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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Algebra and Geometry Through Hamiltonian Systems

Hamiltonian systems are considered to be the prime tool of classical and quantum mechanics. The proper investigation of such systems usually requires deep results from algebra and geometry. Here we present several results which in some sense go the opposite way: the knowledge about the integrable system enables us to obtain results on geometric and algebraic structures which naturally appear in such problems. All the results were obtained by employees of the Chair of Differential Geometry and Applications in Moscow State University in 2011–2012.

Anatoly T. Fomenko, Andrei Konyaev

### Chapter 2. On Hyperbolic Zeta Function of Lattices

This chapter provides an overview of the theory of hyperbolic zeta function of lattices. A functional equation for the hyperbolic zeta function of Cartesian lattice is obtained. Information about the history of the theory of the hyperbolic zeta function of lattices is provided. The relations with the hyperbolic zeta function of nets and Korobov optimal coefficients are considered.

L. P. Dobrovolskaya, M. N. Dobrovolsky, N. M. Dobrovol’skii, N. N. Dobrovolsky

### Chapter 3. The Distribution of Values of Arithmetic Functions

Let us usual

$$\tau _{k}(n)$$

denote the number of ways

$$n$$

may be written as a product of

$$k$$

fixed factors. In this chapter there introduce the notation

$$D_k(x) = \sum _{n \le x} \tau _k(n).$$

We show that the asymptotic formula for

$$D_k(x)$$

is changing with growing values of

$$k$$

and present specific values of

$$k$$

, which is a change.

G. V. Fedorov

### Chapter 4. On the One Method of Constructing Digital Control System with Minimal Structure

We consider the linear digital control system with invariable matrix

$$A.$$

In this report we introduce one method which permit to obtain the characteristic of completely controllability and construct the matrix of control

$$B$$

with minimal structure without calculation of eigenvalues of matrix

$$A.$$

V. V. Palin

### Chapter 5. On Norm Maps and “Universal Norms” of Formal Groups Over Integer Rings of Local Fields

We review and investigate norm maps and “universal norms” of formal groups over integer ring of local and quasi-local fields. Theorem on triviality of universal norm group of one dimensional fornal groups of reduction height 3 over integer ring of local and quasi-local fields is presented. The theorem on triviality of universal norm group is based on the lemma about function that gives the minimal degree of elements of the subgroup

$$F_{K}^{t}$$

of the group

$$F{_K}$$

that contains the norm group

$$N_{L/K} (F^{n}_L)$$

. In the case of formal groups of elliptic curves the function has used by O. N. Vvedenskii and is denoted as

$$\mu (n)$$

. The proof of the lemma is also presented.

Nikolaj M. Glazunov

### Chapter 6. Assignment of Factors Levels for Design of Experiments with Resource Constraints

An optimal procedure for factors levels assignment is proposed. The procedure is based on choice of levels number proportionally to factor significance, guaranteed estimation of entropy, and 1D-parametrization of iteration process for multidimensional mapping fixed point finding. Solution existence and convergence of the procedure is proved.

S. A. Smirnov, O. O. Glushchenko, K. A. Ilchuk, I. L. Makeenko, N. A. Oriekhova

### Chapter 7. How to Formulate the Initial-Boundary-Value Problem of Elastodynamics in Terms of Stresses?

In case when loadings are given on all the boundary of deformable solid, the initial-boundary-value problem for obtaining stress-strain state seems to be more suitable and effective if it is formulated and investigated in terms of stress tensor components. In this chapter typical peculiarities of some (in chronological order) formulations the initial-boundary-value problems in dynamic theory for isotropic linear elastic solid are discussed.

D. V. Georgievskii

### Chapter 8. Finite-Difference Method of Solution of the Shallow Water Equations on an Unstructured Mesh

In the chapter we consider a linearized system of shallow water equations. Since this problem should be solved in domains being seas and oceans (or their parts), then solving this problem should use unstructured meshes to approximate domains under consideration properly. This problem was studied in the papers [

1

4

]. Here we consider finite-difference approximation of these equations, prove convergence of approximate solution to the differential one, and provide a number of numerical experiments confirming theoretical results. We also carried out some numerical experiments for real geographic objects.

G. M. Kobelkov, A. V. Drutsa

### Chapter 9. Dynamics of Vortices in Near-Wall Flows with Irregular Boundaries

Behavior of stationary vortices in near-wall flows with irregular boundaries is investigated. The vortices were shown to locate in the critical points of flow and to be characterized not only by its strength but by the eigenfrequency that specifies precession of the vortex about the flow critical point along the small trajectory. Due to eigenfrequency, the stationary vortex responds selectively on external periodical perturbations. The last cause low-frequency vortex motion with large amplitudes and when the frequency of external perturbations is to be near the vortex eigenfrequency the vortex moves away from the critical point. So, dependency of the amplitude of perturbed vortex motion from the frequency of external perturbations has the resonance character. The resonant perturbations are shown to cause chaotization of local circulation zones generated by stationary vortices.

I. M. Gorban, O. V. Khomenko

### Chapter 10. Strongly Convergent Algorithms for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems

This chapter deals with a variational inequality problem over the set of solutions the equilibrium problem or over the set of solutions the system of equilibrium problems in a real Hilbert space. Several new iterative algorithms are proposed. Strong convergence theorems for algorithms are proved. The convergence of iterative algorithms with the presence of computational errors without traditional summability conditions also studied. To this aim, we use new Mainge’s techniques for analysis non–Fejerian iterative processes (Set–Valued Analysis. 16, 899–912, 2008).

### Chapter 11. Multivalued Dynamics of Solutions for Autonomous Operator Differential Equations in Strongest Topologies

We consider nonlinear autonomous operator differential equations with pseudomonotone interaction functions satisfying

$$(S)$$

-property. The dynamics of all weak solutions on the positive time semi-axis is studied. We prove the existence of a trajectory and a global attractor in a strongest topologies and study their structure. As a possible application, we consider the class of high-order nonlinear parabolic equations.

Mikhail Z. Zgurovsky, Pavlo O. Kasyanov

### Chapter 12. Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System

In this paper we study structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients we prove that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in

$$L^{\infty }(\varOmega )\cap H_0^1(\varOmega )$$

.

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Mikhail Z. Zgurovsky

### Chapter 13. Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations

In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.

Pavlo O. Kasyanov, Luisa Toscano, Nina V. Zadoianchuk

### Chapter 14. Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation

Sufficient conditions for the existence of an inertial manifold for the equation

$$u_{tt}-2\gamma _{s} \varDelta u_t +2\gamma _{w} u_t - \varDelta u = f(u)$$

,

$$\gamma _{s} > 0$$

,

$$\gamma _{w} \ge 0$$

are found. The nonlinear function

$$f$$

is supposed to satisfy Lipschitz property. The proof is based on construction of a new inner product in the phase space in which the conditions of a general theorem on the existence of inertial manifolds for an abstract differential equation in a Hilbert space are satisfied.

Natalia Chalkina

### Chapter 15. On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain

We consider the reaction-diffusion equation with multivalued function of interaction in an unbounded domain. Conditions on the parameters of the problem can not guarantee the uniqueness of the solution of the Cauchy problem. In this work we focus on the study of long-term forecasts of the state functions of reaction-diffusion equation with use of the theory of global attractors for multivalued semiflows. It is obtained the results of the existence and properties of all weak solutions. We obtain the standard a priori estimates for weak solutions of the investigated problem, prove the existence of weak solutions, the existence of global and trajectory attractors for the problem in phase and extended phase spaces respectively. We provide the regularity properties for all globally defined weak solutions and their global and trajectory attractors. The results can be used for the investigation of specific physical models including combustion models in porous media, conduction models of electrical impulses into the nerve endings, climate models.

Nataliia V. Gorban, Pavlo O. Kasyanov

### Chapter 16. On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity

We consider autonomous damped wave equation with discontinuous nonlinearity. The long-term prognosis of the state functions when the conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem are studied. We prove the existence of a global attractor and investigate its structure. It is obtained that trajectory of every weak solution defined on

$$[0;+\infty )$$

tends to a fixed point.

Nataliia V. Gorban, Oleksiy V. Kapustyan, Pavlo O. Kasyanov, Liliia S. Paliichuk

### Chapter 17. On the Regularities of Mass Random Phenomena

This note presents a not very well known result concerning the frequentist origins of probability. This result provides a positive answer to the question of existence of statistical regularities of so called

mass phenomena, using the terminology of A. N. Kolmogorov [

20

]. It turns out, that some closed in weak-

$$*$$

topology family of finitely-additive probabilities plays the role of the statistical regularity of any such phenomenon. If the mass phenomenon is stochastic, then this family degenerates into a usual countably-additive probability measure. The note provides precise definitions, the formulation and the proof of the theorem of existence of statistical regularities, as well as the examples of their application.

Victor I. Ivanenko, Valery A. Labkovsky

### Chapter 18. Optimality Conditions for Partially Observable Markov Decision Processes

This note describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs). The objective criterion is either minimization of total discounted costs or minimization of total nonnegative costs. It is well-known that a POMDP can be reduced to a Completely Observable Markov Decision Process (COMDP) with the state space being the sets of believe probabilities for the POMDP. Thus, a policy is optimal in POMDP if and only if it corresponds to an optimal policy in the COMDP. Here we provide sufficient conditions for the existence of optimal policies for COMDP and therefore for POMDP.

Eugene A. Feinberg, Pavlo O. Kasyanov, Mikhail Z. Zgurovsky

### Chapter 19. On Existence of Optimal Solutions to Boundary Control Problem for an Elastic Body with Quasistatic Evolution of Damage

We study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We use the damage field

$$\zeta =\zeta (t,x)$$

as an internal variable which measures the fractional decrease in the stress-strain response. When

$$\zeta =1$$

the material is damage-free, when

$$\zeta =0$$

the material is completely damaged, and for

$$0<\zeta <1$$

it is partially damaged. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation, whereas the model for the stress in elastic body is given as

$$\varvec{\sigma }=\zeta (t,x) A\mathbf {e}({\mathbf {u}})$$

. The optimal control problem we consider in this paper is to minimize the appearance of micro-cracks and micro-cavities as a result of the tensile or compressive stresses in the elastic body.

Peter I. Kogut, Günter Leugering

### Chapter 20. On Existence and Attainability of Solutions to Optimal Control Problems in Coefficients for Degenerate Variational Inequalities of Monotone Type

In this chapter we study an optimal control problem for a nonlinear monotone variational inequality with degenerate weight function and with the coefficients which we adopt as controls in

$$L^\infty (\varOmega )$$

. Since these types of variational inequalities can exhibit the Lavrentieff phenomenon, we consider the optimal control problem in coefficients in the so-called class of

$$H$$

-admissible solutions. Using a special version of celebrated Compensated Compactness Lemma and the direct method of Calculus of Variations we discuss the solvability of the above optimal control problem and prove attainability of

$$H$$

-optimal pairs via optimal solutions of some non-degenerate perturbed optimal control problems.

Olga P. Kupenko

### Chapter 21. Distributed Optimal Control in One Non-Self-Adjoint Boundary Value Problem

We prove the solvability of the optimal control problem for elliptic equation with nonlocal boundary conditions in a circular sector with terminal quadratic cost functional in the class of distributed controls.

V. O. Kapustyan, O. A. Kapustian, O. K. Mazur

### Chapter 22. Guaranteed Safety Operation of Complex Engineering Systems

A system strategy to estimation of guaranteed survivability and safety operation of complex engineering systems (CES) is proposed. The strategy is based on timely and reliable detection, estimation, and forecast of risk factors and, on this basis, on timely elimination of the causes of abnormal situations before failures and other undesirable consequences occur. The principles that underlie the strategy of the guaranteed safety operation of CES provide a flexible approach to timely detection, recognition, forecast, and system diagnostic of risk factors and situations, to formulation and implementation of a rational decision in a practicable time within an irremovable time constraint. The system control of complex objects is realized. The essence of such control is a systemically coordinated evaluation and adjustment of the operational survivability and safety during the functioning process of an object. The diagnostic unit, which is the basis of a safety control algorithm for complex objects in abnormal situations, is developed as an information platform of engineering diagnostics. By force of systematic and continuous evaluation of critical parameters of object’s functioning in the real time mode, the reasons, which could potentially cause the object’ tolerance failure of the functioning in the normal mode, are timely revealed.

Nataliya D. Pankratova, Andrii M. Raduk

### Backmatter

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