发表机构
School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究从部分轨迹中联合恢复未知系统矩阵和低秩初始状态矩阵，提出秩证书原理与显式重构公式，并用Adam求解非线性最小二乘问题，实验验证了不同采样条件下的恢复性能。
AI 中文摘要
我们研究了在仅观测状态矩阵$A^tX_0$（$t=0,\ldots,T$）的部分坐标的情况下，联合恢复未知系统矩阵$A\in\mathbb R^{n\times n}$和秩为$r$的初始状态矩阵$X_0\in\mathbb R^{n\times m}$的问题。每条轨迹在固定的状态坐标集合上被观测，这些坐标集合可能因轨迹而异。我们推导了可识别性的必要条件，包括由不完整空间覆盖和不足动力学跨度引起的障碍，以及观测数量的下界。随后，我们刻画了测量映射的雅可比矩阵，考虑了低秩分解的内在对称性，并建立了一个秩证书原理，该原理保证了通用的局部可识别性。当初始状态构成$\operatorname{range}(X_0)$的一组基的轨迹被完全观测时，我们获得了全局恢复的充分必要条件，以及显式的重构公式。我们将联合恢复问题表述为关于系统矩阵$A$和$X_0$的低秩因子的非线性最小二乘问题，并使用Adam求解。在合成数据和真实数据上的数值实验评估了不同空间采样率和观测时域下的恢复性能。
英文摘要
We study the joint recovery of an unknown system matrix $A\in\mathbb R^{n\times n}$ and a rank-$r$ initial-state matrix $X_0\in\mathbb R^{n\times m}$ from partial observations of the state matrices $A^tX_0$, for $t=0,\ldots,T$. Each trajectory is observed at a fixed set of state coordinates, which may differ across trajectories. We derive necessary conditions for identifiability, including obstructions arising from incomplete spatial coverage and insufficient dynamical span, as well as lower bounds on the number of observations. We then characterize the Jacobian of the measurement map, account for the intrinsic symmetry of the low-rank factorization, and establish a rank-certificate principle yielding generic local identifiability. When the trajectories whose initial states form a basis of $\operatorname{range}(X_0)$ are fully observed, we obtain necessary and sufficient rank conditions for global recovery, together with explicit reconstruction formulas. We formulate the joint recovery problem as a nonlinear least-squares problem over the system matrix $A$ and the low-rank factors of $X_0$, and solve it using Adam. Numerical experiments on synthetic and real-world data evaluate the recovery performance under different spatial sampling rates and observation horizons.
