We present recent advances in the study of bilinear approximate controllability for both linear and nonlinear partial differential equations (PDEs), with a particular focus on saturation techniques. After introducing the general framework of bilinear control systems, we discuss small-time approximate controllability results obtained via bilinear controls for several models, including the nonlinear Schrödinger equation, the nonlinear heat equation, and the Burgers equation. Under suitable structural assumptions on the controls, the results demonstrate that it is possible to steer the system arbitrarily close to a desired target state, even in the presence of nonlinear dynamics. The proofs rely on the iterative application of saturation properties and on geometric arguments derived from commutator expansions.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006