Controllability for discrete systems with a finite control set
Résumé
In this paper we consider the problem of controllability for a discrete linear control system xk+1 = Axk + Buk, uk ∈ 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 andgt; 1 the reachable set R(0, U) from the origin is R(0, U)(λ) = {∑j=0n ujλj uj ∈ U, n ∈ N} When 0 andlt; λ andlt; 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, Ũ(λ))(λ) for the finite set of integers Ũ(λ) = {0, ±1, ..., ±[λ]} 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.