Geodesic
Exponential Decay of Solution Energy for Equations Associated with Some Operator Models of Mechanics
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 3-14

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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.
Keywords: stability of motion, stability of semigroups, operator equations, operator models in mechanics. 
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     author = {R. O. Hryniv and A. A. Shkalikov},
     title = {Exponential {Decay} of {Solution} {Energy} for {Equations} {Associated} with {Some} {Operator} {Models} of {Mechanics}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a0/}
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R. O. Hryniv; A. A. Shkalikov. Exponential Decay of Solution Energy for Equations Associated with Some Operator Models of Mechanics. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 3-14. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a0/