prove the existence of a unique classical solution which belongs to the weighted Hölder classes.
Nataliya Vasylyeva

### Chapter 10. Convergence Rate of Random Attractors for 2D Navier–Stokes Equation Towards the Deterministic Singleton Attractor

Abstract
In this paper we study the long-time behavior of a 2D Navier–Stokes equation. It is shown that under small forcing intensity the global attractor of the equation is a singleton. When endowed with additive or multiplicative white noise no sufficient evidence was found that the random attractor keeps the singleton structure, but the estimate of the convergence rate of the random attractor towards the deterministic singleton attractor as stochastic perturbation vanishes is obtained.
Hongyong Cui, Peter E. Kloeden

### Chapter 11. The Dynamics of Periodic Switching Systems

Abstract
Switching systems have been recently used to model phenomena from Biology, Economy, Physics, etc. They consist in the iteration of a finite number of maps which can be viewed as the dynamics of a class of triangular maps. We make a special emphasis to periodic switching by showing that the dynamics of these systems can be analyzed from associated dynamical systems. Hence, we introduce the basic background of regular enough piecewise monotone maps and then, as an application, we study the dynamics of two periodic models coming from Biology.
Jose S. Cánovas

### Chapter 12. Co-jumps and Markov Counting Systems in Random Environments

Abstract
Motivated by the analysis of multi-strain infectious disease data, we provide closed-form transition rates for continuous-time Markov chains that arise from subjecting Markov counting systems to correlated environmental noises. Noise correlation induces co-jumps or counts that occur simultaneously in several counting processes. Such co-jumps are necessary and sufficient for infinitesimal correlation between counting processes of the system. We analyzed such infinitesimal correlation for a specific infectious disease model by randomizing time of Kolmogorov’s Backward system of differential equations based on appropriate stochastic integrals.
Carles Bretó

### Chapter 13. On Fractal Dimension of Global and Exponential Attractors for Dissipative Higher Order Parabolic Problems in with General Potential

Abstract
We consider a Cauchy problem for a dissipative fourth order parabolic equation in $$\mathbb {R}^N$$ with a general potential. Using the method by Chueshov and Lasiecka we estimate from above fractal dimension of a global attractor. We also show that it is contained in a finite dimensional exponential attractor.

### Chapter 14. Ergodicity of Stochastic Hydrodynamical-Type Evolution Equations Driven by -Stable Noise

Abstract
The present paper is devoted to the ergodicity of stochastic 2D hydro-dynamical-type evolution equation driven by $$\alpha$$-stable noise with $$\alpha \in (\frac{3}{2},2)$$, which covers stochastic Navier–Stokes equation, magneto-hydrodynamic equation, Boussinesq equation, magnetic Benard equation and so on. The existence and uniqueness of the invariant measure of this stochastic system are established by the strong Feller property and accessibility of the transition semigroup. The novel to overcome those difficulties caused by the trajectory discontinuity and lower regularity of the corresponding Ornstein–Uhlenbeck process for $$\alpha$$-stable noise. As applications of the abstract result, the existence and uniqueness of the invariant measure for the stochastic Boussinesq equation and stochastic 2D Magneto-hydrodynamic equation are given.
Jianhua Huang, Tianlong Shen, Yuhong Li

### Chapter 15. Uniform Global Attractor for a Class of Nonautonomous Evolution Hemivariational Inequalities with Multidimensional “Reaction-Velocity” Law

Abstract
We consider non-autonomous evolution inclusions and hemivariation inequalities with possibly non-monotone multidimensional “reaction-velocity” law. The dynamics of all weak solutions defined on the positive semi-axis of time is investigated. We prove the existence global attractor. New properties of complete trajectories are justified. The pointwise behavior of such problem solutions on attractor is described in the autonomous case.
Michael Z. Zgurovsky, Ciro D’Apice, Umberto De Maio, Nataliia V. Gorban, Pavlo O. Kasyanov, Oleksiy V. Kapustyan, Olha V. Khomenko, José Valero

### Chapter 16. On a Lyapunov Characterization of Input-To-State Stability for Impulsive Systems with Unstable Continuous Dynamics

Abstract
This chapter studies the input-to-state stability (ISS) property for nonlinear control systems with impulsive jumps at fixed moments. Sufficient conditions for the ISS are formulated in terms of a candidate ISS-Lyapunov function equipped with nonlinear rate functions which characterize the evolution of this function along the discontinuous trajectories of the system. For the case of unstable continuous dynamics, we derive new sufficient conditions for ISS under average-type dwell-time that provide a lower bound for the frequency of stabilizing jumps sufficient for the ISS.
Petro Feketa, Alexander Schaum, Thomas Meurer

### Chapter 17. Practical Stability of Discrete Systems: Maximum Sets of Initial Conditions Concept

Abstract
In this chapter we consider practical stability of discrete systems on the basis of maximum sets of initial conditions concept. We propose results concerning nonlinear discrete systems including topological properties of the maximum sets of initial conditions for internal practical stability, the necessary and sufficient conditions of internal practical stability using the optimal Lyapunov function. Further we offer the analytical forms (such as the Minkowski function, the inverse Minkowski function, and the support function) of the maximum sets of initial conditions representation in linear case, optimal ellipsoidal estimations of practical stability domains, and optimal estimations of phase constraints. In the last section we consider the problem of external practical stability of discrete systems.
V. V. Pichkur, Ya. M. Linder

### Chapter 18. Optimal Control for Systems of Differential Equations on the Infinite Interval of Time Scale

Abstract
In the given paper we are dealing with an optimal control problem on the semi-axis. We have stated the connection between solutions to optimal control problems on time scales and to the corresponding problem on real semi-axis. A new method is proposed for constructing a minimizing sequence for a problem on the semi-axis.
O. Stanzhytskyi, V. Mogylova, O. Lavrova

### Chapter 19. Approximate Feedback Control for Hyperbolic Boundary-Value Problem with Rapidly Oscillating Coefficients in the Case of Non-convex Objective Functional

Abstract
The article deals with the optimal control problem consisting of the hyperbolic boundary-value problem with rapidly oscillating coefficients and non-convex objective functional. In the general case, finding an exact formula for optimal control in the feedback form for such class of distributed processes in micro-inhomogeneous media does not seem possible. We assume that the corresponding problem with homogenized parameters allows finding optimal control in the feedback form. The main result of the work is to prove that obtained regulator realizes an approximate optimal control for the original problem.
Olena Kapustian

### Chapter 20. Decomposition of Intersections with Fuzzy Sets of Operands

Abstract
We investigate the operation of intersection of fuzzy sets with a fuzzy set of operands. This is a natural generalization of the corresponding operation which involves a crisp set of operands. The decomposition approach was used to study the intersections of fuzzy sets with a fuzzy set of operands. The result of this operation is a type-2 fuzzy set (T2FS). We prove several results which enable us to simplify constructing the type-2 membership function. It is shown that the resulting T2FS can be decomposed according to secondary membership grades into a finite collection of type-1 fuzzy sets. Each of these sets is the intersection of the original sets with a crisp set of operands. This crisp set is the corresponding $$\alpha$$-cut of the fuzzy set of operands. Illustrative examples are given.
S. O. Mashchenko, D. O. Kapustian

### Chapter 21. Distribution of Values of Cantor Type Fractal Functions with Specified Restrictions

Abstract
We consider $$Q_s^*$$-representation of numbers $$x \in [0,1]$$. It is an encoding of real numbers by means of the finite alphabet $$A=\{0,1,2,\ldots ,s-1\}$$. The article is devoted to continuous non-monotonic singular functions of Cantor type defined in terms of a given $$Q_s^*$$-representation of numbers. We study their local and global properties: structural, variational, differential, integral, self-similar, and fractal. Level sets of functions as well as topological and metric properties of images of Cantor type sets are examined in detail. In this work we also study the distribution of random variable $$Y=f(X)$$, where f is a non-monotonic singular function of Cantor type and X is a random variable such that its distribution induced by distributions of digits of its $$Q_5^*$$-representation that are independent random variables.
Oleg V. Barabash, Andrii P. Musienko, Valentyn V. Sobchuk, Nataliia V. Lukova-Chuiko, Olga V. Svynchuk

### Chapter 22. Solvability Issue for Optimal Control Problem in Coefficients for Degenerate Parabolic Variational Inequality

Abstract
In this paper we investigate an optimal control problem in coefficients for degenerate parabolic variational inequality. Since these types of problems can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, there several possible statements of such problems depending on the choice of the class of admissible solutions. Here we consider the optimal control problem in the so-called class of H-admissible solutions. Using the classical approach to parabolic variational inequalities, we show that the set of admissible pairs is not empty. We prove some topological properties of the set of H-admissible solutions and show that this set possesses some compactness properties with respect to the appropriate convergence in variable spaces. Using, the direct method in Calculus of variations, we prove the theorem on the existence of H-optimal solutions.
Nina V. Kasimova

### Chapter 23. Group Pursuit Differential Games with Pure Time-Lag

Abstract
The paper is devoted to the group pursuit differential-difference game with pure time-lag. An approach to the solution of this problem based on the method of resolving functions is proposed. For the group problem, the integral presentation of game solution based on the time-delay exponential is proposed at the first time. The guaranteed time of the game termination is found, and corresponding control law is constructed. The results are illustrated by a model example. In such game of two persons, it is possible to avoid meeting with the terminal set with any control of the pursuer. It is shown that if the pursuers are several then the pursuit game can be completed.
Lesia V. Baranovska

### Chapter 24. An Indirect Approach to the Existence of Quasi-optimal Controls in Coefficients for Multi-dimensional Thermistor Problem

Abstract
The paper studies a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field $$u=u(x)$$ and temperature $$\theta (x)$$. The coefficient b of operator $$\mathrm {div}\,\left( b(x)\, \nabla \, \theta (x)\right)$$ is used as the control in $$W^{1,q}(\varOmega )$$ with $$q>N$$. The optimal control problem is to minimize the discrepancy between a given distribution $$\theta _d\in L^1(\varOmega )$$ and the temperature of thermistor $$\theta \in W^{1,\gamma }_0(\varOmega )$$ by choosing an appropriate anisotropic heat conductivity b(x). Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an “approximation approach” and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.
Ciro D’Apice, Umberto De Maio, Peter I. Kogut
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