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We consider the approaches to the construction of methods for solving four-dimensional programming problems for calculating directions for multiple minimizations of smooth functions on a set of a given set of linear equalities. The approach consists of two stages.

At the first stage, the problem of quadratic programming is transformed by a numerically stable direct multiplicative algorithm into an equivalent problem of designing the origin of coordinates on a linear manifold, which defines a new mathematical formulation of the dual quadratic problem. For this, a numerically stable direct multiplicative method for solving systems of linear equations is proposed, taking into account the sparsity of matrices presented in packaged form. The advantage of this approach is to calculate the modified Cholesky factors to construct a substantially positive definite matrix of the system of equations and its solution in the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made in the position of the next processed row of the matrix, which allows the use of static data storage formats.

At the second stage, the necessary and sufficient optimality conditions in the form of Kuhn–Tucker determine the calculation of the direction of descent — the solution of the dual quadratic problem is reduced to solving a system of linear equations with symmetric positive definite matrix for calculating of Lagrange's coefficients multipliers and to substituting the solution into the formula for calculating the direction of descent.

It is proved that the proposed approach to the calculation of the direction of descent by numerically stable direct multiplicative methods at one iteration requires a cubic law less computation than one iteration compared to the well-known dual method of Gill and Murray. Besides, the proposed method allows the organization of the computational process from any starting point that the user chooses as the initial approximation of the solution.

Variants of the problem of designing the origin of coordinates on a linear manifold, a convex polyhedron and a vertex of a convex polyhedron are presented. Also the relationship and implementation of methods for solving these problems are described.

The dynamic process can be in equal degree adequately prototyped by a family of Lagrange's systems. Symmetry group ‘wanders’ on this family: systems are transformed from one into another. In this work we show that under determined condition the first integral can be obtained by a simple calculations on some of such groups. The main purpose of the work is to show usefulness of wandering symmetry concept. The considered example: flat motion of a charged particle in magnetic field in presence of viscous friction. With the help of three wandering symmetry first integral is calculated.

In this paper, using our bifurcation-geometric approach, we study global dynamics and solve the problem of the maximum number and distribution of limit cycles (self-oscillating regimes corresponding to states of dynamical equilibrium) in a planar polynomial mechanical system of the Euler–Lagrange–Liйnard type. Such systems are also used to model electrical, ecological, biomedical and other systems, which greatly facilitates the study of the corresponding real processes and systems with complex internal dynamics. They are used, in particular, in mechanical systems with damping and stiffness. There are a number of examples of technical systems that are described using quadratic damping in second-order dynamical models. In robotics, for example, quadratic damping appears in direct-coupled control and in nonlinear devices, such as variable impedance (resistance) actuators. Variable impedance actuators are of particular interest to collaborative robotics. To study the character and location of singular points in the phase plane of the Euler–Lagrange–Liйnard polynomial system, we use our method the meaning of which is to obtain the simplest (well-known) system by vanishing some parameters (usually, field rotation parameters) of the original system and then to enter sequentially these parameters studying the dynamics of singular points in the phase plane. To study the singular points of the system, we use the classical Poincarй index theorems, as well as our original geometric approach based on the application of the Erugin twoisocline method which is especially effective in the study of infinite singularities. Using the obtained information on the singular points and applying canonical systems with field rotation parameters, as well as using the geometric properties of the spirals filling the internal and external regions of the limit cycles and applying our geometric approach to qualitative analysis, we study limit cycle bifurcations of the system under consideration.

The article discusses the model of the anthropomorphic type of mechanism of the exoskeleton with links of variable length. Four models of parts of variable length are considered comprehensively: the model link of the exoskeleton of variable length with a resilient member and a rigid strong core; the model of the telescopic link; the model link with the masses in the hinge-joint between them; the link model with an arbitrary number of masses. The differential equations of motion in the form of Lagrange equations of the second kind are made. On the basis of analysis of differential equations of motion for multi-link rod of a mechanical system type, exoskeleton revealed their structure, which allowed us to represent them in vector-matrix form. The General pattern of building matrices are established for the first time and the generalization of the expressions for elements of matrices in two-dimensional case are obtained. New recursive and matrix methods of composing of differential equations of motion are given. A unified approach to constructing differential equations of motion of the exoskeleton based on the developed recursive and matrix methods write differential equations of motion of the proposed exoskeleton. Comparison of the time of writing the differential equations of motion proposed methods, in comparison with the Lagrange equations of the second kind, in the system of computer mathematics Mathematica conducted. An analytical study of the model of the exoskeleton carried out. It was found that for mechanisms with n movable links of the Cauchy problem for systems of differential equations of motion for any initial conditions there is no single and unlimited continue. Control of the exoskeleton is accomplished using the torques which are located in the hinge-joints in the joints of the links and simulating control actions. Numerical investigation of a model of the exoskeleton is made, a comparison of results of calculations for exoskeletons with various models of units is held. A numerical study of the empirical evidence about the man and his movements is used. It is established that the choice structure of the exoskeleton model with lumped masses is more preferable to a model with perfectly rigid strong core. As an exoskeleton, providing comfortable movement of people, and you should repeat the properties of the musculoskeletal system.

The paper presents a mathematical model that describes the process of obtaining biogas from livestock waste. This model describes the processes occurring in a biogas plant for mesophilic and thermophilic media, as well as for continuous and periodic modes of substrate inflow. The values of the coefficients of this model found earlier for the periodic mode, obtained by solving the problem of model identification from experimental data using a genetic algorithm, are given.

For the model of methanogenesis, an optimal control problem is formulated in the form of a Lagrange problem, whose criterial functionality is the output of biogas over a certain period of time. The controlling parameter of the task is the rate of substrate entry into the biogas plant. An algorithm for solving this problem is proposed, based on the numerical implementation of the Pontryagin maximum principle. In this case, a hybrid genetic algorithm with an additional search in the vicinity of the best solution using the method of conjugate gradients was used as an optimization method. This numerical method for solving an optimal control problem is universal and applicable to a wide class of mathematical models.

In the course of the study, various modes of submission of the substrate to the digesters, temperature environments and types of raw materials were analyzed. It is shown that the rate of biogas production in the continuous feed mode is 1.4–1.9 times higher in the mesophilic medium (1.9–3.2 in the thermophilic medium) than in the periodic mode over the period of complete fermentation, which is associated with a higher feed rate of the substrate and a greater concentration of nutrients in the substrate. However, the yield of biogas during the period of complete fermentation with a periodic mode is twice as high as the output over the period of a complete change of the substrate in the methane tank at a continuous mode, which means incomplete processing of the substrate in the second case. The rate of biogas formation for a thermophilic medium in continuous mode and the optimal rate of supply of raw materials is three times higher than for a mesophilic medium. Comparison of biogas output for various types of raw materials shows that the highest biogas output is observed for waste poultry farms, the least — for cattle farms waste, which is associated with the nutrient content in a unit of substrate of each type.

A two-dimensional mathematical model was developed for estimating the stresses in welded joints formed during multipass welding of multilayer steels. The basis of the model is the system of equations that includes the Lagrange variational equation of incremental plasticity theory and the variational equation of heat conduction, which expresses the principle of M. Biot. Variational-difference method was used to solve the problems of heat conductivity and calculation of the transient temperature field, and then at each time step – for the quasi-static problem of thermoplasticity. The numerical scheme is based on triangular meshes, which gives a more accuracy in describing the boundaries of structural elements as compared to rectangular grids.

The paper provides a comparative analysis of the results obtained by various approaches to modeling the process of artillery shot. In this connection, the main problem of internal ballistics and its particular case of the Lagrange problem are formulated in averaged parameters, where, within the framework of the assumptions of the thermodynamic approach, the distribution of pressure and gas velocity over the projectile space for a channel of variable cross section is taken into account for the first time. The statement of the Lagrange problem is also presented in the framework of the gas-dynamic approach, taking into account the spatial (one-dimensional and two-dimensional axisymmetric) changes in the characteristics of the ballistic process. The control volume method is used to numerically solve the system of Euler gas-dynamic equations. Gas parameters at the boundaries of control volumes are determined using a selfsimilar solution to the Riemann problem. Based on the Godunov method, a modification of the Osher scheme is proposed, which allows to implement a numerical calculation algorithm with a second order of accuracy in coordinate and time. The solutions obtained in the framework of the thermodynamic and gas-dynamic approaches are compared for various loading parameters. The effect of projectile mass and chamber broadening on the distribution of the ballistic parameters of the shot and the dynamics of the projectile motion was studied. It is shown that the thermodynamic approach, in comparison with the gas-dynamic approach, leads to a systematic overestimation of the estimated muzzle velocity of the projectile in the entire range of parameters studied, while the difference in muzzle velocity can reach 35%. At the same time, the discrepancy between the results obtained in the framework of one-dimensional and two-dimensional gas-dynamic models of the shot in the same range of change in parameters is not more than 1.3%.

A spatial gas-dynamic formulation of the backpressure problem is given, which describes the change in pressure in front of an accelerating projectile as it moves along the barrel channel. It is shown that accounting the projectile’s front, considered in the two-dimensional axisymmetric formulation of the problem, leads to a significant difference in the pressure fields behind the front of the shock wave, compared with the solution in the framework of the onedimensional formulation of the problem, where the projectile’s front is not possible to account. It is concluded that this can significantly affect the results of modeling ballistics of a shot at high shooting velocities.