AI 中文总结
本文针对未知动力学的有限时域连续时间逆LQR问题，利用时变最优增益的内生激励建立联合可识别性条件，开发感知条件的采样数据重构方法，通过数值实验验证了理论与条件指数的诊断价值。
AI 中文摘要
逆最优控制（IOC）旨在从智能体的专家行为观测中推断其潜在的代价泛函。本文研究从闭环状态-输入轨迹出发的有限时域连续时间逆LQR问题，其中系统矩阵与二次代价均为未知。有限时域会产生时变最优增益，这种内生激励是实现联合恢复的结构机制。我们通过三个可计算的条件指数量化该机制，这些指数分别衡量状态丰富度、增益变化丰富度以及结构化代价算子的单射性。利用这些指数，我们为所考虑的逆问题建立了联合可识别性条件，关键在于这些条件能保证恢复真实系统矩阵$(A,B)$与真实代价加权矩阵，而非仅恢复行为等效的替代模型。我们还开发了一种感知条件的采样数据重构方法，该方法可从含噪测量中重构增益$K(·)$与闭环动力学矩阵$A_c(·)$，以闭式形式恢复$(A,B)$，并通过凸半定规划识别二次权重。我们进一步建立了亚高斯观测噪声下完整重构方法的非渐近扰动界与一致性，其显式依赖于上述相同的条件指数。数值实验验证了该理论，并说明了条件指数的诊断价值。
英文摘要
Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper studies the finite-horizon continuous-time inverse LQR problem from closed-loop state--input trajectories, where both the system matrices and the quadratic cost are unknown. The finite horizon induces a time-varying optimal gain, and this endogenous excitation serves as the structural mechanism that makes joint recovery possible. We quantify this mechanism through three computable conditioning indices, which measure state richness, gain-variation richness, and injectivity of a structured cost operator. Using these indices, we establish joint identifiability conditions for the inverse problem considered here. Crucially, these conditions guarantee recovery of the ground-truth system matrices $(A,B)$ and the true cost weighting matrices, rather than merely a behaviorally equivalent surrogate. We also develop a conditioning-aware sampled-data reconstruction method that reconstructs the gain $K(\cdot)$ and the closed-loop dynamics matrix $A_c(\cdot)$ from noisy measurements, recovers $(A,B)$ in closed form, and identifies the quadratic weights through a convex semidefinite program. We further establish the non-asymptotic perturbation bounds and the consistency of the full reconstruction method under sub-Gaussian observation noise, with explicit dependence on the same conditioning indices. Numerical experiments support the theory and illustrate the diagnostic value of the conditioning indices.
