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Archive of Issues

Russia Yekaterinburg
Year
2017
Volume
27
Issue
2
Pages
222-237
 Section Mathematics Title On Hamilton-Jacobi-Isaacs-Bellman equation for neutral type systems Author(-s) Plaksin A.R.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa Abstract For a conflict-controlled dynamical system described by functional differential equations of neutral type in Hale’s form, we consider a differential game with a quality index that estimates the motion history realized up to the terminal time and includes an integral estimation of realizations of players’ controls. The game is formalized in the class of pure positional strategies. Based on a coinvariant derivatives conception we derive a Hamilton–Jacobi functional equation. It is proved, firstly, that the solution of this equation, satisfying certain conditions of smoothness, is the value of the initial differential game, and secondly, that value at points of differentiability satisfies the considered Hamilton–Jacobi equation. Thus this equation can be interpreted as the Hamilton-Jacobi-Isaacs-Bellman equation for neutral type systems. Keywords neutral type systems, differential games, Hamilton-Jacobi equation UDC 517.952, 517.977 MSC 49L20, 49N70 DOI 10.20537/vm170206 Received 17 March 2017 Language Russian Citation Plaksin A.R. On Hamilton-Jacobi-Isaacs-Bellman equation for neutral type systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 222-237. References Isaacs R. Differential games, New York: John Wiley and Sons, 1965, 384 p. Translated under the title Differentsyal'nye igry, 1967, 479 p. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 458 p. Osipov Iu.S. On the theory of differential games of systems with aftereffect, J. Appl. Math. Mech., 1971, vol. 35, issue 2, pp. 262-272. DOI: 10.1016/0021-8928(71)90032-3 Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of a dynamic system), Moscow: Nauka, 1985, 516 p. Gomoyunov M.I., Lukoyanov N.Yu., Plaksin A.R. Existence of the value and saddle point in positional differential games for systems of neutral type, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2016, vol. 22, no. 2, pp. 101-112 (in Russian). DOI: 10.21538/0134-4889-2016-22-2-101-112 Gomoyunov M.I., Lukoyanov N.Yu. On the numerical solution of differential games for neutral-type linear systems, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2017, vol. 23, no. 1, pp. 75-87 (in Russian). DOI: 10.21538/0134-4889-2017-23-1-75-87 Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil'tona-Yakobi (Minimax inequalities and Hamilton-Jacobi equations), Moscow: Nauka, 1991, 216 p. Crandall M.G., Lions P.-L. Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 1983, vol. 277, no. 1, pp. 1-42. DOI: 10.1090/S0002-9947-1983-0690039-8 Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R. Nonsmooth analysis and control theory, New York: Springer, 1998, 278 p. Lukoyanov N.Yu. Functional equations of Hamilton-Jacobi type and differential games with hereditary information, Doklady Mathematics, 2000, vol. 61, issue 2, pp. 301-304. Aubin J.-P., Haddad G. History path dependent optimal control and portfolio valuation and management, Positivity, 2002, vol. 6, issue 3, pp. 331-358. DOI: 10.1023/A:1020244921138 Lukoyanov N.Yu. On optimality conditions for the guaranteed result in control problems for time-delay systems, Proc. Steklov Inst. Math., 2010, vol. 268, suppl. 1, pp. S175-S187. DOI: 10.1134/S0081543810050135 Lukoyanov N.Yu. Funktsional'nye uravneniya Gamil'tona-Yakobi i zadachi upravleniya s nasledstvennoi informatsiei (Functional Hamilton-Jacobi equations and control problems with hereditary information), Yekaterinburg: Ural Federal University, 2011, 243 p. Kaise H. Path-dependent differential games of inf-sup type and Isaacs partial differential equations, 2015 IEEE 54th Annual Conference on Decision and Control (CDC), 2016, pp. 1972-1977. DOI: 10.1109/CDC.2015.7402496 Hale J.K., Cruz M.A. Existence, uniqueness and continuous dependence for hereditary systems, Ann. Mat. Pura Appl., 1970, vol. 85, issue 1, pp. 63-81. DOI: 10.1007/BF02413530 Kim A.V. Functional differential equations. Application of $i$-smooth calculus, Springer Netherlands, 1999, xv+168 p. DOI: 10.1007/978-94-017-1630-7 Filippov A.F. Differential equations with discontinuous righthand sides, Berlin: Springer, 1988, x+304 p. Original Russian text published in Filippov A.F. Differentsial'nye uravneniya s razryvnoi pravoi chast'yu, Moscow: Nauka, 1985, 225 p. DOI: 10.1007/978-94-015-7793-9 Full text