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# Department of Nonlinear Dynamic Systems and Control Processes

Head of the department: Stanislav V. Emelyanov, Academician of the Russian Academy of Science (RAS), Professor, Dr.Sc.

The Department conducts research in the field of the modern theory of dynamical systems and control of nonlinear objects.

The Department research focuses on:

- the theory of nonlinear dynamical systems;
- the nonlinear analysis in control theory and optimization;
- the theory of feedback control under uncertainty;
- the stochastic control;
- the game theory;
- inverse control problems;
- mathematical modeling of controlled systems;
- identification of dynamical systems;
- synergetic dynamics;
- neural networks;
- neuro-dynamics and control;
- chaotic dynamics and control;
- the software development for analysis, synthesis and simulation of nonlinear dynamical systems;
- computer visualization methods for simulation of control systems;
- the development of algorithms for robot control;
- theoretical foundations of computer science (theory of algorithms, mathematical complexity theory, information theory, formal systems and languages, mathematical logic);
- symbolic dynamics in the problems of selection, analysis of complex socio-economic systems.

The importance and topicality of these areas are confirmed by a large number of practical problems in technology, economics, biology, etc. that can be solved within the framework of these theories. Students who study at the Department are involved in research on "Nonlinear dynamical systems: a qualitative analysis and control" and undertake their practice at the Institute of Control Sciences (RAS), Institute for Systems Analysis (RAS) and other academic institutions. For their study and research students are provided with a modern computer lab and a mobile robots polygon.

### Staff members:

- Alexander V. Il’in, Corresponding Member of RAS, Professor, Dr.Sc.
- Vasily V. Fomichev, Professor, Dr.Sc., Deputy head of the Department
- Andrey S. Fursov, Associate Professor, PhD, Scientific Secretary of the Department
- Alexander P. Afanas’ev, Associate Professor, Dr.Sc.
- Eduard R. Smolyakov, Professor, Dr.Sc.
- Vladimir I. Elkin, Professor, Dr.Sc.
- Nikolay A. Magnitsky, Professor, Dr.Sc.
- Lev A. Sholomov, Professor, Dr.Sc.
- Yury S. Popkov, Professor, Dr.Sc.
- Vladimir E. Krivonozhko, Associate Professor, Dr.Sc.
- Olga N. Bobyleva, Assistant Professor, PhD
- Alexey V. Boreskov, Researcher, PhD
- Oleg I. Goncharov, Researcher, PhD
- Ivan V. Kapalin, Researcher, PhD
- Andrey V. Kraev, Researcher
- Sergey I. Minyaev, Programmer
- Marina M. Belova, Engineer
- Elena R. Bagrova, Engineer

### Regular courses:

- Feedback control theory by Assoc. Prof. Fursov, 120 lecture hours and 60 seminar hours, 6th, 7th, 8th semesters.
- Mathematical foundations of computer science by Prof. Sholomov, 72 lecture hours, 5th and 6th semesters.
- Contemporary computer technologies in control theory and optimization by Dr. Kraev, 72 lecture hours and 72 seminar hours, 5th and 6th semesters.
- Mathematical methods in control theory and optimization by Assoc. Prof. Fursov, 18 lecture hours and 18 seminar hours, 5th semester.
- Conflict control and differential games by Prof. Smolyakov, 72 lecture hours, 7th and 8th semesters.
- Contemporary theory of dynamical systems by Prof. Elkin, 72 lecture hours, 7th and 8th semesters.
- Modeling and analysis of complex systems by Assoc. Prof. Krivonozhko, 36 lecture hours, 9th semester.
- Theoretical mechanics by Prof. Golubev, 36 lecture hours and 36 seminar hours, 6th semester.
- Control of technical systems by Prof. Fomichev and Dr. Goncharov, 36 lecture hours, 6th semester.

### Special courses:

- Distributed computing technologies and grid by Assoc. Prof. Afanasev, 72 lecture hours, 2 semesters.
- Information theory by Prof. Sholomov, 72 lecture hours, 2 semesters.
- Theory of underdetermined information by Prof. Sholomov, 72 lecture hours, 2 semesters.
- Introduction to non-linear and chaotic dynamics by Prof. Magnitsky, 72 lecture hours, 2 semesters.
- Generalized optimal, conflict and stochastic control by Prof. Smolyakov, 72 lecture hours, 2 semesters.
- Fundamentals of the robotics by Prof. Fomichev and Prof. Il’in, 72 lecture hours, 2 semesters.
- Massively parallel systems, the CUDA architecture and programming by Dr. Boreskov, 72 lecture hours, 2 semesters.

### Special scientific seminars:

- Chaotic dynamics by Prof. Magnitsky.
- Contemporary methods in control theory by Prof. Fomichev, Assoc. Prof. Fursov, Prof. Il’in, Dr. Kraev.
- Problems of non-linear dynamics: a qualitative analysis and control by Prof. Emelyanov.

## Main scientific directions

Various directions which are connected with the analysis of nonlinear dynamic systems and their properties, together with the study of a wide range of control problems of complex systems are being developed at the Department.

Nowadays, the needs in highly-performing systems of the automation of technological processes are becoming more crucial than ever before. Such processes are usually characterized by significant uncertainties – both of parametric and structural nature. The objectives of the control under uncertainty are formulated in terms of the synthesis of universal regulators (implemented by feedback controls) for the families of dynamic objects - both linear and nonlinear ones. The greatest achievement in this field is the development of algorithms for control of complex systems under uncertainty, the development of the methods of identification of unknown system signals and parameters.

In addition to the theoretical tasks, various applied problems of control of mobile robots are solved. There are different available mobile robots at the Department. Using them, it is possible to test newly developed control methods for complex robotic systems. For example, on the basis of the “Khepera” robot we demonstrate the effective automatic control of individual mobile robots as well as groups of robots who are able to perform given common tasks debugging their interaction. On the basis of the “Koala” robot we demonstrate the effective control based on the information that a robot receives in real time from embedded sensors, cameras, etc. On the basis of a "mechanical arm Katana” we solve the problem of a mechanical manipulator control. A group of graduate and post-graduate students of the Department developed a "chess player" robot which is an actual mechanical system that is able to play chess with a human being using a real chess board.

### Observation, stabilization and inversion of dynamical systems

(Academician of RAS S.V. Emelyanov, Prof. A.V. Il’in, Prof. V.V. Fomichev, Ass. Prof. A.S. Fursov)

In various projects carried out by the Department team in recent years, new algorithms for synthesis of controllers that provide simultaneous stabilization of dynamic objects (continuous and discrete ones), and the problem of simultaneous stabilization of dynamic objects are proposed. These problems are of great importance in control theory, for example, in case when a real object operates under the parametric uncertainty and it is necessary to stabilize the efficiency in any possible uncertainty mode. The latter is especially important for automatic control systems (ACS) of complex objects. The main attention is paid to the development of methods that allow to build numerical algorithms for constructing such controllers.

Also, a significant progress is obtained in the field of inverse dynamic problems. In particular, qualitatively new approaches are proposed to the synthesis of the asymptotic observer (i.e., dynamic systems that reconstruct on-line an unknown system or process parameters), the asymptotic stabilizers (i.e., dynamic systems that authorize the closed loop system by dynamics required) and the asymptotic inverters (i.e., dynamic systems that reconstruct on-line unknown external signals). These problems are of great importance in the theory of automatic control, in the case when one needs to develop a system of automatic control of complex objects under uncertainty, with an imprecise information of system parameters, and often with inaccurate system dynamics.

The original results are obtained by the Department team in solving these problems in the non-classical (but important from a practical point of view) formulation of the problem. Thus, methods for the synthesis of observers, stabilizers and inverters of a minimum order are developed. This is due to the fact that the order (the dimension of the dynamical system) of observers, inverters stabilizers largely determines the complexity of the automatic control system. Reducing this order simplifies the automatic control system. This decreases the number of elements (sensors, controllers) to be used, and therefore cuts down the cost of the automatic control system.

### Chaotic Dynamics

(Prof. N.A. Magnitsky)

This research direction includes the development of a universal bifurcation theory of the dynamical chaos in nonlinear systems of differential equations, including both autonomous and nonautonomous ones, dissipative and conservative nonlinear systems of ordinary differential equations, partial differential equations with a retarded argument. This theory concludes that there exists a unique mechanism of self-organization in a wide range of mathematical models that have applications in many fields of science and technology and describe many physical, chemical, biological, economical and social phenomena and processes. Among the main results obtained in this direction in recent years, one may note the establishment of a universal theory of dynamical chaos in nonlinear dissipative systems of differential equations (the Feigenbaum-Sharkovsky-Magnitsky theory) and the justification of its applicability to the analysis of Hamiltonian and conservative systems, as well as to systems of partial differential equations, including the Ginzburg-Landau equation and the Navier-Stokes equations that describe an oscillatory, excitable and turbulent environment.

### Differential geometry of nonlinear control theory of dynamical systems

(Prof. V.I. Elkin)

Nonlinear dynamical systems are studied using methods of differential geometry and the Lie group theory. A controlled system may be related to differential-geometric and group theoretic objects (affine distribution system of Pfaff, Lie groups, Lie algebras, and others). In terms of the associated objects, various management problems (decomposition, equivalence, reduction to a simpler form, and others) are formulated. As a result, one can solve these control problems with sophisticated and powerful mathematical methods.

### Calculus of variations, optimal control, optimization methods, mathematical modeling, distributed computing

(Assoc. Prof. A.P. Afanasiev).

A so-called needle variation helps to reduce the analysis of specific local variation problems. Local variation problems, in turn, are reduced to special problems of mathematical programming. As a result, optimal control problems that are linear with respect to the control, can be reduced to a combination of mathematical programming problems and Cauchy problems for systems of ordinary differential equations. Thus, one makes the natural decomposition of the problem which is a very promising from a further computing point of view. For systems of ordinary differential equations, we introduce the concept of quasi-periodic solutions, which is a generalization of the recursive path. The knowledge of qualitative features of the behavior of systems of ordinary differential equations allows one to apply an approximate solution from the symbolic approach that, in turn, helps to decompose and parallelize the problem. A great attention is paid to solving problems in highly performing computer software, e.g., in the Math Cloud Designed – a distributed computing environment for solving mathematical problems.

### Underdetermined information theory

(Prof. L.A. Sholomov).

Underdetermined information appears in recognition problems, control problems, decision-making logic synthesis schemes, compression and transmission of data, cryptography, genetics. In specific areas of computer science solving problems with underdetermined data usually requires a design of special models and methods. There is a problem of creating a unified system of concepts and developing an object-independent (i.e., independent of specific applications) underdetermined theory of information and introducing on this basis effective methods of underdetermined data treatment. The development of the underdetermined information theory can not be reduced simply to a modification of the known results of classical information theory. This approach raises fundamental difficulties and reveals new effects.

This field of research includes:

- the introduction and further study of characteristics for information underdetermined common data types,
- modification of methods, principles, algorithms and results of classical information theory applied to underdetermined data,
- consideration of the classical information theory more generally extending the range of the results' applicability,
- the study of equivalent transformations of underdetermined data,
- the development of methods and algorithms for operating underdetermined data (compression methods, extension, decomposition),
- application of results to specific areas of computer science.

### Recent publications:

• 2013

- Emel'yanov S.V., Il'in A.V., Fomichev V.V., Fursov A.S. Simultaneous stabilization of plants of various orders // Differential Equations. 2013. 49. N 5. P. 623-629.
- Emel'yanov S.V., Krishchenko A.P., Fetisov D.A. Controllability of irregular systems // Differential Equations. 2013. 48. N 11. P. 245-248.
- Fursov A.S., Kudritskii A.V., Korovin S.K. A constructive algorithm for a controller simultaneously stabilizing a family of plants // Computational Mathematics and Modeling. 2013. 24. N 2. P. 171-181.
- Goncharov O.I., Fomichev V.V. Observer for multivariable systems of arbitraty relative order // Computational Mathematics and Modeling. 2013. 24. N 2. P. 182-202.
- Il'in A.V., Budanova A.V., Fomichev V.V. Synthesis of observers for asymptotically observable time delay systems // Doklady Mathematics. 2013. 87. N 1. P. 129-132.
- Il'in A.V., Goncharov O.I., Fomichev V.V. Construction of observers for bilinear systems of special type // Doklady Mathematics. 2013. 87. N 2. P. 254-257.
- Il'in A.V., Izobov N.A., Fomichev V.V., Fursov A.S. On the construction of a common stabilizer for families of linear nonstationary systems // Doklady Mathematics. 2013. 87. N 1. P. 124-128.
- Kapalin I.V., Fomichev V.V. Design of minimal stabilizers for scalar systems // Computational Mathematics and Modeling. 2013. 24. N 2. P. 203-220.
- Kraev A.V. Some algorithms for inversion of vector discrete systems // Computational Mathematics and Modeling. 2013. 24. N 2. P. 271-278.
- Smol'yakov E.R. Auxiliary equilibria for differential and static games // Doklady Mathematics. 2013. 88. N 2. P. 1-1.
- Smol'yakov E.R. Relatively complicated equilibria in conflict problems // Computational Mathematics and Modeling. 2013. 24. N 3. P. 353-361.
- Smol'yakov E.R. Theory of search for exact differential equations of dynamic processes // Doklady Mathematics. 2013. 87. N 1. P. 53-57.
- Smol'yakov E.R. Weakened notions of equilibria and optimality in conflict problems // Differential Equations. 2013. 49. N 3. P. 359-368.

• 2012

- Bobyleva O.N., Fomichev V.V., Fursov A.S. Sufficient conditions for the existence of a common stabilizer for a family of linear nonstationary plants // Differential Equations. 2012. 48. N 7. P. 901-908.
- Emel'yanov S.V., Fomichev V.V., Fursov A.S. Simultaneous stabilization of linear dynamic plants by the variable-structure controller // Automation and Remote Control. 2012. 73. N 7. P. 1126-1133.
- Goncharov O.I. Transverse function method in stabilization problems for bilinear systems // Differential Equations. 2012. 48. N 1. P. 104-119.
- Il'in A.V., Korovin S.K., Fomichev V.V. Inversion of linear delay dynamical systems // Differential Equations. 2012. 48. N 3. P. 410-418.
- Il’in V.A., Kuleshov A.A. A criterion for the membership in the class L p with p >= 1 of the generalized solution to the mixed problem for the wave equation // Doklady Mathematics. 2012. 86. N 2. P. 688-690.
- Smol'yakov E.R. Missing notions of equilibria for general game theory // Doklady Mathematics. 2012. 85. N 1. P. 156-159.