Similarity to contraction semigroups: structural properties, criteria, and applications to control theory

We reveal new aspects of the structure of Hilbert space $C_0$-semigroups $\mathcal{T}=(T(t){)}_{t\ge 0}$ similar to semigroups of contractions. In particular, we prove that $\mathcal{T}$ is similar to a semigroup of contractions if and only if $\mathcal{T}$ is similar to a quasi-contraction $C_0$-semigroup and $T(t)$ is similar to a contraction for a single $t>0.$ Moreover, our methods allow us to estimate the corresponding similarity constants and clarify their role in the study of similarity to contractions. Along the way, we obtain similarity conditions involving unbounded operators and imposing minimal assumptions on regularity of $\mathcal{T}$. Such a general setting allows us to find significant applications to control theory, including characterizations of exactly observable and exactly controllable systems. Finally we establish several criteria of the same flavor for similarity to isometric semigroups, and illustrate the developed theory by a number of pertinent examples.