Abstract
In this paper we investigate the existence and uniqueness of weak solutions of the nonautonomous Hamilton–Jacobi–Bellman equation on the domain \((0,\infty ) \times \Omega \). The Hamiltonian is assumed to be merely measurable in time variable and the open set \(\Omega \) may be unbounded with nonsmooth boundary. The set \(\overline{\Omega }\) is called here a state constraint. When state constraints arise, then classical analysis of Hamilton–Jacobi–Bellman equation lacks appropriate notion of solution because continuous solutions could not exist. In this work we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity.
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Funding was provided by Program Gaspard Monge in Optimization and Operation Research (Grant No. 2018-0047H) and Air Force Office of Scientific Research (Grant No. FA 9550-18-1-0254).
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Basco, V., Frankowska, H. Hamilton–Jacobi–Bellman Equations with Time-Measurable Data and Infinite Horizon. Nonlinear Differ. Equ. Appl. 26, 7 (2019). https://doi.org/10.1007/s00030-019-0553-y
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DOI: https://doi.org/10.1007/s00030-019-0553-y