Controllability and stability of matrix difference systems

This work deals with a discrete-time control system having the form of X (n + 1) = AX (n)B. We study the effects of the matrix B. We find the general solution X (n) by solving the difference equation. Then we tackle the two important issues of controllability and stability. We found out that with the A matrix having eigenvalues outside the unit disk, we may select matrix B such that the final solution is stable and more importantly asymptotically stable. Proven theorems along with examples are presented.