Perturbation theory and control in classical or quantum mechanics by a closed formula

We consider a perturbation of an "integrable" Hamiltonian and give an expression for the canonical or unitary transformation which "simplifies" this perturbed system. The problem is to invert a functional defined on the Lie-algebra of observables. We give a bound for the perturbation in order to solve this inversion. And apply this result to a particular case of the control theory, as a first example, and to the "quantum adiabatic transformation", as another example.