Small input controllability

This note provides a purely algebraic proof of the theorem: a constant linear system with matrices (A,B) is null-controllable using bounded inputs iff it is null-controllable (with unbounded inputs) and all eigenvalues of A have nonpositive real parts (continuous time) or magnitude not greater than one (discrete time). Related statements on asymptotic null-controllability are also studied, and an interpretation is provided in terms of (local) nonlinear controllability.