Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control~$u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0 + u(t) f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

翻译：我们的目标是强调数值分裂方法与控制理论之间的一些深刻联系。我们考虑形式为 $\dot{x} = f_0(x) + f_1(x)$ 的演化方程，其中 $f_0$ 编码了不可逆的动态，因此人们只关心仅涉及 $f_0$ 前向流的数值格式。在此背景下，分裂方法可解释为控制仿射系统 $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$ 的轨迹，其关联的控制 $u$ 是有限个 Dirac 质量之和。总体目标则是寻找一个控制，使得 $f_0 + u(t) f_1$ 的流尽可能接近 $f_0+f_1$ 的流。利用这一解释以及控制理论中的经典工具，我们重新审视了关于数值分裂方法的已知结果，并证明了一些新的结果，重点讨论了系数具有附加正性条件的分裂格式。首先，我们证明，如果允许 $f_1$ 的流具有复系数，则存在任意高阶的数值格式，且仅涉及 $f_0$ 的前向流。等价地，对于复值控制，我们证明了李代数秩条件等价于系统的小时间局部可控性。其次，对于实值系数，我们证明了众所周知的阶数限制与控制理论中所谓的“坏”李括号有关，这些括号已知会导致小时间局部可控性的障碍。我们利用我们最近提出的自由李代数基，精确地识别了高阶方法存在的条件。