Abstract Uniqueness of positive solutions to viscous Hamilton–Jacobi–Bellman (HJB) equations of the form , with f a coercive function and λ a constant, in the subquadratic case, that is, , appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that and for some , essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for f. Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive f which satisfies for some positive constant κ. Since , this assumption imposes very mild restrictions on the growth of the potential f. We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide optimality results for the associated ergodic control problem, even in a pathwise sense, and without resorting to the parabolic problem.

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关于梯度中具有次二次非线性的粘性 HJB 方程解的唯一性

摘要 形式为 的粘性 Hamilton-Jacobi-Bellman (HJB) 方程的正解的唯一性，具有 fa 强制函数和 λ 常数，在次二次情况下，即 ，似乎是一个开放问题。Barles 和 Meireles [通讯。偏微分方程 41 (2016)] 显示了独特性，对于某些 ，基本上匹配 Ichihara 的早期结果，他考虑了更一般的哈密顿量，但对 f 具有更好的规律性。如果不强制执行这个假设，据我们所知，文献中没有关于唯一性的结果。在这篇简短的文章中，我们展示了该方程对于满足某个正常数 κ 的任何局部 Lipschitz 连续的强制 f 具有唯一的正解。因为 ，这个假设对潜在 f 的增长施加了非常温和的限制。我们还表明，该解决方案完全表征了相关遍历问题的最优性。我们的方法涉及对用于度量的椭圆方程的无限维线性规划的研究，并且与早期的方法非常不同。它也适用于 Ichihara 研究的更大类别的哈密顿量，我们表明它非常适合为相关的遍历控制问题提供最优结果，即使在路径意义上也是如此，并且无需求助于抛物线问题。