Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 55

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.

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DOI : 10.1051/cocv/2022051
Classification : 34A60, 49J52, 49L20, 49L25, 35Q93
Keywords: Hamilton-Jacobi equations, viscosity solutions, optimal control theory, set-valued analysis, nonsmooth analysis, convex analysis 
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     author = {Misztela, Arkadiusz},
     title = {Reduction of lower semicontinuous solutions of {Hamilton-Jacobi-Bellman} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {28},
     doi = {10.1051/cocv/2022051},
     mrnumber = {4469491},
     zbl = {1497.49024},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2022051/}
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Misztela, Arkadiusz. Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 55. doi: 10.1051/cocv/2022051

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