Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations

TL;DR

This paper establishes frequency domain criteria for approximate and necessary conditions for exact controllability of finite-dimensional linear difference delay equations, providing bounds on minimal controllability times.

Contribution

It extends realization theory to derive new controllability conditions for difference delay equations, including explicit bounds on minimal controllability times.

Findings

Necessary and sufficient frequency domain conditions for approximate controllability.

A necessary condition for $L^1$ exact controllability.

Explicit upper bounds on minimal controllability times.

Abstract

The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-\Lambda_1,\dots,t-\Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q \in [1, +\infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $d\max\{\Lambda_1,\dots,\Lambda_N\}$, where $d$ is the dimension of the state space.