Optimal control theory, differential equations, differential games.
Main publications:
M. I. Zelikin, Control Theory and Optimization I. Homogeneous Spaces and the Riccati Equation in the Calculus of Veriations, Springer-Verlag, Berlin, 2000
M. I. Zelikin, V. F. Borisov, Theory of Chattering Control with applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, MA, 1994
M. I. Zelikin, Optimal control and variational calculus., URSS, Moscow, 2004
M. I. Zelikin, “Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems”, Mat. Sb., 195:6 (2004), 57–70
M. I. Zelikin, “Superconductivity of Plasma and Fireballs”, CMFD, 19, 2006, 45–69
L. V. Lokutsievskiy, A. N. Pechen, M. I. Zelikin, “Time-optimal state transfer for an open qubit”, J. Phys. A, 57:27 (2024), 275302 , 23 pp., arXiv: 2401.10099;
2023
2.
Lev Lokutsievskiy, Mikhail Zelikin, “Derivatives of sub-Riemannian geodesics are $L_p$-Hölder continuous”, ESAIM: COCV, 29 (2023), 70 , 30 pp., arXiv: 2203.04956;
Lev Lokutsievskiy, Gerd Wachsmuth, Mikhail Zelikin, “Non-optimality of conical parts for Newton’s problem of minimal resistance in the class of convex bodies and the limiting case of infinite height”, Calc. Var. Partial Differential Equations, 61:1 (2022), 31 , 18 pp. ;
2020
4.
L. V. Lokutsievskiy, M. I. Zelikin, “The analytical solution of Newton’s aerodynamic problem in the class of bodies with vertical plane of symmetry and developable side boundary”, ESAIM: COCV, 26 (2020), 15 , 36 pp., arXiv: 1905.02028;
E. F. Mishchenko, A. S. Mishchenko, M. I. Zelikin, “On the Adequacy of Mathematical Models in Control Theory, Physics and Ecology, Finished”, Math. Ed., 2020, no. 1(93), 2–14
2019
6.
M. I. Zelikin, “Fractal Theory of Saturn's Ring. II: Electromagnetic Phenomena”, Proc. Steklov Inst. Math., 304 (2019), 153–159
7.
M. I. Zelikin, Yu. S. Osipov, “Minimal submanifolds of spheres and cones”, Proc. Steklov Inst. Math. (Suppl.), 307, suppl. 1 (2019), S172–S178
8.
Yu. S. Osipov, M. I. Zelikin, “Fields of smooth compact minimal surfaces”, Russ. J. Math. Phys., 26:2 (2019), 174–179
9.
E. F. Mishchenko, A. S. Mishchenko, M. I. Zelikin, “On the Adequacy of Mathematical Models in Control Theory, Physics and Ecology”, Math. Ed., 2019, no. 4(92), 2–16
2018
10.
L. V. Lokutsievskiy, M. I. Zelikin, “Hessian measures in the aerodynamic Newton problem”, J. Dyn. Control Syst., 24:3 (2018), 475–495
M. I. Zelikin, “Genius. In the Memory of I. Shafarevich”, Math. Ed., 2018, no. 1(85), 2–4
2017
12.
M. I. Zelikin, L. V. Lokoutsievskiy, S. V. Skopintcev, “On Optimal Harvesting of a Resource on a Circle”, Math. Notes, 102:4 (2017), 521–532
13.
M. I. Zelikin, “An analogue of Pontryagin's maximum principle in problems of minimization of multiple integrals”, Izv. Math., 81:5 (2017), 973–984
14.
M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136
2015
15.
M. I. Zelikin, “Fractal Theory of Saturn's Ring”, Proc. Steklov Inst. Math., 291 (2015), 87–101
16.
S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381
17.
M. I. Zelikin, A. O. Ivanov, B. S. Kashin, A. S. Mishchenko, Yu. V. Nesterenko, A. A. Oshemkov, F. Yu. Popelenskii, V. A. Sadovnichii, A. A. Tuzhilin, V. N. Chubarikov, A. I. Shafarevich, A. N. Shiryaev, “Academician Anatolii Timofeevich Fomenko”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2015, no. 6, 66–68
2013
18.
M. I. Zelikin, D. D. Kiselev, L. V. Lokutsievskii, “Optimal control and Galois theory”, Sb. Math., 204:11 (2013), 1624–1638
19.
M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Stochastic dynamics of lie algebras of Poisson brackets in neighborhoods of nonsmoothness points of Hamiltonians”, Dokl. Math., 87:3 (2013), 259–263
20.
R. Hildebrand, L. V. Lokoutsievskiy, M. I. Zelikin, “Generic fractal structure of finite parts of trajectories of piecewise smooth hamiltonian systems”, Russ. J. Math. Phys., 20:1 (2013), 25–32
V. F. Borisov, V. V. Gael, M. I. Zelikin, “Chattering regimes and Lagrangian manifolds in problems with phase constraints”, Izv. Math., 76:1 (2012), 1–38
22.
M. I. Zelikin, “Theory and applications of the problem of Euler elastica”, Russian Math. Surveys, 67:2 (2012), 281–296
23.
M. I. Zelikin, L. V. Lokutsievskiy, R. Hildebrand, “Geometry of neighborhoods of singular trajectories in problems with multidimensional control”, Proc. Steklov Inst. Math., 277 (2012), 67–83
24.
Yu. S. Osipov, M. I. Zelikin, “Higher-order elastics and elastic hulls”, Russ. J. Math. Phys., 19:2 (2012), 234–243
25.
Yu. S. Osipov, M. I. Zelikin, “Multidimensional generalizations of Jacobi's envelope theorem”, Russ. J. Math. Phys., 19:1 (2012), 101–106
M. I. Zelikin, “Theory of fields of extremals for multiple integrals”, Russian Math. Surveys, 66:4 (2011), 733–765
27.
M. I. Zelikin, “Lev Semenovich Pontryagin: reminiscences and reflections”, Istor.-Mat. Issled. (2), 2011, no. 14(49), 77–84, 357–358
2010
28.
V. F. Borisov, M. I. Zelikin, L. A. Manita, “Optimal synthesis in an infinite-dimensional space”, Proc. Steklov Inst. Math., 271 (2010), 34–52
29.
M. I. Zelikin, V. F. Borisov, “Geometry of extremals with a countable number of contact points with the boundary of the phase constraint”, Dokl. Akad. Nauk, 430:1 (2010), 7–12
30.
Yu. S. Osipov, V. V. Kozlov, L. D. Faddeev, D. V. Anosov, V. S. Vladimirov, R. V. Gamkrelidze, A. A. Gonchar, N. N. Krasovskii, A. V. Kryazhimskii, A. B. Kurzhanskii, S. P. Novikov, S. M. Aseev, A. B. Zhizhchenko, D. V. Treschev, A. A. Agrachev, E. A. Volkov, N. L. Grigorenko, A. A. Davydov, M. I. Zelikin, A. Yu. Kolesov, A. A. Mal'tsev, M. S. Nikol'skii, N. Kh. Rozov, A. G. Sergeev, K. O. Besov, S. P. Konovalov, “In memory of Evgenii Frolovich Mishchenko”, Proc. Steklov Inst. Math., 271 (2010), 1–2
2009
31.
M. I. Zelikin, “Chattering control in infinite-dimensional spaces”, Tezisy dokladov, Suzdal, 2009, 186
2008
32.
M. I. Zelikin, “A criterion for the symmetry of a spectrum”, Dokl. Akad. Nauk, 418:6 (2008), 737–740
2009
33.
V. F. Borisov, M. I. Zelikin, L. A. Manita, “Extremals with accumulation of switchings in infinite-dimensional spaces”, J. Math. Sci., 160:2 (2009), 139–196
34.
M. I. Zelikin, L. V. Lokutsievskii, R. A. Usachev, “Vortex singularities of optimal strategies in search games”, J. Math. Sci., 160:2 (2009), 197–220
2006
35.
J. Appl. Math. Mech., 70:2 (2006), 264–273
2008
36.
M. I. Zelikin, L. A. Manita, “Accumulation of Switchings in Distributed Parameter Problems”, Journal of Mathematical Sciences, 151:6 (2008), 3506–3542
37.
M. I. Zelikin, “Superconductivity of Plasma and Fireballs”, Journal of Mathematical Sciences, 151:6 (2008), 3473–3496
2004
38.
M. I. Zelikin, N. B. Melnikov, “The Bellman function and optimal synthesis in control problems with nonsmooth constraints”, J. Math. Sci. (N. Y.), 121:2 (2004), 2281–2294
39.
M. I. Zelikin, “Optimal problems on Hölder function spaces”, J. Math. Sci. (N. Y.), 121:2 (2004), 2265–2280
2006
40.
M. I. Zelikin, “Geometry of operator cross ratio”, Sb. Math., 197:1 (2006), 37–51
41.
V. I. Arnol'd, L. R. Volevich, A. A. Gonchar, S. S. Demidov, M. I. Zelikin, Yu. S. Ilyashenko, G. G. Magaril-Il'yaev, A. V. Mikhalev, S. P. Novikov, I. V. Yaschenko, “Vladimir Mikhailovich Tikhomirov (on his 70th birthday)”, Russian Math. Surveys, 61:1 (2006), 189–192
2005
42.
V. I. Arnol'd, A. S. Demidov, Yu. S. Ilyashenko, G. G. Magaril-Il'yaev, E. F. Mishchenko, K. Yu. Osipenko, A. B. Sossinski, M. A. Tsfasman, V. A. Uspenskii, M. N. Vyalyi, I. V. Yaschenko, M. I. Zelikin, “Vladimir M. Tikhomirov”, Mosc. Math. J., 5:1 (2005), 295
2004
43.
M. I. Zelikin, “An efficient criterion for optimality in terms of two solutions of the Riccati equation”, Dokl. Akad. Nauk, 397:3 (2004), 300–303
44.
J. Appl. Math. Mech., 68:4 (2004), 617–622
45.
M. I. Zelikin, “Hessian of the solution of the Hamilton–Jacobi equation in the theory of extremal problems”, Sb. Math., 195:6 (2004), 819–831
2005
46.
M. I. Zelikin, V. F. Borisov, “Singular optimal regimes in problems of mathematical economics”, J. Math. Sci. (N. Y.), 130:1 (2005), 4409–4570
2003
47.
M. I. Zelikin, “Hölderian extremal problems”, Patterns and waves (Saint Petersburg, 2002), AkademPrint, St. Petersburg, 2003, 412–432
48.
M. I. Zelikin, “The $\zeta$-function of the cross-ratio on Sato's Grassmannian as an integral invariant of the Kadomtsev-Petviashvili hierarchy”, Dokl. Akad. Nauk, 389:2 (2003), 159–163
M. I. Zelikin, V. F. Borisov, “Optimal chattering feedback control”, J. Math. Sci. (N. Y.), 114:3 (2003), 1227–1344
2002
50.
M. I. Zelikin, “Chattering on optimal trajectories of Hölder problems”, Sb. Math., 193:5 (2002), 669–683
51.
M. I. Zelikin, “The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control”, Proc. Steklov Inst. Math., 236 (2002), 164–185
2001
52.
Differ. Equ., 37:9 (2001), 1217–1223
53.
M. I. Zelikin, “Optimal control in Hölder classes of functions”, Dokl. Akad. Nauk, 378:6 (2001), 733–736
54.
M. I. Zelikin, N. B. Melnikov, R. Hildebrand, “The Topological Structure of the Phase Portrait of a Typical Fiber of Optimal Synthesis for Chattering Problems”, Proc. Steklov Inst. Math., 233 (2001), 116–142
55.
M. I. Zelikin, L. F. Zelikina, R. Hildebrand, “Asymptotics of Optimal Synthesis for One Class of Extremal Problems”, Proc. Steklov Inst. Math., 233 (2001), 87–115
56.
M. I. Zelikin, “Synthesis of Optimal Trajectories That Defines the Reeb Foliation”, Proc. Steklov Inst. Math., 233 (2001), 81–86
57.
A. A. Bolibrukh, A. A. Irmatov, M. I. Zelikin, O. B. Lupanov, V. M. Maynulov, E. F. Mishchenko, M. M. Postnikov, Yu. P. Solov'ev, E. V. Troitskii, “Aleksandr Sergeevich Mishchenko (on his 60th birthday)”, Russian Math. Surveys, 56:6 (2001), 1187–1191
2000
58.
V. F. Borisov, M. I. Zelikin, “Approaching the intermediate thrust modes in control problems of space navigation”, Russ. J. Math. Phys., 7:2 (2000), 127–146
59.
M. I. Zelikin, “To the theory of chattering control”, Spectral and evolution problems (Sevastopol', 1999), Vol. 10, Natl. Taurida Univ. “V. Vernadsky”, Simferopol', 2000, 170–172
60.
M. I. Zelikin, Control theory and optimization. I, Homogeneous spaces and the Riccati equation in the calculus of variations, Encyclopaedia of Mathematical Sciences, 86, Springer-Verlag, Berlin, 2000 , xii+284 pp.
M. I. Zelikin, “The asymptotics of the deviation of a functional from its optimal value when chattering is replaced by a suboptimal regime”, Russian Math. Surveys, 54:3 (1999), 662–664
63.
M. I. Zelikin, L. F. Zelikina, “Exact Constants in Kolmogorov-Type Inequalities”, Proc. Steklov Inst. Math., 227 (1999), 131–139
64.
M. I. Zelikin, “Optimal Synthesis Containing Successively Inserted Chattering Bundles”, Proc. Steklov Inst. Math., 224 (1999), 137–148
65.
M. I. Zelikin, V. A. Il'in, N. N. Krasovskiĭ, et al., “Lyudmila Filippovna Zelikina”, Differential Equations, 35:6 (1999), 856–857
1998
66.
M. I. Zelikin, “Cartan–Siegel homogeneous domains as integral manifolds of the complex Riccati-type equation”, Proc. Steklov Inst. Math., 220 (1998), 126–137
67.
M. I. Zelikin, “A quadratic system of partial differential equations that is associated with the problem of minimizing a multiple integral”, Dokl. Akad. Nauk, 358:1 (1998), 7–9
68.
M. I. Zelikin, Odnorodnye prostranstva i uravnenie Rikkati v variatsionnom ischislenii, Faktorial, M., 1998 , 250 pp.
69.
M. I. Zelikin, L. F. Zelikina, “The structure of optimal synthesis in a neighbourhood of singular manifolds for problems that are affine in control”, Sb. Math., 189:10 (1998), 1467–1484
70.
N. S. Bakhvalov, M. I. Zelikin, A. S. Kalashnikov, V. L. Kamynin, O. A. Oleinik, E. Yu. Panov, N. S. Petrosyan, V. M. Tikhomirov, A. V. Faminskii, V. N. Chubarikov, “Stanislav Nikolaevich Kruzhkov (obituary)”, Russian Math. Surveys, 53:5 (1998), 1071–1078
1997
71.
M. I. Zelikin, “One-parameter families of solutions to a class of PDE optimal control problems”, Optimization methods in partial differential equations (South Hadley, MA, 1996), Contemp. Math., 209, Amer. Math. Soc., Providence, RI, 1997, 339–349
M. I. Zelikin, “Optimum control of solid body rotation”, Dokl. Akad. Nauk, 346:3 (1996), 334–336
1995
75.
M. I. Zelikin, “Totally extremal manifolds for optimal control problems”, Semigroups in algebra, geometry and analysis (Oberwolfach, 1993), de Gruyter Exp. Math., 20, de Gruyter, Berlin, 1995, 339–354
76.
M. I. Zelikin, “Irregularity of optimal control in regular extremal problems”, Fundam. Prikl. Mat., 1:2 (1995), 399–408
1994
77.
M. I. Zelikin, V. F. Borisov, Theory of chattering control, With applications to astronautics, robotics, economics, and engineering, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1994 , xvi+242 pp.
78.
M. I. Zelikin, “On the theory of the matrix Riccati equation. II”, Russian Acad. Sci. Sb. Math., 77:1 (1994), 213–230
1993
79.
M. I. Zelikin, V. F. Borisov, “Regimes with frequented switches in the problems of optimal control”, Proc. Steklov Inst. Math., 197 (1993), 95–186
1991
80.
M. I. Zelikin, V. F. Borosov, “Optimal synthesis containing chattering arcs and singular arcs of the second order”, Nonlinear synthesis (Sopron, 1989), Progr. Systems Control Theory, 9, Birkhäuser Boston, Boston, MA, 1991, 283–296
1992
81.
M. I. Zelikin, “On the theory of the matrix Riccati equation”, Math. USSR-Sb., 73:2 (1992), 341–354
1990
82.
M. I. Zelikin, V. F. Borisov, “Synthesis in problems of optimal control containing trajectories with participating switchings and singular trajectories of the second order”, Math. Notes, 47:1 (1990), 41–49
1989
83.
Soviet Math. Dokl., 39:1 (1989), 188–191
1988
84.
J. Appl. Math. Mech., 52:6 (1988), 731–737 (1990)
85.
M. I. Zelikin, “Group symmetry in degenerate extremal problems”, Russian Math. Surveys, 43:2 (1988), 189–190
86.
M. I. Zelikin, “Synthesis of optimal trajectories on representation spaces of Lie groups”, Math. USSR-Sb., 60:2 (1988), 533–546
1986
87.
M. I. Zelikin, “The search for singular extremals”, Stochastic optimization (Kiev, 1984), Lecture Notes in Control and Inform. Sci., 81, Springer, Berlin, 1986, 738–746
88.
A. G. Vitushkin, M. I. Zelikin, N. N. Krasovskii, S. P. Novikov, Yu. S. Osipov, K. A. Sitnikov, “Nicolai Terent'evich Tynyanskii (obituary)”, Russian Math. Surveys, 41:4 (1986), 173–174
1985
89.
M. I. Zelikin, Optimalnoe upravlenie i variatsionnoe ischislenie, Izd-vo MGU, M., 1985 , 96 pp.
90.
M. I. Zelikin, “Minimization of integrals of differential forms, and degenerate variational problems”, Wiss. Z. Tech. Hochsch. Leuna-Merseburg, 27:5 (1985), 526–534
91.
M. I. Zelikin, “On the singular arcs”, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform., 14:2 (1985), 75–88, R1–R15
1984
92.
M. I. Zelikin, “Necessary conditions for optimality of singular trajectories in problems that are linear with respect to control”, Some problems in modern analysis, Moskov. Gos. Univ. Mekh.-Mat. Fak., Moscow, 1984, 35–41
93.
M. I. Zelikin, “Minimization of integrals of differential forms”, Dokl. Akad. Nauk SSSR, 278:5 (1984), 1057–1059
94.
M. I. Zelikin, “On the theory of linear control problems”, Dokl. Akad. Nauk SSSR, 277:4 (1984), 782–785
M. I. Zelikin, “Optimality conditions for singular trajectories in a problem of curvilinear integral minimization”, Dokl. Akad. Nauk SSSR, 267:3 (1982), 528–531
1981
96.
D. V. Anosov, V. I. Arnol'd, M. I. Zelikin, A. N. Kolmogorov, O. V. Lokutsievskii, Yu. S. Osipov, Ya. G. Sinai, V. M. Tikhomirov, M. V. Jakobson, “Vladimir Mikhailovich Alekseev (obituary)”, Russian Math. Surveys, 36:4 (1981), 201–206
1977
97.
M. I. Zelikin, B. Yu. Sternin, “A system of integral equations that arises in the problem of S. L. Sobolev”, Siberian Math. J., 18:1 (1977), 72–76
1972
98.
M. I. Zelikin, “A certain differential game with incomplete information”, Dokl. Akad. Nauk SSSR, 202 (1972), 998–1000
Singular extremals in the generalised Fuller problem M. I. Zelikin, D. D. Kiselev, L. V. Lokutsievskiy International Conference "Mathematical Theory of Optimal Control"
dedicated to the 90th birthday of Academician R. V. Gamkrelidze June 1, 2017 15:45
Заседание посвящено памяти Дмитрия Викторовича Аносова A. M. Stepin, L. M. Lerman, V. Z. Grines, S. Yu. Pilyugin, M. I. Zelikin, V. P. Leksin, A. V. Klimenko, Yu. S. Ilyashenko Meetings of the Moscow Mathematical Society April 7, 2015
16.
Фрактальная структура оптимального синтеза M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand All-Moscow regular scientific seminar "Control Theory and Optimization" December 2, 2014 11:30
2024
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