Controllability for Discrete Systems with a Finite Control Set

Abstract

In this paper we consider the problem of controllability for a discrete linear control system x k+1=Ax k+Bu k, u k∈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 λ and for λ>1 the reachable set R(0,U) from the origin is¶ ¶When 0<λ<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 for the finite set of integers is related to special classes of algebraic integers. In particular if λ is a Pisot number, then the set is nowhere dense in ℝ for any finite control set U of rationals.