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.

中文翻译：

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无限地平线上凸Hamilton-Jacobi-Bellman方程弱解的表示

在本文中，提供了一类演化Hamilton-Jacobi-Bellman方程在无限视域上的弱解的表示结果，其中哈密顿量在时间上可测且纤维凸。这样的哈密顿量与“忠实”表示相关联，即在状态和控制中涉及两个可在时间上测量的函数和局部 Lipschitz 函数。我们的结果涉及在相对于纤维的哈密顿量的芬谢尔变换的宽松假设下恢复凸哈密顿量的表示。我们应用它们来研究一类时间相关的 Hamilton-Jacobi-Bellman 方程的弱解的唯一性，在无穷远处消失。假设上述 Fenchel 变换域的生存能力条件，