We develop a power penalty approach to the discrete Hamilton–Jacobi–Bellman (HJB) equation in \( \mathbb {R}^N \) in which the HJB equation is approximated by a nonlinear equation containing a power penalty term. We prove that the solution to this penalized equation converges to that of the HJB equation at an exponential rate with respect to the penalty parameter when the control set is finite and the coefficient matrices are M-matrices. Examples are presented to confirm the theoretical findings and to show the efficiency of the new method.

中文翻译：

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离散HJB方程的功率损失法。

我们对\（\ mathbb {R} ^ N \）中的离散Hamilton–Jacobi–Bellman（HJB）方程开发了一种功率惩罚方法，其中HJB方程由包含功率惩罚项的非线性方程近似。我们证明了当控制集为有限且系数矩阵为M-矩阵时，该惩罚方程的解相对于惩罚参数以指数速率收敛于HJB方程的解。举例说明了理论发现并证明了新方法的有效性。