We consider linear control systems of the form $\dot{x}\left(t\right)=Ax\left(t\right)-\mu BCx\left(t\right)$ where $\mu$ is a positive real parameter, $A$ is the state operator and generates a linear $C_0-$semigroup of contractions $S\left(t\right)$ on a Banach space $X$, $B$ and $C$ are respectively the operators of control and observability, which are defined in appropriate spaces in which they are unbounded in some sense. We aim to show the exponential stability of the above system under sufficient conditions which are expressed in terms of admissibility and observability properties. The uniform exponential stabilization using bilinear control is considered as well. Applications to transport and heat equations are also provided.

我们考虑以下形式的线性控制系统 $\dot{X}\left(吨\right)=\mathrm{一种}X\left(吨\right)-\mu 乙CX\left(吨\right)$ 在哪里 $\mu$ 是一个正实参数， $\mathrm{一种}$ 是状态运算符并生成线性 $C_0-$收缩半群 $秒\left(吨\right)$ 在巴拿赫空间 $X$, $乙$ 和 $C$分别是控制和可观察性的算子，它们被定义在适当的空间中，它们在某种意义上是无界的。我们的目标是在充分条件下显示上述系统的指数稳定性，这些条件以可接受性和可观察性属性表示。还考虑了使用双线性控制的均匀指数稳定。还提供了传输和热方程的应用。