### 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 R(0, U)(λ) = {∑_{j=0}^{n} u_{j}λ^{j}: u_{j} ∈ U, n ∈ N} 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 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.

Original language | English (US) |
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Pages (from-to) | 173-193 |

Number of pages | 21 |

Journal | Mathematics of Control, Signals, and Systems |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2001 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Signal Processing
- Applied Mathematics
- Control and Systems Engineering

### Keywords

- Controllability
- Discrete systems
- Finite control set
- Pisot numbers
- Reachability

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## Cite this

*Mathematics of Control, Signals, and Systems*,

*14*(2), 173-193. https://doi.org/10.1007/PL00009881