We investigate the stability properties of strongly continuous semigroups generated by operators of the form $A-B{B}^{\ast }$, where $A$ is the generator of a contraction semigroup and $B$ is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient conditions for nonuniform stability of the semigroup generated by $A-B{B}^{\ast }$ in terms of selected observability-type conditions on the pair $(B^{\ast },A)$. The core of our approach consists of deriving resolvent estimates for the generator expressed in terms of these observability properties. We apply the abstract results to obtain rates of energy decay in one-dimensional and two-dimensional wave equations, a damped fractional Klein–Gordon equation and a weakly damped beam equation.

我们研究由以下形式的算子生成的强连续半群的稳定性性质$A-乙{乙}^{*}$， 在哪里$A$是收缩半群的生成元并且$乙$是一个可能无界的运算符。在边界或空间域内具有阻尼的双曲偏微分方程的研究中自然会出现此类系统。作为我们的主要结果，我们提出了由下式生成的半群的非均匀稳定性的一般充分条件$A-乙{乙}^{*}$就货币对的选定可观测性条件而言$（{乙}^{*},A）$。我们方法的核心包括导出以这些可观测性属性表示的生成器的解析估计。我们应用抽象结果来获得一维和二维波动方程、阻尼分数克莱因-戈登方程和弱阻尼梁方程中的能量衰减率。