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Exponential Stability of Semigroups Related to Operator Models in Mechanics

Authors:	Griniv R O Shkalikov A A

Institution:	(1) Institute of Applied Problems in Mathematics and Mechanics, Lviv, Ukraina;(2) M. V. Lomonosov Moscow State University, Russia

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.

Keywords:	self-adjoint operator C ₀-semigroup exponential stability energy space dissipative operator Hilbert space generalized spectrum
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