标题： 与收缩半群的相似性：结构性质、判据及在控制理论中的应用 显示英文标题

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

摘要： 我们揭示了希尔伯特空间$C_0$-半群结构的新方面，这些半群类似于收缩半群$\mathcal T = (T(t))_{t\ge 0}$。特别地，我们证明了$\mathcal T$与收缩半群相似当且仅当$\mathcal T$与准收缩$C_0$-半群相似，并且$T(t)$对于单个$t>0.$来说与收缩相似。此外，我们的方法允许我们估计相应的相似常数并阐明它们在研究与收缩相似性中的作用。 在这一过程中，我们得到了涉及无界算子的相似性条件，并对$\mathcal T$的正则性施加了最小的假设。 这种一般的设置使我们能够找到控制理论中的重要应用，包括精确可观测和精确可控系统的特征描述。 最后，我们建立了若干类似的标准，用于相似于等距半群的情况，并通过多个相关示例说明了所发展的理论。 显示英文摘要

摘要： 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.