A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations $A_k{X}_k+Y_k{B}_k=M_k,C_k{X}_{k+1}+Y_k{D}_k=N_k(k=1,2)$ over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.