Resolvent conditions for the control of unitary groups and their approximations

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 $x\mapsto \Vert (\mathcal{A}-\lambda )x\Vert +\Vert \mathcal{C}x\Vert$ is a norm on $\mathcal{D}(\mathcal{A})$ equivalent to $x\mapsto \Vert (\mathcal{A}-\lambda )x\Vert +\Vert x\Vert$ 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.