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# Department of Optimal Control

Head of the department: Yury Osipov, Academician of RAS, Professor, Dr.Sc.

Optimal control is a field in mathematics that develops tools for formalizing and methods for solving problems of choosing the best (in a preliminary prescribed sense) strategy to control a dynamic process. The dynamic process may be described by means of differential, integral, functional, or finite-difference equations which depend on functions of parameters called controls; these parameters are to be determined. Problems considered in the mathematical theory of optimal control had come from the practical needs, primarily from mechanics, robotics, economics, biophysics, nuclear physics, electrical engineering, metallurgy etc.

The Department was established in 1970 due to the initiative of the outstanding Soviet mathematician, Academician Lev Pontryagin who headed the Department until 1988.

The research in the field of mathematical theory of optimal control, inverse control problems, theory of differential games, theory of differential inclusions, numerical methods and algorithms for solving control problems is being carried out at the Department, the related modern software is being developed for solving different applied control problems. The Department maintains fruitful collaboration with potential suppliers of the research’s results.

The education on the specialization “Optimal Control” includes an in-depth study of modern mathematics and new highly performing software. Students are also involved in the research conducted at the Department under the individual supervision of its members.

### Staff Members:

- Sergey Aseev, Corresponding Member of RAS, Professor, Dr.Sc.
- Fedor Vasilyev, Professor, Dr.Sc.
- Nikolay Grigorenko, Professor, Dr.Sc., Vice Head of the Department
- Andrey Dmitruk, Professor, Dr.Sc.
- Vladislav Zhukovsky, Professor, Dr.Sc.
- Michael Nikolsky, Professor, Dr.Sc.
- Michael Potapov, Professor, Dr.Sc.
- Vladimir Boykov, Associate Professor, PhD
- Dmitry Kamzolkin, Associate Professor, PhD
- Yury Kiselev, Associate Professor, PhD
- Nikolay Melnikov, Associate Professor, Dr.Sc.
- Michael Orlov, Associate Professor, PhD
- Sergey Samsonov, Associate Professor, PhD, Scientific Secretary of the Department
- Evgeny Khailov, Associate Professor, PhD
- Sergey Avvakumov, Associate Professor
- Alexander Kulevsky, Senior Lecturer
- Elena Rovenskaya, Research Fellow, PhD
- Boris Budak, Assistant Professor, PhD
- Ludmila Artemieva, Assistant Professor, PhD
- Alexey Smirnov, Assistant Professor
- Tatiana Goryakova, Engineer

### Regular Courses:

- Variations calculus by Assoc. Prof. Khailov, 32 lecture hours and 32 seminar hours, 5th semester.
- Convex analysis by Prof. Dmitruk, 32 lecture hours, 7th semester.
- Optimization methods by Prof. Vasiliev, 68 lecture hours, 7th and 8th semesters.
- Optimization methods by Prof. Potapov, 68 lecture hours and 36 seminar hours, 5th and 6th semesters.
- Optimal control by Assoc. Prof. Orlov, 68 lecture hours, 5th and 6th semesters.
- Nonlinear controlled processes by Assoc. Prof. Kiselev, 32 lecture hours, 9th semester.
- Optimal control theory: additional chapters by Assoc. Prof. Melnikov, 32 lecture hours, 9th semester.
- Differential games by Prof. Nikolsky, 36 lecture hours, 7th semester.
- Positional differential games by Dr. Kamzolkin, 32 lecture hours, 9th semester.
- Dynamical regularization method by Dr. Budak, 36 lecture hours, 8th semester.
- Application of optimal control in economics by Prof. Aseev, 32 lecture hours, 9th semester.
- Mathematical models of dynamic controlled processes by Prof. Grigorenko, 68 lecture hours, 7th and 8th semesters.
- Numerical methods for optimal control by Assoc. Prof. S.Samsonov, 32 lecture hours, 6th semester.
- Softwares for applications by Dr. Smirnov, 32 lecture hours and 32 seminar hours, 7th semester.
- Modern modeling problems in the Program complex for automated dynamic analysis of multi-component mechanical systems (EULER) by Assoc. Prof. Boykov, 68 lecture hours, 7th and 8th semesters.

### Special Courses:

- Modern theory of necessary and sufficient conditions of higher order for a local minimum in extreme problems with constraints by Prof. Dmitruk, 32 lecture hours, 8th semester.
- Introduction to geometrical control theory by Assoc. Prof. Melnikov, 32 lecture hours, 9th semester.
- Econometrics by Dr. Rovenskaya, 32 lecture hours, 6th semester.
- Dynamic programming in multi-steps games by Prof. Zhukovsky, 32 lecture hours, 7th semester.
- Global optimization methods by Prof. Shchedrin, 32 lecture hours, 6th semester.
- Differential games with several players and their applications by Prof. N.Grigorenko, 32 lecture hours, 8th semester.
- Cooperative games under uncertainty by Prof. Zhukovsky, 32 lecture hours, 8th semester.
- Digression problem: linear theory and its applications by Dr. Lukyanova, 32 lecture hours, 7th semester.

### Scientific Seminars:

- Qualitative aspects of optimal control by Assoc. Prof. Kiselev, Dr. Avvakumov, Assoc. Prof. Orlov.
- Game-type control problems by Prof. Nikolskiy, Prof. Grigorenko.
- Extremal problems and functional analysis by Prof. Dmitruk, Prof. Osmolovskiy.
- Optimal control problems in economics by Prof. Aseev, Assoc. Prof. Khailov, Assoc. Prof. Samsonov, Dr. Kulevskiy, Dr. Smirnov.
- Optimization methods by Prof. Vasiliev, Prof. Potapov, Prof. Antipin, Dr. Budak.
- Risks in complex control systems by Prof. Zhukovsky.
- Controlled process under uncertainty by Dr. Rovenskaya, Dr. D.Kamzolkin.
- Variation methods in the theory of multi-electron systems by Assoc. Prof. Melnikov.

## Main Scientific Directions:

### Optimization Methods

(Prof. F. Vasiliev, Prof. M. Potapov, Prof. A. Antipin, and Dr. B. Budak)

This research direction investigates stationary and dynamical multi-criteria optimization problems, problems and methods of searching equilibriums in saddle games, as well as control, observation and stabilization problems for dynamical processes described by partial differential equations. The main efforts are directed to the development and theoretical substantiation of new adequate numerical methods for finding approximate solutions to these problems with an acceptable accuracy which are stable to any noise in input data.

Refs:

- F.P.Vasil’ev, L.A.Artem’eva and A.S.Antipin, A stabilization method for solving parametric multicriteria equilibrium programming problems // MSU Comput. Math. & Cybern., vol. 33, no. 4, pp. 181-188, 2009.
- F.P.Vasil’ev, A.S.Antipin and L.A.Artem’eva, Regularized continuous extragradient method for a multi-criteria equilibrium programming problem // Diff. Eqs., vol. 46, no. 11, pp. 1584-1600, 2010.
- M.M.Potapov, Variational approach to stable numerical approximation of optimal Dirichlet boundary controls for the wave equation // Advances in Math. Research, 12 (Edt. A.R. Baswell). NY: Nova Science Publishers, pp. 95-119, 2012.

### Investment Optimization under Uncertainty

(Professors A. Kryazhimskiy, N. Grigorenko, and Dr. D. Kamzolkin)

This research was performed in cooperation with the BHP Billiton mining company and deals with the open pit mine optimization problems. The mathematical model of the mining process includes the ore body block model, cash flow valuation and initial capital investments into processing and mining equipment. This model helps to solve the NPV maximization and mining phase optimization problems using the optimal control theory, game theory and theory of stochastic differential equations. Those problems were solved using two approaches: the cash flow valuation and the Real Options approach.

Refs:

- N.L.Grigorenko, D.V.Kamzolkin, L.N.Luk’yanova and D.G.Pivovarchuk, On a class of control problems under uncertainty // Trudy Inst. Matem. i Mekh. UrO RAN, vol. 17, no. 2, 2011.
- N.L.Grigorenko, D.V.Kamzolkin and D.G.Pivovarchuk, Optimization of a two-step investment in a production process // Proc. Steklov Inst. Math., p. 262, 2008.

### Variational Methods in Computational Physics of Magnetic Materials

(Dr. N. Melnikov)

Magnetic metals and alloys have a wide range of applications of a different scale: from fuselage tooling in aerospace industry and turbine blades’ design in power engineering to reading heads of memory devices. Capacity of modern computers allows one to calculate properties of these materials and to synthesize new materials with required properties based on their microscopic description.

The main problem of the `first-principles’ modeling is the need to take into account interaction between particles. The most difficult problem is to deal with a system where the interaction is not small but, on the opposite, is essential for correct description of system’s properties.

It is this type of problem where one faces the magnetism of metals. Mathematical formalism of such problems yields nonlinear functional equations which require special numerical methods. One of the most successful approaches to the microscopic description is based on a functional integral method which replaces the pair interaction between particles by an interaction of particles with an external stochastic field. As a part of the joint research program with the Institute of Metal Physics RAS, we develop effective numerical methods for calculating functional integrals that take into account both dynamics and nonlocal character of the interaction. In order to calculate macroscopic characteristics, we use averaging methods that are based on appropriate variational principles. Our calculations allow one to study magnetic properties over a wide range of temperatures.

Refs:

- B.I.Reser and N.B.Melnikov, Problem of temperature dependence in the dynamic spin-fluctuation theory for strong ferromagnets // J. Phys.: Condens. Matter., vol. 20, no. 28, pp. 5205-5215, 2008.
- N.B.Melnikov, B.I.Reser and V.I.Grebennikov, Spin-fluctuation theory beyond Gaussian approximation // J. Phys. A: Math. Theor., vol. 43, no. 19, pp. 5004-5023, 2010.
- N.B.Melnikov, B.I.Reser and V.I.Grebennikov, Extended dynamic spin-fluctuation theory of metallic magnetism // J. Phys.: Condens. Matter., vol. 23, no. 27, pp. 6003-6014, 2011.

### Application of the Optimal Control Theory to Biology Processes Analysis

(Dr. Yu. Kiselev and Dr. M. Orlov)

Optimal control problems in microbiology are considered. A generalization of the Droop quota model for special biological structure is studied. A microbial trichome grows by assimilating nutrients from its environment and converting them into catalytic macro-molecular machinery. This machinery may be divided into assimilatory and proliferative one. The former type is involved in the nutrient uptake whereas the latter type enables the trichome to grow. The cells in the trichome are faced with an allocation problem: given the availability of nutrients in the environment, decide how many macro-molecular building blocks should be allocated for the synthesis of the assimilatory machinery and how many to the synthesis of the proliferative one. A two-dimensional non-linear optimal control problem is formulated. An optimal allocation regime with a singular segment based on Pontryagin maximum principle is derived.

The other problem involves a simple model of microbial growth and internal stores kinetics. A bacterial cell must distribute its molecular building blocks among various types of nutrient uptake systems. If a microbe has to maximize its average growth rate then this allocation of building blocks must be adjusted to the environmental availabilities of various nutrients. These adjustments can be found from growth balancing considerations.

Refs:

- H. van den Berg, Yu.N.Kiselev and M.V.Orlov, Optimal allocation of building blocks between nutrient uptake systems in a microbe// J. Math. Biol., vol. 44, pp. 276-296, 2002.
- H. van den Berg, Yu.N.Kiselev, and M.V.Orlov, Studying mathematical models of resources allocation among a cell’s assimilator mechanisms// J. Math. Sci., vol. 116, no. 6, pp. 3683-3732, 2003.
- H. van den Berg, Yu.N.Kiselev, and M.V.Orlov, The malthusian parameter in microbial ecology and evolution: an optimal control treatment// Comput. Math. & Mod., vol. 4, no. 19, pp. 406-428, 2008.

### Optimal Strategies for Sustainable Socio-environmental Development

(Prof. A. Kryazhimskiy and Dr. E. Rovenskaya)

This research is performed in collaboration with the International Institute for Applied System Analysis (Austria). The goal is to investigate how the decline of the environmental quality caused by increasing production affects an optimal economic growth both qualitatively and quantitatively. In order to achieve this goal we elaborate appropriate demographic-economic-environmental global and regional models that reflect complex linkages between the population growth, economic development and environmental pollutions.

In this framework, public health is an example of a competing goal of utility-maximizing agents as there is a trade-off between the growing consumption and its environmental consequences. In the optimum, agents accept a certain environmental degradation resulting in the decrease of public health as an exchange for ongoing consumption. In some cases, an unsustainable solution may appear to be optimal – this phenomenon explains the current rapidly growing pollution in some countries and regions.

Refs:

- S.M.Aseev and A.V.Kryazhimskiy, The Pontryagin maximum principle and optimal economic growth problems // Procs. Steklov Inst. Math., vol. 257, no. 1, pp. 1-255, 2007.
- A.V.Kryazhimskiy, Y.Minullin and L.Schrattenholzer, Global long-term energy-economy-environment scenarios with an emphasis on Russia // Persp. in Energy, vol. 9, pp. 119-137, 2005.
- A.V.Kryazhimskiy, M.Obersteiner and A.Smirnov, Infinite-horizon dynamic programming and application to management of economies effected by random natural hazards // Appl. Math. and Comput., vol. 204, no. 2, pp. 609-620, 2008.
- U.Lehmijoki and E.Rovenskaya, Environmental mortality and long run growth // In: Dynamic Systems, Economic Growth, and the Environment Series: Dynamic Modeling and Econometrics in Economics and Finance, J. Crespo Cuaresma, T. Palokangas, A. Tarasyev (Edt.), pp. 239-258, 2010.

### Attainability Domains Analysis

(Prof. A.Kryazhimskiy, Prof. S.Aseev, and A. Smirnov)

A possible way to perform a comprehensive analysis of the model’s dynamics is to use its attainability domain. This approach helps to observe all the possible states of the model, compare different strategies, and perform sensitivity and uncertainty analysis which is especially useful for models from natural and social sciences.

In collaboration with the International Institute of Applied Systems Analysis (Austria) this approach was applied to analyze the simplified DICE model which represents the economic impact of the climate change and evaluates costs of the greenhouse gases (GHG) emission reduction. It was revealed that the utility surface is practically flat in a large neighborhood of its optimal point and thus in the framework of this model there exist a variety of suboptimal controls that steer the system to different final states with practically no loss in the utility.

Refs:

- A.I.Smirnov, Attainability analysis of the DICE model // IIASA Interim Report, IR-05-049, 2005.

### Mathematical Modeling of Multi-component Mechanical Systems

(Dr. V. Boykov)

Computer modeling and various algorithms for controlling mechanical systems under conditions which are close to the reality are developed by means of a special original software EULER. Being based on modern results from the control theory and on new computer technologies, EULER serves as an advanced tool for engineering. In particular, it allows one to test how different elements of a device work under certain (sometimes critical) conditions.

Refs:

- V.G.Boykov and A.A.Yudakov, Simulation of the solid and elastic bodies dynamics in software EULER // Inform. Techn. & Comp. Systems, vol. 1, pp. 42-52, 2011 (in Russian).
- V.G.Boykov. Simulation of the mechanic systems in software EULER // SAPR and Graphics, vol. 1, pp. 38-48, 1998 (in Russian).

### Recent papers concerning ecological problems:

- E.V.Grigorieva and E.N. Khailov, Minimization of pollution concentration on a given time interval for the water cleaning plant // J. Control Sci. and Engineering, N ID 712794, pp. 1-10, 2010.
- E.V.Grigorieva and E.N. Khailov, Optimal control of pollution stock through ecological interaction of the manufacturer and the state // Revista de Matematica: Teoria y Aplicaciones., vol. 18, no. 1, pp. 77-109, 2011.
- E.A.Rovenskaya, Remarks on fair wealth accumulation in Russia // Environment, Development and Sustainability, vol. 13, no. 5, pp. 923-937, 2011.

### Recent publications:

• 2013

- Grigoremko N.L., Kamzolkin D.V., Lukianova L.N. A class of models describing the dynamics of production and infrastructure-planning indicators // Computational Mathematics and Modeling. 2013. 24. N 4. P. 543-551.
- Grigorieva E.V., Khailov E.N., Korobeinikov A. An optimal control problem in HIV treatment // Discrete and Continuous Dynamical Systems. 2013. 2. N 4. P. 299-310.
- Grigorieva E.V., Bondarenko N.V., Khailov E.N., Korobeinikov A. Analysis of optimal control problems for the process of wastewater biological treatment // Revista de Matematica: Teoria y Aplicaciones. 2013. 20. N 2. P. 103-118.
- Grigorieva E.V., Korobeinikov A., Khailov E.N. Parametrization of the attainable set for a nonlinear control model of a biochemical process // Mathematical Biosciences and Engineering. 2013. 10. N 4. P. 1067-1094.
- Kiselev Yu.N., Orlov M.V., Orlov S.M. Investigating a two-sector model with an integrating type functional cost // Moscow University Comput. Mathematics and Сybern. 2013. 37. N 4. P. 172-179.
- Orlov M.V., Puchkova A.I. Comparing two modes of control during the growth of a colony of microorganisms // Moscow University Comput. Mathematics and Сybern. 2013. 37. N 2. P. 61-64.

• 2012

- Potapov M.M. Variational approach to stable numerical approximation of optimal dirichlet boundary controls for the wave equation // Advances in Mathematics Research. N 12. N.Y., USA: Nova Science Publishers, 2012. P. 95-119.