带共同自由终端时间的连续时间协方差控制：有限时域解与无限时域极限
Continuous-Time Covariance Steering with Common Free-Final Time: Finite-Horizon Solutions and Infinite-Horizon Limits
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中文总结 AI 辅助
本文针对连续时间随机线性系统的协方差转向问题，建立SDE约束SOCP的确定性重述，推导最优性条件与有限终端时间的充分条件，分析有限时域解的无限时域极限，开发算法并在三类问题上验证性能。
中文摘要 AI 辅助
本文研究连续时间随机线性系统的状态协方差在给定初始和终端协方差矩阵间转向的最优共同自由终端时间问题。首先建立SDE约束自由终端时间随机最优控制问题（SOCP）的确定性重述；针对该SOCP，给出最优性必要条件，并证明最优共同终端时间为有限的充分条件。随后刻画有限时域最优解随终端时间趋于无穷的渐近行为、该极限 regime 对应的不变集与轨迹，推导哈密顿量对共同终端时间的敏感性。最后，利用这些敏感性，开发信赖域线搜索算法及无限时域情形检测方法，并在三类问题上验证其性能：(a) 说明性协方差转向示例；(b) 基于协方差控制的航天器机动演示；(c) 高斯混合到高斯混合的转向。
英文摘要
This article studies the optimal common free-final time problem for steering the state covariance of a continuous-time stochastic linear system between prescribed initial and terminal covariance matrices. We first establish a deterministic reformulation of the SDE-constrained free-final time stochastic optimal control problem (SOCP). For the ensuing SOCP, we provide necessary conditions for optimality and establish sufficient conditions for the optimal common final time to be finite. Subsequently, we characterize the asymptotic behavior of the finite-horizon optimal solutions as the final time tends to infinity, the invariant sets and trajectories associated with this limiting regime, and derive the sensitivity of the Hamiltonian with respect to the common final time. Finally, leveraging these sensitivities, we develop a trust-region line-search algorithm together with an infinite-horizon case detection method, and demonstrate its performance on three different problems: (a) an illustrative covariance-steering example, (b) a demonstration of spacecraft maneuver using covariance control, and (c) Gaussian mixture to Gaussian mixture steering.
