Head of the department: Kurzhanski Alexander, Distinguished Professor, Dr.Sc., Academician of RAS (the Russian Academy of Sciences).

Systems analysis, as treated here, is the science of applying mathematical and computer modeling to problems of decision making under realistic information. This science was initiated to solve new classes of applied problems, from dynamics and control to those of environment and demography, economics and finances, as well as interdisciplinary problems in between the above, or within such areas as biomedical and societal problems, energy studies, transportation, technological innovation, etc.

The Department was established in 1992. Education in this specialization embraces all the steps involved in solving problems “to the end”, namely, the model selection, its identification and validation, its further mathematical analysis and design of constructive solutions - from theory to algorithms and related software, including, if possible, an experiment in computer visualization, and, finally, the construction of computerized decision support systems. Students are tutored in relevant areas of modern mathematical tools - nonlinear analysis and set-valued calculus, advanced courses in differential equations, probability and statistics, stochastic processes, optimization methods, including dynamic programming, and a broad array of numerical techniques. An important element of learning is the study of applied sciences as well as of advanced informational and computer technologies - modern operating environments and software tools including those for parallel calculations and computer graphics. The Department pursues an active research on new theoretical and applied problems in the above mentioned areas. It has extensive international links and also offers studies of two foreign languages.

### Staff members:

- Arutyunov Aram, Professor, Dr.Sc.
- Bratus Alexander, Professor, Dr.Sc.
- Lotov Alexander, Professor, Dr.Sc.
- Shananin Alexander, Professor, Dr.Sc.
- Daryina Irina, Associate Professor, PhD
- Roublev Ilya, Associate Professor, PhD
- Smirnov Sergey, Associate Professor, PhD
- Minaeva Yulia, Assistant Professor
- Rudeva Anastasia, Assistant Professor, PhD
- Tochilin Pavel, Associate Professor, PhD, Scientific Secretary of the Department
- Vostrikov Ivan, Assistant Professor
- Lapushkina Tatiana, Technician

### Regular courses:

- Multivalued Calculus by Prof. Arutyunov, 70 lecture hours, 5th and 6th semesters.
- Optimal Control (Linear Systems) by Assoc. Prof. Roublev, 36 lecture hours, 5th semester.
- Stochastic Analysis by Assoc. Prof. Smirnov, 70 lecture hours, 6th and 7th semesters.
- Optimal Control (Nonlinear Systems) by Assoc. Prof. Roublev, 32 lecture hours and 16 seminar hours, 6th semester.
- Dynamic Systems and Biomathematics by Prof. Bratus, 68 lecture hours, 6th semester.
- Dynamic Programming and Controlled Processes by Prof. Kurzhanski, Assist. Prof. Vostrikov, 70 lecture hours and 70 seminar hours, 7th and 8th semesters.
- Identification Theory by Assoc. Prof. Daryina, 70 lecture hours, 7th and 8th semesters.
- Mathematical Models in Economics by Prof. Shananin and Assist. Prof. Rudeva, 36 lecture hours and 36 seminar hours, 7th semester.
- Theory of Stability by Assoc. Prof. Tochilin, 70 lecture hours, 8th semester.
- Multiobjective Optimization by Prof. Lotov, 32 lecture hours, 8th semester.

### Special scientific seminars:

- Applied problems of systems analysis by Prof. Kurzhanski.
- Multiobjective Optimization and Reachability Set Methods by Prof. Lotov.
- Special Topics in Mathematical Modeling for Economics by Prof. Shananin.
- Modeling of Financial Markets by Assoc. Prof. Smirnov.
- Mathematic Modeling in Biomedical Systems by Prof. A.Bratus.

### Master programs (MSc)

Mathematical Methods of Systems Analysis, Dynamics and Control. Specialty 010500 «Applied Mathematics and Computer Science». Director of the Program: Distinguished Professor, Academician Alexander B. Kurzhanski.

### Regular courses of Master Program:

- Additional Chapters of Dynamic Programming and Control Processes by Prof. Kurzhanski, 70 lecture hours and 70 seminar hours, 1st and 2nd semesters.
- Applied Problems of Systems Analysis (Problems of Biomathematics) by Prof. Bratus, 32 lecture hours, 1st semester.
- Theory of Stabilization by Assoc. Prof. Tochilin, 70 lecture hours, 1st semester.
- Generalized Functions and Control Problems, 32 lecture hours, 1st semester.
- Applied Problems of Systems Analysis (Models for Economics) by Prof. Shananin, 32 lecture hours, 2nd semester.
- Elements of Financial Mathematics by Assoc. Prof. Smirnov, 70 lecture hours, 2nd and 3rd semesters.
- Modeling and Control of Traffic Flows, 32 lecture hours, 2nd semester.
- Applied Problems of Systems Analysis (Environmental Models) by Prof. Kurzhanski and Assist. Prof. Vostrikov, 36 lecture hours and 36 seminar hours, 3rd semester.
- Variational Principles of Science, 70 lecture hours, 3rd and 4th semesters.

## Main scientific directions

The Department promotes an active research on mathematical models of modern cyber-physical systems together with the related information environment in which they immerse. Under investigation are topics motivated by new problems in automation and motion control, including reliability and safety of controlled processes, problems that involve remote and team control, computer and communication networks and coherent topics. The results of this research may be used in navigation and mechatronics, for new developments in industry, transportation and economics, in problems of resource management and environment control. Research studies are also related to new theoretical challenges in mathematical analysis, modeling and optimization for economics and finances, medical and biological processes.

The undergoing research activities are aimed at creating new solution approaches by improving existing and developing new mathematical tools. Their realization thus provides an environment for organizing the decision making process through attractive computational methods and advanced information technologies.

In recent years the research at the Department was supported by Grants from the Russian Foundation for Basic Research, the Federal Target Program “Research and Pedagogical Personnel for Innovative Russia” and the Federal Research Program for Leading Scientific Schools.

### Feedback control for complex systems under uncertainties and incomplete data

(Professor A.B. Kurzhanski, Assoc. Professors I.V. Roublev, P.A. Tochilin, Assist. Professor I.V. Vostrikov)

Synthesized control laws that provide system feedback based on available measurements (observations) that are usually incomplete, are investigated. The related motions are often subject to uncertain disturbances of various nature. The proposed solutions take into account these features. For problems of target control, they arrive in the form of set-valued functions («trajectory tubes») and are applicable to models, where the unknown items may be of both probabilistic and set-membership description. Namely, they may be either random, or unknown but bounded, with given bounds, or a combination of both types. New computational approaches and related procedures that allow treating problems of orders higher than in common approaches (as they are based on parallel processing) are proposed.

These classes of problems include systems with hybrid dynamics, nonlinearities, discontinuities and impulsive inputs that ensure a fast performance and also systems of coordinated team control (see Fig.1) and related topics.

### Problems of the medium — term analysis of the Russian economy

(Prof. A.A. Shananin, Assist. Prof. A. Rudeva)

As a result of a relative stabilization of the Russian economy in early XXI century, an interest in the construction of medium-term macroeconomic development forecasts has actively revived. In order to find an effective solution to this problem, a new methodology was required. This methodology would reflect the peculiarities of Russia's production system and be able to produce quick forecasts under changes in structural shifts and external economic conditions. The Department is developing the theory and the mathematical model for medium-term analysis of the current state of the Russian economy and forecasting the dynamics of macroeconomic indicators and their dependence on governmental economic policy. Fig.2 shows a block diagram of the mid-term analysis model of the Russian economy.

### Modeling the dynamics of interest rates term structure

(Assoc. Prof. S.N. Smirnov)

Modeling the stochastic dynamics of interest rates term structure is one of the important topics in mathematical finance. Absence of arbitrage opportunities is a vital property of such models since arbitrage opportunities make problems of pricing interest rates derivatives senseless. Parametric (low-dimensional) models widely used in practice usually yield an unrealistic picture. There are theoretical results which show that the class of arbitrage-free models with a finite number of parameters is very narrow. Infinite-dimensional (non-parametric) models have turned out to yield much more realistic pictures. A new model is introduced and tested on real market data exhibiting a robust performance even in a crisis period.

Another important problem of financial engineering consists in liquidating a large market position on an order-driven market using limit order book data. The purpose of the ongoing research is to construct a more realistic model of market liquidity.

### Multiobjectiive analysis of complex systems by computer visualization of the Pareto frontier

(Prof. A.V. Lotov)

Over the past two decades, the team of scientists led by Prof. A.V.Lotov has developed a new computer technique that supports the search for efficient decisions in the presence of conflicting decision criteria (objectives). The technique, named the Interactive Decision Maps, applies an interactive visualization of the Pareto (non-dominated) frontier in the case of three to eight objectives. The technique is based on preliminary approximating the feasible objective set (or a broader set that has the same Pareto frontier) and its subsequent interactive display in the form of bi-objective slices. Interactive exploration of the systems of slices helps the user to specify the preferred balanced feasible combination of objective values (feasible goal) and obtain the related efficient decision.

The technique was used for decision support in economic and environmental decision problems, in the computer-based design of technical and engineering systems of different nature, financial planning, selecting from large databases, etc. One of the important applications is the web-based support of public decision making (the participative decision making).

### Distributed replicator systems and mathematical models of therapy for malignant cells and viruses

(Prof. A.S. Bratus)

The mathematical theory of self replication for complicated macromolecules was suggested in 1971 by Manfred Eigen and his co-workers. The remarkable features of that theory rely on the main principles of evolution formulated by Charles Darwin: heredity, variability and natural selection. Such a model was named the hyper-cycling one and its theory was extended to a broader class named as the replicator systems which are realized mathematically by a nonlinear system of ODE’s with an additional condition of constancy for the sum of all concentrations. The main topic for investigation at the Department is the stability and limit behavior of replicator systems with spatial distribution of macromolecules under the action of homogeneous diffusion processes. Also considered are mathematical models for optimal therapy of malignant cells and viruses based on nonlinear ODE and PDE systems with control terms. The effectiveness of medicine is described in terms of a therapy function. The problems emphasize the search of optimal feedback strategies through solving corresponding Hamilton-Jacobi-Bellman equations.

### Selected recent publications:

• 2014

- Kurzhanski A., Varaiya P. Dynamics and Control of Trajectory Tubes. Theory and Computation. Birkhäuser, 2014. 445 p.
- Henkin G., Shananin A. Cauchy–Gelfand problem for quasilinear conservation law // Bull. des Sciences Mathematiques. 2014. 138, N 7. P. 783–804.
- Kurzhanski A., Mesyats A. Control of ellipsoidal trajectories: Theory and numerical results // Comput. Mathematics and Math. Phys. 2014. 54, N 3. P. 418–428.
- Kurzhanski A., Daryin A. The specifics of closed-loop impulse control // Proc. of the 19th IFAC World Congress, 2014. Cape Town, South Africa, 2014. P. 1655–1660.
- Kurzhanski A., Mesyats A. The mathematics of team control // Proc. of the 21st International Symposium on Mathematical Theory of Networks and Systems. Groningen, Netherlands, 2014. P. 1755–1758.
- Bratus А., Zapolski K., Admiralski Yu. Hybrid cellular automation method for homogeneous tumor growth modeling // Russian Journal of Numerical Analysis and Mathematical Modelling. 2014. 29. N 5. P. 319–329.
- Bratus А., Fimmel E., Kovalenko S. On assessing quality of therapy in non-linear distributed mathematical models for brain tumor growth dynamics // Math. Biosciences. 2014. N 248. P. 88–96.

• 2013

- Kurzhanski A.B., Daryin A.N. Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty // Comput. Mathematics and Math. Phys. 2013. 53. N 1. P. 34-43.
- Kurzhanski A.B., Daryin A.N. Nonlinear feedback types in impulse and fast control // 9-th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013). Toulouse, France: Elsevier Science Publ., 2013. P. 235-240.
- Kurzhanski A.B., Tochilin P.A. Tracking within a time interval on the basis of data supplied by finite observers // Differential Equations. 2013. 49. N 5. P. 630–639.
- Kurzhanski A.B., Tochilin P.A. Output feedback guaranteed tracking control through finite observers // Proc. of the 52nd IEEE Conf. on Decision and Control. Florence, Italy, 2013. P. 4448–4453.
- Sinyakov V.V., Roublev I.V. Approximation of reachability sets for nonlinear unicycle control system using the comparison principle // 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013). Toulouse, France: Elsevier Science Publ., 2013. P. 688-692.
- Daryin A.N., Minaeva Yu.Yu. Numerical algorithm for solving the problem of synthesis of impulse controls under uncertainty // J. Computer and Systems Sci. Intern. 2013. 52. N 3. P. 365-376.

• 2012

- Bratus A.S., Goncharov A.S., Todorov I.T. Optimal control in a mathematical model for leukemia therapy with phase constraints // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 4. P. 178-182.
- Daryin A.N., Kurzhanski A.B. A method for calculating the invariant sets of high dimensional linear systems under uncertainty // Doklady Mathematics. 2012. 86. N 2. P. 684-687.
- Kurzhanski A.B. On the problem of control for ellipsoidal motions // Proc. of the Steklov Inst. of Math. 2012. 277. P. 168–177.
- Kurzhanski A.B., Mesyats A.I. Optimal control of ellipsoidal motions // Differential Equations. 2012. 48. N 11. P. 1-1.
- Mazurenko S.S. A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion // Doklady Mathematics. 2012. 86. N 1. P. 476-479.
- Vostrikov I.V. A dynamic programming method for controlled linear systems with delays // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 2. P. 66-73.