We study the well-posedness of Hamilton–Jacobi–Bellman equations on subsets of \({\mathbb {R}}^d\) in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker–Varadhan theory of large deviations for occupation-time measures. To allow for this flexibility, we assume that the internal Hamiltonian and cost functional have controlled growth, and that they satisfy an equi-continuity estimate uniformly over compact sets in the space of controls. In addition to establishing the comparison principle for the Hamilton–Jacobi–Bellman equation, we also prove existence, the viscosity solution being the value function with exponentially discounted running costs. As an application, we verify the conditions on the internal Hamiltonian and cost functional in two examples.

我们研究了\（{{mathbb {R}} ^ d \）的子集上Hamilton–Jacobi–Bellman方程的适定性在没有边界条件的情况下。哈密​​顿量是两个部分的最高值：取决于外部控制变量的内部哈密顿量和对控制进行惩罚的成本函数。本文的关键特征是控制功能可以是无界且不连续的。通过这种方式，我们可以处理在Donsker–Varadhan理论中针对占用时间量度的大偏差所出现的功能。为了获得这种灵活性，我们假定内部哈密顿量和成本函数具有受控的增长，并且它们在控制空间内的紧凑集合上均等地满足等连续性估计。除了建立汉密尔顿-雅各比-贝尔曼方程的比较原理外，我们还证明了存在性，粘度溶液是价值函数，其运行成本呈指数折扣。作为一个应用程序，我们在两个示例中验证内部哈密顿量和成本函数的条件。