We introduce the problem of finding minimum distance to the range of Toeplitz operator in l1 space of (bounded-sum-of-absolute-values) infinite sequences. However abstract and specialized it sounds, it is nothing else then just the common problem of solving overdetermined (Toeplitz) linear system Ax=b extended to infinite matrices, with the size of the residue measured using l1 norm. We give a rigorous analysis of existence and uniqueness of an optimal solution and propose an efficient numerical algoritm. Application of this result to a design of an optimal feedback controller (both for SISO and MIMO systems) minimizing worst-case magnification of peaks in the input variables will be given. The presented results may be of interest as such, but the value of the presentation is also in showing that rather abstract and rigorous theoretical tools from Toeplitz operator theory can be useful in control theory.