Volume 28, 2022
|Number of page(s)||32|
|Published online||18 August 2022|
Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations
Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
* Corresponding author: email@example.com
Accepted: 16 July 2022
This article is devoted to the study of lower semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians in a gradient variable. Such Hamiltonians appear in the optimal control theory. We present a necessary and sufficient condition for a reduction of a Hamiltonian satisfying optimality conditions to the case when the Hamiltonian is positively homogeneous and also satisfies optimality conditions. It allows us to reduce some uniqueness problems of lower semicontinuous solutions to Barron-Jensen and Frankowska theorems. For Hamiltonians, which cannot be reduced in that way, we prove the new existence and uniqueness theorems.
Mathematics Subject Classification: 34A60 / 49J52 / 49L20 / 49L25 / 35Q93
Key words: Hamilton-Jacobi equations / viscosity solutions / optimal control theory / set-valued analysis / nonsmooth analysis / convex analysis
© 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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