Abstract
In this paper, we consider equations of the form \(\user1{\ddot x}\user2{ + }B\user1{\dot x}\user2{ + }A\user1{x} = 0\), where \(\user1{x}\user2{ = }\user1{x}\left( \user1{t} \right)\) is a function with values in the Hilbert space \(\mathcal{H}\), the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in \(\mathcal{H}\). The linear operator \(\mathcal{T}\) generating the C 0-semigroup in the energy space \({\mathcal{H}}_1 \times {\mathcal{H}}\) 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.
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Griniv, R.O., Shkalikov, A.A. Exponential Stability of Semigroups Related to Operator Models in Mechanics. Mathematical Notes 73, 618–624 (2003). https://doi.org/10.1023/A:1024052419431
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DOI: https://doi.org/10.1023/A:1024052419431