关于非线性系统的展开式、误差估计和收敛性问题
On expansions for nonlinear systems, error estimates and convergence issues
摘要：
非自治微分方程解的显式表达式在诸如控制理论或数值算子分裂等许多应用领域中具有重要意义。特别是，内在的公式允许解耦的时间依赖的功能，从几何依赖的功能的解决方案已被广泛研究。首先，我们给出了一个说教的正式的线性微分方程，包括著名的马格努斯展开（与第一类坐标）和Sussmann的无穷乘积展开（与第二类坐标）的经典扩展审查。受量子力学的启发，我们引入了一个新的混合展开，旨在隔离作用的时不变漂移的作用随时间变化的扰动。其次，在正则向量场驱动的非线性常微分方程的背景下，我们给出了形式展开式的精确解和有限近似之间的误差估计的严格证明。特别是，我们得到新的估计集中在随时间变化的扰动的作用。对于标量输入系统，我们得到新的估计只涉及一个弱的Sobolev范数的输入。第三，我们研究了这些展开式的局部收敛。我们回顾已知的幂零动力学和线性动力学的积极成果。然而，我们也表现出任意小的解析向量场的收敛的马格纳斯展开失败，即使在非常弱的意义。本文提出一个关于Sussmann无穷乘积展开式收敛性的公开问题。最后，我们推导出近似的直接内在表示的状态，并讨论它们的联系与选择一个适当的坐标变化。
英文摘要：
Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied. First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation. Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input. Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion. Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.
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