  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.

Classical Analysis and ODEs;Optimization and Control