线性指数二次高斯协方差转向
Linear Exponential Quadratic Gaussian Covariance Steering
- Iowa State University（爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出并分析连续时间LEQG协方差转向问题，证明最优控制器由对称矩阵参数化，推广风险中性结果，并证明匹配噪声情形下解的存在唯一性。
AI中文摘要：
我们针对给定截止时间（有限时间范围）内的连续时间线性指数二次高斯（LEQG）协方差转向问题进行了公式化与分析。该问题的解可视为线性二次设置中高斯端点之间的风险敏感薛定谔桥。与风险中性情形不同，LEQG协方差转向控制器——仍为线性状态反馈——无法再以闭式形式写出。我们证明最优控制器由一个对称矩阵参数化，该矩阵求解一个代数方程，编码了对风险敏感参数的隐式依赖。我们解释了该最优控制器的结构如何显著推广了风险中性情形的现有结果。基于这些结果，对于匹配噪声和输入通道情形，我们证明了在已知风险中性最优解邻域内LEQG协方差转向问题解的存在唯一性。我们给出了一个说明性数值示例。
英文摘要：
We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schrödinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller--still a linear state feedback--can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example.
