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Developing B. N. Pshenichnyi’s scientific ideas in optimization and mathematical control theory. (English. Russian original) Zbl 1298.90004

Cybern. Syst. Anal. 48, No. 2, 157-179 (2012); translation from Kibern. Sist. Anal. 2012, No. 2, 3-28 (2012).
Summary: The paper overviews the studies initiated by fundamental works of B. N. Pshenichnyi in the field of necessary extremum conditions, numerical optimization methods, optimal control theory, and differential games. Among them are new scientific fields such as processes with fractional derivatives and impulsive control systems.

MSC:

90-03 History of operations research and mathematical programming
93-03 History of systems and control theory
01A70 Biographies, obituaries, personalia, bibliographies
91A23 Differential games (aspects of game theory)

Biographic References:

Pshenichnyi, B. N.
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References:

[1] I. V. Sergienko and A. A. Chikrii, ”The scientific heritage of B.N. Pshenichnyi,” Cybern. Syst. Analysis, 38, No. 2, 153–174 (2002). · Zbl 1023.01014 · doi:10.1023/A:1016309910620
[2] B. N. Pshenichnyi, Necessary Extremum Conditions [in Russian], Nauka, Moscow (1969). · Zbl 0764.90079
[3] B. N. Pshenichnyi, Convex Analysis and Extremum Problems [in Russian], Nauka, Moscow (1980).
[4] B. N. Pshenichnyi, ”Convex multi-valued mappings and mappings conjugate to them,” Kibernetika, No. 3, 94–102 (1972).
[5] V. M. Kuntsevich and B. N. Pshenichnyi, ”Minimal invariant sets of dynamic systems with bounded disturbances,” Cybern. Syst. Analysis, 32, No. 1, 58–64 (1996). · Zbl 0888.93044 · doi:10.1007/BF02366582
[6] V. M. Kuntsevich and B. N. Pshenichnyi, ”Invariant and stationary sets of nonlinear discrete systems under bounded disturbances,” Probl. Upravl. Inform., No. 1–2, 35–45 (1996).
[7] B. N. Pshenichnyi and E. E. Kirik, ”Nonlinear programming methods and network flows,” Cybern. Syst. Analysis, 30, No. 6, 846–854 (1994). · Zbl 0837.90043 · doi:10.1007/BF02366443
[8] L. S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Fizmatgiz, Moscow (1961).
[9] A. D. Ioffe and V.M. Tikhomirov, Theory of Extremum Problems [in Russian], Nauka, Moscow (1974). · Zbl 0191.13101
[10] B. N. Pshenichnyi, ”Necessary conditions of the extremum for differential inclusions,” Cybernetics, 12, No. 6, 871–886 (1976). · doi:10.1007/BF01070420
[11] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley (1984). · Zbl 0641.47066
[12] B.Sh. Mordukhovich, Approximations Methods in Optimization and Control Problems [in Russian], Nauka, Moscow (1988). · Zbl 0643.49001
[13] L.S. Pontryagin, Selected Works [in Russian], Vol. 2, Nauka, Moscow (1988).
[14] N.N. Krasovskii, Game Problems on the Encounter of Motions [in Russian], Nauka, Moscow (1970).
[15] B.N. Pshenichnyi and V.V. Ostapenko, Differential Games [in Russian], Naukova Dumka, Kyiv (1992).
[16] B.N. Pshenichnyi, ”Linear differential games,” Avtom. Telemekh., No. 1, 65–78 (1968).
[17] B.N. Pshenichnyi, ”{\(\epsilon\)}-strategies in differential games,” in: Topics in Differential Games, North Holland Publ. Co, New York–London–Amsterdam (1973), pp. 45–99.
[18] A.A. Chikrii, ”Differential games with several pursuers,” in: Trans. Banach Intern. Math. Center (Warsaw), 14, Math. Control Theory (1985), pp. 81–107.
[19] B.N. Pshenichnyi and Yu.N. Onopchuk, ”Linear differential games with integral constraints,” Tech. Kibern., No. 1, 13–22 (1968).
[20] B.N. Pshenichnyi, A.A. Chikrii, and J.S. Rappoport, ”Group pursuit in differential games,” J. Leipzig Techn. High School, No. 1, 13–27 (1982).
[21] B.N. Pshenichnyi and A.A. Chikrii, ”The problem of evading contact in differential games,” USSR Comp. Math. and Math. Physics, Vol. 14, Issue 6, 46–56 (1974).
[22] B.N. Pshenichnyi and A.A. Chikrii, ”The differential game of evasion,” Izv. AN SSSR, Tech. Kibern., No. 1, 3–10 (1977).
[23] A.A. Chikrii, ”The problem of avoidance for controlled dynamic objects,” J. Mathematics, Game Theory and Algebra, 7, No. 2/3, 81–94 (1998). · Zbl 0944.91007
[24] M.S. Nikol’skii, The First Direct Pontryagin Method in Differential Games [in Russian], Izd. MGU, Moscow (1984).
[25] N.L. Grigorenko, Mathematical Methods for the Control of Several Dynamic Processes [in Russian], Izd. MGU, Moscow (1990).
[26] A.A. Chikrii, Conflict Controlled Processes, Kluwer Academ. Publ., Boston–London–Dordrecht (1997). · Zbl 0868.93001
[27] A.A. Chikrii and S.D. Eidel’man, ”Generalized Mittag–Leffler matrix functions in game problems for evolutionary equations of fractional order,” Cybern. Syst. Analysis, 36, No. 3, 315–338 (2000). · Zbl 1007.91010 · doi:10.1007/BF02732983
[28] A.A. Chikrii and S.D. Eidelman, ”Control game problems for quasilinear systems with Riemann–Liouville fractional derivatives,” Cybern. Syst. Analysis, 37, No. 6, 836–864 (2001). · Zbl 1117.91012 · doi:10.1023/A:1014529914874
[29] A.A. Chikrii, ”Quasilinear controlled processes under conflict ” J. Math. Sci., 80, No. 1, 1489–1518 (1996). · Zbl 0914.90285 · doi:10.1007/BF02363923
[30] A.A. Chikrii and K.G. Dzyubenko, ”Bilinear Markovian processes of search for moving objects,” Probl. Upravl. Inform., No. 1, 92–107 (1997).
[31] B.N. Pshenichnyi and Yu.M. Danilin, Numerical Methods in Extremum Problems [in Russian], Nauka, Moscow (1975).
[32] B.N. Pshenichnyi, The Linearization Method [in Russian], Nauka, Moscow (1983).
[33] F.L. Chernous’ko and A.A. Melikyan, Game Problems of Control and Search [in Russian], Nauka, Moscow (1978).
[34] A.I. Subbotin and A.G. Chentsov, Guaranty Optimization in Control Problems [in Russian], Nauka, Moscow (1981). · Zbl 0542.90106
[35] V.F. Dem’yanov and L.V. Vasil’ev, Nondifferentiable Optimization [in Russian], Nauka, Moscow (1981).
[36] A.B. Kurzhanskii, Control and Observation under Uncertainty [in Russian], Nauka, Moscow (1977). · Zbl 0461.93001
[37] V.M. Kuntsevich and M.M. Lychak, Synthesis of Optimal and Adaptive Control Systems. A Game Approach [in Russian], Naukova Dumka, Kyiv (1985). · Zbl 0667.93002
[38] G.Ts. Chikrii, ”Searching for a fixed target by a moving object,” J. Appl. Math. Mech., 48, No. 4, 410–413 (1984). · Zbl 0584.49016 · doi:10.1016/0021-8928(84)90007-8
[39] V.M. Kuntsevich, Control under Uncertainty: Guaranteed Results in Control and Identification Problems [in Russian], Naukova Dumka, Kyiv (2006).
[40] V.M. Kuntsevich and B.Ò. Polyak, ”Invariant sets of nonlinear discrete systems with bounded disturbances and control problems,” J. Autom. Inform. Sci., Vol. 41, Issue 11, 1–16 (2009). · doi:10.1615/JAutomatInfScien.v41.i11.10
[41] V.M. Kuntsevich and A.V. Kuntsevich, ”Design of robust stable controls for nonlinear objects,” Automation and Remote Control, Vol. 69, No. 12, 2088–2100 (2008). · Zbl 1155.93406 · doi:10.1134/S0005117908120072
[42] A.V. Kuntsevich and V.M. Kuntsevich, ”Invariant sets for families of linear and nonlinear discrete systems with bounded disturbances,” Automation and Remote Control, Vol. 73, No. 1, 83–96 (2011). · Zbl 1307.93240 · doi:10.1134/S0005117912010067
[43] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ser. Comprehensive Studies in Mathematics, 330, I. Basic Theory; 331, II. Applications, Springer, Berlin–Heidelberg–New York (2006).
[44] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Springer, Berlin (1997). · Zbl 0888.49001
[45] J.M. Borwein and Q.J. Zhu, Techniques of Variational Analysis, Springer, New York (2005). · Zbl 1076.49001
[46] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, Massachusetts (1990).
[47] A.A. Chikrii, ”Method of alternating directions in nonlinear differential games of evasion,” Cybernetics, 20, No. 1, 71–82 (1984). · Zbl 0578.90108
[48] B.N. Pshenichny, ”The flight problem,” Cybernetics, 11, No. 4, 642–651 (1975). · doi:10.1007/BF01070443
[49] M.S. Nikolskii, ”On a linear evation problem,” Dokl. AN SSSR, 15, No. 5, 1462–1466 (1974). · Zbl 0319.90083
[50] L.S. Pontryagin and E.F. Mishchenko, ”Evasion problem in linear differential games,” Diff. Uravneniya, 7, No. 3, 436–445 (1971). · Zbl 0246.90064
[51] P.B. Gusyatnikov, ”One linear object evading several more inert pursuers,” Diff. Uravneniya, 12, No. 2, 213–226 (1976).
[52] A.I. Blagodatskikh and N.N. Petrov, Conflict Interaction of Groups of Controlled Objects [in Russian], Izd. Udmurt. Univ., Izhevsk (2009). · Zbl 1233.49002
[53] A.A. Chikrii, Conflict-Controlled Processes [in Russian], Naukova Dumka, Kyiv (1992). · Zbl 1127.91310
[54] V.I. Zhukovskii and A.A. Chikrii, Linear-Quadratic Differential Games [in Russian], Naukova Dumka, Kyiv (1994).
[55] Yu.G. Krivonos, I.I. Matichin, and A.A. Chikrii, Dynamic Games with Discontinuous Trajectories [in Russian], Naukova Dumka, Kyiv (2005). · Zbl 1242.91022
[56] A.A. Chikrii, ”Optimization of game interaction of fractional-order controlled systems,” Int. J. Optimization Methods and Software, 23, No. 1, 39–72 (2008). · Zbl 1133.91323 · doi:10.1080/10556780701281309
[57] A.A. Chikrii, ”Game dynamic problems for systems with fractional derivatives,” in: Pareto Optimality, Game Theory and Equilibria, 17, Springer, New York (2008), pp. 349–387. · Zbl 1153.91357
[58] A.A. Belousov, Yu.I. Berdyshev, A.G. Chentsov, and À.À. Chikrii, ”Solving the dynamic traveling salesman game problem,” Cybern. Syst. Analysis, 46, No. 5, 718–723 (2010). · Zbl 1298.91041 · doi:10.1007/s10559-010-9252-8
[59] A.A. Chikrii and S.F. Kalashnikova, ”Pursuit of a group of evaders by a single controlled object,” Cybern. Syst. Analysis, 23, No. 4, 437–445 (1987). · Zbl 0648.90109 · doi:10.1007/BF01078897
[60] M.S. Nikol’skii, ”Direct method in linear differential games with general integral constraints,” Diff. Uravneniya, 8, No. 6, 964–971 (1972). · Zbl 0279.90050
[61] D. Zonnevend, ”A pursuit method,” DAN SSSR, 204, No. 6, 1296–1299 (1972). · Zbl 0274.90077
[62] A.Ya. Azimov, ”A pursuit method in linear differential games with integral constraints,” Izv. AN SSSR, Tech. Kibern., No. 2, 31–35 (1974). · Zbl 0277.90104
[63] I.S. Rappoport, ”A problem of pursuit by several controlled objects under integral constraints,” DAN UkrSSR, Ser. A, No. 3, 221–224 (1979). · Zbl 0426.90100
[64] A.A. Chikrii and V.V. Bezmagorychnyi, ”Method of resolving functions in linear differential games with integral constraints,” Avtomatika, No. 4, 26–36 (1993).
[65] A.A. Chikrii and A. A. Belousov, ”On linear differential games with integral constraints,” Trans. Inst. Math. and Mechanics of the Ural Branch of the Russian Academy of Sciences, 15, No. 4, 290–301 (2009). · Zbl 1236.49081
[66] A.A. Belousov, ”Differential games with integral constraints for controls in the norm L 1,” Teoriya Optym. Rishen’, No. 10, 10–15 (2011).
[67] I.I. Matichin, ”Control of systems with fractional derivatives under conflict,” DAN Ukrainy, No. 8, 38–42 (2011).
[68] A.A. Chikrii and I.I. Matichin, ”Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann–Liouville, Caputo, and Miller–Ross,” J. Autom. Inform. Sci., Vol. 40, Issue 6, 1–11 (2008).
[69] A.A. Chikrii and I.I. Matychyn, ”Game problems for fractional-order systems,” in: New Trends in Nanotechnology and Fractional Calculus Applications, 11, Springer, Dordrecht–Heidelberg–London–New York (2010), pp. 233–241. · Zbl 1206.49041
[70] A.A. Chikrii and I.I. Matichin, ”Game problems for linear systems of fractional order,” Trans. Inst. of Math. and Mechanics of the Ural Branch of the Russian Academy of Sciences, 15, No. 3, 262–278 (2009). · Zbl 1303.91042
[71] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kyiv (1987). · Zbl 0837.34003
[72] O. Hellman, An Introduction to the Theory of Optimal Search [in Russian], Nauka, Moscow (1985).
[73] B.N. Pshenichnyi and A.A. Chikrii, A Discrete Search Model [in Russian], Prepr. AS UkrSSR 84–12, Inst. Cybernetics, Kyiv (1984).
[74] K.G. Dziubenko and A.A. Chikrii, ”An approach problem for a discrete system with random disturbances,” Cybern. Syst. Analysis, 46, No. 2, 271–281 (2010). · Zbl 1204.49029 · doi:10.1007/s10559-010-9204-3
[75] J. Albus, A. Meystel, A.A. Chikrii, A.A. Belousov, and A.I. Kozlov, ”Analytical method for solution of the game problem of soft landing for moving objects,” Cybern. Syst. Analysis, 37, No. 1, 75–91 (2001). · Zbl 1003.93027 · doi:10.1023/A:1016620201241
[76] J. Albus and A. Meystel, ”The eagle snatch,” in: Proc. Int. Conf., National Institute of Standards and Technology, Gaithersburg, USA (1996), pp. 129–136.
[77] J. Albus, A. Meystel, and A. Chikrii, ”Soft landing of moving objects,” in: Rep. National Institute of Standards and Technology, Gaithersburg, USA. · Zbl 1003.93027
[78] A.A. Chikrii, ”On an analytic method in dynamic games of approach,” in: Trans. V. A. Steklov Math. Inst. Russian Academy of Sciences, 271, 76–92 (2010).
[79] A.A. Chikrii, ”Set-valued mapping in game dynamic problems,” in: Proc. 8th Congress ISAAC–2011, Moscow (2011). · Zbl 1297.49067
[80] A.A. Chikrii and I.I. Matichin, ”On linear conflict controlled processes with fractional derivatives,” Trans. Inst. Math. and Mechanics of the Ural Branch of the Russian Academy of Sciences, 17, No. 2, 256–270 (2011).
[81] A.A. Chikrii, ”Guaranteed result in game problems of motion control,” Trans. Inst. of Math. and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, No. 5, 223–232 (2010).
[82] A.A. Chikrii and I.S. Rappoport, ”On inverse image theorem for (L{\(\times\)}B) -measurable multi-valued mappings,” Dokl. NAN Ukrainy, No. 11, 54–58 (2011). · Zbl 1249.49053
[83] A.A. Chikrii, I.I. Matychyn, K. Gromaszek, and A. Smolarz, ”Control of fractional-order dynamic systems under uncertainty,” in: J. Sikora (ed.), Modelling and Optimization, Publ. Lublin Univ. of Technology, Lublin (2011), pp. 3–56.
[84] G.Ts. Chikrii, ”One approach to solution of complex game problems for some quasilinear evolutionary systems,” Int. J. of Mathematics, Game Theory and Algebra, 14, No. 4, 307–314 (2004). · Zbl 1182.91038
[85] G.Ts. Chikrii, ”Using the effect of information delay in differential pursuit games,” Cybern. Syst. Analysis, 43, No. 2, 233–245 (2007). · Zbl 1253.91028 · doi:10.1007/s10559-007-0042-x
[86] G.Ts. Chikrii, ”On one problem of approach for damped oscillations,” J. Autom. Inform. Sci., Vol. 41, Issue 10, 1–9 (2009). · doi:10.1615/JAutomatInfScien.v41.i10.10
[87] G.Ts. Chikrii, ”A game problem of soft meeting of two objects of different types,” Teoriya Optym. Rishen’, No. 10, 31–37 (2011).
[88] B.N. Pshenichnyi, E.I. Nenakhov, and V.N. Kuz’menko, ”Mixed method for solving the general convex programming problem,” Cybern. Syst. Analysis, 34, No. 4, 577–587 (1998). · Zbl 0953.90047 · doi:10.1007/BF02667002
[89] V.N. Kuz’menko and E.I. Nenakhov, ”Solution algorithm for quadratic problem in the PNK-method,” Teoriya Optym. Rishen’, No. 10, 76–83 (2011),
[90] E.I. Nenakhov, ”On matrix optimization problems,” Teoriya Optym. Rishen’, No. 9, 79–85 (2010),
[91] E.I. Nenakhov and L.A. Sobolenko, ””Linearization method and nonsmooth optimization,” Systemni Doslidzhennya ta Informatsiini Tekhnologii, No. 3, 90–104 (2009).
[92] V.V. Ostapenko and L.A. Sobolenko, ”Packing various ellipsoids in a parallelepiped with the minimum sum of sides,” in: Trans. 11th Intern. Sci.-Tehn. Conf. ”Systems analysis and information technologies” (May 26–30, 2009, Kyiv), UNK ”IPSA” NTUU ”KPI,” Kyiv (2009).
[93] E.E. Kirik and B.N. Pshenichnyi, ”Theory and methods of network design,” in: A Review of Applied and Industrial Mathematics [in Russian], 2, Issue 1 (1995), pp. 46–69.
[94] O.E. Kirik, ”Algorithms of iterative quadratic programming for optimal flow distribution problems,” Systemni Doslidzhennya ta Informatsiini Tekhnologii, No. 1, 101–113 (2008).
[95] O.E. Kirik, ”Problem of the existence of solution in flow distribution problems,” Systemni Doslidzhennya ta Informatsiini Tekhnologii, No. 4, 119–128 (2011).
[96] O.E. Kirik and V.V. Ostapenko, ”Optimal distribution of water resources in irrigation systems of network structure,” Systemni Doslidzhennya ta Informatsiini Tekhnologii, No. 4, 79–90 (2010).
[97] S.D. Eidel’man and A.A. Chikrii, ”Dynamic game problems of approach for fractional-order equations,” Ukr. Math. J., 52, No. 11, 1787–1806 (2000). · Zbl 1032.91029 · doi:10.1023/A:1010439422856
[98] A.A. Chikrii and S.D. Eidelman, ”Game problems for fractional quasilinear systems,” Int. J. Computers and Mathematics with Applications, 44, 835–851(2002). · Zbl 1038.91021 · doi:10.1016/S0898-1221(02)00197-9
[99] Ye.V. Klimenko and A.A. Chikrii, ”Search methods for moving evaders,” Int. J. Facta Universitatis, Yugoslavia, University of Nis, 1, No. 4, 451–460 (1994). · Zbl 0840.49014
[100] K.B. Polinkevich and Yu.N. Onopchuk, ”Conflicts in the regulation of the main function of the respiratory system of the organism and mathematical models of conflict resolution,” Cybernetics, 22, No. 3, 385–390 (1986).
[101] A.Z. Kolchinskaya, B.N. Pshenichnyi, Yu.N. Onopchuk, et al., ”Modeling the dynamics of the mass transfer of gases in the human body,” Kibern. Vych. Tekhnika, Issue 40, 54–61 (1978).
[102] Yu.N. Onopchuk and A.G. Misyura, ”Methods of mathematical modeling and control in theoretical studies and solution of applied problems in sports medicine and physiology,” Sport. Meditsina, No. 1, 181–189 (2008).
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