An important stage in problem solving process for aerospace and aerostructures designing is calculating their main characteristics optimization. The results of the four constrained optimization problems related to the design of various technical systems: such as determining the best parameters of welded beams, pressure vessel, gear, spring are presented. The purpose of each task is to minimize the cost and weight of the construction. The object functions in optimization practical problem are nonlinear functions with a lot of variables and a complex layer surface indentations. That is why using classical approach for extremum seeking is not efficient. Here comes the necessity of using such methods of optimization that allow to find a near optimal solution in acceptable amount of time with the minimum waste of computer power. Such methods include the methods of Swarm Intelligence: spiral dynamics algorithm, stochastic diffusion search, hybrid seeker optimization algorithm. The Swarm Intelligence methods are designed in such a way that a swarm consisting of agents carries out the search for extremum. In search for the point of extremum, the particles exchange information and consider their experience as well as the experience of population leader and the neighbors in some area. To solve the listed problems there has been designed a program complex, which efficiency is illustrated by the solutions of four applied problems. Each of the considered applied optimization problems is solved with all the three chosen methods. The obtained numerical results can be compared with the ones found in a swarm with a particle method. The author gives recommendations on how to choose methods parameters and penalty function value, which consider inequality constraints.

The author presents the spectral method of determining relatively optimal control in case of incomplete infor- mation about the state vector for multidimensional nonlinear continuous stochastic systems, which are governed by Itô’s stochastic differential equations. The quality criterion is given as the mean of the function determined on the system tracks. One should find the equation that depends on state vector time and coordinates, of which there is exact information from measuring system. Solving the problem of finding optimal control is based upon the actual sufficient optimum condition and the ratios derived from them. These ratios, which determine nonlinear continuous stochastic systems optimal control in case of incomplete state vector information (Fokker-Planck-Kolmogorov and Bellman equation systems and the tying ratios that allow to determine control structure) with the help of a spectral transformation usually lead to the system of nonlinear equations for the coefficients of optimal control and optimal state vector probability density coordinates expansion into a basic system functions series. This nonlinear equations system solving method does not depend on the chosen basis, it is solved either with iterative methods or with reducing it to the equivalent method of unconditional optimization with the following usage of zero-order method, including metaheuristic methods global extremum search. In this article, determin- ing optimal control goes down to improving control spectral characteristics in space (in the coefficient space of dividing control according to the orthonormal system functions). The author dwells upon the issue of taking so called geometrical control constraints into account as a special case. Using the spectral form of the mathematical description it is necessary to reduce spectral characteristics of functions, operators and functionals to some chosen orders, and therefore moving to finite-dimensional problems of optimization. Reduction order choice and basis system choice determine the approximate solution accuracy for optimal control problem.

The article deals with a new approximation method for enhanced accuracy measurement system errors distribution. The method is based upon the mistie analysis of this system and a more robust design data. The method is considered on the example of comparison of Automatic Dependent Surveillance - Broadcast (ADS-B) with ground radar warning system used at present. The peculiarity of the considered problem is that the target parameter (aircraft swerve) value may drastically change in the scale of both measurement systems errors during observation. That is why it is impossible to determine the position of the aircraft by repeatedly observing it with ground radar warning system. It is only possible to compare the systems’ one-shot measurements, which are called errors here. The article considers that the distribution of robust measurement system errors probability density (the system that has been continuously in operation) is known, the histogram of errors is given and it is needed to obtain an asymptotic estimate of errors occurrence distribution for a new improved measurement system.This approach is based on cumulant analysis of measurement systems error distribution functions. The approach allows us to carry out the reduction of corresponding infinite series properly. The author shows that due to measurement systems independency, their errors distribution cumulants are connected by a simple ratio, which allow to calculate the values easily. To reconstruct distribution initial form one should use Edgeworth’s asymptotic series, where a normal distribution derivative is used as a basis function. The latter is proportional to Hermitian polynomial, thus the series can be considered as an orthogonal decomposition.The author reveals the results of coordinate error component distribution calculation; the error is measured when the normal line lies towards aircraft path, using error statistics experimental information obtained in ”RI of Aeronavigation” Branch of FSUE SRI of Civil Aviation.

The article considers the usage of metaheuristic methods of constrained global optimization: “Big Bang - Big Crunch”, “Fireworks Algorithm”, “Grenade Explosion Method” in parameters of dynamic systems estimation, described with algebraic-differential equations. Parameters estimation is based upon the observation results from mathematical model behavior. Their values are derived after criterion minimization, which describes the total squared error of state vector coordinates from the deduced ones with precise values observation at different periods of time. Paral- lelepiped type restriction is imposed on the parameters values. Used for solving problems, metaheuristic methods of constrained global extremum don’t guarantee the result, but allow to get a solution of a rather good quality in accepta- ble amount of time. The algorithm of using metaheuristic methods is given. Alongside with the obvious methods for solving algebraic-differential equation systems, it is convenient to use implicit methods for solving ordinary differen- tial equation systems. Two ways of solving the problem of parameters evaluation are given, those parameters differ in their mathematical model. In the first example, a linear mathematical model describes the chemical action parameters change, and in the second one, a nonlinear mathematical model describes predator-prey dynamics, which characterize the changes in both kinds’ population. For each of the observed examples there are calculation results from all the three methods of optimization, there are also some recommendations for how to choose methods parameters. The obtained numerical results have demonstrated the efficiency of the proposed approach. The deduced parameters ap- proximate points slightly differ from the best known solutions, which were deduced differently. To refine the results one should apply hybrid schemes that combine classical methods of optimization of zero, first and second orders and heuristic procedures.

In this article an attempt is made to explain the nature of differences in measurements of forces and moments, which influence an aircraft at take-off and landing when testing on different types of stands. An algorithm for numerical simulation of unsteady separated flow around airfoil is given. The algorithm is based on the combination of discrete vortex method and turbulent boundary layer equations. An unsteady flow separation modeling has been used. At each interval vortex method was used to calculate the potential flow around airfoils located near a screen. Calculated pressures and velocities were then used in boundary layer calculations to determine flow separation points and separated vortex in- tensities. After that calculation were made to determine free vortex positions to next time step and the process was fulfilled for next time step. The proposed algorithm allows using numeric visualization to understand physical picture of flow around airfoil moving close to screen. Three different ways of flow modeling (mirror method, fixed or movable screens) were tested. In each case the flow separation process, which determines pressure distribution over airfoil surface and influ- ences aerodynamic performance, was viewed. The results of the calculations showed that at low atitudes of airfoil over screen mirror method over predicts lift force compared with movable screen, while fixed screen under predicts it. The data obtained can be used when designing equipment for testing in wind tunnels.

The experience of developing unmanned fighting vehicles indicates that the main challenge in this field reduces itself to creating the systems which can replace the pilot both as a sensor and as the operator of the flight. This problem can be partial- ly solved by introducing remote control, but there are certain flight segments where it can only be executed under fully inde- pendent control and data support due to various reasons, such as tight time, short duration, lack of robust communication, etc. Such stages also include close-range air combat maneuvering (CRACM) - a key flight segment as far as the fighter's purpose is concerned, which also places the highest demands on the fighter's design. Until recently the creation of an unmanned fighter airplane has been a fundamentally impossible task due to the absence of sensors able to provide the necessary data support to control the fighter during CRACM. However, the development prospects of aircraft hardware (passive type flush antennae, op- tico-locating panoramic view stations) are indicative of producing possible solutions to this problem in the nearest future. There- fore, presently the only fundamental impediment on the way to developing an unmanned fighting aircraft is the problem of cre- ating algorithms for automatic trajectory control during CRACM. This paper presents the strategy of automatic trajectory con- trol synthesis by a two-stage dynamic system aiming to reach the conditions specified with respect to an object in pursuit. It contains certain results of control algorithm parameters impact assessment in regards to the pursuit mission effectiveness. Based on the obtained results a deduction is drawn pertaining to the efficiency of the offered method and its possible utilization in au- tomated control of an unmanned fighting aerial vehicle as well as organizing group interaction during CRACM.

For the time being, a combat-capable trainer aircraft has already been used as a light attack aircraft. The quality of mission effectiveness evaluation depends on the degree of relevance of mathematical models used. It is known that the mission efficiency is largely determined by maneuvering capabilities of the aircraft which are realized most fully in extreme angle of attack flight modes. The article presents the study of the effect of Reynolds number, angle of attack and position on the external sling on the parameters characterizing the state of separated-vortex flow, which was conducted using software complexes such as Solid Works and Ansys Fluent. There given the dependences of the observed parameters for stationary and nonstationary cases of light attack aircraft movement. The article considers the influence of time constants, which characterize the response rate and delaying of separated flow development and attached flow recovery on the state of separated-vortex flow. The author mentions how the speed of angle of attack change influences lift coefficient of a light attack aircraft with external slings due to response rate and delaying of separated flow development and attached flow recovery. The article describes the mathematical model invented by the authors. This is the model of the movements of light attack aircraft with external slings within a vertical flight maneuver, considering the peculiarities of separated-vortex flow. Using this model, there has been obtained the parameters of light attack aircraft output path from the pitch using large angles of attack. It is demonstrated that not considering the peculiarities of the separated-vortex flow model of light attack aircraft movements leads to certain increase of height loss at the pullout of the maneuver, which accordingly makes it possible to decrease the height of the beginning of the pullout.

Maximum principle for subsonic flow is fair for stationary irrotational subsonic gas flows. According to this principle, if the value of the velocity is not constant everywhere, then its maximum is achieved on the boundary and only on the boundary of the considered domain. This property is used when designing form of an aircraft with a maximum critical value of the Mach number: it is believed that if the local Mach number is less than unit in the incoming flow and on the body surface, then the Mach number is less then unit in all points of flow. The known proof of maximum principle for subsonic flow is based on the assumption that in the whole considered area of the flow the pressure is a function of density. For the ideal and perfect gas (the role of diffusion is negligible, and the Mendeleev-Clapeyron law is fulfilled), the pressure is a function of density if entropy is constant in the entire considered area of the flow. Shows an example of a stationary subsonic irrotational flow, in which the entropy has different values on different stream lines, and the pressure is not a function of density. The application of the maximum principle for subsonic flow with respect to such a flow would be unreasonable. This example shows the relevance of the question about the place of the points of maximum value of the velocity, if the entropy is not a constant. To clarify the regularities of the location of these points, was performed the analysis of the complete Euler equations (without any simplifying assumptions) in 3-D case. The new proof of the maximum principle for subsonic flow was proposed. This proof does not rely on the assumption that the pressure is a function of density. Thus, it is shown that the maximum principle for subsonic flow is true for stationary subsonic irrotational flows of ideal perfect gas with variable entropy.

A number of papers deals with the heat conductivity of composite materials: Zarubin et al used new approaches to the problem of evaluation of the effective heat conductivity coefficients of composite material with ball inclusions. We used variational analysis for a simplified model in a vicinity of inclusion. Contemporary computers allow implementing another approach to solving the problem of the effective heat conductivity: it may be modelled by the Brownian motion of virtual heat particles. The main idea is to obtain the exact formula for the heat conductivity for a homogeneous material and subsequently obtain a statistical evaluation of this formula for a composite material.In the present paper we compare two methods for finding the effective heat conductivity coefficients of composite materials by modeling the process of heat conduction via the Brownian motion of virtual heat particles. We consider a composite with ball inclusions of a material with heat conductivity and heat capacity coefficientsdiffering from those of the matrix material. In a computational experiment, we simulate the process of heatconduction through a flat layer of the composite material, which has been heated on one side at the initial moment. In order to find the confidence interval for the effective heat conductivity coefficient, we find, by means of statistics, either the displacement of the center of heat energy, or the probability of a virtual particle to pass through the layer during a certain time. We compare our results with theoretical assessments suggested by other authors.

We study local symmetries of the generalized Proudman-Johnson equation. Symmetries of a partial differential equation may be used to find its invariant solutions. In particular, if <р is a characteristic of a symmetry for a PDE Н = О then the <р-invariant solution of the PDE is a solution to the compatible over-determined system Н = О, < р = О. We show that the Lie algebra of local symmetries for the generalized Proudman-Johnson equation is infinite-dimensional. Reductions of equation with respect to the local symmetries provide ordinary differential equations that describe invariant solutions. For a certain value of the parameter entering the equation we find some cases when the reduced ODE is integrable by quadratures and thus allows one to construct exact solutions. Differential coverings (or Wahlquist-Estabrook prolongation structures, or zero-curvature representations, or integrable extensions, etc.) are of great importance in geometry of PDEs. The theory of coverings is a natural framework for dealing with inverse scattering constructions for soliton equations, Bäcklund transformations, recursion operators, nonlocal symmetries and nonlocal conservation laws, Darboux transformations, and deformations of nonlinear PDEs. In the last section we show that in the case of a certain value of the parameter entering the equation it has a differential covering. This property is referred to as a Lax integrability.

The behavior of the soliton type solutions to the KdV-Burgers equation is studied numerically in the case of non- homogeneous dissipative media. A soliton moves from left to right and it does not change its form. The solitons with great- er amplitude are narrower and move faster. The aim of the presented research is to study the behavior of the soliton that, while moving in nondissipative medium encounters a barrier (finite or infinite) with finite constant dissipation; one may imagine an impulse of light meeting on its way a partially absorbing layer. The modelling included the case of a finite dis- sipative layer similar to a wave passing through the air-glass-air as well as a wave passing from a nondissipative layer into a dissipative one (similar to the passage of light from air to water). The present paper is a continuation of the authors’ pub- lications. New results include a numerical model of the wave’s behavior for different types of the media non-homogeneity. The dissipation predictably results in reducing the soliton’s amplitude, but some new effects occur in the case of finite piecewise constant barrier on the soliton path: after the wave leaves the dissipative barrier it retains, on the whole, a soliton form yet some small and rapidly decreasing oscillations arises in front of the soliton. These oscillations are getting larger and spread as the soliton is moving of the barrier; the distance between the soliton and the oscillation grows. That is, the oscillations are faster than the soliton. The modelling used the Maple software PDETools packet; these activities were time and resources consuming.

This problem is related to the safety problem in the area of forest fires. It is well known that is possible to extinguish a fire, for example, by means of a powerful air stream. Such flow arises from the explosive shock wave. To enhance the im- pact of the blast wave can be used an explosive charge of annular shape. The shock wave, produced by the explosion, in- creased during moves to the center and can serve as a means of transportation dust in the seat of the fire. In addition, emerging after the collapse of a converging shock wave strong updraft can raise dust on a greater height and facilitate fire extinguishing, precipitating dust over a large area. This updraft can be dangerous for aircraft that are in the sky above the fire. To determine the width and height of the danger zone performed the numerical simulation of the ring of the explosion and the subsequent movement of dust and gas mixtures. The gas is considered ideal and perfect. The explosion is modeled as an instantaneous increase in the specific internal energy in an annular zone on the value of the specific heat of explosives. The flow is consid- ered as two-dimensional, and axisymmetric. The axis of symmetry perpendicular to the Earth surface. This surface is considered to be absolutely rigid and is considered as the boundary of the computational domain. On this surface is exhibited the condition of no motion. For the numerical method S. K. Godunov is used a movable grid. One system of lines of this grid is moved in accordance with movement of the shock wave. Others lines of this grid are stationary. The calculations were per- formed for different values of the radii of the annular field and for different sizes of rectangular cross-sectional of the annular field. Numerical results show that a very strong flow is occurring near the axis of symmetry and the particles rise high above the surface. These calculations allow us to estimate the sizes of the zone of danger in specific situations.

For the current Lie algebra on the three-dimensional torus with non-standard Lie bracket some properties, in the case when the sum of adjoint and coadjoint operators on infinite-dimensional Lie algebra with scalar product has a finite norm are established. For the Landau-Lifshitz equation in the three-dimensional torus it is established that the operatorm mS = (ad+ ad* ) / 2mhas a finite norm, though it is not true the operators of the adjoint action adm and coadjoint ac-mtion ad ∗ . It follows that the coefficients of expansion of the solution in an orthonormal basis of eigenvectors of the La- place operator satisfy Lipschitz conditions. Thus, for the Landau-Lifshitz equation on the three-dimensional torus situationis similar to the equations of an ideal fluid and Korteweg de Vries. On the other hand, if for the equations of fluid dynamicsand Korteweg de Vries, this fact has been established in a general way, for the Landau-Lifshitz equation in the three- dimensional torus it is obtained specifically through the calculation of structural constants and the matrix of the coadjoint action for the current algebra with non-standard Lie bracket.

The geometric properties of conjugated circular arcs connecting two points on the plane with set directions of tan- gent vectors are studied in the work. It is shown that pairs of conjugated circular arcs with the same conditions in frontier points create one-parameter set of smooth curves tightly filling all the plane. One of the basic properties of this set is the fact that all coupling points of circular arcs are on the circular curve going through the initially given points. The circle radius depends on the direction of tangent vectors. Any point of the circle curve, named auxiliary in this work, determines a pair of conjugated arcs with given boundary conditions. One more condition of the auxiliary circle curve is that it divides the plane into two parts. The arcs going from the initial point are out of the circle limited by this circle curve and the arcs coming to the final point are inside it. These properties are the basis for the method of conjugated circular arcs tracing pro- posed in this article. The algorithm is rather simple and allows to fulfill all the needed plottings using only the divider and ruler. Two concrete examples are considered. The first one is related to the problem of tracing of a pair of conjugated arcs with the minimal curve jump when going through the coupling point. The second one demonstrates the possibility of trac- ing of the smooth curve going through any three points on the plane under condition that in the initial and final points the directions of tangent vectors are given. The proposed methods of conjugated circular arcs tracing can be applied in solving of a wide variety of problems connected with the tracing of cam contours, for example pattern curves in textile industry or in computer-aided-design systems when programming of looms with numeric control.

Measures of noncompactness are numerical characteristics of bounded subsets of metric space, equal to zero on relatively compact subsets. The quantitative characteristic of measure of noncompactness of metric space subset was introduced by K. Kuratovskiy in 1930 in connection with problems of general typology. Different measures of noncompactness exist. Measures of noncompactness are a simple and useful instrument for any problem solving. So the theory of measures of noncompactness is still developing and it finds more and more new applications in different branches of mathematics. In this article measures of noncompactness are used to study inequalities, more exactly the extension of an equality, studied in many works and having wide application. For example in the works by Yu.A. Dubinskiy, J.-L. Lions and E. Magenes this inequality is proved for embedding operators in Banach spaces (a particular case of metric spaces). Then it is used to prove the solvability of nonlinear elliptic and parabolic equations. In contrast to these authors in this work the compactness of the embedding operator is not assumed in the study of the inequality. Furthermore, in metric space for the analogue of the inequality, written via any numerical characteristics of bounded subsets (not necessarily measures of noncompactness), the needed and sufficient conditions of the correctness of this analogue are received. In case if numerical characteristic of a set is a measure of noncompactness, the conclusion of this result is a new criterion of compactness of the operator (not necessarily linear) under the condition of compactness of another one.The results of this work generalize some results achieved by the author previously.

The problem of optimal control of epidemic through vaccination and isolation, taking into account latent period is considered. The target function is minimized-functionality summarizing costs on epidemic prevention and treatment and also considering expenses on infected people left at the end of control T who may be a new source of epidemic. On the left endpoint of the integration segment initial data is given-quantity of infected and confirmed people at the moment t, the right endpoint is free. The dynamic constraints are written by way of a system of simple differential equations describing the speed of changes of number of subjected to infection and number of already infected. Besides the inhomogeneous community is considered, consisting of four age groups (babies, preschool children, school children and adults). The speed of vaccination (number of vaccinated per a time unit) and isolation speed are used as the control functions. There are some restrictions on control above and below. The latent period is described by the constant h and is part of the equation describing the contamination speed of people as a retarding in argument t, i.e. a person being in a latent period infects others not being aware of his disease. For problem solving Pontryagin maximum principle is used where it can be seen that the control is piecewise constant. The result of numerical implementation of discrete problem of optimal control is given. The conclusions are made that the latent period significantly influence the incidence rate and as consequence the costs on epidemic suppression. The programme based on the programming language Delphi gives an opportunity to estimate the scale of epidemic at different initial data and restrictions on control as well as to find an optimal control minimizing costs on epimedic suppression.

This article studies the elastic filtration oil drive of oil in a layer based on the estimation of risks of environmental oil pollution because of accidental releases. The model of oil spillage and resorption by the precoat is based on continuity equation and Darcy rule as well as on equations of state taking into account fluid compressibility due to pressure. Filtering area is a line between the precoat and air. Oil filtering area is limited by soil surface below and by free surface above, its equation is known beforehand and is to be defined. The case of soil pollution from the point source, which is the point of fracture of pipeline or borehole, is considered. Upper and approximate estimates of the oil pollution radius due to different types of underlying terrains and to oil characteristics as well as to environmental conditions. The dynamics of oil free surface depending on spillage radii is calculated and presented. The estimates of temporary duration of oil filtering by the precoat in terms of light ends and soil type are made. The thickness of the oil film and the square of the spill upon condition of constant speed of oil spillage, horizontal position of underlying terrain and the proximity of pressure to normal are determined. For the numerical implementation of the model different cases of oil spillage were considered. Under given values of air temperature, soil porosity and filtration speed the pollution radii according to light end, the time from the moment of accident till the leakage suppression and the speed of oil spillage was calculated.

The patterns of early repayment in multi-period credit transactions from internal rate of return view are analyzed in this article. At the early repayment of debt a borrower pays to a moneylender a sum of money equal to current balance (the status of credit account). After that the contracted is finished. However in reality the borrower pays to a moneylender a sum of all unpaid payment, and the moneylender in his turn gives back to the borrower a part of this sum. To calculate it the rule 78 is often used. The borrower’s extra costs connected with the application of the rule 78 may be considered as extra punitive sanctions at the early repayment of debt. However the sanction value achieves its maximum when the maturity is between 57 and 66 percent of the total of the loan term what is contrary to good sense.

The situation changes drastically if the sanctions are watched from the point of view of the internal interest rate, which may also determine the cost of credit. In this case, as it was shown, the interest rate can greatly exceed the originally announced rate. The value of this exceedance monotonously increases with the increase in the number of outstanding payments in case of early repayment of the debt. From this it follows that the sanctions associated with the use of rule 78, monotonically increase too. This is consistent with the idea of sanctions from the lender's perspective. Namely, the sanctions must become the smaller, the less time remains prior to the initial date of repayment of the loan. 

The article considers the problem of the dynamic risk management of the investment portfolio using future contracts. The management starts with the concept of effective inhomogeneous portfolios, which contain futures together with underlying asserts. The effective portfolios are defined as the ones of the minimal dispersion with the expected return greater or equal to the specified value. Risk is measured by the probability of losing of a certain part of the portfolio value. The control parameters are the number of futures for each asset of portfolio, which is defined from the condition of effectiveness of portfolio and risk acceptability on each step.The effective adaptive strategies of portfolio risk management together with comparative analysis on a concrete example are presented. The proposed approach provides the forecast correction of the expected income and its variance for the assets with the emergence of new data. The financial time series are determined by volatility clustering, i.e. relative or absolute price changes tend to keep high or low magnitude for some time, with the result that clusters are created - periods of high or low volatility. Then adaptive estimate of correlational relationships between asset prices are essential because the degree of correlational relationship also changes in time. So the correlation of future and spot price changes considerably increases while approaching to performance of contracts. For taking into account of data instability of dispersion and correlation simple methods of volatility forecasting and correlation of relative changes of price data based on exponential smoothing are implemented.