Representation of Weak Solutions of Convex Hamilton-Jacobi-Bellman Equations on Infinite Horizon

DOI10.1007/S10957-020-01763-1arXiv2208.08725MaRDI QIDQ6408194

Publication date: 18 August 2022

Abstract: In the present paper, it is provided a representation result for the weak solutions of a class of evolutionary Hamilton-Jacobi-Bellman equations on infinite horizon, with Hamiltonians measurable in time and fiber convex. Such Hamiltonians are associated with a - faithful - representation, namely involving two functions measurable in time and locally Lipschitz in the state and control. Our results concern the recovering of a representation of convex Hamiltonians under a relaxed assumption on the Fenchel transform of the Hamiltonian with respect to the fiber. We apply them to investigate the uniqueness of weak solutions, vanishing at infinity, of a class of time-dependent Hamilton-Jacobi-Bellman equations. Assuming a viability condition on the domain of the aforementioned Fenchel transform, these weak solutions are regarded as an appropriate value function of an infinite horizon control problem under state constraints.

Mathematics Subject Classification ID

Nonsmooth analysis (49J52) Weak solutions to PDEs (35D30) Hamilton-Jacobi equations in mechanics (70H20) Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games (49L25) Optimality conditions for problems involving ordinary differential equations (49K15) Convexity properties of solutions to PDEs with constant coefficients (35E10) Hamilton-Jacobi equations (35F21) Hamilton-Jacobi equations in optimal control and differential games (49L12)

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