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

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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.

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Notes

  1. We denote \(\int _{a}^{\infty }w(s)\,\mathrm{d}s:=\lim _{b\rightarrow \infty }\int _{a}^{b}w(s)\,\mathrm{d}s\), provided this limit exists.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC, 3010, Australia

    Vincenzo Basco

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  1. Vincenzo Basco
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Correspondence to Vincenzo Basco.

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Communicated by Hélène Frankowska.

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Basco, V. Representation of Weak Solutions of Convex Hamilton–Jacobi–Bellman Equations on Infinite Horizon. J Optim Theory Appl 187, 370–390 (2020). https://doi.org/10.1007/s10957-020-01763-1

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