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Perov, Anatoliy Ivanovich
Statistics Math-Net.Ru
Total publications: 74
Scientific articles: 71

Number of views:
This page:4551
Abstract pages:19388
Full texts:8657
References:1453
Professor
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.

Subject:

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.

Biography

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.

https://www.mathnet.ru/eng/person8955
List of publications on Google Scholar
https://zbmath.org/authors/ai:perov.anatolij-i
https://mathscinet.ams.org/mathscinet/MRAuthorID/197254
https://orcid.org/0000-0002-5198-1115

Publications in Math-Net.Ru Citations
2025
1. A. I. Perov, I. D. Kostrub, “The Vandermonde matrix in the general case”, Dokl. RAN. Math. Inf. Proc. Upr., 521 (2025),  81–87  mathnet  elib
2024
2. A. I. Perov, I. D. Kostrub, “The Vandermonde matrix in the commutative case”, Dokl. RAN. Math. Inf. Proc. Upr., 517 (2024),  33–37  mathnet  elib; Dokl. Math., 109:3 (2024), 216–220
2023
3. I. D. Kostrub, A. I. Perov, “Spectral Test for Exponential Stability”, Mat. Zametki, 113:4 (2023),  489–498  mathnet  mathscinet; Math. Notes, 113:4 (2023), 480–487  scopus
2020
4. A. I. Perov, I. D. Kostrub, “Differential equations in Banach algebras”, Dokl. RAN. Math. Inf. Proc. Upr., 491 (2020),  73–77  mathnet  zmath  elib; Dokl. Math., 101:2 (2020), 139–143 8
5. A. I. Perov, I. D. Kostrub, “On differential equations in Banach algebras”, Russian Universities Reports. Mathematics, 25:132 (2020),  410–421  mathnet 1
2018
6. A. I. Perov, I. D. Kostrub, O. I. Kleshchina, E. E. Dikarev, “Absolute logarithmic norm”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, no. 4,  70–85  mathnet; Russian Math. (Iz. VUZ), 62:4 (2018), 60–73  isi  scopus 2
7. 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  elib
8. 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  elib
9. 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  elib
2017
10. 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 2
11. 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 6
2016
12. A. I. Perov, V. K. Kaverina, “Non-linear differential equations of higher order”, Meždunar. nauč.-issled. žurn., 2016, no. 6-5(48),  99–102  mathnet
13. 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 6
2014
14. A. I. Perov, “New features of stability of linear systems of differential equations with constant coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 2014, no. 9,  49–58  mathnet; Russian Math. (Iz. VUZ), 58:9 (2014), 41–48  scopus 3
2013
15. A. I. Perov, “On one stability criterion for linear systems of differential equations with periodic coefficients”, Avtomat. i Telemekh., 2013, no. 2,  22–37  mathnet  zmath; Autom. Remote Control, 74:2 (2013), 183–195  isi  scopus 3
2010
16. 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 13
17. 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
2009
18. A. I. Perov, “On the Hadamard lemma and the Lipschitz condition”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 12,  36–48  mathnet  mathscinet  zmath; Russian Math. (Iz. VUZ), 53:12 (2009), 30–40
2002
19. A. I. Perov, “New Stability Criteria for Constant-Coefficient Linear Systems”, Avtomat. i Telemekh., 2002, no. 2,  22–33  mathnet  mathscinet  zmath; Autom. Remote Control, 63:2 (2002), 189–199  isi  scopus 3
2001
20. 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 2
21. 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 1
2000
22. A. I. Perov, “Sufficient conditions for the stability of linear systems with constant coefficients in critical cases. II”, Avtomat. i Telemekh., 2000, no. 10,  49–59  mathnet  mathscinet  zmath; Autom. Remote Control, 61:10 (2000), 1630–1638 1
23. A. I. Perov, “The discrete matrix Luré equation (the singular case)”, Avtomat. i Telemekh., 2000, no. 3,  46–57  mathnet  mathscinet  zmath; Autom. Remote Control, 61:3 (2000), 397–407 1
1999
24. 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 2
1998
25. A. I. Perov, “On a theorem of Ostrowski”, Izv. Vyssh. Uchebn. Zaved. Mat., 1998, no. 11,  77–82  mathnet  mathscinet  zmath; Russian Math. (Iz. VUZ), 42:11 (1998), 73–78
26. 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 2
1997
27. A. I. Perov, “Sufficient Conditions of Stability of Linear Systems with Constant Coefficients in Critical Cases. I”, Avtomat. i Telemekh., 1997, no. 12,  80–89  mathnet  mathscinet  zmath; Autom. Remote Control, 58:12 (1997), 1906–1913 4
28. 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
1995
29. A. I. Perov, “Lur'e discrete matrix equation (regular case)”, Avtomat. i Telemekh., 1995, no. 3,  21–29  mathnet  mathscinet  zmath; Autom. Remote Control, 56:3 (1995), 324–330
1994
30. A. I. Perov, “On an inequality of La Salle”, Avtomat. i Telemekh., 1994, no. 10,  188–189  mathnet  mathscinet  zmath; Autom. Remote Control, 55:10 (1994), 1550–1551
1990
31. 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 1
1988
32. A. I. Perov, “Optimal corrective control of a linear discrete-time system”, Avtomat. i Telemekh., 1988, no. 1,  163–166  mathnet  zmath
33. 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
34. 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
1986
35. A. I. Perov, A. A. Tananika, “A generalization of Wirtinger's inequality”, Differ. Uravn., 22:6 (1986),  1074–1076  mathnet  mathscinet  zmath
1984
36. A. I. Perov, T. I. Smagina, V. L. Khatskevich, “A variational approach to a problem on periodic solutions”, Sibirsk. Mat. Zh., 25:1 (1984),  106–119  mathnet  mathscinet  zmath; Siberian Math. J., 25:1 (1984), 88–100  isi 1
1983
37. 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
1982
38. A. I. Perov, V. L. Khatskevich, “Some local existence theorems for periodic solutions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1982, no. 7,  50–60  mathnet  mathscinet  zmath; Soviet Math. (Iz. VUZ), 26:7 (1982), 62–75
1978
39. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. IV”, Differ. Uravn., 14:7 (1978),  1192–1202  mathnet  mathscinet  zmath
40. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. III”, Differ. Uravn., 14:2 (1978),  232–244  mathnet  mathscinet  zmath
41. A. I. Perov, “Potential operators”, Mat. Zametki, 24:6 (1978),  793–799  mathnet  mathscinet  zmath; Math. Notes, 24:6 (1978), 921–925
1977
42. S. V. Pervachev, A. I. Perov, “A multi-dimensional algorithm of sliding adaptive reception”, Avtomat. i Telemekh., 1977, no. 6,  57–63  mathnet  zmath; Autom. Remote Control, 38:6 (1977), 821–826
43. 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 2
1976
44. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. II”, Differ. Uravn., 12:7 (1976),  1223–1237  mathnet  mathscinet  zmath
1974
45. A. I. Perov, Yu. V. Trubnikov, “Monotone differential equations. I”, Differ. Uravn., 10:5 (1974),  804–815  mathnet  mathscinet  zmath
1972
46. 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 3
47. A. I. Perov, Q. H. Ta, “The almost periodic solutions of homogeneous differential equations”, Differ. Uravn., 8:3 (1972),  453–458  mathnet  mathscinet  zmath 1
1970
48. A. I. Perov, V. G. Zadorozhniy, “Multidimensional linear differential equations with periodic coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 1970, no. 5,  64–73  mathnet  mathscinet  zmath
1969
49. A. I. Perov, T. K. Katsaran, “Nonlinear multidimensional differential equations”, Dokl. Akad. Nauk SSSR, 189:3 (1969),  479–482  mathnet  mathscinet  zmath
50. A. I. Perov, “New complete solvability conditions and questions of reducibility”, Dokl. Akad. Nauk SSSR, 186:1 (1969),  26–29  mathnet  mathscinet  zmath
51. A. I. Perov, “A certain generalization of the Frobenius theorem”, Differ. Uravn., 5:10 (1969),  1881–1884  mathnet  mathscinet  zmath 1
1968
52. A. I. Perov, “On the theory of multidimensional differential equations with constant coefficients”, Differ. Uravn., 4:7 (1968),  1289–1298  mathnet  mathscinet  zmath
53. A. I. Perov, T. K. Katsaran, “Theorems of Favard and Bohr–Neugebauer for multidimensional differential equations”, Izv. Vyssh. Uchebn. Zaved. Mat., 1968, no. 5,  62–70  mathnet  mathscinet  zmath
1967
54. A. I. Perov, “A question on the structure of the limit set”, Dokl. Akad. Nauk SSSR, 176:3 (1967),  526–529  mathnet  mathscinet  zmath
55. A. I. Perov, “Bounded solutions of differential equations of second order”, Differ. Uravn., 3:3 (1967),  524–528  mathnet  mathscinet  zmath
1966
56. 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
57. 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
58. A. I. Perov, A. P. Makhmudov, “On a certan boundary value problem”, Differ. Uravn., 2:3 (1966),  365–370  mathnet  mathscinet
59. 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 1
60. A. I. Perov, “A multi-dimensional linear differential equation of the second order with constant coefficient”, Izv. Vyssh. Uchebn. Zaved. Mat., 1966, no. 6,  117–124  mathnet  mathscinet  zmath
61. A. I. Perov, “On multidimensional differential equations of the first order”, Sibirsk. Mat. Zh., 7:2 (1966),  344–352  mathnet  mathscinet  zmath; Siberian Math. J., 7:2 (1967), 279–286
1965
62. A. I. Perov, “Some remarks on differential inequalities”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, no. 4,  104–112  mathnet  mathscinet  zmath 5
1964
63. 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
64. A. I. Perov, “Topological characteristics of solutions of higher-dimensional differential equations”, Dokl. Akad. Nauk SSSR, 157:4 (1964),  791–794  mathnet  mathscinet  zmath
65. A. I. Perov, “On multi-dimensional differential equations with constant coefficients”, Dokl. Akad. Nauk SSSR, 154:6 (1964),  1266–1269  mathnet  mathscinet  zmath
1963
66. V. B. Melamed, A. I. Perov, “A generalization of a theorem of M. A. Krasnosel'skii on the complete continuity of the Fréchet derivative of a completely continuous operator”, Sibirsk. Mat. Zh., 4:3 (1963),  702–704  mathnet 1
1962
67. 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
1960
68. 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 2
1958
69. 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 14
70. A. I. Perov, “A two-point boundary value problem”, Dokl. Akad. Nauk SSSR, 122:6 (1958),  982–985  mathnet  mathscinet  zmath
71. A. I. Perov, “On uniqueness theorems for ordinary differential equations”, Dokl. Akad. Nauk SSSR, 120:4 (1958),  704–707  mathnet  mathscinet  zmath 1
2006
72. 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 1
1998
73. 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, no. 2,  179–184  mathnet
1968
74. 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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