Approximate and exact controllability of linear difference equations

TL;DR

This paper investigates the controllability of linear difference equations with multiple delays, providing new characterizations especially when delays are incommensurable, and offers explicit criteria for low-dimensional cases.

Contribution

It extends controllability analysis to incommensurable delays and provides explicit criteria for two-dimensional systems with two delays.

Findings

Approximate and exact controllability are equivalent when delays are commensurable.

New characterizations of controllability without the commensurability assumption.

Explicit controllability criteria for 2D systems with two delays.

Abstract

In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum\_{j=1}^N A\_j x(t - \Lambda\_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $\Lambda\_1, \dotsc, \Lambda\_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.