Equivalence of minimax and viscosity solutions of path-dependent Hamilton-Jacobi equations
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arXiv
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| Natura: | Preprint |
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2022
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| author | Gomoyunov, Mikhail Plaksin, Anton |
| author_facet | Gomoyunov, Mikhail Plaksin, Anton |
| contents | 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 super-gradients. 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 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 guage-type functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_09275 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equivalence of minimax and viscosity solutions of path-dependent Hamilton-Jacobi equations Gomoyunov, Mikhail Plaksin, Anton Optimization and Control Analysis of PDEs Functional Analysis 35R15, 35F21, 35D40, 49L25 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 super-gradients. 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 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 guage-type functional. |
| title | Equivalence of minimax and viscosity solutions of path-dependent Hamilton-Jacobi equations |
| topic | Optimization and Control Analysis of PDEs Functional Analysis 35R15, 35F21, 35D40, 49L25 |
| url | https://arxiv.org/abs/2204.09275 |