Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form $\ddot x+B\dot x+Ax=0$, where $x=x(t)$ is a function with values in the Hilbert space $\mathscr H$ , the operator $B$ is symmetric, and the operator $A$ is uniformly positive and self-adjoint in $\mathscr H$. The linear operator $\mathscr T$ generating the $C_0$-semigroup in the energy space $\mathscr H_1\times\mathscr H$ is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator $A$ dominates $B$ in the sense of quadratic forms.