Control theory and splitting methods

Our goal is to highlight some deep connections 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 non-reversible dynamics, motivating schemes that involve only 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 $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ 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 on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.