On near-controllability of discrete-time bilinear systems using a minimum-time control

In this paper, we consider a class of n-dimensional discrete-time bilinear systems which can be nearly controllable. We first show that, to achieve near-controllability, at least n+1 control inputs are required. That is, the minimum time to steer the class of systems between any given pair of states is no less than n + 1 time steps. We then prove by applying the root locus theory that the systems can be nearly controllable with exactly n + 1 control inputs only if a corresponding matrix has no Jordan block with dimension greater than two and, meanwhile, has no more than one Jordan block with dimension two in its Jordan canonical form. Finally, we give examples to demonstrate the results of this paper.