Nikolai Bobylev, in full Nikolai Antonovich Bobylev, obtained important results in various fields of nonlinear analysis, optimization, and control theory. He developed the homotopic method for studying extremum problems based on the minimum invariance principle. (It is also known as the deformation method). This method led to significant advances in classical fields of mathematics. In particular, different inequalities were proved, strengthened, and generalized; exact constants for inequalities were derived; new approaches were proposed to analyze the stability of gradient, potential, and Hamiltonian systems. The deformation method turned out to be efficient and useful in problems of mathematical physics, variational calculus, and mathematical programming (the stability analysis of solutions, new sufficient criteria for minima, an algorithm for studying degenerate extremals, and the connection between uniqueness theorems for boundary value problems and the criteria for integral functionals minima). The well-known Ulam problem on the well-posedness (correctness) of variational problems was solved based on this method.
Another line of Bobylev’s research was the theory of topological invariants and its applications to the problems of chaotic dynamics. He developed an infinite-dimensional version of the Poincaré topological index of a stable equilibrium, which has numerous applications. In particular, Bobylev established that the Ginzburg—Landau equations, describing the behavior of a superconductor in an external magnetic field, have a previously unknown unstable solution corresponding to the saddle point of the integral of the superconductor’s total energy. He proposed a technique for localizing limit cycles in systems with the chaotic behavior of trajectories based on nonlinear functional analysis methods.
The relatedness theorems presented by Bobylev and M.A. Krasnosel’skii are an effective tool for investigating nonlinear problems of oscillation theory. These theorems connect the topological characteristics of zeros of different vector fields arising in the analysis of a particular problem. They were applied to study the convergence of approximate methods for constructing periodic oscillations of automatic control systems and systems with delay and to estimate the number of oscillating modes. In addition, Bobylev studied the convergence and applicability range of different numerical methods for solving nonlinear problems (the harmonic balance method, the mechanical quadrature method, the collocation method, the Galerkin method, factoring methods, and gradient methods).
Bobylev solved several practical problems of control theory. In particular, he developed an original approach to constructing piecewise linear Lyapunov functions for continuous-time systems and estimated the stability radius for large classes of finite- and infinite-dimensional dynamic systems.
Professor, Dr. Sci. (Phys.—Math.), and Head of Laboratory No. 61 (Mathematical Methods for Studying Complex Systems) at the Institute of Control Sciences (ICS), the USSR (Russian) Academy of Sciences, Bobylev carried out considerable scientific-organizational work. He was a member of the editorial boards of the journals Automation and Remote Control and Differential Equations, a member of Dissertation Councils at ICS and the Institute for Information Transmission Problems (IITP), the USSR (Russian) Academy of Sciences, and a member of the Expert Council for Control, Computer Science, and Informatics of the Higher Attestation Commission of Russia. Bobylev paid much attention to raising new scientific staff. Under his guidance, 12 candidate’s dissertations in physics and mathematics were defended. He led the Seminar on Nonlinear Analysis Methods in Control Theory (ICS RAS). Also, Bobylev taught at Moscow State University and the Moscow Institute of Physics and Technology (courses on modern nonlinear analysis, functional analysis, and their applications).
His main monographs are as follows:
- Homotopy of Extremal Problems. Theory and Applications, Berlin: Walter de Gruyter, 2007. — 303 p. (coauthors S.V. Emelyanov, S.K. Korovin, and A.V. Bulatov);
- Metody nelineinogo analiza v zadachakh upravleniya i optimizatsii (Nonlinear Analysis Methods in Control and Optimization Problems), Moscow: Editorial URSS, 2002. — 432 p. (coauthors S.V. Emelyanov, S.K. Korovin, and A.V. Bulatov);
- Gomotopii ekstremal’nykh zadach (Homotopies of Extremal Problems), Moscow: Nauka, 2001. — 350 p. (coauthors S.V. Emelyanov, S.K. Korovin, and A.V. Bulatov);
- Geometrical Methods in Variational Problems, Mathematics and Its Applications, vol. 485, Springer, 1999, — 539 p. (coauthors S.V. Emelyanov and S.K. Korovin);
- Geometricheskie metody v variatsionnykh zadachakh (Geometrical Methods in Variational Problems), Moscow: Magistr, 1998. — 658 p. (coauthors S.V. Emelyanov and S.K. Korovin);
- Approximation Procedures in Nonlinear Oscillation Theory, Berlin: Walter de Gruyter, 1994. — 269 p. (coauthors Yu.M. Burman and S.K. Korovin);
- Metody nelineinogo analiza v zadachakh negladkoi optimizatsii (Nonlinear Analysis Methods in Nonsmooth Optimization Problems), Moscow: Nauka, 1992. — 207 p.;
- Matematicheskaya teoriya sistem (Mathematical Theory of Systems), Moscow: Nauka, 1986. — 165 p. (coauthors V.G. Boltyanskii, S.Yu. Vsekhsvyatskii, V.V. Kalashnikov, V.S. Kozyakin, V.B. Kolmanovskii, A.A. Kravchenko, A.M. Krasnosel’skii, and A.V. Pokrovskii).
Many books are presented in the Institute’s database:
https://www.ipu.ru/d7ipu/books_library_grid?combine=Бобылев
The list of journal papers by Bobylev can be found at Math-Net.Ru:
https://www.mathnet.ru/php/person.phtml?&personid=19256&option_lang=eng
Also, his publications are available at:
https://elibrary.ru/author_items.asp?authorid=25&pubrole=100&show_refs=1&show_option=0
Articles about N.A. Bobylev
1. | I. V. Gaishun, S. V. Emel’yanov, N. A. Izobov, V. A. Il’in, S. K. Korovin, E. F. Mishchenko, E. I. Moiseev, N. Kh. Rozov, V. A. Sadovnichii, T. K. Shemjakina, Nikolai Antonovich Bobylev, Differ. Uravn., 39:4 (2003), 570—572. |
2. | Nikolai Antonovich Bobylev (October 28, 1947 — December 17, 2002), Avtomat. i Telemekh., 2003, no. 2, 189; Autom. Remote Control, 64:2 (2003), 343. |
In addition, see the Wikipedia page devoted to Bobylev:
https://ru.wikipedia.org/wiki/Бобылев,_Николай_Антонович
Scopus Author ID: 7005909255