An algebraic condition to reachability of time varying discrete-time linear systems

We show that the reachability and the observability of discrete time, time varying linear systems is equivalent to a structured Kalman rank condition, under the difference algebraic independence of the time varying coefficients of the structure matrices. In the case, when the system is not reachable, we only can state that the dimension of the reachability subspace reaches its maximum at infinitely many times. The observability of a bilinear system is also considered