dynamical systems; integral manifolds; singular perturbations; dynamics of multibody and gyroscopic systems; mathematical modelling; automatic control; chemical kinetics.
Subject:
Theorems on existence and properties of weakly attractive slow integral mamifolds were proved and, as consequence, the known problem on mathematical justification of the precession theory of gyroscopic systems was solved and some paradoxes of this theory were explained. A notion of a fast integral manifold was proposed and the possibility to reduce singularly perturbed systems to a block triagular form was stated. This permits to develop a new method of decomposition of control systems this slow and fast variables and to apply this method to solution of some control theory problems. Such approach was developed on continuous and discrete systems with several time scales (with N. V. Voropaeva), stochastic systems (with E. Ya. Gorelova), periodic control systems (with E. N. Zharikova), time delay systems (with E. Fridman), distributed parameter systems (with S. V. Ozerskii). The slow integral manifolds theory was extended to systems in the events that the usual conditions of known Tikhonov theorem are violated. The slow integral manifolds branching problems were studied with K. Schneider. A new approach to investigation of so called canard-trajectories, which is based on an idea of glueing of attractive and reppelent slow integral manifolds, was proposed. A notion of black swan (multidimensional analogy of a canard) was proposed and theorem on existence and properties of such integral surface were stated (with E. A. Shchepakina). Jointly with G. N. Gorelov a canard-trajectory was found at first in parabolic systems and used to solve the chemical kinetics problems (critical condition of thermal explosion in the case of autocatalytic combustion were obtained). Jointly with V. M. Gol'dshtein a detailed algorithm of chemical kinetics systems, based on a combined using of integral manifolds method and Mishchenko–Rozov asymptotic formulae, was worked out. Two monographs and number of papers (with V. I. Babushok, V. M. Gol'dshtein, G. N. Gorelov, A. Ziniviev, E. A. Shchepakina, G. S. Yablonslii et al) are devoted to investigation of chemical kinetics problems. Jointly with K. Scneider and E. A. Shchepakina a new type of combustion travelling waves is described (canard travelling wave). Jointly with V. V. Strygin some problems of dynamics and stability of rigid bodies systems and satellites were solved.
Biography
Graduated from Faculty of Mathematics and Mechanics of Voronezh State University in 1971 (department of algebra and topological methods of analysis). Ph. D. thesis was defended in 1981. D. Sci. thesis was defended in 1991. A list of my works contains more than 100 titles. Since 1990 I have led the research seminar at Samara on nonlinear modelling and control.
Full Member of Russian Academy of Natural Sciences, 1996; Holder of Honorary Badge of Academy "For Merits in Development of Science and Economy", 1997; Editor-in-Chief of scientific journal "Transaction of Russian Academy of Natural Sciences. Series:Mathematics. Mathematical Modeling. Informatics and Control", 1997. In 2001 I was awarded the prize of the Samara region for a best work in mathematics. Member of AMS since 1984.
Main publications:
Mortell M.P., O'Malley R.E., Pokrovskii A., Sobolev V. Singular perturbations and hysteresis. SIAM, 2005.
Shchepakina E., Sobolev V., Mortell M.P. Singular perturbations: Introduction to system order reduction methods with applications. Springer, 2014.
V. A. Sobolev, “Decomposition of singularly perturbed optimal tracking problems with a given reference trajectory”, Sib. Zh. Ind. Mat., 26:3 (2023), 112–124; J. Appl. Industr. Math., 17:3 (2023), 640–650
2022
2.
V. A. Sobolev, “Reduction of the optimal tracking problem in the presence of noise”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 28:3-4 (2022), 32–39
2021
3.
V. A. Sobolev, E. A. Tropkina, E. A. Shchepakina, L. Zhang, “Decomposition of traveling waves problems”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 27:3 (2021), 22–30
4.
V. A. Sobolev, E. A. Tropkina, E. A. Shchepakina, L. Zhang, J. Wang, “Critical travelling waves in one model of the "reaction-diffusion" type”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 27:2 (2021), 16–24
2018
5.
V. A. Sobolev, E. A. Shchepakina, E. A. Tropkina, “Parametrization of invariant manifolds of slow motions”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 24:4 (2018), 33–40
2015
6.
A. A. Archibasov, A. Korobeinikov, V. A. Sobolev, “Asymptotic expansions of solutions in a singularly perturbed model of virus evolution”, Zh. Vychisl. Mat. Mat. Fiz., 55:2 (2015), 242–252; Comput. Math. Math. Phys., 55:2 (2015), 240–250
M. Osintcev, V. Sobolev, “Order reduction of control and estimation problems for flexible joint manipulator”, Matem. Mod., 26:7 (2014), 72–86
8.
M. O. Osintsev, V. A. Sobolev, “Reduction of dimension of optimal estimation problems for dynamical systems with singular perturbations”, Zh. Vychisl. Mat. Mat. Fiz., 54:1 (2014), 50–64; Comput. Math. Math. Phys., 54:1 (2014), 45–58
M. O. Osintsev, V. A. Sobolev, “Reduction of Dimensionality of Optimal Estimation and Control Problems for Systems of Low Dissipativity Solid Bodies”, Avtomat. i Telemekh., 2013, no. 8, 121–137; Autom. Remote Control, 74:8 (2013), 1334–1347
V. A. Sobolev, E. A. Tropkina, “Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models”, Zh. Vychisl. Mat. Mat. Fiz., 52:1 (2012), 81–96; Comput. Math. Math. Phys., 52:1 (2012), 75–89
V. A. Sobolev, D. M. Shchepakin, “Integral manifolds and the reduction principle”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2011, no. 5(86), 81–92
2006
12.
N. V. Voropaeva, V. A. Sobolev, “Decomposition of a linear-quadratic optimal control problem with fast and slow variables”, Avtomat. i Telemekh., 2006, no. 8, 3–11; Autom. Remote Control, 67:8 (2006), 1185–1193
E. N. Smetannikova, V. A. Sobolev, “Regularization of cheap periodic control problems”, Avtomat. i Telemekh., 2005, no. 6, 59–73; Autom. Remote Control, 66:6 (2005), 903–916
E. N. Smetannikova, V. A. Sobolev, “A Periodic Singularly Perturbed Problem for the Matrix Riccati Equation”, Differ. Uravn., 41:4 (2005), 500–507; Differ. Equ., 41:4 (2005), 529–537
E. V. Kitaeva, V. A. Sobolev, “Numerical determination of bounded solutions to discrete singularly perturbed equations and critical combustion regimes”, Zh. Vychisl. Mat. Mat. Fiz., 45:1 (2005), 56–87; Comput. Math. Math. Phys., 45:1 (2005), 52–82
E. N. Zharikova, V. A. Sobolev, “Optimal Periodic Control Systems Subject to Singular Perturbations”, Avtomat. i Telemekh., 1997, no. 7, 151–168; Autom. Remote Control, 58:7 (1997), 1188–1202
V. A. Sobolev, E. A. Shchepakina, “Duck trajectories in a problem of combustion theory”, Differ. Uravn., 32:9 (1996), 1175–1184; Differ. Equ., 32:9 (1996), 1177–1186
L. I. Kononenko, V. A. Sobolev, “Asymptotic decomposition of slow integral manifolds”, Sibirsk. Mat. Zh., 35:6 (1994), 1264–1278; Siberian Math. J., 35:6 (1994), 1119–1132
V. A. Sobolev, E. A. Shchepakina, “Self-ignition of dusty media”, Fizika Goreniya i Vzryva, 29:3 (1993), 133–136; Combustion, Explosion and Shock Waves, 29:3 (1993), 378–381
V. A. Sobolev, “Singular perturbations in a linear-quadratic problem of optimal control”, Avtomat. i Telemekh., 1991, no. 2, 53–64; Autom. Remote Control, 52:2 (1991), 180–189
V. A. Sobolev, K. I. Chernyshov, “A singularly perturbed differential equation with a Fredholm operator multiplying the derivative”, Differ. Uravn., 25:2 (1989), 247–258; Differ. Equ., 25:2 (1989), 181–190
1988
24.
Yu. V. Miheev, V. A. Sobolev, È. M. Fridman, “Asymptotic analysis of digital control systems”, Avtomat. i Telemekh., 1988, no. 9, 83–88; Autom. Remote Control, 49:9 (1988), 1175–1180
V. A. Sobolev, L. M. Fridman, “Decomposition of systems operating at different rates with discontinuous controls”, Avtomat. i Telemekh., 1988, no. 3, 29–34; Autom. Remote Control, 49:3 (1988), 288–292
1985
26.
V. V. Strygin, V. A. Sobolev, E. Ya. Gorelova, È. M. Fridman, “Integral manifolds of singularly perturbed systems and some of their applications”, Differ. Uravn., 21:10 (1985), 1723–1726
Contact us: