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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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/

[1] Thompson W. (Lord Kelvin), Tait P., Treatise on Natural Philosophy, Part 1, Cambridge University Press, 1869

[2] Chetaev N. G., Ustoichivost dvizheniya, Nauka, M., 1990 | MR | Zbl

[3] Chen G., Russel D. L., “A mathematical model for linear elastic systems with structural damping”, Quart. Appl. Math., 39 (1982), 433–454 | DOI | MR | Zbl

[4] Chen S., Triggiani R., “Proof of extensions of two conjectures on structural damping for elastic systems”, Pacific J. Math., 136:1 (1989), 15–55 | DOI | MR | Zbl

[5] Chen S., Triggiani R., “Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0\alpha1/2$”, Proc. Amer. Math. Soc., 110:2 (1990), 401–415 | DOI | MR | Zbl

[6] Huang F., “On the mathematical model for linear elastic systems with analytic damping”, SIAM J. Control Optim., 26:3 (1988), 714–724 | DOI | MR | Zbl

[7] Huang F., “Some problems for linear elastic systems with damping”, Acta Math. Sci., 10:3 (1990), 316–326 | DOI | MR

[8] Griniv R. O., Shkalikov A. A., “Operatornye modeli v teorii uprugosti i gidromekhanike i assotsiirovannye s nimi analiticheskie polugruppy”, Vestnik MGU, ser. matem., mekh., 1999, no. 5, 5–14 | MR | Zbl

[9] Griniv R. O., Shkalikov A. A., “Eksponentsialnaya ustoichivost polugrupp, svyazannykh s nekotorymi operatornymi modelyami v mekhanike”, Matem. zametki, 73:5 (2003), 657–664 | DOI | MR | Zbl

[10] Veselić K., “Energy decay of damped systems”, Z. Angew. Math. Mech., 84:12 (2004), 856–863 | DOI | MR | Zbl

[11] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[12] Lyubich Yu. I., “Klassicheskoe i lokalnoe preobrazovanie Laplasa v abstraktnoi zadache Koshi”, UMN, 21(129):3 (1966), 3–51 | MR | Zbl

[13] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York–Berlin, 1983 | DOI | MR | Zbl

[14] Engel K. J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin–Heildelberg–New York, 2000 | MR

[15] Paidoussis M. P., Issid N. T., “Dynamic stability of pipes conveying fluid”, J. Sound Vibration, 33 (1974), 267–294 | DOI

[16] Zefirov V. N., Kolesov V. V., Miloslavskii A. I., “Issledovanie sobstvennykh chastot pryamolineinogo truboprovoda”, Izv. AN SSSR, seriya mekh. tv. tela, 1985, no. 1, 179–188

[17] Datko R., “Extending a theorem of A. M. Liapunov to Hilbert space”, J. Math. Anal. Appl., 32 (1970), 610–616 | DOI | MR | Zbl

[18] van Casteren J. A., “Operators similar to unitary or selfadjoint ones”, Pacific J. Math., 104:1 (1983), 241–255 | DOI | MR | Zbl

[19] Naboko S. N., “Ob usloviyakh podobiya unitarnym i samosopryazhennym operatoram”, Funkts. analiz i ego pril., 18:1 (1984), 16–27 | MR | Zbl

[20] Malamud M. M., “Kriterii podobiya zamknutogo operatora samosopryazhennomu”, Ukr. matem. zh., 37:1 (1985), 49–56 | MR | Zbl

[21] Rolewicz S., “On uniform $N$-equistability”, J. Math. Anal. Appl., 115 (1986), 431–441 | DOI | MR

[22] Littman W., “A generalization of a theorem of Datko and Pazy”, Lecture Notes in Control and Inform. Sci., 130, Springer-Verlag, Berlin–New York, 1989, 318–323 | DOI | MR

[23] Gomilko A. M., “Ob usloviyakh na proizvodyaschii operator ravnomerno ogranichennoi $C_0$-polugruppy operatorov”, Funkts. analiz i ego pril., 33:4 (1999), 66–69 | DOI | MR | Zbl

[24] Malamud M. M., “O podobii treugolnogo operatora diagonalnomu”, Zap. nauchn. semin. POMI, 270, 2000, 201–241 | Zbl

[25] Gearhart L., “Spectral theory for contraction semigroups on Hilbert spaces”, Trans. Amer. Math. Soc., 236 (1978), 385–394 | DOI | MR | Zbl

[26] Shkalikov A. A., “Operator pencils arising in elasticity and hydrodynamics: the instability index formula”, Operator Theory: Adv. and Appl., 87, Birkhäuser, 1996, 358–385 | MR | Zbl