Reachability and controllability of time-variant discrete-time positive linear systems

In this paper necessary and sufficient conditions (and criteria) for null-controllability, reachability and controllability of time-variant discrete-time positive linear systems are established. These properties appear to be entirely structural properties, that is, they do depend on the zero-nonzero pattern of the pair (A(k), B[k)) > 0 and do not depend on the values of its entries. An interesting phenomenon has been revealed namely the time needed to reach the origin for a null-controllable system as well as the time to reach a (non-negative) state from the origin for a reachable system can be less, equal or greater than the dimension of the system. This phenomenon has no equivalent in the case of time-invariant discrete-time positive linear systems where this time is always less or equal to the system dimension. Examples are provided