The topic of the discussion is a trend, typical for development of the modern science, namely the process of "universal mathematization", i.e. of a wider and deeper penetration of mathematics into the practice of scientific knowledge. The emphasis is put on the phenomenon of "the incredible effectiviness" of mathematization of science. It is the matter of the central philosophical question of mathematics - its interconnection with the real world. The examples of the critical situation in mathematics and physics are given, which are the result of their extreme formalization, i.e. the maximal abstraction. The opposite views on the practical use of mathematics are presented. The main problem of the work is connected with the specific features of the process of "the all winning mathematization" of the modern control theory, one of the relatively young, successful and quickly developing areas of knowledge. It has been demonstrated that mathematics is of key importance for formation and development of the control theory. The author considers the problems of the methodological crisis of "the extreme mathematization" of the control theory. Among the main indications of the crisis the author points out the following: firstly, it is an increasing gap between the theory and practice, and, secondly, - its decreasing ability to help the specialists, involved in applied mathematics, with solving of the problems of a practical control. It is demonstrated that "the key" to overcoming of the existing crisis of the control theory is achievement of an organic unity of the mathematical strictness and physical sense.

The Lagrange problem arising in the analysis of large deviations in the states of dynamical systems was analyzed with the use of Wentzell-Freidlin method. For the linear case and Hurwitz state matrix of an unperturbed system the author obtained relations for quasipotential extremals, providing estimates of the probabilities of events for the initial conditions close to zero. On this basis, the author proposes an algorithm for prediction of the critical states of the dynamical systems, with the perturbed vector, "white noise", multiplied by a small parameter. Examples of application of the method to the task of controlling the angle of heel for a marine vessel in rough seas, and the angle of attack of an aircraft are presented. The case of lack of Hurwitz is analyzed on the example of the financial mathematics - instruments known as the Black-Scholes model The results show that the quasipotential extremals can be effectively used for the tasks of stabilization as a means of additional quality assurance management: passage of the stochastic system state through a small neighborhood of a quasipotential extremal signalizes about an abnormal movement of the controlled process. More accurate conclusions about the danger rate and the need to switch to the crisis management can be made with the estimated probability of a crisis and action functional, calculated on the basis of the quasipotential (equations included). In case of instability the question can be solved directly on the action functional, as an example of a financial mathematics model shows.

The article is devoted to the influence of various reduction methods of the mathematical models on the robust stability of the automation systems of the technological processes. As is known, these automation systems are designed on the basis of the reduced models. The automation systems of a high order are usually characterized by an increased complexity and high sensitivity to the parameters of deviations from the calculated values. Since the known parameters of the systems are always inexact, a high sensitivity (or low robustness) of the systems lowers sharply the quality of the automated technological processes. Reduction of the dynamic models of the technological processes is frequently applied to eliminate this drawback. The reduction of the models can be carried out by various methods such as elimination of the poorly influencing bonds and small time constants due to removal of the quickly changing variables (cutting of the quick dynamics), etc. The reduced models also allow us to design simpler regulators and more robust automation systems of the lower orders. Besides, a solution to the problem of the automation systems design becomes much easier. However, in a real automation system the cut off quick dynamics influences the properties of the reduced system. Exactly the influence of this cut off quick dynamics on the robust stability of the reduced automation systems is the subject of this work. The robust stability is estimated with the help of V. L. Kharitonov criteria. For solving of the problem the models are reduced by three various methods. The reduced automation systems are created by the analytical design method. The maximal admissible deviations of their parameters for the robust stability are studied with account of the quick dynamics cut off at the stage of designing. It was established, that the cut off quick dynamics rendered an essential influence on the robust stability of the reduced automation systems. The received results can be applied for development of the automation systems for the technological processes in the chemical, power engineering, aviation, machine-building and other sectors.

In many fields of knowledge there are tasks, which require filming of the behavior of a dynamic object with the use of high-speed cameras. Usually, the existing industrial systems do not combine a large work area and a high speed of filming of the equipment's field of view. In this regard, the quality parameters of the resulting footage, such as motion blur, illuminance, depth of field, angular resolution, suffer on complex trajectories. The goal of our work was to design a new robotic system structure, based on a study of the classical approaches of the automated high-speed flight follower systems. An original structure with two mirrors, leveling several disadvantages of the known structures, was proposed. The paper describes the mathematical dependencies of the size parameters, inverse kinematics and an algorithm for minimizing the velocity of the elements of the system. It also offers a method for estimation of the parameters of the work area. A robotic system based on the proposed optical layout will allow filming objects moving not only along the linear, but also more complex trajectories, in a wide range of speeds, combining the advantages of both the optical system with a moving camera and the panning mirror systems. The proposed optical layout allows us to reduce the angular velocities of the primary mirror during tracking, and it can be used not only for filming, but also for projecting images on dynamic objects.

The article presents results of the feasibility study of a new robotic platform as an active six-axis-driven support for protection of the facilities, technical equipment and structures from the extraordinary mechanical influences. It was demonstrated that such a system can be realized, if the mechanical part of the platform of the robot is a six-movable parallel arm with a new topology structure. In order to substantiate the use of a platform robot as an actively controllable support, a working model was created of the robot platform with the hydraulic drives, which ensured functioning of the robot platform as a controllable support. In order to evaluate the reaction of the robot to the changing external perturbations, the indicator of the motor reaction was used. For analysis of the motor reactions of the real-sided spring return actuator with throttling holes in the piston and without throttling apertures, an experimental study was done on the current model of the robot platform. The experimental results confirmed the possibility of using the robot as a platform for an actively controllable support and allowed to formulate recommendations for application of the new platform as an active support. An example is presented of the use of the robot platform system in an active seismological protection system for buildings.

The article presents substantiation of the concept for construction of the Ergatic Airspace System for monitoring of the environment. The system includes multi-purpose manned and unmanned aerial vehicles, a land mobile system for monitoring of the stationary and mobile land objects, satellite navigation and communication systems (GLONASS, Gonets, Yamal, Arktika, RSCC Space Communication, lridium, Galileo), a station of ground mobile control of the monitoring system, a station of air control and remote control of the multi-purpose unmanned aerial vehicles for solving the problems of the environment protection. The principles were substantiated for complexation of the Ergatic Airspace System of monitoring on the basis of the manned and unmanned aerial vehicles, satellite navigation systems, mobile satellite communications and mobile system for monitoring of the stationary and mobile land objects. The developed layout sample of the unified land mobile station for control and diagnostics of the state of equipment for the multipurpose unmanned aerial vehicles allows to solve the following problems: autonomous automatic control of LUAV during environmental monitoring; remote command-leader control of LUAV groups in special flights; reference maneuvering in case of changing targets of monitoring; combined control, when LUAV groups go to the area of application autonomously under the program, while application of the target load is produced by the remote command control Complex use of GLONASS/GPS technologies and multifunctional satellite communications and data transmission systems Gonets-DIM, Yamal, Arktika, RSCC Space Communication, Iridium, and Galileo in the Ergatic Airspace System of monitoring provides effective solutions to the problems of monitoring and environment protection in the Russian Federation, CIS States and partner countries.

During operation of many strapdown inertial navigation systems (SINS) the orientation vector of a rigid body is periodically calculated by the method of approximate solution of the Bortz approximate linear differential equation (in the practice of construction of SINS for small angles of rotation the nonlinear term in the Bortz equation is neglected). Note, that the full nonlinear Bortz equation for the vector orientation of the rigid body is an analog of the quaternion linear equation; the vector and the quaternion of the rigid body orientation are linked by known relations. In the article on the basis of the obtained exact solution of the Bortz approximate linear equation (valid for small angles of rotation of a rigid body) due to the quadratures the task is solved for determination of the quaternion orientation of a rigid body with an arbitrary angular velocity and small angle of rotation of the rigid body. Proceeding from this solution, the following approach to the construction of a new algorithm for computation of SINS orientation is proposed: 1) By the set components of the angular velocity of a rigid body on the basis of mutually - unambiguous changes of the variables at each time point, a new angular velocity of a rigid body is calculated; 2) Using the new angular velocity and the initial position of a rigid body, with the help of the quadratures we find the exact solution of the Bortz approximate linear equation (vector of orientation) with a zero initial condition; 3) The value of the quaternion orientation of a rigid body (SINS) is determined by the vector of orientation. During construction of the orientation algorithm of SINS at each subsequent step the change of the variables takes into account the previous step of the algorithm in such a way that each time the initial value of the vector orientation of a rigid body will be equal to zero.

The article presents a research of the characteristics of the optoelectronic ring angular velocity transducer based on the optical tunneling effect. Nowadays MEMS gyroscopes are used in the aircraft control and navigation systems. A big number of MEMS angular velocity transducers, based on the properties of a vibrating element, are already available. There are several ways to evaluate sensitivity of these angular velocity transducers, including capacitive, electrostatic and piezoresistive means. The main disadvantage of these angular velocity transducers is nonlinear sensitivity. In this article those problems are solved by the research of the optoelectronic ring angular velocity transducer with the optical tunneling readout schematics, which ensures a measurement range of several hundred degrees per second. The response amplitude of a ring resonator depends on various factors, such as the diameter, type of material, radial thickness, and axial length. Reflectivity of the angular velocity transducer is changed by the initial gap, amplitude of the vibrating ring resonator and incident beam angle. The output signal is obtained by varying the amplitude of the secondary radial displacement by means of the optical tunneling. With application of a few hundred degrees per second of the angular velocity, the output sensitivity of the desired angular velocity transducer is more linear than with the use of the capacitive means. So, the desired optoelectronic ring angular velocity transducer can be used in the navigation, moving object control and trajectory tracking systems.

The topic of the article is movement of a group of autonomous mobile objects during their functioning in the environments with stationary obstacles. The task is solved within the framework of the decentralized control systems. Thus the exchange of information between the mobile objects in a group is minimized. During the planning of the movement and control of the mobile objects the obstacles are transformed into repellers by means of synthesizable controls. The method of the potential fields and the method of control of the mobile objects with the use of unstable modes are the closest to the proposed method. The main difference of the developed method from the method of the potential fields is that a mobile object moves to the field of the forces depending not only on the relative positioning of the robot and an obstacle, but also on the additional dynamic variables. The dynamic way of formation of the repellent forces allows one to operate robots within the system of a decentralized control. The main difference of the offered approach from the method of position and trajectory control with the use of the unstable modes is the way of introduction of the unstable states. In the method of position and trajectory control the change of the parameters of the reference equation of a control system is used. In the offered method additional dynamic variables are used. Stable and unstable states of this variable depend on the state variables of a robot and the objects, next to it. In the local control systems of each mobile object the only values used are those of the own coordinates and speeds, and also coordinates and speeds of the neighboring objects. At that, a centralized algorithm of control is absent. Obstacles in the local algorithms are presented as mobile objects, which makes it possible to unify the control systems for the heterogeneous groups. An analysis was carried out, during which the existence and asymptotic stability of the steady movement modes were proved. The carried out numerical modeling confirmed the results of the analysis and synthesis.

The authors of the paper propose an approach to the control of the underwater vehicles, the mathematical model of which is known with a certain degree of accuracy, namely, its linear part. The proposed approach is based on the use of the estimates of the phase coordinate of the control object, and the estimates of the unknown external disturbances, which represent a collection of the external influences, and incomplete information about the model of the control object. The authors of this paper propose the filtering algorithms, based on separation of the residuals of the high and low frequency components and allow us to recover not only the phase coordinates of the movement, but the unknown perturbations. The paper also offers control algorithms for the underwater vehicles based on the use of not only estimates of the movement coordinates, but also the estimates of the unknown disturbances, obtained due to the filtering algorithms. Compared with the classical control laws, these algorithms allow us to significantly improve the quality control of the modern underwater vehicles. The paper contains a smoothing algorithm of the signal applied to the controller. This can significantly reduce the load on the governing bodies and thereby improve their performance. Conducted on the example of a non-planar motion of an underwater vehicle described by a complex system of nonlinear differential equations under the action of a slowly varying wave disturbances and modeling, the experiments confirm RA-combat capability proposed in the paper algorithms for filtering and control.