In this paper we construct a method for solving the problem of optimizing the maximal deviation of the real plan (or the trajectory, the technological process . . . ) from some ideal one. The mathematical model of the dynamical system being considered is a system of linear differential equations with a control function. The method is based on some ideas of the so-called support method proposed by R. Gabasov and F.M. Kirillova.After introducing support controls and establishing their relation with controllability of the system, we derive a criterion (which can be easily verified) for a support control to be optimal. Then we briefly describe an iteration for improving the existing control if it has not been optimal yet. Finally we present an illustrative example.