发表机构
Sorbonne Université; Université Paris Cité; CNRS; Inria; Laboratoire Jacques-Louis Lions (LJLL); FAU Erlangen-Nürnberg; Fundación Deusto; Universidad Autónoma de Madrid（索邦大学; 巴黎西岱大学; 法国国家科学研究中心; 法国国家信息与自动化研究所; 雅克-路易·利翁斯实验室; 埃尔朗根-纽伦堡大学; 德乌斯托基金会; 马德里自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对可控单输入线性系统，显式计算可控性Gramian特征值的下一阶系数及谱子空间变分，建立局部实解析展开，并推导最坏情况能量、Gramian行列式、条件数及Ornstein--Uhlenbeck分布的精细渐近，对称情形下谱数据可重构系统。
AI 中文摘要
对于维数为$n$的可控单输入线性系统，其可控性Gramian特征值按递减顺序排列，已知具有连续的小时间阶$T, T^3, \ldots, T^{2n-1}$。我们通过显式计算每个特征值的下一阶系数以及其特征方向和嵌套谱子空间的一阶变分，细化了这一主导阶层次结构。所得公式在标准正交Krylov基下具有内蕴表达式：特征值修正描述了动力学沿每个Krylov方向的作用，而谱子空间的变分则描述了与下一方向的耦合。我们还建立了在$T=0$附近关于$(T, A, b)$的局部联合实解析依赖性，针对除以各自主导幂次$T$的特征值、一致取向的特征向量以及谱投影算子。这提供了收敛展开，其余项在可控对的紧致族上一致。这些结果为从原点到达单位目标所需的最坏情况最小能量、Gramian行列式和条件数，以及在移动主坐标下的Ornstein--Uhlenbeck高斯分布提供了精细的渐近行为。能量等于最小Gramian特征值的倒数，其特征方向在足够小的时间下识别出最耗能的目标。我们进一步推导了前向Fokker--Planck方程的精确Gramian加权能量恒等式，以及分别利用Lyapunov方程和精确Fourier范数公式结合分级Gramian分解的尖锐各向异性短时间平滑渐近。对于对称动力学，谱数据重构$A$并确定$b$（至多相差全局符号）。
英文摘要
For a controllable single-input linear system in dimension $n$, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders $T, T^3, \ldots, T^{2n-1}$. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near $T=0$ for the eigenvalues divided by their leading powers of $T$, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct $A$ and determine $b$ up to its global sign.
