We consider a Cauchy problem for a Hamilton–Jacobi equation with coinvariant derivatives of an order α ∈ (0, 1). Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order α. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov–Krasovskii functional.
Keywords: Hamilton–Jacobi equations, coinvariant derivatives, minimax solutions, Caputo fractional derivatives
@article{COCV_2022__28_1_A23_0,
author = {Gomoyunov, Mikhail Igorevich},
title = {Minimax solutions of {Hamilton{\textendash}Jacobi} equations with fractional coinvariant derivatives},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022017},
mrnumber = {4395151},
zbl = {1486.35136},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022017/}
}
TY - JOUR AU - Gomoyunov, Mikhail Igorevich TI - Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022017/ DO - 10.1051/cocv/2022017 LA - en ID - COCV_2022__28_1_A23_0 ER -
%0 Journal Article %A Gomoyunov, Mikhail Igorevich %T Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022017/ %R 10.1051/cocv/2022017 %G en %F COCV_2022__28_1_A23_0
Gomoyunov, Mikhail Igorevich. Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 23. doi: 10.1051/cocv/2022017
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This work was supported by RSF (project no. 19-71-00073).
