Abstract:
For a dynamical system with discrete and distributed time delays, a control problem under disturbance or counteraction is considered. The problem is formalized in the context of the game-theoretical approach in the class of control strategies with memory. The problem is associated with a functional Hamilton–Jacobi type equation with coinvariant derivatives. The minimax and viscosity approaches to a generalized solution of this equation are discussed. It is shown that, under the same condition at the right endpoint, the minimax and viscosity solutions coincide, thereby uniquely defining the functional of optimal guaranteed result in the control problem
\Bibitem{Luk09}
\by N.~Yu.~Lukoyanov
\paper Minimax and viscosity solutions in optimization problems for hereditary systems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2009
\vol 15
\issue 4
\pages 183--194
\mathnet{http://mi.mathnet.ru/timm435}
\elib{https://elibrary.ru/item.asp?id=12952764}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2010
\vol 269
\issue , suppl. 1
\pages S214--S225
\crossref{https://doi.org/10.1134/S0081543810060179}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84962468865}
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This publication is cited in the following 12 articles:
Elena Bandini, Christian Keller, “Path-Dependent Hamilton–Jacobi Equations with u-Dependence and Time-Measurable Hamiltonians”, Appl Math Optim, 91:2 (2025)
M. I. Gomoyunov, N. Yu. Lukoyanov, “Minimax solutions of Hamilton–Jacobi equations in dynamic optimization problems for hereditary systems”, Russian Math. Surveys, 79:2 (2024), 229–324
Daniel Hernández-Hernández, Hidehiro Kaise, “Path-dependent zero-sum deterministic games with intermediate Hamiltonians”, NACO, 2024
M.I. Gomoyunov, A.R. Plaksin, “Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations”, Journal of Functional Analysis, 285:11 (2023), 110155
Anton Plaksin, “Viscosity Solutions of Hamilton–Jacobi Equations for Neutral-Type Systems”, Appl Math Optim, 88:1 (2023)
Mikhail Igorevich Gomoyunov, “Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives”, ESAIM: COCV, 28 (2022), 23
M. I. Gomoyunov, “O kriteriyakh minimaksnykh reshenii uravnenii Gamiltona - Yakobi s koinvariantnymi proizvodnymi drobnogo poryadka”, Tr. IMM UrO RAN, 27, no. 3, 2021, 25–42
Anton Plaksin, “Viscosity Solutions of Hamilton–Jacobi–Bellman–Isaacs Equations for Time-Delay Systems”, SIAM J. Control Optim., 59:3 (2021), 1951
Mikhail I. Gomoyunov, Nikolai Yu. Lukoyanov, Anton R. Plaksin, “Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised”, Appl Math Optim, 84:S1 (2021), 1087
Anton Plaksin, “Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems”, J Optim Theory Appl, 187:1 (2020), 22
Bayraktar E., Keller Ch., “Path-Dependent Hamilton–Jacobi Equations in Infinite Dimensions”, J. Funct. Anal., 275:8 (2018), 2096–2161
N. Yu. Lukoyanov, “On Hamilton–Jacobi formalism in time-delay control systems”, Tr. IMM UrO RAN, 16, no. 5, 2010, 269–277
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