The authors look for asymptotic solutions of the form \[ x=x(t,\varepsilon)=\sum_{i=0}^\infty\varepsilon^i[x_i(t)+\chi_i(\tau)],\quad (t,\varepsilon)\in [a,b]\times (0,\varepsilon_0],\quad \tau=(t-a)/\varepsilon, \] to the singularly perturbed system \(\varepsilon(d/dt)x=Ax+\varepsilon f(t,x,\varepsilon)+\varphi(t),\) with prescribed values of \(l(x)\). Here, \(A\) is a constant \(n\times n\)-matrix with eigenvalues contained in the left complex half-plane, \(f\), \(\varphi\) are infinitely smooth functions of their arguments and \(l\) is a bounded linear functional taking values in \({\mathbb R}^m\).