自由端具有局部粘弹性阻尼的悬臂Timoshenko梁的能量衰减问题的研究
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摘要
本论文研究自由端具有局部粘弹性阻尼的悬臂梁的能量指数衰减问题。通过建立适当的Hilbert空间,把描述梁的振动的偏微分方程抽象化为相应的Hilbert空间中的发展方程,并利用算子半群理论研究抽象方程的系统算子所生成的C_0—压缩半群,进而探讨其系统相关的稳定性问题。在一定的光滑假设和系统的结构条件下,得到系统能量指数衰减的结论。
In this paper,we consider the exponential decay of energy of the cantilever Timoshenko beam with local viscoelastic damping on the free end.By establishing suitable Hilbert spaces,we can rewritten the partial differential equations which describing the beam' vibration as a evolution equation in Hilbert space,and we study the Co-semigroup generated by the system operator with the theory of semigroups of operators.Under some certain smooth assumptions and structural conditions,we obtain the exponential decay of the beam energy.
In this paper,we consider the exponential decay of energy of the cantilever Timoshenko beam with local viscoelastic damping on the free end.By establishing suitable Hilbert spaces,we can rewritten the partial differential equations which describing the beam' vibration as a evolution equation in Hilbert space,and we study the Co-semigroup generated by the system operator with the theory of semigroups of operators.Under some certain smooth assumptions and structural conditions,we obtain the exponential decay of the beam energy.
引文
[1]A.R.ADAMS,Sobolev space,Acadamic Press,New York,1975.
[2]H.T.BANKS,R.C.SMITH,AND Y.WANG,The modeling of piezoceramic patch interactions with shells,plates,and beams,Quart.Appl.Math.,LⅢ(1995),pp.353-381.
[3]C.BARDOS,G.LEBEAU,AND J.RAUCH,Sharp sufficient conditions for the observation,control,and stabilization of waves from the boundary,SIAM J.Control Optim.,Vol.30,No.5(1992),pp.1024-1065.
[4]A.BENADDI,AND B.P.RAO,Energy decay rate of wave equations with indefinite damping,J.Diff.Eqs.,161(2000),pp.337-357.
[5]G.CHEN,Control and stabilization for the wave equation in a bounded domain,SIAM J.Control Optim.,Vol.17,No.1(1979),pp.66-81.
[6]G.CHEN,Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain,J.Math.Pures Appl.,Tome 58,No.3(1979),pp.249-273.
[7]G.CHEN,A note on the boundary stabilization of the wave equation,SIAM J.Control Optim.,Vol.19,No.1(1981),pp.106-113.
[8]G.CHEN,Control and stabilization for the wave equation in a bounded domain,part Ⅱ,SIAM J.Control Optim.,Vol.19,No.1(1981),pp.114-122.
[9]G.CHEN,S.A.FULLING,F.J.NARCOWICH,AND S.SUN,Exponential decay of energy of evolution equations with locally distributed damping,SIAM J.Appl.Math.,Vol.51,No.1(1991),pp.266-301.
[10]G.CHEN,AND J.X.ZHOU,Vibration and damping in distributed systems.Vol.Ⅰ:Analysis,estimation,attenuation,and design,CRC Press Inc.,Boca Raton,1993.
[11]G.CHEN,AND J.X.ZHOU,Vibration and damping in distributed systems.Vol.Ⅱ:WKB and wave methods,visualization and experimentation,CRC Press Inc.,Boca Raton,1993.
[12]S.P.CHEN,K.S.LIU AND Z.Y.LIU,Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping,SIAM J.Appl.Math.,Vol.59,No.2(1998),pp.651-668.
[13]R.M.CHRISTENSEN,Theory of viscoelasticity,2nd ed.,Acadamic Press Inc.,1982.
[14]C.M.DAFERMOS,An abstract Volterra equation with applications to linear viscoelasticity,J.Diff.Eqs.7(1970),pp.554-569.
[15]R.H.FABIANO,Uniformly stable approximation for a linear viscoelastic model,SIAM J.Math.Anal.,Vol.21,No.2(1990),pp.374-393.
[16]L.M.GEARHART,Spectral theory for contraction semigroups on Hilbert space,Trans.Amer.Math.Soc.,263(1978),pp.385-394.
[17]K.B.HANNSGEN,Stabilization of the viscoelastic Timoshenko beam,International series of numerical mathematics,Vol.91,pp.Birkh(a|¨)user Verlag,Basel,1989.
[18]L.F.Ho Exact controllability of the one-dimensional wave equation with locally distributed control,SIAM J.Control Optim.,Vol.28,No.3(1990),pp.337-348.
[19]胡海昌,弹性力学的变分原理及其应用,科学出版社,北京,1981年5月。
[20]F.L.HUANG,Strong asymptotic stability of linear dynamical systems in Banach spaces,J.Diff.Eqs.,104(1993),pp.307-324.
[21]F.L.HUANG,Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces.Ann.of Diff.Eqs..1(1)(1985),pp.43-56.
[22]F.L.HUANG,On the mathematical model for linear elastic systems with analytic damping,SIAM J.Control Optim.,Vol.26,No.3(1988),pp.714-724.
[23]T.KATO,Perturbation theory for linear operators,Springer-Verlag,New York,1980.
[24]S.KAWASHIMA,Global solutions to the equation of viscoelasticity with fading memory,J.Diff.Eqs.,101(1993),pp.388-420.
[25]J.U.KIM Exponential decay of the energy of a one-dimensional nonhomogeneous medium,SIAM J.Control Optim.,Vol.29,No.2(1991),pp.368-380.
[26]J.U.KIM Exact internal controllability of a one-dimensional aeroelastic plate,Appl.Math.Optim.,Vol.24(1991),pp.99-111.
[27]J.LAGNESE,Control of wave process with distributed controls supported on a subregion,SIAM J.Control Optim.,Vol.21(1983),pp.68-85.
[28]K.S.LIU,Locally distributed control and damping for the conservative system,SIAM J.Control Optim.,Vol.35,No.5(1997),pp.1574-1590.
[29]K.S.LIU,Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces,J.Diff.Eqs.,141(1997),pp.340-355.
[30]K.S.LIU AND Z.Y.LIU,Exponential decay of energy of vibrating strings with local viscoelasticity,Z.angew Math.Phys.,2002,to appear.
[31]K.S.LIU AND Z.Y.LIU,Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,SIAM J.Control Optim.,Vol.36,No.3(1998),pp.1086-1098.
[32]K.S.LIU AND Z.Y.LIU,On the type of Co-semigroup associated with the abstract linear viscoelastic system,Z.angew Math.Phys.,47(1996),pp.1-15.
[33]K.S.LIU AND B.P.RAO,Exponential stability of high-dim wave equation with local K-V Damping,manuscript.
[34]Z.Y.LIU AND C.PENG,Exponential stability of a viscoelastic Timoshenko beam,Adv.Math.Sci.Appl.,8(1998),No.l,pp.343-351.
[35]K.S.LIU,H.L.ZHAO,Exponential stability for Timoshenko beam system with local viscoelasticity of Boltzmann type,秦化淑、褚健主编,第二十一届中国控制会议论文集,PP.376-381,浙江大学出版社,2002年8月。
[36]Z.Y.LIU,S.M.ZHENG,Semigroups associated with dissipative systems,Chapman &Hall/CRC Research Notes in Mathematics 398,Chapman & Hall/CRC,London,1999.
[37]A.PAZY,Semigroups of linear operators and applications to partial differential equations,Springer,New York,1983.(有中译本:线性算子半群及对偏微分方程的应用,黄发伦、郑权译,四川大学出版社,成都,1988年5月。)
[38]J.PR(U|¨)SS,On the spectrum of Co-semigroups,Trans.Amer.Math.Soc.,284(1984),pp.1299-1307.
[39]J.E.M.RIVERA,Asymptotic behavior in linear viscoelasticity,Quart.Appl.Math.,Vol.LⅡ,No.4(1994),pp.629-648.
[40]J.E.M.RIVERA AND A.P.SALVERRA,Asymptotic behavior of energy in partially viscoelastic materials,Quart.Appl.Math.,Vol.LⅨ,No.3(2001),pp.557-578.
[41]D.L.RUSSELL,Mathematical models for the elastic beam and their control-theoretic implications,in Semigroups,theory and applications,Vol.Ⅱ,pp.177-216,H.Brezis,M.G.Crandall & F.Kappel ed.,Pitman research notes in mathematics series 152,Longman Scientific & Techical,1986.
[42]E.TAYLOR AND D.C.LAY,Introduction to functional analysis,2ed.,John Wiley &Sons,New York,1980.
[43]S.TIMOSHENKO,Vibration problem inengineering,Von Nostrand,New York,1955.
[44]K.WASHIZU,Variational method in elasticity and plasticity,2nd ed.,Perganon Press,Elinsford,New York,1975.(有中译本:鹫津久一郎,弹性和塑性力学中的变分法,科学出版社,北京,1984年1月。)
[45]杨挺青等,黏弹性理论与应用,科学出版社,北京,2004年9月。
[46]H.L.ZHAO,K.S.LIU,AND C.G.ZHANG,Stability for the Timoshenko Beam System with Local Kelvin-Voigt Damping,Acta Mathematica Sinica,EnglishSeries June,2005,Vol.21,No.3,pp.655-666
[47]章春国、赵宏亮、刘康生,具有局部分布反馈与边界反馈耦合控制的非均质Timoshenko梁的指数镇定,数学年刊(A辑),2003年第四期。
[48]赵宏亮,具局部粘弹性的弹性系统的稳定性,浙江大学博士学位论文,2002年7月。
[49]E.ZUAZUA,Exponential decay for the semilinear wave equation with localized damping,Comm.Part.Diff.Eq.,15(1990),pp.205-235.
[2]H.T.BANKS,R.C.SMITH,AND Y.WANG,The modeling of piezoceramic patch interactions with shells,plates,and beams,Quart.Appl.Math.,LⅢ(1995),pp.353-381.
[3]C.BARDOS,G.LEBEAU,AND J.RAUCH,Sharp sufficient conditions for the observation,control,and stabilization of waves from the boundary,SIAM J.Control Optim.,Vol.30,No.5(1992),pp.1024-1065.
[4]A.BENADDI,AND B.P.RAO,Energy decay rate of wave equations with indefinite damping,J.Diff.Eqs.,161(2000),pp.337-357.
[5]G.CHEN,Control and stabilization for the wave equation in a bounded domain,SIAM J.Control Optim.,Vol.17,No.1(1979),pp.66-81.
[6]G.CHEN,Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain,J.Math.Pures Appl.,Tome 58,No.3(1979),pp.249-273.
[7]G.CHEN,A note on the boundary stabilization of the wave equation,SIAM J.Control Optim.,Vol.19,No.1(1981),pp.106-113.
[8]G.CHEN,Control and stabilization for the wave equation in a bounded domain,part Ⅱ,SIAM J.Control Optim.,Vol.19,No.1(1981),pp.114-122.
[9]G.CHEN,S.A.FULLING,F.J.NARCOWICH,AND S.SUN,Exponential decay of energy of evolution equations with locally distributed damping,SIAM J.Appl.Math.,Vol.51,No.1(1991),pp.266-301.
[10]G.CHEN,AND J.X.ZHOU,Vibration and damping in distributed systems.Vol.Ⅰ:Analysis,estimation,attenuation,and design,CRC Press Inc.,Boca Raton,1993.
[11]G.CHEN,AND J.X.ZHOU,Vibration and damping in distributed systems.Vol.Ⅱ:WKB and wave methods,visualization and experimentation,CRC Press Inc.,Boca Raton,1993.
[12]S.P.CHEN,K.S.LIU AND Z.Y.LIU,Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping,SIAM J.Appl.Math.,Vol.59,No.2(1998),pp.651-668.
[13]R.M.CHRISTENSEN,Theory of viscoelasticity,2nd ed.,Acadamic Press Inc.,1982.
[14]C.M.DAFERMOS,An abstract Volterra equation with applications to linear viscoelasticity,J.Diff.Eqs.7(1970),pp.554-569.
[15]R.H.FABIANO,Uniformly stable approximation for a linear viscoelastic model,SIAM J.Math.Anal.,Vol.21,No.2(1990),pp.374-393.
[16]L.M.GEARHART,Spectral theory for contraction semigroups on Hilbert space,Trans.Amer.Math.Soc.,263(1978),pp.385-394.
[17]K.B.HANNSGEN,Stabilization of the viscoelastic Timoshenko beam,International series of numerical mathematics,Vol.91,pp.Birkh(a|¨)user Verlag,Basel,1989.
[18]L.F.Ho Exact controllability of the one-dimensional wave equation with locally distributed control,SIAM J.Control Optim.,Vol.28,No.3(1990),pp.337-348.
[19]胡海昌,弹性力学的变分原理及其应用,科学出版社,北京,1981年5月。
[20]F.L.HUANG,Strong asymptotic stability of linear dynamical systems in Banach spaces,J.Diff.Eqs.,104(1993),pp.307-324.
[21]F.L.HUANG,Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces.Ann.of Diff.Eqs..1(1)(1985),pp.43-56.
[22]F.L.HUANG,On the mathematical model for linear elastic systems with analytic damping,SIAM J.Control Optim.,Vol.26,No.3(1988),pp.714-724.
[23]T.KATO,Perturbation theory for linear operators,Springer-Verlag,New York,1980.
[24]S.KAWASHIMA,Global solutions to the equation of viscoelasticity with fading memory,J.Diff.Eqs.,101(1993),pp.388-420.
[25]J.U.KIM Exponential decay of the energy of a one-dimensional nonhomogeneous medium,SIAM J.Control Optim.,Vol.29,No.2(1991),pp.368-380.
[26]J.U.KIM Exact internal controllability of a one-dimensional aeroelastic plate,Appl.Math.Optim.,Vol.24(1991),pp.99-111.
[27]J.LAGNESE,Control of wave process with distributed controls supported on a subregion,SIAM J.Control Optim.,Vol.21(1983),pp.68-85.
[28]K.S.LIU,Locally distributed control and damping for the conservative system,SIAM J.Control Optim.,Vol.35,No.5(1997),pp.1574-1590.
[29]K.S.LIU,Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces,J.Diff.Eqs.,141(1997),pp.340-355.
[30]K.S.LIU AND Z.Y.LIU,Exponential decay of energy of vibrating strings with local viscoelasticity,Z.angew Math.Phys.,2002,to appear.
[31]K.S.LIU AND Z.Y.LIU,Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,SIAM J.Control Optim.,Vol.36,No.3(1998),pp.1086-1098.
[32]K.S.LIU AND Z.Y.LIU,On the type of Co-semigroup associated with the abstract linear viscoelastic system,Z.angew Math.Phys.,47(1996),pp.1-15.
[33]K.S.LIU AND B.P.RAO,Exponential stability of high-dim wave equation with local K-V Damping,manuscript.
[34]Z.Y.LIU AND C.PENG,Exponential stability of a viscoelastic Timoshenko beam,Adv.Math.Sci.Appl.,8(1998),No.l,pp.343-351.
[35]K.S.LIU,H.L.ZHAO,Exponential stability for Timoshenko beam system with local viscoelasticity of Boltzmann type,秦化淑、褚健主编,第二十一届中国控制会议论文集,PP.376-381,浙江大学出版社,2002年8月。
[36]Z.Y.LIU,S.M.ZHENG,Semigroups associated with dissipative systems,Chapman &Hall/CRC Research Notes in Mathematics 398,Chapman & Hall/CRC,London,1999.
[37]A.PAZY,Semigroups of linear operators and applications to partial differential equations,Springer,New York,1983.(有中译本:线性算子半群及对偏微分方程的应用,黄发伦、郑权译,四川大学出版社,成都,1988年5月。)
[38]J.PR(U|¨)SS,On the spectrum of Co-semigroups,Trans.Amer.Math.Soc.,284(1984),pp.1299-1307.
[39]J.E.M.RIVERA,Asymptotic behavior in linear viscoelasticity,Quart.Appl.Math.,Vol.LⅡ,No.4(1994),pp.629-648.
[40]J.E.M.RIVERA AND A.P.SALVERRA,Asymptotic behavior of energy in partially viscoelastic materials,Quart.Appl.Math.,Vol.LⅨ,No.3(2001),pp.557-578.
[41]D.L.RUSSELL,Mathematical models for the elastic beam and their control-theoretic implications,in Semigroups,theory and applications,Vol.Ⅱ,pp.177-216,H.Brezis,M.G.Crandall & F.Kappel ed.,Pitman research notes in mathematics series 152,Longman Scientific & Techical,1986.
[42]E.TAYLOR AND D.C.LAY,Introduction to functional analysis,2ed.,John Wiley &Sons,New York,1980.
[43]S.TIMOSHENKO,Vibration problem inengineering,Von Nostrand,New York,1955.
[44]K.WASHIZU,Variational method in elasticity and plasticity,2nd ed.,Perganon Press,Elinsford,New York,1975.(有中译本:鹫津久一郎,弹性和塑性力学中的变分法,科学出版社,北京,1984年1月。)
[45]杨挺青等,黏弹性理论与应用,科学出版社,北京,2004年9月。
[46]H.L.ZHAO,K.S.LIU,AND C.G.ZHANG,Stability for the Timoshenko Beam System with Local Kelvin-Voigt Damping,Acta Mathematica Sinica,EnglishSeries June,2005,Vol.21,No.3,pp.655-666
[47]章春国、赵宏亮、刘康生,具有局部分布反馈与边界反馈耦合控制的非均质Timoshenko梁的指数镇定,数学年刊(A辑),2003年第四期。
[48]赵宏亮,具局部粘弹性的弹性系统的稳定性,浙江大学博士学位论文,2002年7月。
[49]E.ZUAZUA,Exponential decay for the semilinear wave equation with localized damping,Comm.Part.Diff.Eq.,15(1990),pp.205-235.