Preamble

DIFFERENTIAL EQUATIONS, DECEMBER 1957, VOLUME III, NO. 12
SOVIET SCIENCE: EVGENY ALEKSEEVICH BARBASHIN (On his 50th Birthday)

January 17, 1968, marks the 50th birthday of Academician of the Academy of Sciences of the BSSR, Evgeny Alekseevich Barbashin. Born into a civil servant's family in the village of Uinsk, Perm Region, Barbashin was orphaned at the age of six and was raised in an orphanage from 1927. After graduating from high school in 1935, he enrolled in the Faculty of Physics and Mathematics at Ural University. In 1940, he began his postgraduate studies at Moscow State University under the supervision of V. V. Nemytskii. Barbashin immediately became an active participant in the renowned seminar on qualitative theory led by V. V. Stepanov and V. V. Nemytskii, where he was the first to develop the theory of dynamical systems without the assumption of uniqueness.

Following the defense of his candidate dissertation in 1943, Barbashin was appointed Associate Professor in the Department of Higher Mathematics at the Ural Polytechnic Institute. In 1948, he entered the doctoral program at the V. A. Steklov Mathematical Institute of the USSR Academy of Sciences, successfully defending his doctoral dissertation, "The Method of Sections in the Theory of Dynamical Systems," in 1951. In this work, he was the first to link the qualitative study of dynamical systems as a whole with problems of motion stability. This connection led to several new results in Lyapunov stability theory, including the proof of the existence of a smooth Lyapunov function in the domain of attraction of an asymptotically stable singular point.

In 1952, Barbashin returned to Sverdlovsk to the Department of Higher Mathematics at the Ural Polytechnic Institute. There, he began extensive scientific and organizational work to establish the now well-known Sverdlovsk school of differential equations. Under his leadership, a large and capable collective of Sverdlovsk mathematicians emerged, conducting significant research across a broad spectrum of problems, including the theory of differential equations, stability theory, oscillation theory, and the theory of optimal control. This group consistently balanced the investigation of fundamental theoretical questions with the solution of important applied problems.

During this period, Barbashin’s own scientific interests focused primarily on two major areas: the global stability of substantially nonlinear systems and the stability of nonlinear systems based on first approximations. His work resulted in the proof of critical theorems justifying the Lyapunov function method for new classes of problems. He developed methods for the effective construction of Lyapunov functions for specific types of nonlinear systems and established general methods for proving stability theorems by first approximation. These methods proved applicable not only to ordinary differential equations but also to more general systems, such as integro-differential equations. The Sverdlovsk seminar and the research groups that grew from it maintained close ties and collaborated extensively with other scientific centers across the country, including Leningrad University, the Institute of Problems in Mechanics of the USSR Academy of Sciences, and the Minsk mathematical community.

From 1961 to 1966, Barbashin served as a staff member of the Sverdlovsk Branch of the Mathematical Institute of the USSR Academy of Sciences, where he headed the Department of Mathematical Analysis. He played a leading role in establishing and strengthening this major new mathematical center in the Urals. During this time, his research shifted significantly toward the theory of automatic control. He published a series of works on program control, providing a comprehensive look at the relationship between programmed motion tasks, general stability theory, optimal process theory, and function approximation theory. Furthermore, he developed original and effective methods for stabilizing automatic control systems, including dynamic systems with irregular or discontinuous characteristics, which are critical for applications but difficult to study mathematically.

Most recently, Barbashin developed a new conceptual framework for the fundamental definitions of motion stability, treating stability in relation to a system's reaction to various classes of external perturbations. This approach allowed for the application of the apparatus of generalized functions to stability problems. Many of his significant results from this period were compiled in his monograph, Introduction to the Theory of Stability (Nauka Publishing, 1967). Throughout his career, Barbashin has mentored a large number of researchers, including more than 25 Candidates of Sciences.

In 1966, Barbashin was elected an Academician of the Academy of Sciences of the BSSR and moved to Minsk. He currently serves on the editorial board of the journal Differential Equations, heads the Laboratory of Applied Mathematics and Mechanics at the Institute of Mathematics of the BSSR Academy of Sciences, and chairs the Department of Applied Mathematics at the V. I. Lenin Belarusian State University. In 1967, he was awarded the Order of the Red Banner of Labour for his scientific achievements. Evgeny Alekseevich Barbashin remains in the prime of his career, and he and his students will undoubtedly continue to produce profound research in the theory of differential equations.

— N. P. ERUGIN, N. N. KRASOVSKII

List of Published Works by E. A. Barbashin

1. On certain features arising in a dynamical system upon violation of uniqueness. DAN SSSR, 1943.
2. Local features of ordinary points of a system of differential equations. DAN SSSR, 41, No. 5, 183–186.
3. On the behavior of points under homeomorphic transformations of space. DAN SSSR, 51, No. 1, 3–6.
4. On $\sigma$-coverings of space. Matem. Sb., 18 (60), No. 3, 423–428.
5. On the classification of integral manifolds of a system of equations in total differentials. DAN SSSR, 55, No. 4, 283–286.
6. Toward a theory of generalized dynamical systems. Uch. Zap. Moskovskogo Un-ta.
7. On dynamical systems possessing a velocity potential. DAN SSSR, 61, No. 2, 185–187.
8. On homomorphisms of dynamical systems. DAN SSSR, 61, No. 3, 429–432.
9. Toward a theory of systems of multi-valued mappings of a topological space. Uch. Zap. Ural’skogo Un-ta, 7, 54–60.
10. Dispersive dynamical systems. UMN, 5:4 (38), 138–139.
11. On homomorphisms of dynamical systems. Matem. Sb., 27 (69), No. 3, 455–470.
12. On a condition for the existence of a secant surface. DAN SSSR, 70, No. 3.
13. On the existence of smooth solutions for certain linear partial differential equations. DAN SSSR, 72, No. 3, 445–447.
14. The method of sections in the theory of dynamical systems. Matem. Sb., 29 (71), No. 2, 233–280.
15. On homomorphisms of dynamical systems. II. Matem. Sb., 29 (71), No. 3, 501–518.
16. On the stability of the solution of a certain third-order nonlinear equation. PMM, 16, Iss. 5, 629–632.
17. On the stability of motion as a whole. DAN SSSR, 453–456 (with N. N. Krasovskii).
18. On the behavior of points under homeomorphic transformations of space (generalization of Birkhoff's theorem). Trudy Ural’skogo Politekhnicheskogo In-ta, 51, 4–11.
19. On the existence of Lyapunov functions in the case of asymptotic stability as a whole. PMM, 18, Iss. 3, 345 (with N. N. Krasovskii).
20. On the question of stability by first approximation. PMM, 19, Iss. 5 (with M. A. Skalkina).
21. On two schemes for proving stability theorems by first approximation. DAN SSSR, No. 1, 9–11.
22. On conditions for preserving the stability property of solutions to integro-differential equations. Izv. Vuzov. Matematika, No. 1, 25–34.
23. On the works of members of the Sverdlovsk seminar on qualitative methods in the theory of differential equations. Trudy Vsesoyuznogo Matematicheskogo S’ezda, 31–36; AN SSSR.
24. Toward a topological definition of integral invariants. Izv. Vuzov. Matematika, No. 3 (4), 8–12 (with V. A. Baidosov).
25. On the stability of solutions to a system of integro-differential equations. Nauchn. Dokl. Vysshei Shkoly. Fiz. Matem., No. 3, 18–22 (with L. Kh. Liberman).
26. On the stability of solutions to integro-differential equations. Trudy Ural’skogo Politekhnicheskogo In-ta, Sb.
27. On conditions for the existence of limit cycles. PMM, 23, Iss. 5, 826–835 (with V. A. Tabueva).
28. On the oscillations of a pendulum in the presence of dry friction. Izv. Vuzov. Matematika, No. 5 (12), 48–57 (with V. A. Tabueva).
29. On a problem in the theory of dynamic programming. PMM, Iss. 6.
30. On the mapping of a dynamical system into a dynamical system analytic with respect to time. Izv. Vuzov. Matematika, No. 1 (14), 11–15 (with F. A. Sholokhovich).
31. On estimating the mean-square deviation of a given trajectory. Avtomatika i Telemekhanika, No. 7, 941–950.
32. On estimating the maximum deviation from a given trajectory. Avtomatika i Telemekhanika, No. 10, 1341–1351.
33. On conditions for the uniqueness of limit cycles. Izv. Vuzov. Matematika, No. 3 (16), 43–47 (with E. V. Vdovina).
34. Toward a theory of dynamical systems with multi-valued and discontinuous characteristics. DAN SSSR, No. 1, 9–11 (with Yu. I. Alimov).
35. Qualitative study of equations describing the motion of interacting points on a circle. Izv. Vuzov. Matematika, No. 2 (21), 137–146 (with V. S. Serebryakova).
36. On circular motions of coupled pendulums. Izv. Vuzov. Matematika, 4 (23), 111–118 (with V. S. Serebryakova).
37. On the construction of periodic motions. PMM, Iss. 2, 276–283.
38. On the construction of Lyapunov functions for nonlinear systems. Trudy Mezhdunarodnogo Kongressa Mezhd. Federatsii po Avtomaticheskomu Upravleniyu. Theory of Continuous Systems. Special Math Problems. AN SSSR.
39. On the approximate realization of motion along a given trajectory. Avtomatika i Telemekhanika, No. 6, 681–687.
40. Program control of systems with random parameters. PMM, Iss. 5.
41. Toward a theory of relay differential equations. Izv. Vuzov. Matematika, No. 1 (26), 3–13 (with Yu. I. Alimov).
42. On a method for stabilizing a third-order control system with a high gain coefficient. Avtomatika i Telemekhanika, No. 10 (with V. A. Tabueva).
43. On a method for stabilizing third-order control systems with a high gain coefficient. Avtomatika i Telemekhanika, No. 5, 608–614 (with V. A. Tabueva).
44. On the stability of a variable-structure control system upon violation of sliding conditions. Avtomatika i Telemekhanika, No. 7, 882–890 (with V. A. Tabueva and R. M. Eidinov).
45. On the stability of solutions to integro-differential equations. Izv. Vuzov. Matematika, No. 3 (34), 7–14 (with L. P. Bisyarina).
46. Construction of periodic motion as a problem in program control theory. Trudy Mezhdunar. Simpoziuma po Nelineinym Kolebaniyam, Vol. III. Kiev.
47. Application of variable-structure automatic control systems for the regulation of a class of linear static objects. Avtomatika i Telemekhanika, No. 1, 27–32 (with I. N. Pechorina and R. M. Eidinov).
48. A theorem on the stability of the solution of a third-order differential equation with a discontinuous characteristic. PMM, Iss. 4, 664–671 (with V. A. Tabueva).
49. On a method for stabilizing a control system with limited permissible values of regulator parameters. Izv. AN SSSR. Tekhnicheskaya Kibernetika, No. 2, 121–128 (with V. M. Badkov).
50. Realization of processes and construction of signals at the system output. Trudy 4 Vsesoyuznogo Matem. S’ezda, Vol. II, Section Reports. Nauka Publishing, 410–415.
51. Seminar on the theory of variable-structure control systems. UMN, No. 6.
52. Theory of variable-structure control systems. Vestnik AN SSSR.
53. On the stabilization of control systems. PMM, Iss. 4, 761–765 (with E. I. Gerashchenko).
54. Theorems on the asymptotic stability of solutions to certain third-order differential equations with discontinuous characteristics. PMM, Iss. 3, 528 (with V. A. Tabueva).
55. On a principle for constructing stabilization systems. Avtomatika i Telemekhanika, No. 6, 995–1003 (with E. I. Gerashchenko).
56. On periodic motions of a double pendulum. Izv. Vuzov. Matematika, (48), 19–26 (with M. D. Kocheva).
57. On forcing sliding modes in automatic control systems. Differentsial’nye Uravneniya, 1, No. 11, 25–32 (with E. I. Gerashchenko).
58. Program control and the theory of optimal systems. Trudy II Mezhdunarodnogo Kongressa Mezhdunarodnoi Federatsii po Avtomaticheskomu Upravleniyu. Optimal Systems. Statistical Methods. Nauka Publishing.
59. Stability of variable-structure control systems. Trudy II Vsesoyuznogo S’ezda po Teoreticheskoi i Prikladnoi Mekhanike, Iss. II. Nauka Publishing, 7–14 (with V. A. Tabueva and R. M. Eidinov).
60. On stability with respect to impulsive actions. Differentsial’nye Uravneniya.
61. Certain questions in the synthesis of variable-structure automatic control systems. Trudy III Vsesoyuznogo Soveshchaniya po Avtomaticheskomu Upravleniyu. Nauka (with E. I. Gerashchenko, V. A. Tabueva, R. M. Eidinov).
62. Introduction to the Theory of Stability. Monograph. Nauka Publishing.
63. Methods for analyzing the stability of variable-structure automatic control systems. Trudy III Mezhdunarodnogo Kongressa Mezhdunarodnoi Federatsii po Avtomaticheskomu Upravleniyu. Nauka (with E. I. Gerashchenko, V. A. Tabueva, R. M. Eidinov).
64. On a class of pendulum motion in a medium with high resistance. Differentsial’nye Uravneniya, No. 3, 371–379 (with V. A. Tabueva).
65. On a method for stabilizing control systems. Differentsial’nye Uravneniya, No. 5, 733–741 (with N. G. Yarov).
66. Conditions for the existence of recurrent trajectories in dynamical systems with a cylindrical phase space. Differentsial’nye Uravneniya, No. 10, 1627–1633.
67. Classification of trajectories of a dynamical system with a cylindrical phase space. Differentsial’nye Uravneniya, No. 12.