# 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 $A$ and an operator $C$ bounded from the domain $D(A)$ 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 $iA$ with respect to the observation operator $C$ are characterized by some spectral inequalities on $A$ and $C$. E.g. both properties hold if and only if $x↦∥(A−λ)x∥+∥Cx∥$ is a norm on $D(A)$ equivalent to $x↦∥(A−λ)x∥+∥x∥$ uniformly with respect to $λ∈R$.

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 $iA$ and $if(A)$ for positive $A$ 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