# Resolvent conditions for the control of unitary groups and their approximations

### Luc Miller

Université Paris-Ouest La Défense, Nanterre, France

## Abstract

A self-adjoint operator $\mathcal A$ and an operator $\mathcal C$ bounded from the domain $\mathcal D(\mathcal A)$ with the graph norm to another Hilbert space are considered. The admissibility or the exact observability in finite time of the unitary group generated by $i\mathcal A$ with respect to the observation operator $\mathcal C$ are characterized by some spectral inequalities on $\mathcal A$ and $\mathcal C$. E.g. both properties hold if and only if $x\mapsto\ \|(\mathcal A-\lambda)x \|+\|\mathcal C x \|$ is a norm on $\mathcal D(\mathcal A)$ equivalent to $x\mapsto\|(\mathcal A -\lambda)x\|+\|x\|$ uniformly with respect to $\lambda\in\mathbb R$.

This paper generalizes and simplifies some results on the control of unitary groups obtained using these so-called *resolvent conditions*, also known as *Hautus tests*. It proves new theorems on the equivalence (with respect to admissibility and observability) between first and second order equations, between groups generated by $i\mathcal A$ and $if(\mathcal A)$ for positive $\mathcal A$ and convex $f$, and between a group and its Galerkin approximations. E.g. they apply to the control of linear Schrödinger, wave and plates equations and to the uniform control of their finite element semi-discretization.

## Cite this article

Luc Miller, Resolvent conditions for the control of unitary groups and their approximations. J. Spectr. Theory 2 (2012), no. 1, pp. 1–55

DOI 10.4171/JST/20