For the third order evolution differential equations, the algebra of differential invariants with respect to the pseudogroup of point transformations is constructed. Such equations arise in the study of filtration processes (the Rapoport - Leas equation), the theory of nonlinear waves (the Korteweg - de Vries equation).

In the process of developing oil fields, periodic disturbances are used to increase the flow rate of wells and the final oil recovery of formations by changing the mobility of oil and the filtration-capacity parameters of the porous medium. In an inhomogeneous medium, in the presence of dissipation and / or dispersion, harmonic oscillations created in the wellbores transform into sawtooth waves with periodic shock fronts, which change the dynamics of phase transitions and can lead to cavitation in liquid phases. Taking this circumstance into account will make it possible to adequately predict the dynamics of phase transitions and optimize the control of the oil production process.

A model distribution of anomalous masses in the southwestern part of Elysium Planitia under the surface of the site of the InSight mission seismometer has been obtained. Numerical simulations are per- formed with the S-approximation method by expansions of the data on the Martian topography and gravita- tional field (the MRO120D model) in spherical harmonics to the 90th degree and order. Inhomogeneity of the density structure in the form of zones, containing blocks of higher and lower density in the subsurface lay- ers of the study area, has been revealed.

The point stabilization problem for a robot-wheel is considered. The problem consists in synthesizing control torque in the form of feedback that brings the wheel from an arbitrary initial position on a straight line to a given one, with the control torque and the maximum velocity of wheel motion being constrained. To meet the phase and control constraints, an advanced feedback law in the form of nested saturation functions is suggested. Two of the four coefficients employed in the saturation functions are uniquely determined by the limit value of the control torque and the maximum allowed wheel velocity, while the selection of the other two coefficients can be used to optimize the performance of the controller. In this study, the optimality is meant in the sense that the phase portrait of the closed-loop system is similar to that of a stable node, with the asymptotic rate of decrease of the distance to the target point being as high as possible. The discussion is illustrated by numerical examples.

In the tasks of predicting the volumes of methane emissions from thermokarst lakes in the Arctic territories, as one of the causes of modern global warming, it is necessary to use, along with climatic characteristics, data on the dynamics of lake areas, which are usually obtained using satellite imagery. Due to the large number of cloudy days in the northern territories, it is possible to obtain only a small number of cloudless images, which leads to significant omissions in the time series of lake areas. To restore the missing values of the area of the lakes, it is proposed to use a new approach to the restoration of missing values based on the methods and algorithms of entropy-randomized machine learning. The work is supposed to restore the missing values in the experimental data on the areas of thermokarst lakes using time series of average annual temperature and annual precipitation. As experimental data on the thermokarst lakes areas and climatic parameters (temperature and amount of precipitation), we used the results of studies conducted in the Arctic zone of Western Siberia from 1973 to 2007. Studies were conducted in nine test sites selected in different permafrost zones (continuous, discontinuous and insular). Data on the average annual temperature and annual precipitation for each test site were obtained by reanalysis. The developed algorithm for recovering missing values within the framework of this approach is implemented using the MATLAB R2019a tools. The missing values are calculated for the selected nine test sites. To illustrate, the time series of the values of the area of lakes, temperature and precipitation in one of the test sites are shown. An analysis of the omissions recovery errors was carried out, which showed that the developed algorithm allows us to restore the missing values of the lake areas from the data on changes in temperature and precipitation with practically acceptable accuracy.