non-convexvalued multifunctions; continuous selectors; evolution equations; evolution inclusions; evolution operators; control systems; optimal control; calculas of variations.
Subject:
Main scientific interests are various problems of non-convex multivalued analysis. Theory of differential inclusions with non-convex right-hand side in a Banach space was created. For the first time the existence and relaxation theorems for continuous selectors passing through extreme points of multivalued mappings with closed convex decomposable values in spaces of Bochner integrable functions were proved. The last means the density of the set of extreme continuous selectors in the set of all continuous selectors. To this end the definition of seminorm with properties of relaxation and scalar compactnes in the spaces of Bocner integrable functions was introduced. Rich in content examples such seminorms having numerious applications were constracted. Similar theorems for continuous selectors passing through fixed points of multivalued mappings with non-convex closed decomposable values depending on a parametr were proved. Numerous applications of selectors theorems for study of non-convex problems in calculus of variations and control systems were examined. In particular "bang-bang" principle for nonlinear evolution control systems with non-convex mixed constraints for control and an analog of N. N. Bogolyubov theorem concerning an approximation of value of convexified integrand by values of non-convex original integrand along solutions of such control sdystems were proved.
Biography
Graduated from Kazan aviation institute in 1963 (specialized field is dynamic of flight). Ph.D. thesis was defended in 1967. D.Sci. thesis was defended in 1984. A list of my pyblications contains more than 80 titles including 2 monographs.
In 1986 I was awarded the prize of Siberian Branch of USSR Academy of Sciences in the field of fundamental sciences for a series of papers on the theory of differential inclusions. I am member of editirial board of International Mathematical Journals: Set-valued Analysis; Discussiones Mathematicae. Differential Inclusions, Control and Optimization; Stability and Control: Theory and Applications.
Main publications:
A. Tolstonogov. Differential inclusions in a Banach space. Kluwer Academic Publishers, 2000.
A. A. Tolstonogov, D. A. Tolstonogov. $L_p$-continuous extreme selectors of multifunctions with decomposable values: existence theorems // Set-valued Analysis, v. 4, no. 2 (1996), 173–203.
A. A. Tolstonogov, D. A. Tolstonogov. $L_p$-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems // Set-valued Analysis, v. 4, no. 3 (1996), 237–269.
A. A. Tolstonogov, “Evolution inclusions with state-dependent maximal monotone operators”, Trudy Inst. Mat. i Mekh. UrO RAN, 30:3 (2024), 241–254
2023
2.
A. A. Tolstonogov, “Existence and relaxation of solutions for a differential inclusion with maximal monotone operators and perturbations”, Dokl. RAN. Math. Inf. Proc. Upr., 514:1 (2023), 65–68; Dokl. Math., 108:3 (2023), 477–480
3.
A. A. Tolstonogov, “Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory”, Mat. Sb., 214:6 (2023), 110–135; Sb. Math., 214:6 (2023), 853–877
A. A. Tolstonogov, “Distances between maximal monotone operators”, Sibirsk. Mat. Zh., 64:4 (2023), 815–829
2022
5.
A. A. Tolstonogov, “The $g$-convergence of maximal monotone Nemytskii operators”, Sibirsk. Mat. Zh., 63:6 (2022), 1369–1381; Siberian Math. J., 63:6 (2022), 1169–1180
2021
6.
A. A. Tolstonogov, “Maximal monotonicity of a Nemytskii operator”, Funktsional. Anal. i Prilozhen., 55:3 (2021), 51–61; Funct. Anal. Appl., 55:3 (2021), 217–225
A. A. Tolstonogov, “Differential Inclusions with Mixed Semicontinuity Properties in a Banach Space”, Funktsional. Anal. i Prilozhen., 54:3 (2020), 48–62; Funct. Anal. Appl., 54:3 (2020), 188–199
10.
A. A. Tolstonogov, “Bogolyubov's theorem for a controlled system related to a variational inequality”, Izv. RAN. Ser. Mat., 84:6 (2020), 165–196; Izv. Math., 84:6 (2020), 1192–1223
A. A. Tolstonogov, “Differential Inclusions in a Banach Space with Composite Right-Hand Side”, Trudy Inst. Mat. i Mekh. UrO RAN, 26:1 (2020), 212–222; Proc. Steklov Inst. Math. (Suppl.), 313, suppl. 1 (2021), S201–S210
A. A. Tolstonogov, “Densities of measures as an alternative to derivatives for measurable inclusions”, Funktsional. Anal. i Prilozhen., 53:4 (2019), 52–62
14.
A. A. Tolstonogov, “Local existence conditions for sweeping process solutions”, Mat. Sb., 210:9 (2019), 107–128; Sb. Math., 210:9 (2019), 1305–1325
A. A. Tolstonogov, “Space of continuous set-valued mappings with closed unbounded values”, Trudy Inst. Mat. i Mekh. UrO RAN, 24:1 (2018), 200–208; Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), S216–S222
2017
16.
A. A. Tolstonogov, “Existence and relaxation of solutions to differential inclusions with unbounded right-hand side in a Banach space”, Sibirsk. Mat. Zh., 58:4 (2017), 937–953; Siberian Math. J., 58:4 (2017), 727–742
A. A. Tolstonogov, “Solutions of evolution inclusions generated by a difference of subdifferentials”, Trudy Inst. Mat. i Mekh. UrO RAN, 21:2 (2015), 236–251; Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 199–215
2014
18.
N. I. Pogodaev, A. A. Tolstonogov, “The variational stability of an optimal control problem for Volterra-type equations”, Sibirsk. Mat. Zh., 55:4 (2014), 818–839; Siberian Math. J., 55:4 (2014), 667–686
19.
A. A. Tolstonogov, “Differential inclusions with unbounded right-hand side. Existence and relaxation theorems”, Trudy Inst. Mat. i Mekh. UrO RAN, 20:3 (2014), 246–262; Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 190–207
A. A. Tolstonogov, “Variational stability of optimal control problems involving subdifferential operators”, Mat. Sb., 202:4 (2011), 123–160; Sb. Math., 202:4 (2011), 583–619
A. A. Tolstonogov, “Control systems of subdifferential type depending on a parameter”, Izv. RAN. Ser. Mat., 72:5 (2008), 149–188; Izv. Math., 72:5 (2008), 985–1022
A. A. Tolstonogov, “Relaxation in control systems of subdifferential type”, Izv. RAN. Ser. Mat., 70:1 (2006), 129–162; Izv. Math., 70:1 (2006), 121–152
A. A. Tolstonogov, “Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion”, Mat. Sb., 196:2 (2005), 117–138; Sb. Math., 196:2 (2005), 263–285
A. A. Tolstonogov, “The bang-bang principle for controlled systems of subdifferential type”, Trudy Inst. Mat. i Mekh. UrO RAN, 11:1 (2005), 189–200; Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 1, S222–S233
2004
26.
A. A. Tolstonogov, “Properties of attainable sets of evolution inclusions and control systems of subdifferential type”, Sibirsk. Mat. Zh., 45:4 (2004), 920–945; Siberian Math. J., 45:4 (2004), 763–784
A. A. Tolstonogov, “On solutions of an evolution control system depending on parameters”, Mat. Sb., 194:9 (2003), 113–140; Sb. Math., 194:9 (2003), 1383–1409
A. A. Tolstonogov, “Approximation of attainable sets of an evolution inclusion of subdifferential type”, Sibirsk. Mat. Zh., 44:4 (2003), 883–904; Siberian Math. J., 44:4 (2003), 695–712
A. A. Tolstonogov, “Strongly Exposed Points of Decomposable Sets in Spaces of Bochner Integrable Functions”, Mat. Zametki, 71:2 (2002), 298–306; Math. Notes, 71:2 (2002), 267–275
A. A. Tolstonogov, “Properties of solutions to second order evolution control systems. II”, Sibirsk. Mat. Zh., 43:5 (2002), 1149–1167; Siberian Math. J., 43:5 (2002), 926–941
A. A. Tolstonogov, “Properties of solutions to second order evolution control systems. I”, Sibirsk. Mat. Zh., 43:4 (2002), 907–923; Siberian Math. J., 43:4 (2002), 731–745
A. A. Tolstonogov, “Properties of the set of admissible “state-control” pairs for first-order evolution control systems”, Izv. RAN. Ser. Mat., 65:3 (2001), 201–224; Izv. Math., 65:3 (2001), 617–640
A. A. Tolstonogov, “Existence of an optimal control without convexity assumptions in a first-order evolution system”, Mat. Sb., 192:9 (2001), 125–142; Sb. Math., 192:9 (2001), 1381–1398
2000
35.
A. A. Tolstonogov, “A theorem of existence of an optimal control for the Goursat–Darboux problem without convexity assumptions”, Izv. RAN. Ser. Mat., 64:4 (2000), 163–182; Izv. Math., 64:4 (2000), 807–826
A. A. Tolstonogov, “Relaxation in non-convex optimal control problems described by first-order evolution equations”, Mat. Sb., 190:11 (1999), 135–160; Sb. Math., 190:11 (1999), 1689–1714
A. A. Tolstonogov, “$L_p$-continuous selectors of fixed points of multivalued mappings with decomposable values. III. Applications”, Sibirsk. Mat. Zh., 40:6 (1999), 1380–1396; Siberian Math. J., 40:6 (1999), 1173–1187
A. A. Tolstonogov, “$L_p$-continuous selectors of fixed points of multivalued mappings with decomposable values. II. Relaxation theorems”, Sibirsk. Mat. Zh., 40:5 (1999), 1167–1181; Siberian Math. J., 40:5 (1999), 991–1003
A. A. Tolstonogov, “$L_p$-continuous selectors of fixed points of multivalued mappings with decomposable values. I. Existence theorems”, Sibirsk. Mat. Zh., 40:3 (1999), 695–709; Siberian Math. J., 40:3 (1999), 595–607
A. A. Tolstonogov, “Continuous selectors of multivalued mappings with nonclosed
decomposable values, and their properties”, Dokl. Akad. Nauk, 352:3 (1997), 311–313
1996
41.
A. A. Tolstonogov, “Continuous selections of multivalued maps with non-convex non-closed decomposable values”, Mat. Sb., 187:5 (1996), 121–142; Sb. Math., 187:5 (1996), 745–766
V. V. Goncharov, A. A. Tolstonogov, “Continuous selections for a family of nonconvex-valued mappings with noncompact domain”, Sibirsk. Mat. Zh., 35:3 (1994), 537–553; Siberian Math. J., 35:3 (1994), 479–494
A. A. Tolstonogov, V. V. Goncharov, “On sublinear functionals defined on the space of Bochner integrable functions”, Sibirsk. Mat. Zh., 35:1 (1994), 194–206; Siberian Math. J., 35:1 (1994), 178–188
A. A. Tolstonogov, Ya. I. Umansky, “Solutions of evolution inclusions. II”, Sibirsk. Mat. Zh., 33:4 (1992), 163–174; Siberian Math. J., 33:4 (1992), 693–702
A. A. Tolstonogov, “Extremal selectors of multivalued mappings and the “bang-bang”
principle for evolution inclusions”, Dokl. Akad. Nauk SSSR, 317:3 (1991), 589–593; Dokl. Math., 43:2 (1991), 481–485
A. A. Tolstonogov, “Properties of integral solutions of differential inclusions with $m$-accretive operators”, Mat. Zametki, 49:6 (1991), 119–131; Math. Notes, 49:6 (1991), 636–644
V. V. Goncharov, A. A. Tolstonogov, “Joint continuous selections of multivalued mappings with nonconvex values, and their applications”, Mat. Sb., 182:7 (1991), 946–969; Math. USSR-Sb., 73:2 (1992), 319–339
V. V. Goncharov, A. A. Tolstonogov, “Continuous selectors and properties of solutions of differential
inclusions with $m$-accretive operators”, Dokl. Akad. Nauk SSSR, 315:5 (1990), 1035–1039; Dokl. Math., 42:3 (1991), 861–865
50.
A. A. Tolstonogov, “Scorza–Dragoni's theorem for multi-valued mappings with variable domain of definition”, Mat. Zametki, 48:5 (1990), 109–120; Math. Notes, 48:5 (1990), 1151–1158
A. A. Tolstonogov, “Solutions of a differential inclusion with an unbounded right-hand side”, Sibirsk. Mat. Zh., 29:5 (1988), 212–225; Siberian Math. J., 29:5 (1988), 857–868
A. A. Tolstonogov, “Theorem on continuous extension of a multivalued function and its applications”, Mat. Zametki, 42:4 (1987), 581–593; Math. Notes, 42:4 (1987), 721–727
1984
53.
A. A. Tolstonogov, “Integral funnel equation of a differential inclusion in a Banach
space and properties of its solutions”, Dokl. Akad. Nauk SSSR, 276:5 (1984), 1074–1078
54.
A. A. Tolstonogov, “Properties of the space of proper functions”, Mat. Zametki, 35:6 (1984), 803–812; Math. Notes, 35:6 (1984), 422–427
A. A. Tolstonogov, I. A. Finogenko, “On solutions of a differential inclusion with lower semicontinuous nonconvex right-hand side in a Banach space”, Mat. Sb. (N.S.), 125(167):2(10) (1984), 199–230; Math. USSR-Sb., 53:1 (1986), 203–231
A. A. Tolstonogov, “The solution set of a differential inclusion in a Banach space. II”, Sibirsk. Mat. Zh., 25:1 (1984), 159–173; Siberian Math. J., 25:1 (1984), 131–143
A. A. Tolstonogov, P. I. Chugunov, “The solution set of a differential inclusion in a Banach space. I”, Sibirsk. Mat. Zh., 24:6 (1983), 144–159; Siberian Math. J., 24:6 (1983), 941–954
A. A. Tolstonogov, “A dependence on a parameter of the solution of a differential inclusion with nonconvex right-hand side”, Differ. Uravn., 18:9 (1982), 1559–1570
59.
A. A. Tolstonogov, “Equation of the solution funnel of a differential inclusion”, Mat. Zametki, 32:6 (1982), 841–852; Math. Notes, 32:6 (1982), 908–914
A. A. Tolstonogov, “On the structure of the solution set for differential inclusions in a Banach space”, Mat. Sb. (N.S.), 118(160):1(5) (1982), 3–18; Math. USSR-Sb., 46:1 (1983), 1–15
A. A. Tolstonogov, “On density and co-density of the solution set of a differential inclusion in a Banach space”, Dokl. Akad. Nauk SSSR, 261:2 (1981), 293–296
A. A. Tolstonogov, “On comparison theorems for differential inclusions in locally convex space. II. Properties of solutions”, Differ. Uravn., 17:6 (1981), 1016–1024
63.
A. A. Tolstonogov, “On comparison theorems for differential inclusions in locally convex space. I. Existence of solutions”, Differ. Uravn., 17:4 (1981), 651–659
64.
A. A. Tolstonogov, “Differential inclusions in a Banach space with nonconvex right-hand side. Existence of solutions”, Sibirsk. Mat. Zh., 22:4 (1981), 182–198; Siberian Math. J., 22:4 (1981), 625–637
A. A. Tolstonogov, I. A. Finogenko, “On functional-differential inclusions in a Banach space with a nonconvex right-hand side”, Dokl. Akad. Nauk SSSR, 254:1 (1980), 45–49
1979
66.
A. A. Tolstonogov, “On properties of solutions of differential inclusions in a Banach space”, Dokl. Akad. Nauk SSSR, 248:1 (1979), 42–46
Yu. E. Linke, A. A. Tolstonogov, “Properties of spaces of sublinear operators”, Sibirsk. Mat. Zh., 20:4 (1979), 792–806; Siberian Math. J., 20:4 (1979), 554–564
A. A. Tolstonogov, “On the theorems of Radon–Nikodym and A. A. Ljapunov for a multivalued measure”, Dokl. Akad. Nauk SSSR, 225:5 (1975), 1023–1026
73.
A. A. Tolstonogov, “The maximum principle for differential inclusions”, Differ. Uravn., 11:10 (1975), 1838–1842
74.
A. A. Tolstonogov, “Topological structure of continuous set-valued mappings”, Sibirsk. Mat. Zh., 16:4 (1975), 837–852; Siberian Math. J., 16:4 (1975), 642–654
1970
75.
A. A. Tolstonogov, “The properties of the maximum function with constraints”, Zh. Vychisl. Mat. Mat. Fiz., 10:3 (1970), 597–606; U.S.S.R. Comput. Math. Math. Phys., 10:3 (1970), 56–67
I. V. Gaishun, A. A. Gryzlov, V. Ya. Derr, N. A. Izobov, A. I. Karpov, E. K. Makarov, G. V. Merzlyakova, N. N. Petrov, S. N. Popova, L. I. Rodina, N. Kh. Rozov, I. N. Sergeev, A. A. Tolstonogov, V. N. Ushakov, A. G. Chentsov, “In memory of Evgenii Leonidovich Tonkov (27.06.1940–28.09.2014)”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 4, 146–154
1988
77.
Yu. E. Linke, A. A. Tolstonogov, P. L. Ul'yanov, “Workshop “Contemporary Problems of the Theory of Functions””, Uspekhi Mat. Nauk, 43:4(262) (1988), 250–251
V. M. Matrosov, A. A. Tolstonogov, R. Z. Abdullin, “VI Всесоюзная конференция по качественной теории дифференциальных уравнений”, Differ. Uravn., 23:4 (1987), 729–730
1982
79.
G. P. Akilov, S. S. Kutateladze, A. A. Tolstonogov, “VI All-Union School on Theory of Operators in Functional Spaces”, Uspekhi Mat. Nauk, 37:2(224) (1982), 268–269