Volume 28, 2022
|Number of page(s)||36|
|Published online||17 March 2022|
Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives*
N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences,
2 Ural Federal University, Ekaterinburg, Russia.
** Corresponding author: firstname.lastname@example.org
Accepted: 23 February 2022
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.
Mathematics Subject Classification: 35F21 / 35D99 / 26A33
Key words: Hamilton–Jacobi equations / coinvariant derivatives / minimax solutions / Caputo fractional derivatives
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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