Perov, Anatoliy Ivanovich

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Total publications: 63
Scientific articles: 60

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Doctor of physico-mathematical sciences (1970)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date: 30.09.1933
E-mail: ,
Keywords: the method of guiding functions in the theory of nonlinear oscillations; stability in critical cases; matrix equations of Liapounov and Rikkatti; periodic optimization.


The guiding functions method of approving the uniqueness theorems and of obtaining the estimates of periodical, almost periodical and bounded solutions of nonlinear systems of ordinary differential equations was developed together with M. A. Krasnoselskii.The formulas of the norm and logarithmic norm of the linear operator (matrix) on the subspace were obtained for the case where the norm of the vector is defined as the maximum of the absolute values of its coordinates. The proof of the stabilizing solvability theorem for the discrete matrix Lourie (Riccati) equation based on the methods of the cone theory was obtained, that gives the new proof of the discrete matrix Kalman–Popov–Yakubovitch theorem. The position and the meaning of different inequalities (Wirtinger, Steklov, Bohr–Favard, Bernstein, Stechkin, Nikolskii etc.) for smooth periodical functions in the theory of nonlinear oscillations is demonstrated.


Graduated from Voronezh State University (Departement of Physics & Mathematics, Chair of Functional Alalysis and Operator Equations) in 1956. Candidate (PhD) thesis — 1959 (supervisor M. A. Krasnoselskii). Doctor thesis — 1970. The number of publications — 140.

1999 — academician of Academy of Nonlinear Sciences 2000 — Soros Professor.

Main publications:
  • Krasnoselskii M. A., Perov A. I. Ob odnom printsipe suschestvovaniya ogranichennykh, periodicheskikh i pochti-periodicheskikh reshenii u sistemy obyknovennykh differentsialnykh uravnenii // DAN SSSR, 1958, 123, # 2, 235–238.
  • Krasnoselskii M. A., Perov A. I., Povolotskii A. I., Zabreiko P. P. Vektornye polya na ploskosti. Moskva. Fizmatgiz. 1963. 248 s.
  • Perov A. I. Variatsionnye metody v teorii nelineinykh kolebanii. Voronezh. Izd-vo VGU. 1981. 196 s.
  • Trubnikov Yu. V., Perov A. I. Differentsialnye uravneniya s monotonnymi nelineinostyami. Minsk. Nauka i tekhnika. 1986. 200 s.
  • Perov A. I. Formuly dlya normy lineinogo operatora na podprostranstve // DAN, 1999, 368, # 5, 601–603.
List of publications on Google Scholar

Publications in Math-Net.Ru
1. A. I. Perov, I. D. Kostrub, O. I. Kleshchina, E. E. Dikarev, “Absolute logarithmic norm”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, 4,  70–85  mathnet; Russian Math. (Iz. VUZ), 62:4 (2018), 60–73  isi  scopus
2. A. I. Perov, I. D. Kostrub, “Signs of regularity and stability of differential equations of higher order”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018),  674–678  mathnet
3. A. I. Perov, V. K. Kaverina, “Research of the nonautonomous system of ode by the ideas of the method of guiding functions”, Tambov University Reports. Series: Natural and Technical Sciences, 23:123 (2018),  510–516  mathnet
4. A. I. Perov, “Kolmogorov matrix, and a continuous markov chain with a finite number of states”, Tambov University Reports. Series: Natural and Technical Sciences, 23:123 (2018),  503–509  mathnet
5. A. I. Perov, “Conditions for invertibility and Hurwitz stability”, Funktsional. Anal. i Prilozhen., 51:4 (2017),  84–89  mathnet  elib; Funct. Anal. Appl., 51:4 (2017), 310–315  isi  scopus
6. A. I. Perov, I. D. Kostrub, “On the Spectral Abscissa and the Logarithmic Norm”, Mat. Zametki, 101:4 (2017),  562–575  mathnet  mathscinet  elib; Math. Notes, 101:4 (2017), 677–687  isi  scopus
7. A. I. Perov, V. K. Kaverina, “Non-linear differential equations of higher order”, Meždunar. nauč.-issled. žurn., 2016, 6-5(48),  99–102  mathnet
8. A. I. Perov, I. D. Kostrub, “On bounded solutions to weakly nonlinear vector-matrix differential equations of order $n$”, Sibirsk. Mat. Zh., 57:4 (2016),  830–849  mathnet  elib; Siberian Math. J., 57:4 (2016), 650–665  isi  elib  scopus
9. A. I. Perov, “New features of stability of linear systems of differential equations with constant coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 2014, 9,  49–58  mathnet; Russian Math. (Iz. VUZ), 58:9 (2014), 41–48  scopus
10. A. I. Perov, “On one stability criterion for linear systems of differential equations with periodic coefficients”, Avtomat. i Telemekh., 2013, 2,  22–37  mathnet  zmath; Autom. Remote Control, 74:2 (2013), 183–195  isi  scopus
11. A. I. Perov, “Multidimensional Version of M. A. Krasnosel'skii's Generalized Contraction Principle”, Funktsional. Anal. i Prilozhen., 44:1 (2010),  83–87  mathnet  mathscinet  zmath; Funct. Anal. Appl., 44:1 (2010), 69–72  isi  scopus
12. A. I. Perov, “A canonical system of two differential equations with periodic coefficients and the Poincaré–Denjoy theory of differential equations on a torus”, Sibirsk. Mat. Zh., 51:2 (2010),  373–387  mathnet  mathscinet; Siberian Math. J., 51:2 (2010), 301–312  isi  scopus
13. A. I. Perov, “On the Hadamard lemma and the Lipschitz condition”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, 12,  36–48  mathnet  mathscinet  zmath; Russian Math. (Iz. VUZ), 53:12 (2009), 30–40
14. A. I. Perov, “New Stability Criteria for Constant-Coefficient Linear Systems”, Avtomat. i Telemekh., 2002, 2,  22–33  mathnet  mathscinet  zmath; Autom. Remote Control, 63:2 (2002), 189–199  isi  scopus
15. A. I. Perov, “Jorke's Theorem and Wirtinger's Inequality”, Mat. Zametki, 70:2 (2001),  237–245  mathnet  mathscinet  zmath; Math. Notes, 70:2 (2001), 213–220  isi
16. N. A. Atomanichenko, A. I. Perov, “Bounds of determinants of matrices satisfying the Hadamard and Brauer conditions”, Zh. Vychisl. Mat. Mat. Fiz., 41:2 (2001),  179–185  mathnet  mathscinet  zmath  elib; Comput. Math. Math. Phys., 41:2 (2001), 161–166
17. A. I. Perov, “Sufficient conditions for the stability of linear systems with constant coefficients in critical cases. II”, Avtomat. i Telemekh., 2000, 10,  49–59  mathnet  mathscinet  zmath; Autom. Remote Control, 61:10 (2000), 1630–1638
18. A. I. Perov, “The discrete matrix Luré equation (the singular case)”, Avtomat. i Telemekh., 2000, 3,  46–57  mathnet  mathscinet  zmath; Autom. Remote Control, 61:3 (2000), 397–407
19. A. I. Perov, “Estimates for the elements of inverse matrices under the conditions of regularity criteria”, Zh. Vychisl. Mat. Mat. Fiz., 39:6 (1999),  867–875  mathnet  mathscinet  zmath; Comput. Math. Math. Phys., 39:6 (1999), 833–841
20. A. I. Perov, “On a theorem of Ostrowski”, Izv. Vyssh. Uchebn. Zaved. Mat., 1998, 11,  77–82  mathnet  mathscinet  zmath; Russian Math. (Iz. VUZ), 42:11 (1998), 73–78
21. A. I. Perov, “On Brauer's theorem and Cassini's ovals”, Uspekhi Mat. Nauk, 53:1(319) (1998),  225–226  mathnet  mathscinet  zmath; Russian Math. Surveys, 53:1 (1998), 221–222  isi  scopus
22. A. I. Perov, “Sufficient Conditions of Stability of Linear Systems with Constant Coefficients in Critical Cases. I”, Avtomat. i Telemekh., 1997, 12,  80–89  mathnet  mathscinet  zmath; Autom. Remote Control, 58:12 (1997), 1906–1913
23. A. I. Perov, “Sufficient conditions for the stability of linear systems with constant coefficients in critical cases”, Differ. Uravn., 33:10 (1997),  1349–1357  mathnet  mathscinet; Differ. Equ., 33:10 (1997), 1356–1363
24. A. I. Perov, “Lur'e discrete matrix equation (regular case)”, Avtomat. i Telemekh., 1995, 3,  21–29  mathnet  mathscinet  zmath; Autom. Remote Control, 56:3 (1995), 324–330
25. A. I. Perov, “On an inequality of La Salle”, Avtomat. i Telemekh., 1994, 10,  188–189  mathnet  mathscinet  zmath; Autom. Remote Control, 55:10 (1994), 1550–1551
26. A. I. Perov, A. A. Tananika, “On a geometric result in problems of periodic optimization”, Differ. Uravn., 26:4 (1990),  718–721  mathnet  mathscinet  zmath
27. A. I. Perov, “Optimal corrective control of a linear discrete-time system”, Avtomat. i Telemekh., 1988, 1,  163–166  mathnet  zmath
28. A. I. Perov, E. G. Glushko, I. G. Tyuleneva, “Eigenvalues and eigenfunctions of an integro-differential operator”, Differ. Uravn., 24:3 (1988),  516–519  mathnet  mathscinet; Differ. Equ., 24:3 (1988), 373–376
29. A. I. Perov, V. L. Khatskevich, “Regularity classes of bilinear forms, and nonlinear elliptic boundary value problems”, Differ. Uravn., 24:3 (1988),  464–476  mathnet  mathscinet; Differ. Equ., 24:3 (1988), 330–339
30. A. I. Perov, A. A. Tananika, “A generalization of Wirtinger's inequality”, Differ. Uravn., 22:6 (1986),  1074–1076  mathnet  mathscinet  zmath
31. O. I. Nikitin, A. I. Perov, “On the question of approximate determination of periodic solutions of differential equations”, Differ. Uravn., 19:11 (1983),  2001–2004  mathnet  mathscinet  zmath
32. A. I. Perov, V. L. Khatskevich, “Some local existence theorems for periodic solutions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1982, 7,  50–60  mathnet  mathscinet  zmath; Soviet Math. (Iz. VUZ), 26:7 (1982), 62–75
33. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. IV”, Differ. Uravn., 14:7 (1978),  1192–1202  mathnet  mathscinet  zmath
34. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. III”, Differ. Uravn., 14:2 (1978),  232–244  mathnet  mathscinet  zmath
35. A. I. Perov, “Potential operators”, Mat. Zametki, 24:6 (1978),  793–799  mathnet  mathscinet  zmath; Math. Notes, 24:6 (1978), 921–925
36. S. V. Pervachev, A. I. Perov, “A multi-dimensional algorithm of sliding adaptive reception”, Avtomat. i Telemekh., 1977, 6,  57–63  mathnet  zmath; Autom. Remote Control, 38:6 (1977), 821–826
37. A. I. Perov, V. V. Yurgelas, “The convergence of a certain iteration process”, Zh. Vychisl. Mat. Mat. Fiz., 17:4 (1977),  859–870  mathnet  mathscinet  zmath; U.S.S.R. Comput. Math. Math. Phys., 17:4 (1977), 45–56
38. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. II”, Differ. Uravn., 12:7 (1976),  1223–1237  mathnet  mathscinet  zmath
39. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. I”, Differ. Uravn., 10:5 (1974),  804–815  mathnet  mathscinet  zmath
40. A. I. Perov, I. Yu. Ègle, “On the Poincaré–Denjoy theory of multidimensional differential equations”, Differ. Uravn., 8:5 (1972),  801–810  mathnet  mathscinet  zmath
41. A. I. Perov, Q. H. Ta, “The almost periodic solutions of homogeneous differential equations”, Differ. Uravn., 8:3 (1972),  453–458  mathnet  mathscinet  zmath
42. A. I. Perov, V. G. Zadorozhniy, “Multidimensional linear differential equations with periodic coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 1970, 5,  64–73  mathnet  mathscinet  zmath
43. A. I. Perov, T. K. Katsaran, “Nonlinear multidimensional differential equations”, Dokl. Akad. Nauk SSSR, 189:3 (1969),  479–482  mathnet  mathscinet  zmath
44. A. I. Perov, “New complete solvability conditions and questions of reducibility”, Dokl. Akad. Nauk SSSR, 186:1 (1969),  26–29  mathnet  mathscinet  zmath
45. A. I. Perov, “A certain generalization of the Frobenius theorem”, Differ. Uravn., 5:10 (1969),  1881–1884  mathnet  mathscinet  zmath
46. A. I. Perov, “On the theory of multidimensional differential equations with constant coefficients”, Differ. Uravn., 4:7 (1968),  1289–1298  mathnet  mathscinet  zmath
47. A. I. Perov, T. K. Katsaran, “Theorems of Favard and Bohr–Neugebauer for multidimensional differential equations”, Izv. Vyssh. Uchebn. Zaved. Mat., 1968, 5,  62–70  mathnet  mathscinet  zmath
48. A. I. Perov, “A question on the structure of the limit set”, Dokl. Akad. Nauk SSSR, 176:3 (1967),  526–529  mathnet  mathscinet  zmath
49. A. I. Perov, “Bounded solutions of differential equations of second order”, Differ. Uravn., 3:3 (1967),  524–528  mathnet  mathscinet  zmath
50. A. I. Perov, “The zero-manifolds of solutions of multidimensional differential equations of the second order”, Dokl. Akad. Nauk SSSR, 168:6 (1966),  1254–1256  mathnet  mathscinet  zmath
51. A. I. Perov, “Investigation in the neighborhood of a singular point of a multi-dimensional differential equation in the analytic case”, Dokl. Akad. Nauk SSSR, 166:3 (1966),  544–547  mathnet  mathscinet  zmath
52. A. I. Perov, A. P. Makhmudov, “On a certan boundary value problem”, Differ. Uravn., 2:3 (1966),  365–370  mathnet  mathscinet
53. A. I. Perov, A. V. Kibenko, “On a certain general method for investigation of boundary value problems”, Izv. Akad. Nauk SSSR Ser. Mat., 30:2 (1966),  249–264  mathnet  mathscinet  zmath
54. A. I. Perov, “A multi-dimensional linear differential equation of the second order with constant coefficient”, Izv. Vyssh. Uchebn. Zaved. Mat., 1966, 6,  117–124  mathnet  mathscinet  zmath
55. A. I. Perov, “Some remarks on differential inequalities”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, 4,  104–112  mathnet  mathscinet  zmath
56. A. I. Perov, “On a higher-dimensional linear differential equation of second order”, Dokl. Akad. Nauk SSSR, 159:4 (1964),  755–758  mathnet  mathscinet  zmath
57. A. I. Perov, “Topological characteristics of solutions of higher-dimensional differential equations”, Dokl. Akad. Nauk SSSR, 157:4 (1964),  791–794  mathnet  mathscinet  zmath
58. A. I. Perov, “On multi-dimensional differential equations with constant coefficients”, Dokl. Akad. Nauk SSSR, 154:6 (1964),  1266–1269  mathnet  mathscinet  zmath
59. A. I. Perov, “A boundary-value problem for a system of two differential equations”, Dokl. Akad. Nauk SSSR, 144:3 (1962),  493–496  mathnet  mathscinet  zmath
60. A. I. Perov, “Periodical, almost periodical and bounded solutions to differential equation $dx/dt=f(t,x)$”, Dokl. Akad. Nauk SSSR, 132:3 (1960),  531–534  mathnet  mathscinet  zmath
61. M. A. Krasnosel'skii, A. I. Perov, “On a certain principle of the existence of bounded, periodical and almost periodical solutions to systems of ordinary differential equations”, Dokl. Akad. Nauk SSSR, 123:2 (1958),  235–238  mathnet  mathscinet  zmath
62. A. I. Perov, “A two-point boundary value problem”, Dokl. Akad. Nauk SSSR, 122:6 (1958),  982–985  mathnet  mathscinet  zmath
63. A. I. Perov, “On uniqueness theorems for ordinary differential equations”, Dokl. Akad. Nauk SSSR, 120:4 (1958),  704–707  mathnet  mathscinet  zmath

64. R. A. Arhontova, A. G. Baskakov, I. A. Bakhtin, E. S. Belyaeva, Yu. G. Borisovich, A. S. Grechenko, V. R. Zachepa, A. A. Karatsuba, A. F. Kudinov, V. V. Obukhovskii, A. I. Perov, A. S. Potapov, E. M. Semenov, V. P. Trofimov, N. N. Udodenko, “Kirill Andreevich Rodosskii (obituary)”, Uspekhi Mat. Nauk, 61:5(371) (2006),  173–175  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 61:5 (2006), 971–973  isi
65. E. A. Asarin, I. A. Bakhtin, N. A. Bobylev, V. A. Bondarenko, V. Sh. Burd, E. A. Gorin, S. V. Emel'yanov, P. P. Zabreiko, L. A. Ivanov, V. S. Kozyakin, A. M. Krasnosel'skii, N. A. Kuznetsov, A. B. Kurzhanskii, A. Yu. Levin, È. M. Muhamadiev, A. I. Perov, Yu. V. Pokornyi, A. V. Pokrovskii, D. I. Rachinskii, V. V. Strygin, Ya. Z. Tsypkin, V. V. Chernorutski, “Memory of M. A. Krasnosel'skii”, Avtomat. i Telemekh., 1998, 2,  179–184  mathnet
66. A. V. Kibenko, A. I. Perov, “Second Voronezh Winter Mathematical School”, Uspekhi Mat. Nauk, 23:4(142) (1968),  274–275  mathnet

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