Hamilton–Jacobi–Bellman Equations with Time-Measurable Data and Infinite Horizon

  • V. Basco
  • H. FrankowskaEmail author


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.

Mathematics Subject Classification

34A60 49J15 49L25 70H20 



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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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.CNRS, Institut de Mathématiques de Jussieu - Paris Rive GaucheSorbonne UniversitéParisFrance

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