Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations
Электронный научный архив УРФУ
Информация об архиве | Просмотр оригинала| Поле | Значение | |
| Заглавие |
Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations
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| Автор |
Gomoyunov, M. I.
Plaksin, A. R. |
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| Тематика |
MINIMAX SOLUTIONS
PATH-DEPENDENT HAMILTON–JACOBI EQUATIONS VARIATIONAL PRINCIPLE VISCOSITY SOLUTIONS |
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| Описание |
In the paper, we consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton–Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and supergradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton–Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein–Preiss variational principle with an appropriate gauge-type functional. © 2023 Elsevier Inc.
We would like to thank Prof. Andrea Cosso for a discussion on the subject of this paper and for pointing us to the paper [27]. |
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| Дата |
2024-04-05T16:32:29Z
2024-04-05T16:32:29Z 2023 |
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| Тип |
Article
Journal article (info:eu-repo/semantics/article) |info:eu-repo/semantics/submittedVersion |
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| Идентификатор |
Gomoyunov, MI & Plaksin, AR 2023, 'Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations', Journal of Functional Analysis, Том. 285, № 11, 110155. https://doi.org/10.1016/j.jfa.2023.110155
Gomoyunov, M. I., & Plaksin, A. R. (2023). Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations. Journal of Functional Analysis, 285(11), [110155]. https://doi.org/10.1016/j.jfa.2023.110155 0022-1236 Final All Open Access, Green https://www.scopus.com/inward/record.uri?eid=2-s2.0-85170275130&doi=10.1016%2fj.jfa.2023.110155&partnerID=40&md5=d406ca1c0ed9f3420f88dbaadf058133 https://arxiv.org/pdf/2204.09275 http://elar.urfu.ru/handle/10995/130773 10.1016/j.jfa.2023.110155 85170275130 001080404600001 |
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| Язык |
en
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| Права |
Open access (info:eu-repo/semantics/openAccess)
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| Формат |
application/pdf
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| Издатель |
Academic Press Inc.
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| Источник |
Journal of Functional Analysis
Journal of Functional Analysis |
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