The problem of control in coefficients for the elliptic-type equation is considered. For this problem, set by Lions [1], it is not too difficult a matter to obtain necessary optimality conditions under the assumption that the optimal control does exist; yet, solvability of this problem is hard to establish. Nonetheless, it makes sense to search for an admissible control such that the value of the functional calculated on this control is as close as is wished to its lower bound. In this paper, a method for finding minimizing sequences is proposed. The approach used is based on sequential extension of the problem which is analogous to the scheme of complement of spaces [2]. For the extended problem, necessary conditions for sequential optimum property, which characterize minimizing sequences, are established. With the example of Varga [3], we show that, following this method, one can find the minimizing sequences even if the optimal control does not exist.