Concerning feedbacks of monotonically stabilized pulse system

In the pulse control system $\mathbf{x}_{n+1}=A\mathbf{x}_n+R\mathbf{u}_n, \mathbf{u}_n = S*\mathbf{x}_n$ the matrix $A$ is given. There are determined the conditions which the control matrix should satisfy in order that there might exist such $R$ a matrix $S$ that the pair of matrices $R$ and $S$ should secure the monotonic diminution of the given positively definite quadratic form $Q(x)$ at any solution of the system. For the matrix $R$ satisfying the conditions obtained the particular type of its twin matrix $S$ is determined. The same conditions are determined for the matrix $S$ and the particular type of its twin matrix $R$ is determined.