The purpose of this paper is to extend the bang-bang principle of the optimal time-control problem to the case in which the modelled equation \[ (d/dt)x(t)=A(t)x(t)+B(t)u(t),\quad x(0)=x_ 0\in {\mathcal D}(A(0)), \] has a reduced dimension with respect to the real physical problem. Here X and U are normed function spaces (both real or both complex), \(B\in {\mathcal L}(U,X)\), A is the generator of a linear \(C_ 0\)-semigroup on X with domain \({\mathcal D}(A(.))\) and u belongs to the space of U-valued Lebesgue- integrable functions on [0,t] which are also square integrable.