Магистерское и послевузовское образование

Special Courses

Annotation: this course is about solving the problems of linear and non-linear dynamics and control. The principal problem of the control theory is that of control synthesis in the form of feedback, with complete or incomplete measurements of the state. At the foundation of the course lies the Habilton-Jacobi-Bellman formalism for variational problems motivated by the control problems described by ordinary differential equations. The course shows applications of the theory to motion control problems, the problems of economical and financial dynamics.

Distinguished Professor of Moscow State (Lomonosov) Univeristy, Full Member of Russian Academy of Sciences Alexander B. Kurzhanski
(1st and 2nd semester, 2 hours/week lectures, 2 hours/week seminars, 2 hours/week computer practicum)

Systems Analysis for Environmental Problems

Annotation: this course is about mathematical equations of atmospheric diffusion which allow modeling of the problems of environment. The theory of observation and guaranteed estimation for PDEs, control problems and general identification problems of distributed systems are described within the systems theory framework. Mathematical problems motivated by the problems of ecological monitoring are presented. The course includes methods of regularization of ill-posed problems based on works of A. N. Tychonoff and J.-L. Lions. We propose an unified approach for such problem taking into account the dynamics of the infinite-dimensional processes under consideration. A special attention is given to inverse problems of estimation of solutions of infinite-dimensional systems with finite-dimensional observations.

Distinguished Professor of Moscow State (Lomonosov) Univeristy, Full Member of Russian Academy of Sciences Alexander B. Kurzhanski
(3rd semester, 2 hours/week lectures, 2 hours/week seminars, 2 hours/week computer practicum)

Multi-objective Problems in Decision-Making

Annotation: the course reflects the state-of-the-art in decision-making theory, elaborates on the nature of multi-objective choice and human possibility in the multi-object problems of decision-making. Along with the fundamental notions of multi-objective optimization, the course includes method of multi-objective optimization, modern graphical interactive methods, methods for approximation of Pareto frontier for nonlinear systems.

Professor A. V. Lotov.

(3rd semester, 2 hours/week lectures, lab work in Internet)

Identification of Dynamical Systems

Annotation: this course is about the problem of identification of unknown parameters of dynamical systems described by differential or difference equations on the basis of a priori information about these parameters and observations of the system. The course focuses on the theories of stochastic and guaranteed estimation and their combinations as approaches to identification problems. The course shows how solution of identification problems may be further used for control, forecast and related issues.

Associate Professor I. A. Digailova

(1st semester, 2 hours/week lectures)

Mathematical Models of Economical Dynamics

Annotation: the course described the mathematical tools which are using in analysis of mathematical models of economical processes. The course reflects the experience and the point of view of a research group in the field of mathematical modeling of Western, Soviet and later Russian economics.

Professor A. A. Shananin
(4nd semester, 2 hours/week lectures)

Mathematical Models in Financial Dynamics

Annotation: The course starts with the basics of martingale techniques necessary for problems of stochastic financial mathematics. In the expected utility framework (by von Neumann and Morgenstern) we formalize the problem of optimal investment and the problem of pricing and hedging of options. The course uses an original approach — a game-theoretical interpretation of the problem of guaranteed pricing of options.

Associate Professor S. N. Smirnov

(2nd and 3rd semester, 2 hours/week lectures, 2 hours/week seminars)

Hybrid Systems (Dynamics and Control)

Annotation: this course is about a new class of models for processes of dynamics and control, the so-called hybrid systems. Such systems combine models with continuous time with discrete models. The system may follow a continuous motion described by one model from a set of models, and it may switch from one model to another. The processes of switching may be governed, for example, by a finite automaton. Such approach captures the behavior of rather complex systems. The course includes examples of hybrid systems and their formalization, as well as general approaches to develop a theory of hybrid systems. The problems of stability, control and observation are considered. A special attention is given to numerical aspects which play an important role in the development of solution methods for such problems.

Distinguished Professor of Moscow State (Lomonosov) Univeristy, Full Member of Russian Academy of Sciences Alexander B. Kurzhanski
(3th semester, 2 hours/week lectures, 2 hours/week computer labs)

Mathematical Theory of Communications

Annotation: the course is an introduction to mathematical models of modulation and demodulation of signals and of transmission of information through linear channels (including those with delay) with adaptive white noise and colored noises. A special attention is given to mathematical problems of optimal choice of coding under given channel characteristics, and to the issue of network synchronization.

Associate Professor I. A. Digailova
(4th semester, 2 hours/week lectures)

Stability Theory and Problems of Stabilization

Annotation: The course starts with principal results of the Lyapunov's stability theory. Theorems on stability, asymptotical stability and instability are given. The direct method of Lyapunov is studied, along with the theory of Lyapunov's functions and the theorem of Barbashin-Krasovski, inverse theorems on stability, first-order stability. Stability of linear systems is studied in detail: Routh-Hurwtz criterion, Lyapunov's equation, periodical systems and Floquet's theory. The applications of vector and nonsmooth Lyapunov's functions are indicated. Stability of difference schemes and time-delay equations is discussed.

Problems of stabilization are studied, including the linear-quadratic problem. Then nonlinear systems are considered, the «control» Lyapunov's functions. Then the theory of stabilization of linear systems is exposed, the conditions for the system to be stabilizable, the method of algebraic Riccati equation.

Stabilization under unknown disturbances is duscussed, along with principal fundamental notions of stochastic stability and stabilization. Examples of real-world applications for the theory above are given.

Professor V. D. Furasov
(4rd semester, 2 hours/week lectures)

Dynamical Systems and Models in Biology

Annotation: This course covers the following topics. Behavior of time-invariant dynamical systems. Liouville's theorem, rectification lemma, first integrals, motion in potential field. Lyapunov's stability of stationary points. Lyapunov functions. Lyapunov's theorems. Limit sets. Bendixson-Poincare theorem. Poincare indices and mapping. Central manifold theorem. Andronov-Hopf bifurcation.Mathematical models of interacting populations: Lotka-Volterra predator-prey model, competition of species, Gause-Kolmogorov model. Food chains. Cyclical competition of species. Mathematical model of pre-biological evolution. Model of interaction of pollution and environment. Models of therapy and epidemic propagation. Fischer-Kolmogorov equation.

Professor A. S. Bratus

(1rd and 2nd semester, 2 hours/week lectures)

The Basics of Set-valued Analysis

Annotation: This course begins with the basics of convex analysis. Firstly fundamental results are proven, such as Karatheodory's theorem and separation theorems. Treated further on are convex conjugate functions, the theorem of Fenchel-Moreau, support and indicator functions and their conjugates. The definitions of set-valued mapping are then given. Considered are various classes of set-valued mappings and their basic properties. These are followed by the treatment of differential inclusions - definition of solution and existence theorems. The course concludes with applications of differential inclusions to control theory.

Professor A. V. Arutyunov.
(1rd semester, 2 hours/week lectures)

Scientific Seminar: Mathematical Modeling of Complex Systems

Annotation: The seminar covers the models of complex systems, taking the following problems as examples: control of data flow in computer networks; control of traffic in a highway network; stability and control of modern electrical power systems. One discusses mathematical tools which allow

Distinguished Professor of Moscow State (Lomonosov) Univeristy, Full Member of Russian Academy of Sciences Alexander B. Kurzhanski
(1st-4th semesters, 2 hours/week)