  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.
