关于正规半群精确可观性的Hautus检验
On the Hautus test for exact observability of normal semigroups
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中文总结 AI 辅助
本文针对正规半群的精确可观性Hautus检验，构造反例反驳修正后的Russell-Weiss猜想，同时证明自伴算子情形下Hautus检验与精确可观性的蕴含关系，及正规算子有限维观测时的相关结论。
中文摘要 AI 辅助
设$T$是希尔伯特空间$X$上的指数稳定强连续半群，$A$为其生成元，$X_1$为$A$的定义域，范数定义为$\|v\|_1:= \|Av\|$，$C$是取值于希尔伯特空间$Y$的容许观测算子。Russell与Weiss曾猜想，若$(A,C)$满足无穷维Hautus检验，则$(A,C)$是精确可观的，在找到反例后该猜想被修改为附加假设$T$相似于压缩半群。本文通过构造反例反驳该修正猜想，其中$A$是正规算子，$X$具有$A$的特征向量构成的标准正交基，且$C:X_1 \ o Y$是希尔-施密特算子；由于$A$正规且$T$指数稳定，故$T$是压缩半群。此外，本文证明当$A$为自伴算子时，Hautus检验蕴含精确可观性；最后表明若$A$正规且$C:X_1 \ o Y$是紧算子，则Hautus检验蕴含$A$具有紧预解式，因此对正规$A$与有限维$Y$，Hautus检验蕴含精确可观性。
英文摘要
Let $T$ be an exponentially stable strongly continuous semigroup on a Hilbert space $X$, $A$ its generator, $X_1$ the domain of $A$ with the norm $\|v\|_1 := \|Av\|$, and $C$ an admissible observation operator taking values in a Hilbert space $Y$. Russell and Weiss conjectured that if $(A,C)$ satisfies the infinite-dimensional Hautus test, then $(A,C)$ is exactly observable. After a counterexample was found their conjecture was modified to include the additional assumption that $T$ is similar to a contraction semigroup. We disprove this modified conjecture by constructing a counterexample in which $A$ is normal, $X$ has an orthonormal basis of eigenvectors of $A$ and $C: X_1 \to Y$ is Hilbert-Schmidt. Since $A$ is normal and $T$ is exponentially stable, it follows that $T$ is a contraction semigroup. In contrast, we prove that the Hautus test implies exact observability for self-adjoint $A$. Finally, we show that if $A$ is normal and $C:X_1 \to Y$ is compact, then the Hautus test implies that $A$ has compact resolvent. Consequently, for normal $A$ and finite-dimensional $Y$ the Hautus test implies exact observability.
