Exponential Decay of Solution Energy for Equations Associated with Some Operator Models of Mechanics

We consider the equation $\ddot x+B\dot x+Ax=0$ in a Hilbert space $\mathcal{H}$, where $A$ is a uniformly positive self-adjoint operator and $B$ is a dissipative operator. The main result is the proof of a theorem stating the exponential energy decay for solutions of this equation (or the exponential stability of the semigroup associated with the equation) under the additional assumption that $B$ is sectorial and is subordinate to $A$ in the sense of quadratic forms.