Chern characteristic classes and their computation for complex algebraic manifolds, linear optimal control systems, Pontryagin maximum principle, optimal control with phase constraints, sliding optimal states
Algebraic topology and characteristic classes, optimal control and calculus of variations
Main publications:
R.V. Gamkrelidze, “Tsikly Cherna kompleksnykh algebraicheskikh mnogoobrazii”, Izvestiya AN SSSR seriya matematicheskaya, 20:4 (1956), 685-706
V.G. Boltyanskii, R.V. Gamkrelidze, L.S. Pontryagin, “K teorii optimalnykh protsessov”, DAN SSSR, 110:1 (1956), 7-10
R.V. Gamkrelidze, “Teoriya optimalnykh po bystrodeistviyu protsessov v lineinykh sistemakh”, Izvestiya AN SSSR seriya matematicheskaya, 22 (1958), 449-474
R.V. Gamkrelidze, “Optimalnye protsessy upravleniya pri ogranichennykh fazovykh koordinatakh”, Izvestiya AN SSSR seriya matematicheskaya, 24 (1960), 315-356
R. V. Gamkrelidze, A. V. Ovchinnikov, “Dual formulation of the lemma on substitution homomorphisms”, Dokl. Math., 106:1 (2022), 218–219
2.
R. V. Gamkrelidze, “History of the Discovery of the Pontryagin Maximum Principle”, Proc. Steklov Inst. Math., 304 (2019), 1–7
3.
R. V. Gamkrelidze, Osnovy optimalnogo upravleniya, Izd. 3, ispr., URSS, 2019 , 200 pp.
4.
R. V. Gamkrelidze, “Dual Formulation of the Pontryagin Maximum Principle in Optimal Control”, Proc. Steklov Inst. Math., 291 (2015), 61–67
5.
R. V. Gamkrelidze, “Differential-geometric and invariance properties of the equations of Maximum Principle (MP)”, Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series, 5, Springer, 2014, 167–175
R. V. Gamkrelidze, “Mathematical Work of L. S. Pontryagin”, Proc. Steklov Inst. Math., 224 (1999), 8–19
16.
A. Agrachev, R. Gamkrelidze, “Symplectic methods for optimization and control”, Geometry of feedback and optimal control, Monogr. Textbooks Pure Appl. Math., 207, Dekker, New York, 1998, 19–77
17.
R. V. Gamkrelidze, “The mathematical work of L. S. Pontryagin”, J. Math. Sci. (New York), 100:5 (2000), 2447–2457
18.
A. A. Agrachev, R. V. Gamkrelidze, “Feedback-invariant optimal control theory and differential geometry. I. Regular extremals”, J. Dynam. Control Systems, 3:3 (1997), 343–389
A. A. Agrachev, R. V. Gamkrelidze, “On the orbits of groups of diffeomorphisms and flows”, Proc. Steklov Inst. Math., 209 (1995), 1–10
20.
A. A. Agrachëv, R. V. Gamkrelidze, “The shuffle product and symmetric groups”, Differential equations, dynamical systems, and control science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994, 365–382
21.
A. A. Agrachëv, R. V. Gamkrelidze, “Local controllability and semigroups of diffeomorphisms”, Acta Appl. Math., 32:1 (1993), 1–57
A. A. Agrachev, R. V. Gamkrelidze, “Volterra series and permutation groups”, J. Math. Sci., 71:3 (1994), 2409–2433
24.
A. A. Agrachev, R. V. Gamkrelidze, “Symplectic geometry and conditions necessary conditions for optimality”, Math. USSR-Sb., 72:1 (1992), 29–45
25.
Soviet Math. Dokl., 41:2 (1990), 227–231 (1991)
26.
A. A. Agrachëv, R. V. Gamkrelidze, “Symplectic geometry for optimal control”, Nonlinear controllability and optimal control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990, 263–277
27.
A. A. Agrachëv, R. V. Gamkrelidze, A. V. Sarychev, “Local invariants of smooth control systems”, Acta Appl. Math., 14:3 (1989), 191–237
A. A. Agrachev, R. V. Gamkrelidze, “Quadratic mappings and smooth vector functions: Euler characteristics of level sets”, J. Soviet Math., 55:4 (1991), 1892–1928
29.
A. A. Agrachev, R. V. Gamkrelidze, “Quasi-extremality for controllable systems”, J. Soviet Math., 55:4 (1991), 1849–1864
30.
R. V. Gamkrelidze, A. A. Agrachev, S. A. Vakhrameev, “Ordinary differential equations on vector bundles, and chronological calculus”, J. Soviet Math., 55:4 (1991), 1777–1848
31.
Soviet Math. Dokl., 37:2 (1988), 297–300
32.
L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze, E. F. Mishchenko, Selected works, v. 4, Classics of Soviet Mathematics, The mathematical theory of optimal processes, Gordon & Breach Science Publishers, New York, 1986 , xxiv+360 pp.
33.
A. A. Agrachëv, R. V. Gamkrelidze, “Morse and Maslov indices for the extremals of controllable systems”, Dokl. Akad. Nauk SSSR, 287:3 (1986), 521–524
34.
A. A. Agrachëv, R. V. Gamkrelidze, “The index of extremality and quasiextremal controls”, Dokl. Akad. Nauk SSSR, 284:4 (1985), 777–781
R. V. Gamkrelidze, “Sliding modes in optimal control theory”, Proc. Steklov Inst. Math., 169 (1986), 185–198
36.
D. V. Anosov, R. V. Gamkrelidze, E. F. Mishchenko, M. M. Postnikov, “The mathematical work of L. S. Pontryagin”, Proc. Steklov Inst. Math., 166 (1986), 1–15
37.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, 4-e izd., Nauka, M., 1983 , 392 pp.
38.
A. A. Agrachev, S. A. Vakhrameev, R. V. Gamkrelidze, “Differential geometric and group theoretic methods in optimal control theory”, J. Soviet Math., 28:2 (1985), 145–182
39.
A. A. Agrachev, R. V. Gamkrelidze, “Chronological algebras and nonstationary vector fields”, J. Soviet Math., 17:1 (1981), 1650–1675
40.
R. V. Gamkrelidze, “Exponential representation of solutions of ordinary differential equations”, Equadiff IV, Proc. Czechoslovak Conf. Differential Equations and their Applications (Prague, 1977), Lecture Notes in Math., 703, Springer, Berlin, 1979, 118–129
L. Pontriaguine, V. Boltianski, R. Gamkrélidzé, E. Michtchenko, Théorie mathématique des processus optimaux, Traduit du russe par Djilali Embarek; Unaltered reprint, Mir, Moscow, 1978 , 317 pp.
42.
A. A. Agrachev, R. V. Gamkrelidze, “The exponential representation of flows and the chronological calculus”, Math. USSR-Sb., 35:6 (1979), 727–785
43.
A. A. Agrachev, R. V. Gamkrelidze, “A second order optimality principle for a time-optimal problem”, Math. USSR-Sb., 29:4 (1976), 547–576
44.
R. V. Gamkrelidze, Osnovy optimalnogo upravleniya, Izd-vo Tbilisskogo un-ta, Tbilisi, 1975 , 230 pp. ; Р. В. Гамкрелидзе, Основы оптимального управления, Изд-во Тбилисского ун-та, Тбилиси, 1977 , 254 с. ; R. V. Gamkrelidze, Principles of optimal control theory, Mathematical Concepts and Methods in Science and Engineering, 7, Plenum Press, New York, 1978 , xii+175 pp.
45.
R. V. Gamkrelidze, G. L. Kharatishvili, “A differential game of evasion with nonlinear control”, SIAM J. Control, 12 (1974), 332–349
L. Pontriaguine, V. Boltianski, R. Gamkrelidze, E. Michtchenko, Théorie mathématique des processus optimaux, Mir, Moscow, 1974 , 317 pp.
47.
R. V. Gamkrelidze, Kharatishvili G. L., “Conditions nécessaires du premier ordre dans les problèmes d'extremum”, Actes du Congrès International des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, 169–176
48.
R. V. Gamkrelidze, “Necessary first order conditions, and the axiomatics of extremal problems”, Proc. Steklov Inst. Math., 112 (1971), 156–186
49.
R. V. Gamkrelidze, G. L. Kharatishvili, “Extremal problems in linear topological spaces”, Math. USSR-Izv., 3:4 (1969), 737–794
50.
Gamkrelidze Revaz V., “Extremal problems in finite-dimensional spaces”, J. Optimization Theory Appl., 1 (1967), 173–193
R. V. Gamkrelidze, G. L. Haratišvili, “The theory of the first variation in extremal problems”, Sakharth. SSR Mecn. Akad. Moambe, 46 (1967), 27–31
53.
R. V. Gamkrelidze, “On some extremal problems in the theory of differential equations with applications to the theory of optimal control”, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 106–128
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, A Pergamon Press Book, The Macmillan Co., New York, 1964 , vii+338 pp.
56.
L. S. Pontrjagin, V. G. Boltjanskiĭ, R. V. Gamkrelidze, E. F. Miščenko, “The mathematical theory of optimal processes”, Proc. 4th All-Union Math. Congr. (Leningrad, 1961), Izdat. Akad. Nauk SSSR, Leningrad, 1963, 214–218
57.
R. V. Gamkrelidze, “On sliding optimal states”, Dokl. Akad. Nauk SSSR, 143 (1962), 1243–1245
V. G. Boltyanski, R. V. Gamkrelidze, E. F. Mishchenko, L. S. Pontryagin, “The maximum principle in the theory of optimal processes of control (with discussion)”, Automatic and Remote Control, Proc. First Internat. Congr. Internat. Fed. Automat. Control (Moscow, 1960), Vol. I, Butterworths, London, 1961, 454–459
59.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, Fizmatlit, M., 1961 , 391 pp. ; L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, ed. L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc., New York–London, 1962 , viii+360 pp. ; L. S. Pontrjagin, V. G. Boltjanskij, R. V. Gamkrelidze, E. F. Miščenko, Mathematische Theorie optimaler Prozesse, R. Oldenbourg, Munich–Vienna, 1964 , 340 pp.
60.
V. G. Boltjanskiĭ, R. V. Gamkrelidze, L. S. Pontryagin, “The theory of optimal processes. I. The maximum principle”, Amer. Math. Soc. Transl. (2), 18 (1961), 341–382
61.
R. V. Gamkrelidze, “Optimal control processes for bounded phase coordinates”, Izv. Akad. Nauk SSSR Ser. Mat., 24:3 (1960), 315–356
62.
V. G. Boltyanskii, R. V. Gamkrelidze, L. S. Pontryagin, “Theory of optimal processes. I. The maximum principle”, Izv. Akad. Nauk SSSR Ser. Mat., 24:1 (1960), 3–42
63.
R. V. Gamkrelidze, “Optimum-rate processes with bounded phase coordinates”, Dokl. Akad. Nauk SSSR, 125 (1959), 475–478
64.
R. V. Gamkrelidze, “On the general theory of optimal processes”, Dokl. Akad. Nauk SSSR, 123 (1958), 223–226
65.
R. V. Gamkrelidze, “Theory of processes in linear systems which are optimal with respect to rapidity of action”, Izv. Akad. Nauk SSSR Ser. Mat., 22:4 (1958), 449–474
66.
R. V. Gamkrelidze, “On the theory of optimal processes in linear systems”, Dokl. Akad. Nauk SSSR (N.S.), 116 (1957), 9–11
V. G. Boltyanskiĭ, R. V. Gamkrelidze, L. S. Pontryagin, “On the theory of optimal processes”, Dokl. Akad. Nauk SSSR (N.S.), 110 (1956), 7–10
68.
R. V. Gamkrelidze, “Chern's cycles of complex algebraic manifolds”, Izv. Akad. Nauk SSSR Ser. Mat., 20:5 (1956), 685–706
69.
R. V. Gamkrelidze, “Computation of the Chern cycles of algebraic manifolds”, Doklady Akad. Nauk SSSR (N.S.), 90 (1953), 719–722
70.
A. A. Agrachev, R. V. Gamkrelidze, V. V. Kozlov, Yu. I. Zhuravlev, A. V. Mikhalev, A. V. Ovchinnikov, D. O. Orlov, V. L. Popov, V. G. Romanov, A. L. Semenov, V. G. Chirskii, V. A. Shamolin, “To the 55th anniversary of Professor M. V. Shamolin”, Geometry, Mechanics, and Differential Equations, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 210, VINITI, Moscow, 2022, 3–5
71.
A. I. Aptekarev, D. E. Apushkinskaya, O. V. Besov, V. I. Burenkov, V. V. Vlasov, R. V. Gamkrelidze, M. L. Gol'dman, G. V. Demidenko, V. G. Zadorozhniy, V. V. Kozlov, G. G. Lazareva, S. P. Novikov, V. M. Savchin, V. A. Sadovnichii, A. L. Skubachevskii, A. P. Soldatov, D. V. Treschev, M. A. Ragusa, H.-O. Walther, “Vladimir Mikhailovich Filippov”, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, CMFD, 67, no. 3, PFUR, M., 2021, 423–426
72.
Optimalnoe upravlenie i differentsialnye uravneniya, Sbornik statei. K 110-letiyu so dnya rozhdeniya akademika Lva Semenovicha Pontryagina, Trudy MIAN, 304, ed. R. V. Gamkrelidze, S. M. Aseev, M. S. Nikolskii, MIAN, M., 2019 , 328 pp.
73.
A. A. Agrachev, R. V. Gamkrelidze, E. S. Golod, A. B. Zhizhchenko, Yu. I. Zhuravlev, V. V. Kozlov, A. V. Mikhalev, A. V. Ovchinnikov, N. Kh. Rozov, A. L. Semenov, V. G. Chirskii, V. A. Shamolin, “Anniversary of Professor M. V. Shamolin”, Dynamical systems, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 134, VINITI, Moscow, 2017, 3–5
74.
D. V. Anosov, S. M. Aseev, R. V. Gamkrelidze, S. P. Konovalov, M. S. Nikol'skii, N. Kh. Rozov, “Evgenii Frolovich Mishchenko (on the 90th anniversary of his birth)”, Russian Math. Surveys, 67:2 (2012), 385–402
75.
D. V. Anosov, V. A. Vassiliev, V. S. Vladimirov, R. V. Gamkrelidze, A. A. Gonchar, V. A. Il'in, V. V. Kozlov, A. V. Kryazhimskiy, S. M. Nikol'skii, A. N. Parshin, V. M. Filippov, I. R. Shafarevich, “Lev Dmitrievich Kudryavtsev (obituary)”, Russian Math. Surveys, 67:3 (2012), 569–571
76.
Matematicheskaya teoriya upravleniya i differentsialnye uravneniya, Sbornik statei. K 90-letiyu so dnya rozhdeniya akademika Evgeniya Frolovicha Mischenko, Trudy MIAN, 277, ed. R. V. Gamkrelidze, A. G. Sergeev, MAIK «Nauka/Interperiodika», M., 2012 , 287 pp.
77.
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
78.
D. V. Anosov, F. L. Chernous'ko, S. V. Emel'yanov, R. V. Gamkrelidze, V. A. Il'in, S. K. Korovin, A. B. Kurzhanskii, G. I. Marchuk, V. M. Matrosov, E. F. Mishchenko, E. I. Moiseev, Yu. S. Osipov, V. A. Pliss, “Nikolai Nikolaevich Krasovskii (A tribute in honor of his eighty-fifth birthday)”, Differ. Equ., 45:10 (2009), 1401–1407
79.
Differ. Equ., 40:11 (2004), 1517–1532
80.
R. V. Gamkrelidze, “Discovery of the maximum principle in optimal control”, Mathematics and war (Karlskrona, 2002), Birkhäuser, Basel, 2003, 160–173
81.
V. I. Arnol'd, A. A. Bolibrukh, R. V. Gamkrelidze, V. P. Maslov, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, Ya. G. Sinai, A. M. Stepin, L. D. Faddeev, “Dmitrii Viktorovich Anosov (on his 60th birthday)”, Russian Math. Surveys, 52:2 (1997), 437–445
82.
Optimalnoe upravlenie i differentsialnye uravneniya, Sbornik statei. K semidesyatiletiyu so dnya rozhdeniya akademika Evgeniya Frolovicha Mischenko, Tr. MIAN, 211, ed. E. F. Mischenko, R. V. Gamkrelidze, Nauka, Fizmatlit, M., 1995 , 526 pp.
83.
Itogi nauki i tekhn. Ser. Algebra. Topol. Geom., 29, ed. R. V. Gamkrelidze, VINITI, M., 1991 , 193 pp.
84.
Nekotorye voprosy teorii kolebanii i teorii optimalnogo upravleniya, Tr. MIAN SSSR, 197, ed. R. V. Gamkrelidze, E. F. Mischenko, Nauka, M., 1991 , 168 pp.
85.
V. G. Boltyanskii, R. V. Gamkrelidze, A. A. Mal'tsev, E. F. Mishchenko, S. P. Novikov, Yu. B. Rudyak, “Mikhail Mikhailovich Postnikov (on his 60th birthday)”, Russian Math. Surveys, 44:6 (1989), 201–203
86.
V. I. Vorotnikov, L. N. Zaikov, E. K. Ligachev, A. N. Yakovlev, G. I. Marchuk, V. A. Grigor'ev, G. A. Yagodin, V. A. Kotel'nikov, P. N. Fedoseev, E. P. Velikhov, K. V. Frolov, A. A. Logunov, A. L. Yanshin, I. M. Makarov, N. G. Basov, N. N. Bogolyubov, V. S. Vladimirov, I. M. Gel'fand, A. A. Gonchar, N. N. Krasovskii, M. M. Lavrent'ev, V. A. Mel'nikov, Yu. A. Mitropol'skii, E. F. Mishchenko, S. M. Nikol'skii, S. P. Novikov, V. P. Platonov, A. V. Pogorelov, Yu. V. Prokhorov, L. I. Sedov, S. L. Sobolev, A. N. Tikhonov, L. D. Faddeev, R. V. Gamkrelidze, “Academician Lev Semenovich Pontryagin (obituary)”, Russian Math. Surveys, 44:1 (1989), 1–2
87.
R. V. Gamkrelidze, “L. S. Pontryagin, “The method of coordinates” (a review)”, Uspekhi Mat. Nauk, 33:6(204) (1978), 247–248
88.
Itogi nauki. Ser. Mat. Algebra. Topol. Geom. 1967, ed. R. V. Gamkrelidze, VINITI, M., 1969 , 228 pp.
89.
P. S. Aleksandrov, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, “Lev Semenovich Pontryagin (on the occasion of his sixtieth birthday)”, Russian Math. Surveys, 23:6 (1968), 143–152
90.
Itogi nauki. Algebra. Topol. 1962, ed. R. V. Gamkrelidze, VINITI, M., 1963 , 224 pp.
91.
R. V. Gamkrelidze, “L. S. Pontryagin Seminar on Mathematical Problems of Theory of Oscillations and Automatic Control”, Uspekhi Mat. Nauk, 12:3(75) (1957), 267–272
Memoirs about L. S. Pontryagin Yu. S. Ledyaev, R. V. Gamkrelidze Conference "Optimal Control and Applications" dedicated to the 105th anniversary of Lev Semenovich Pontryagin September 24, 2013 18:00
Инвариантное представление принципа максимума R. V. Gamkrelidze Conference "Differential Equations and Optimal Control", dedicated to the 90th anniversary of E. F. Mishchenko April 16, 2012 11:05
Hamiltonian format of Pontryagin's maximum principle R. V. Gamkrelidze International Conference "Differential Equations and Topology" dedicated to the Centennial Anniversary of L. S. Pontryagin June 18, 2008 10:20
Books in Math-Net.Ru
Optimal control and differential equations, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin, Trudy Mat. Inst. Steklova, 304, ed. R. V. Gamkrelidze, S. M. Aseev, M. S. Nikol'skii, 2019, 328 с. http://mi.mathnet.ru/book1743
Mathematical control theory and differential equations, Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko, Trudy Mat. Inst. Steklova, 277, ed. R. V. Gamkrelidze, A. G. Sergeev, 2012, 287 с. http://mi.mathnet.ru/book1463
Optimal control and differential equations, Collection of articles. To the seventieth birthday of academician E. F. Mishchenko, Trudy Mat. Inst. Steklov., 211, ed. E. F. Mishchenko, R. V. Gamkrelidze, 1995, 526 с. http://mi.mathnet.ru/book1065
Selected topics in the theory of oscillations and optimal control theory, Trudy Mat. Inst. Steklov., 197, ed. R. V. Gamkrelidze, E. F. Mishchenko, 1991, 168 с. http://mi.mathnet.ru/book1109
Itogi Nauki i Tekhniki. Ser. Algebra. Topol. Geom., 29, ed. R. V. Gamkrelidze, 1991, 193 с. http://mi.mathnet.ru/book40
Itogi Nauki. Ser. Mat. Algebra. Topol. Geom. 1967, ed. R. V. Gamkrelidze, 1969, 228 с. http://mi.mathnet.ru/book44