In the present paper, unlike the papers [17,20,24,26] using the matrix method there are presented the new relations to calculate feedback matrix coefficients for a discrete periodic optimal regulator problem with respect to output. On the base of these relations the analogue of Levine – Athans algorithm  (in stationary case) is created, that on an example  does not improve the exactness of solutions. Therefore, using the gradient methods  we offer a calculating algorithm that improves the exactness of solution of the problem considered in . Further, the inverse problem is formulated for a static feedback discrete periodic system with respect to output variable on the base of linear matrix inequalities (LMI) [16,18,22,23]. A new calculating algorithm is offered for its solution. The systems of inequalities are composed by means of relations defining the feedback chain matrix of the linear regulators for the discrete periodic systems that are obtained in section 2 of the paper. The efficiency of the offered algorithm is illustrated on an example .