Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/111428
Title: Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised
Authors: Gomoyunov, M. I.
Lukoyanov, N. Y.
Plaksin, A. R.
Issue Date: 2021
Publisher: Springer
Springer Science and Business Media LLC
Citation: Gomoyunov M. I. Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised / M. I. Gomoyunov, N. Y. Lukoyanov, A. R. Plaksin. — DOI 10.1371/journal.pone.0258161 // Applied Mathematics and Optimization. — 2021. — Vol. 84. — P. 1087-1117.
Abstract: Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton–Jacobi equation with a right-end boundary condition. Minimax solutions of this problem are studied. The existence and uniqueness result is obtained under assumptions that are weaker than those considered earlier. In contrast to previous works, on the one hand, we do not require any properties concerning positive homogeneity of the Hamiltonian in the impulse variable, and on the other hand, we suppose that the Hamiltonian satisfies a Lipshitz continuity condition with respect to the path variable in the uniform (supremum) norm. The progress is related to the fact that a suitable Lyapunov–Krasovskii functional is built that allows to prove a comparison principle. This functional is in some sense equivalent to the square of the uniform norm of the path variable and, at the same time, it possesses appropriate smoothness properties. In addition, the paper provides non-local and infinitesimal criteria of minimax solutions, their stability with respect to perturbations of the Hamiltonian and the boundary functional, as well as consistency of the approach with the non-path-dependent case. Connection of the problem statement under consideration with some other possible statements (regarding the choice of path spaces and derivatives used) known in the theory of path-dependent Hamilton–Jacobi equations is discussed. Some remarks concerning viscosity solutions of the studied Cauchy problem are given. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Keywords: CO-INVARIANT DERIVATIVES
MINIMAX SOLUTIONS
PATH-DEPENDENT HAMILTON–JACOBI EQUATIONS
VISCOSITY SOLUTIONS
BOUNDARY CONDITIONS
OPTIMAL CONTROL SYSTEMS
STABILITY CRITERIA
COMPARISON PRINCIPLE
CONTINUITY CONDITIONS
DIFFERENTIAL GAMES
EXISTENCE AND UNIQUENESS RESULTS
FUNCTIONAL DIFFERENTIAL EQUATIONS
KRASOVSKII FUNCTIONAL
OPTIMAL CONTROL PROBLEM
VISCOSITY SOLUTIONS
HAMILTONIANS
URI: http://hdl.handle.net/10995/111428
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85108605401
PURE ID: 23717072
ISSN: 0095-4616
DOI: 10.1371/journal.pone.0258161
Appears in Collections:Научные публикации, проиндексированные в SCOPUS и WoS CC

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