Controllability for discrete systems with a finite control set

In this paper we consider the problem of controllability for a discrete linear control system x(k+1) = Ax(k) + Bu-k, u(k) is an element of U, where (A,B) is controllable and U is a finite set. We prove the existence of a finite set U ensuring density for the reachable set from the origin under the necessary assumptions that the pair (A, B) is controllable and A has eigenvalues with modulus greater than or equal to 1. In the case of A only invertible we obtain density on compact sets. We also provide uniformity results with respect to the matrix A and the initial condition. In the one-dimensional case the matrix A reduces to a scalar lambda and for lambda > 1 the reachable set R(0, U) from the origin is R(0, U)(lambda) = {Sigma (n)(j=0) u(j)lambda (j): u(j) is an element of U, n is an element of N} When 0 < lambda < 1 and U = {0, 1, 3}, the closure of this set is the subject of investigation of the well-known {0, 1, 3}-problem. It turns out that the nondensity of R(0, (U) over tilde(lambda))(lambda) for the finite set of integers (U) over tilde(lambda) = {0, +/-1,...,+/-[lambda]} is related to special classes of algebraic integers. In particular if 1 is a Pisot number, then the set is nowhere dense in R for any finite control set U of rationals.