2025/08/31 by Jesús Oliva-Maza, Oliva-Maza, J., Yury V. Tomilov +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Control and Stability of Dynamical Systems #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2509.01007
We reveal new aspects of the structure of Hilbert space C0-semigroups \mathcal T = (T(t))t≥ 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 C0-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.