Relative controllability of linear difference equations

TL;DR

This paper investigates the relative controllability of linear difference equations with multiple delays, providing algebraic criteria, analyzing the impact of delay rationality, and establishing bounds on controllability time.

Contribution

It introduces a formula-based approach to characterize controllability, extends classical criteria to delayed systems, and explores the influence of delay rationality on controllability.

Findings

Controllability characterized by algebraic properties of matrix coefficients.

Delay rationality affects the controllability of the system.

An upper bound on minimal controllability time depending on system dimension and delays.

Abstract

In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time $T$ in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces $L^p$ and $\mathcal C^k$. We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are "less rationally dependent" than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay.