Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method

The present work proposes a gradient based iterative method to find the solutions of the general Sylvester discrete-time periodic matrix equations ∑j=1m(AijXiBij+CijXi+1Dij+EijYiFij+GijYi+1Hij)=Mi,i=1,2,…. It is proven that the proposed iterative method can obtain the solutions of the periodic matrix equations for any initial matrices. Finally a numerical example is included to demonstrate the validity and applicability of the iterative method.